import Mathlib.Analysis.SpecialFunctions.ExpDeriv import Mathlib.Analysis.SpecialFunctions.Exp import Mathlib.Analysis.Calculus.Taylor import Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas import Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno import Mathlib.Analysis.InnerProductSpace.Laplacian import Mathlib.Probability.Distributions.Gaussian.Real import Mathlib.Probability.Distributions.Gaussian.Multivariate import Mathlib.MeasureTheory.Integral.DominatedConvergence import Mathlib.MeasureTheory.Integral.DivergenceTheorem import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic import AutoSamplingTheory.SDE /-! # SALD and VA-SALD faithful-paper contracts The first faithful target is the original VA-SALD paper source under `/home/nitanda_sub/mark/repos/sald/paper`, excluding `sald_version_2.tex`. -/ namespace AutoSamplingTheory namespace SALD open scoped Topology def saldPaperRoot : String := "/home/nitanda_sub/mark/repos/sald/paper" def saldMainSource : SourceAnchor := localTexAnchor "sald-original-main-body" (saldPaperRoot ++ "/main_body.tex") "main_body.tex" "Main-body SALD and VA-SALD theorem statements; sald_version_2.tex is excluded from faithful reproduction." def saldAppendixSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "appendix.tex" "Appendix proofs for SALD, VA-SALD, and discrete-time bounds." def saldIterationSource : SourceAnchor := localTexAnchor "sald-original-vp" (saldPaperRoot ++ "/iteration_complexity.tex") "iteration_complexity.tex" "VP reverse marginal verification under dissipativity." def saldGronwallSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "lem:gronwall" "appendix.tex lines 47-71: Gronwall inequality and integrating-factor proof." def saldGronwallExponentRewriteSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:lem:gronwall:exponent-rewrite" "appendix.tex lines 63-69: multiply by exp(-int_0^t1 a) and rewrite the product of exponentials as exp(-int_t^t1 a)." def saldDvVariationSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "lem:dv_variation" "appendix.tex lines 73-79: Donsker--Varadhan variational formula cited from Boucheron et al." def saldPiSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "def:PI" "appendix.tex lines 86-94: Poincare inequality vocabulary." def saldPiVelocityNormSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "lem:velocity-norm-bound" "appendix.tex lines 96-151: weighted Sobolev space, weak PDE, Riesz representation, and velocity norm bound under PI." def saldKlFiLsiSource : SourceAnchor := localTexAnchor "sald-original-main-body" (saldPaperRoot ++ "/main_body.tex") "eq:LSI-KL-FI" "main_body.tex lines 202-215: LSI definition plus KL/FI vocabulary and comparison." def saldContinuousSdeSource : SourceAnchor := localTexAnchor "sald-original-main-body" (saldPaperRoot ++ "/main_body.tex") "eq:SALD" "main_body.tex lines 13-21: continuous-time SALD SDE and its Fokker--Planck equation." def saldFokkerPlanckSource : SourceAnchor := localTexAnchor "sald-original-main-body" (saldPaperRoot ++ "/main_body.tex") "eq:FP-eq" "main_body.tex lines 18-21: probability law rho_s of SALD satisfies the Fokker--Planck equation." def saldAlphaComplexitySource : SourceAnchor := localTexAnchor "sald-original-main-body" (saldPaperRoot ++ "/main_body.tex") "def:alpha-complexity" "main_body.tex lines 218-228: pointwise alpha-complexity density and integrated alpha-complexity." def saldForwardKlSource : SourceAnchor := localTexAnchor "sald-original-main-body" (saldPaperRoot ++ "/main_body.tex") "thm:forward-KL" "main_body.tex line 240: continuous-time SALD forward KL theorem." def saldForwardKlProofSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:forward-KL" "appendix.tex lines 164-252: proof of Theorem thm:forward-KL." def saldForwardKlDerivativeSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:forward-KL:derivative" "appendix.tex lines 168-228: KL differentiation, SALD Fokker--Planck equation, transport velocity, LSI, and time change." def saldForwardKlDvEnergySource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:forward-KL:dv-energy" "appendix.tex lines 230-241: Donsker--Varadhan applied with Z=alpha*||v_t||^2." def saldForwardKlGronwallSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:forward-KL:gronwall" "appendix.tex lines 244-252: Gronwall application and separation of LSI and alpha-complexity exponent terms." def saldForwardKlEndpointScheduleSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:forward-KL:endpoint-schedule" "main_body.tex lines 9-13 and 238-247 plus appendix.tex lines 218-252: inverse slowdown endpoint identities needed to rewrite K(0), K(T), and the slowed target." def saldForwardKlDependencyChainSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:forward-KL:lsi-dv-gronwall-chain" "appendix.tex lines 210-252: LSI conversion, inverse-schedule time change, DV velocity-energy bound, and Gronwall coefficient/exponent bookkeeping." def saldForwardKlDiscreteSource : SourceAnchor := localTexAnchor "sald-original-main-body" (saldPaperRoot ++ "/main_body.tex") "thm:forward-KL-discrete" "main_body.tex line 301: discrete-time SALD forward KL theorem." def saldForwardKlDiscreteLipSource : SourceAnchor := localTexAnchor "sald-original-main-body" (saldPaperRoot ++ "/main_body.tex") "eq:lip_SALD_1,eq:lip_SALD_2" "main_body.tex lines 273-298: discrete SALD score Lipschitz assumptions and Gamma/Delta definitions." def saldForwardKlDiscreteInterpolationSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "eq:frozen_interp_terminal_disc_prop_additive_final" "appendix.tex lines 260-266: continuous Euler--Maruyama interpolation for discrete SALD." def saldFrozenDeltaCrossLipSaldSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "lem:frozen_delta_cross_lip_sald" "appendix.tex lines 268-330: frozen score-defect cross-term bound for discrete SALD." def saldForwardKlDiscreteProofSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:forward-KL-discrete" "appendix.tex lines 332-592: proof of Theorem thm:forward-KL-discrete." def saldForwardKlDiscreteDerivativeSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:forward-KL-discrete:derivative" "appendix.tex lines 334-491: EM interpolation Fokker--Planck equation, frozen defect, Young, and LSI yield the pre-DV discrete differential inequality." def saldForwardKlDiscreteConditionalFpSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:forward-KL-discrete:conditional-fp" "appendix.tex lines 347-385: conditional frozen drift bar b_{k,s}, interpolation Fokker--Planck equation, and Laplacian split relative to the slowed target." def saldForwardKlDiscreteDvVelocitySource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:forward-KL-discrete:dv-velocity" "appendix.tex lines 493-523: Donsker--Varadhan applied to the moving velocity term under the EM interpolation law." def saldForwardKlDiscreteGronwallSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:forward-KL-discrete:gronwall" "appendix.tex lines 526-592: time change and general-schedule Gronwall bound for the discrete theorem; the linear-slowdown theorem bound is stated in main_body.tex lines 309-323." def saldForwardKlDiscreteAccumulatedErrorSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:forward-KL-discrete:accumulated-error" "appendix.tex lines 557-590 and main_body.tex lines 309-323: endpoint identifications, exponent split, and barGamma/barDelta accumulated-error collection for the discrete theorem." def saldForwardKlDiscreteCoefficientChainSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:forward-KL-discrete:coefficient-chain" "appendix.tex lines 454-592 and main_body.tex lines 309-323: frozen one-step defect coefficients, DV velocity coefficient, Gronwall accumulation, and linear-slowdown error collection." def saldGuidedResidualSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "prop:guided_path_residual" "appendix.tex lines 619-704: residual identity for the guided path." def saldGuidedResidualProofSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:prop:guided_path_residual" "appendix.tex lines 630-704: normalizer derivative, guided-path derivative, divergence cancellation, and mean-zero residual." def saldGeneralMovingTargetSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "thm:general-moving-target-SALD" "appendix.tex lines 724-949: general moving-target SALD theorem and proof." def saldGeneralMovingTargetDerivativeSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:general-moving-target-SALD:derivative" "appendix.tex lines 765-884: KL derivative, general VA-SALD Fokker--Planck equation, transport velocity, residual field m=v-c, Young inequality, LSI, and time change." def saldGeneralMovingTargetDvGronwallSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:general-moving-target-SALD:dv-gronwall" "appendix.tex lines 886-934: Donsker--Varadhan bound for m_t and Gronwall application." def saldGeneralMovingTargetResidualDvSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:general-moving-target-SALD:residual-dv" "appendix.tex lines 885-907: Donsker--Varadhan applied with Z=alpha*||m_t||^2." def saldGeneralMovingTargetPureContractionSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:general-moving-target-SALD:pure-contraction" "appendix.tex lines 936-945: specialization c_t=v_t, hence m_t=0 and the residual alpha-complexity vanishes." def saldUnifiedForwardKlSource : SourceAnchor := localTexAnchor "sald-original-main-body" (saldPaperRoot ++ "/main_body.tex") "thm:unified-forward-KL" "main_body.tex line 372: unified VA-SALD forward KL theorem." def saldUnifiedForwardKlProofSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:unified-forward-KL" "appendix.tex lines 949-951: specialization of thm:general-moving-target-SALD by setting c_t <- u_t." def saldVaSaldItoSource : SourceAnchor := localTexAnchor "sald-original-main-body" (saldPaperRoot ++ "/main_body.tex") "eq:SALD_Ito" "main_body.tex lines 76-99: VA-SALD SDE written with c_t=u_t and guided score nabla log pi_t." def saldGuidedResidualMainSource : SourceAnchor := localTexAnchor "sald-original-main-body" (saldPaperRoot ++ "/main_body.tex") "eq:residual-term" "main_body.tex lines 358-363: guided residual identity motivating the correction field." def saldCorrectionFieldSource : SourceAnchor := localTexAnchor "sald-original-main-body" (saldPaperRoot ++ "/main_body.tex") "eq:poisson-eq" "main_body.tex lines 364-368: correction field w_t satisfies div(pi_t*w_t)=pi_t*(g_t-E_pi_t[g_t]), making u_t+w_t a transport velocity for pi_t." def saldUnifiedTransportBridgeSource : SourceAnchor := localTexAnchor "sald-original-main-body" (saldPaperRoot ++ "/main_body.tex") "proof:thm:unified-forward-KL:transport-bridge" "main_body.tex lines 359-368: combine the guided residual equation with the correction-field divergence equation to obtain u_t+w_t as a transport velocity for pi_t." def saldGeneralMovingTargetDiscreteSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "thm:general-moving-target-SALD-discrete" "appendix.tex lines 1313-1603: discrete-time general VA-SALD theorem and proof." def saldGeneralMovingTargetDiscreteEmSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "eq:SALD_general_EM" "appendix.tex lines 957-996: Euler--Maruyama update and continuous frozen interpolation for discrete-time general VA-SALD." def saldGeneralMovingTargetDiscreteDeltaSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "eq:general_discrete_delta_def" "appendix.tex lines 998-1023: general VA-SALD frozen-field error delta_pi^VA." def saldFrozenDeltaCrossLipGeneralSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "lem:frozen_delta_cross_lip" "appendix.tex lines 1026-1307: frozen delta cross-term bound for general VA-SALD." def saldGeneralMovingTargetDiscreteDerivativeSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:general-moving-target-SALD-discrete:derivative" "appendix.tex lines 1354-1598: discrete general VA-SALD KL derivative, frozen-delta split, LSI, DV, and time change." def saldGeneralMovingTargetDiscreteConditionalDriftSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:general-moving-target-SALD-discrete:conditional-drift" "appendix.tex lines 1368-1377: define the conditional drift bar b_{k,s} as a conditional expectation of the frozen guide drift and score term given hat X_s=x." def saldGeneralMovingTargetDiscreteWeakFpSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:general-moving-target-SALD-discrete:weak-conditional-fp" "appendix.tex lines 1379-1387: the Fokker--Planck equation associated with eq:general_moving_target_SALD_frozen_interp has drift term -div(hat rho_s*bar b_{k,s}) and diffusion term +(sigma_eta^2/2)*Delta hat rho_s." def saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:general-moving-target-SALD-discrete:weak-fp-to-kl-derivative" "appendix.tex lines 1358-1387: differentiate KL(hat rho_s||tilde pi_s), then substitute the weak conditional Fokker--Planck identity at the log-density-ratio test." def saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource : SourceAnchor where key := "mathlib-probability-kernel-condexp" kind := SourceKind.mathlib pathOrUrl := ".lake/packages/mathlib/Mathlib/Probability/Kernel/Condexp.lean" label := "ProbabilityTheory.condExpKernel" note := "Mathlib conditional-expectation kernel for standard Borel finite measures; candidate backend for the regular conditional-law layer in appendix.tex lines 1368-1377." def saldGeneralMovingTargetDiscreteCondDistribIntegralMathlibSource : SourceAnchor where key := "mathlib-condDistrib-compProd-integral" kind := SourceKind.mathlib pathOrUrl := ".lake/packages/mathlib/Mathlib/Probability/Kernel/CondDistrib.lean" label := "ProbabilityTheory.compProd_map_condDistrib; MeasureTheory.Measure.integral_compProd" note := "Mathlib disintegration and composition-product integral identities used to turn the canonical condDistrib component integral for X_k^eta | hat X_s into a sample-law weak pairing in appendix.tex lines 1368-1377." def saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource : SourceAnchor where key := "mathlib-parametric-integral-map-law" kind := SourceKind.mathlib pathOrUrl := ".lake/packages/mathlib/Mathlib/Analysis/Calculus/ParametricIntegral.lean" label := "parametric integral and map-law weak time derivative" note := "Mathlib parametric-integral and Measure.map calculus candidates for the missing weak generator-to-law derivative interface in appendix.tex lines 1379-1387." def saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource : SourceAnchor where key := "mathlib-divergence-theorem-bochner" kind := SourceKind.mathlib pathOrUrl := ".lake/packages/mathlib/Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean" label := "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable" note := "Candidate Mathlib divergence/integration-by-parts backend for identifying the conditional-drift weak action with -div(hat rho_s * bar b_{k,s}) in appendix.tex lines 1379-1387." def saldGeneralMovingTargetDiscreteKlLogRatioMathlibSource : SourceAnchor where key := "mathlib-kl-log-ratio-regularity" kind := SourceKind.mathlib pathOrUrl := ".lake/packages/mathlib/Mathlib/InformationTheory/KullbackLeibler/Basic.lean" label := "InformationTheory.klDiv_ne_top_iff; MeasureTheory.stronglyMeasurable_llr; MeasureTheory.llr_def" note := "Mathlib finite-KL and log-likelihood-ratio regularity used to discharge the measurable/integrable log-ratio weak-test side of appendix.tex lines 1358-1366." def saldGeneralMovingTargetDiscreteYoungSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:general-moving-target-SALD-discrete:frozen-residual-young" "appendix.tex lines 1469-1511: frozen/residual decomposition and the two sigma_eta^2/8 Young splits before the LSI handoff." def saldGeneralMovingTargetDiscreteResidualDvSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:general-moving-target-SALD-discrete:residual-dv" "appendix.tex lines 1544-1552: Donsker--Varadhan applied to the residual field m_t under the EM interpolation law." def saldGeneralMovingTargetDiscreteGronwallSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:general-moving-target-SALD-discrete:gronwall" "appendix.tex lines 1316-1347 and 1600: final Gronwall bound for the discrete general VA-SALD theorem." def saldGeneralMovingTargetDiscreteGronwallSideConditionSource : SourceAnchor := localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "proof:thm:general-moving-target-SALD-discrete:gronwall-side-conditions" "appendix.tex lines 1573-1600 and theorem display lines 1316-1347: constant-schedule time change, endpoint laws, coefficient regularity, and exact Gronwall-display matching." def firstFaithfulLabels : List String := [ "lem:gronwall", "lem:dv_variation", "def:PI", "eq:LSI-KL-FI", "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] /-- Lean-facing calculus interface for the appendix Gronwall lemma. This is contract data, not a theorem. The eventual proof should instantiate these fields using Mathlib's interval-integral and derivative APIs while preserving the paper's signs and endpoint expression. -/ structure GronwallCandidateContract where timeDomain : String aRegularity : String bRegularity : String kRegularity : String differentialInequality : String boundAtT1 : String integratingFactorDerivative : String integratedFactorInequality : String mathlibRoute : List String := [] source : SourceAnchor status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Endpoint-safe calculus ledger for the appendix Gronwall proof. The source proof differentiates an integrating factor on a closed interval, integrates a pointwise derivative inequality, and rewrites the exponential factor. This record isolates those proof obligations before a future Lean theorem selects a concrete `HasDerivWithinAt`/absolute-continuity backend. -/ structure GronwallEndpointCalculusContract where sourceBlock : SourceAnchor intervalInterface : String derivativeMode : String integratingFactor : String productDerivativeStep : String orderIntegrationStep : String endpointEvaluation : String exponentAlgebra : String reusableInstantiations : List String := [] dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Lower-level ledger for the final exponent rewrite in `lem:gronwall`. This keeps the interval-additivity and exponential-product algebra separate from the derivative/FTC part of the Gronwall proof, because later SALD theorem blocks reuse the same rewrite when splitting Gronwall exponents. -/ structure GronwallExponentRewriteContract where sourceBlock : SourceAnchor startingFactor : String intervalAdditivity : String negIntegralRewrite : String expProductRewrite : String integralTermRewrite : String noSignAssumptions : String dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Formal scalar algebra for the final exponent rewrite in `lem:gronwall`. The interval-integral equality `i0 = it + it1` is still a separate analytic obligation; this lemma only closes the real additive negation part. -/ theorem gronwallNegIntegralRewriteScalar (i0 it it1 : Real) (h : i0 = it + it1) : -i0 + it = -it1 := by rw [h] simp [add_comm] /-- Formal scalar `Real.exp` product algebra for the Gronwall rewrite. Once interval additivity has produced `i0 = it + it1`, this proves the pointwise exponential factor used in `appendix.tex:65-69`. -/ theorem gronwallExpProductRewriteScalar (i0 it it1 : Real) (h : i0 = it + it1) : Real.exp (-i0) * Real.exp it = Real.exp (-it1) := by rw [← Real.exp_add] congr 1 exact gronwallNegIntegralRewriteScalar i0 it it1 h /-- Interval-integral additivity bridge for the final Gronwall exponent rewrite. The source uses this with `0 <= t <= t1`; Mathlib's oriented interval integral version only needs interval-integrability on the adjacent pieces. This theorem supplies the scalar equality consumed by `gronwallNegIntegralRewriteScalar`. -/ theorem gronwallIntervalIntegralAdditivityScalar (a : Real → Real) (t t1 : Real) (h0t : IntervalIntegrable a MeasureTheory.volume 0 t) (htt1 : IntervalIntegrable a MeasureTheory.volume t t1) : (∫ u in (0 : Real)..t1, a u) = (∫ u in (0 : Real)..t, a u) + ∫ u in t..t1, a u := by exact (intervalIntegral.integral_add_adjacent_intervals h0t htt1).symm /-- Compiled bridge from interval-integral additivity to the Gronwall `Real.exp` product rewrite. This closes only the pointwise exponential factor. Rewriting the whole `b_t` integral remains the separate congruence obligation recorded below. -/ theorem gronwallExpProductRewriteIntervalIntegral (a : Real → Real) (t t1 : Real) (h0t : IntervalIntegrable a MeasureTheory.volume 0 t) (htt1 : IntervalIntegrable a MeasureTheory.volume t t1) : Real.exp (-(∫ u in (0 : Real)..t1, a u)) * Real.exp (∫ u in (0 : Real)..t, a u) = Real.exp (-(∫ u in t..t1, a u)) := by exact gronwallExpProductRewriteScalar (∫ u in (0 : Real)..t1, a u) (∫ u in (0 : Real)..t, a u) (∫ u in t..t1, a u) (gronwallIntervalIntegralAdditivityScalar a t t1 h0t htt1) /-- Push the Gronwall exponent rewrite through the outer source integral. This formalizes only the congruence step from `appendix.tex:67` to `appendix.tex:69`, assuming the adjacent interval-integrability needed by the pointwise bridge. The full Gronwall lemma still needs the derivative/FTC and order-integration backend recorded in the obligations below. -/ theorem gronwallExpProductRewriteIntegralCongr (a b : Real → Real) (t1 : Real) (h0t : ∀ t ∈ Set.uIcc (0 : Real) t1, IntervalIntegrable a MeasureTheory.volume 0 t) (htt1 : ∀ t ∈ Set.uIcc (0 : Real) t1, IntervalIntegrable a MeasureTheory.volume t t1) : (∫ t in (0 : Real)..t1, (Real.exp (-(∫ u in (0 : Real)..t1, a u)) * Real.exp (∫ u in (0 : Real)..t, a u)) * b t) = ∫ t in (0 : Real)..t1, Real.exp (-(∫ u in t..t1, a u)) * b t := by apply intervalIntegral.integral_congr intro t ht have hpoint := gronwallExpProductRewriteIntervalIntegral a t t1 (h0t t ht) (htt1 t ht) calc (fun x => Real.exp (-(∫ u in (0 : Real)..t1, a u)) * Real.exp (∫ u in (0 : Real)..x, a u) * b x) t = (Real.exp (-(∫ u in (0 : Real)..t1, a u)) * Real.exp (∫ u in (0 : Real)..t, a u)) * b t := rfl _ = Real.exp (-(∫ u in t..t1, a u)) * b t := by rw [hpoint] _ = (fun x => Real.exp (-(∫ u in x..t1, a u)) * b x) t := rfl /-- Product derivative for the Gronwall integrating factor. This is the Lean form of `appendix.tex:58-60`, after the derivative of `A(t)=int_0^t a` has been supplied by the interval-integral FTC backend. -/ theorem gronwallIntegratingFactorProductDerivative (A K a k' : Real → Real) (t : Real) (hA : HasDerivAt A (a t) t) (hK : HasDerivAt K (k' t) t) : HasDerivAt (fun x => Real.exp (A x) * K x) (a t * Real.exp (A t) * K t + Real.exp (A t) * k' t) t := by have hraw := ((Real.hasDerivAt_exp (A t)).comp t hA).mul hK apply hraw.congr_deriv simp only [Function.comp_apply] ring /-- Scalar order core for `appendix.tex:60-61`. Once the source differential inequality `K' <= -a*K+b` is available and the integrating factor is known to be nonnegative, this closes the real algebra turning the product derivative into the source upper bound `I*b`. -/ theorem gronwallIntegratingFactorDerivativeInequalityScalar (a I K dK b : Real) (hI : 0 ≤ I) (hdK : dK ≤ -a * K + b) : a * I * K + I * dK ≤ I * b := by have hmul : I * dK ≤ I * (-a * K + b) := by exact mul_le_mul_of_nonneg_left hdK hI calc a * I * K + I * dK ≤ a * I * K + I * (-a * K + b) := by linarith _ = I * b := by ring /-- Pointwise derivative inequality for the Gronwall integrating factor. This packages the source line 58-61 step after an antiderivative derivative `d/dt int_0^t a = a(t)` is supplied. The subsequent integration from `0` to `t1` and endpoint evaluation remain separate obligations. -/ theorem gronwallIntegratingFactorDerivativeLe (a b K : Real → Real) (t k' : Real) (hA : HasDerivAt (fun x => ∫ u in (0 : Real)..x, a u) (a t) t) (hK : HasDerivAt K k' t) (hineq : k' ≤ -a t * K t + b t) : ∃ d : Real, HasDerivAt (fun x => Real.exp (∫ u in (0 : Real)..x, a u) * K x) d t ∧ d ≤ Real.exp (∫ u in (0 : Real)..t, a u) * b t := by refine ⟨a t * Real.exp (∫ u in (0 : Real)..t, a u) * K t + Real.exp (∫ u in (0 : Real)..t, a u) * k', ?_, ?_⟩ · exact gronwallIntegratingFactorProductDerivative (fun x => ∫ u in (0 : Real)..x, a u) K a (fun _ => k') t hA hK · exact gronwallIntegratingFactorDerivativeInequalityScalar (a t) (Real.exp (∫ u in (0 : Real)..t, a u)) (K t) k' (b t) (le_of_lt (Real.exp_pos _)) hineq /-- FTC-backed version of `gronwallIntegratingFactorDerivativeLe`. This discharges the local derivative of `int_0^t a` using Mathlib's interval-integral fundamental theorem at the point `t`. -/ theorem gronwallIntegratingFactorDerivativeLeOfIntegral (a b K : Real → Real) (t k' : Real) (haInt : IntervalIntegrable a MeasureTheory.volume 0 t) (haMeas : StronglyMeasurableAtFilter a (nhds t) MeasureTheory.volume) (haCont : ContinuousAt a t) (hK : HasDerivAt K k' t) (hineq : k' ≤ -a t * K t + b t) : ∃ d : Real, HasDerivAt (fun x => Real.exp (∫ u in (0 : Real)..x, a u) * K x) d t ∧ d ≤ Real.exp (∫ u in (0 : Real)..t, a u) * b t := by exact gronwallIntegratingFactorDerivativeLe a b K t k' (intervalIntegral.integral_hasDerivAt_right haInt haMeas haCont) hK hineq /-- Order-integration backend for the Gronwall integrating-factor proof. This is the source step from `appendix.tex:62-63`: once the derivative of `F(t)=exp(int_0^t a)*K(t)` is represented by `f'` on the source interval and `f' <= g`, interval-integral FTC and monotonicity give the integrated inequality. Endpoint-safe production of `F` and `f'` remains a separate side condition. -/ theorem gronwallOrderIntegrationOfHasDerivAt (F f' g : Real → Real) (t1 : Real) (ht1 : 0 ≤ t1) (hF' : ∀ t ∈ Set.uIcc (0 : Real) t1, HasDerivAt F (f' t) t) (hf'Int : IntervalIntegrable f' MeasureTheory.volume 0 t1) (hgInt : IntervalIntegrable g MeasureTheory.volume 0 t1) (hle : ∀ t ∈ Set.Icc (0 : Real) t1, f' t ≤ g t) : F t1 - F 0 ≤ ∫ t in (0 : Real)..t1, g t := by have hFTC : (∫ t in (0 : Real)..t1, f' t) = F t1 - F 0 := by exact intervalIntegral.integral_eq_sub_of_hasDerivAt hF' hf'Int rw [← hFTC] exact intervalIntegral.integral_mono_on ht1 hf'Int hgInt hle /-- Endpoint-safe order-integration backend for the Gronwall proof. This version matches the closed-interval issue in `appendix.tex:62-63` more closely than `gronwallOrderIntegrationOfHasDerivAt`: it only differentiates on the open interval and uses Mathlib's right-derivative FTC with continuity on `[0,t1]`. The pointwise order bound is still stated on the closed interval so the interval-integral monotonicity step has the same source endpoints. -/ theorem gronwallOrderIntegrationOfHasDerivRight (F f' g : Real → Real) (t1 : Real) (ht1 : 0 ≤ t1) (hFcont : ContinuousOn F (Set.Icc (0 : Real) t1)) (hF' : ∀ t ∈ Set.Ioo (0 : Real) t1, HasDerivWithinAt F (f' t) (Set.Ioi t) t) (hf'Int : IntervalIntegrable f' MeasureTheory.volume 0 t1) (hgInt : IntervalIntegrable g MeasureTheory.volume 0 t1) (hle : ∀ t ∈ Set.Icc (0 : Real) t1, f' t ≤ g t) : F t1 - F 0 ≤ ∫ t in (0 : Real)..t1, g t := by have hFTC : (∫ t in (0 : Real)..t1, f' t) = F t1 - F 0 := by exact intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le ht1 hFcont hF' hf'Int rw [← hFTC] exact intervalIntegral.integral_mono_on ht1 hf'Int hgInt hle /-- Endpoint evaluation for the integrated Gronwall factor. After integrating the derivative inequality, the source uses `exp(int_0^0 a)=1` to turn the left endpoint into `K_0`. -/ theorem gronwallEndpointEvaluationScalar (a K : Real → Real) (t1 J : Real) (hint : Real.exp (∫ u in (0 : Real)..t1, a u) * K t1 - Real.exp (∫ u in (0 : Real)..(0 : Real), a u) * K 0 ≤ J) : Real.exp (∫ u in (0 : Real)..t1, a u) * K t1 ≤ K 0 + J := by have hsub : Real.exp (∫ u in (0 : Real)..t1, a u) * K t1 - K 0 ≤ J := by simpa using hint have hle := sub_le_iff_le_add.mp hsub linarith /-- Scalar multiplication by the inverse integrating factor. This is the endpoint algebra immediately before the final source exponent rewrite in `appendix.tex:65-69`. -/ theorem gronwallEndpointMultiplyByExpNegScalar (A K1 K0 J : Real) (h : Real.exp A * K1 ≤ K0 + J) : K1 ≤ Real.exp (-A) * K0 + Real.exp (-A) * J := by have hmul : Real.exp (-A) * (Real.exp A * K1) ≤ Real.exp (-A) * (K0 + J) := by exact mul_le_mul_of_nonneg_left h (le_of_lt (Real.exp_pos (-A))) calc K1 = Real.exp (-A) * (Real.exp A * K1) := by rw [← mul_assoc, ← Real.exp_add] simp _ ≤ Real.exp (-A) * (K0 + J) := hmul _ = Real.exp (-A) * K0 + Real.exp (-A) * J := by ring /-- Move the endpoint inverse integrating factor through the source `b_t` integral and apply the final Gronwall exponent rewrite. This is the compiled version of the passage from `appendix.tex:65` to `appendix.tex:69` after the adjacent interval-integrability of `a` has been supplied. No sign assumption on `a` or `b` is used. -/ theorem gronwallEndpointIntegralRewrite (a b : Real → Real) (t1 : Real) (h0t : ∀ t ∈ Set.uIcc (0 : Real) t1, IntervalIntegrable a MeasureTheory.volume 0 t) (htt1 : ∀ t ∈ Set.uIcc (0 : Real) t1, IntervalIntegrable a MeasureTheory.volume t t1) : Real.exp (-(∫ u in (0 : Real)..t1, a u)) * (∫ t in (0 : Real)..t1, Real.exp (∫ u in (0 : Real)..t, a u) * b t) = ∫ t in (0 : Real)..t1, Real.exp (-(∫ u in t..t1, a u)) * b t := by rw [← intervalIntegral.integral_const_mul] have hmulAssoc : (∫ t in (0 : Real)..t1, Real.exp (-(∫ u in (0 : Real)..t1, a u)) * (Real.exp (∫ u in (0 : Real)..t, a u) * b t)) = ∫ t in (0 : Real)..t1, (Real.exp (-(∫ u in (0 : Real)..t1, a u)) * Real.exp (∫ u in (0 : Real)..t, a u)) * b t := by apply intervalIntegral.integral_congr intro t ht ring rw [hmulAssoc] exact gronwallExpProductRewriteIntegralCongr a b t1 h0t htt1 /-- Global integrating-factor assembly for the appendix Gronwall proof. This theorem threads the proof-producing local Gronwall helpers into the paper's displayed bound under explicit global calculus and interval-integral hypotheses. It is intentionally slightly below the source lemma: the remaining obligation is to derive these Mathlib side conditions from the paper's concise "continuous" and "differentiable on `[0,t_1]`" hypotheses. -/ theorem gronwallIntegratingFactorBoundOfDerivatives (a b K K' : Real → Real) (t1 : Real) (ht1 : 0 ≤ t1) (hA : ∀ t ∈ Set.uIcc (0 : Real) t1, HasDerivAt (fun x => ∫ u in (0 : Real)..x, a u) (a t) t) (hK : ∀ t ∈ Set.uIcc (0 : Real) t1, HasDerivAt K (K' t) t) (hineq : ∀ t ∈ Set.Icc (0 : Real) t1, K' t ≤ -a t * K t + b t) (hfInt : IntervalIntegrable (fun t => a t * Real.exp (∫ u in (0 : Real)..t, a u) * K t + Real.exp (∫ u in (0 : Real)..t, a u) * K' t) MeasureTheory.volume 0 t1) (hgInt : IntervalIntegrable (fun t => Real.exp (∫ u in (0 : Real)..t, a u) * b t) MeasureTheory.volume 0 t1) (h0t : ∀ t ∈ Set.uIcc (0 : Real) t1, IntervalIntegrable a MeasureTheory.volume 0 t) (htt1 : ∀ t ∈ Set.uIcc (0 : Real) t1, IntervalIntegrable a MeasureTheory.volume t t1) : K t1 ≤ Real.exp (-(∫ u in (0 : Real)..t1, a u)) * K 0 + ∫ t in (0 : Real)..t1, Real.exp (-(∫ u in t..t1, a u)) * b t := by let F : Real → Real := fun t => Real.exp (∫ u in (0 : Real)..t, a u) * K t let f' : Real → Real := fun t => a t * Real.exp (∫ u in (0 : Real)..t, a u) * K t + Real.exp (∫ u in (0 : Real)..t, a u) * K' t let g : Real → Real := fun t => Real.exp (∫ u in (0 : Real)..t, a u) * b t have hF' : ∀ t ∈ Set.uIcc (0 : Real) t1, HasDerivAt F (f' t) t := by intro t ht simpa [F, f'] using (gronwallIntegratingFactorProductDerivative (fun x => ∫ u in (0 : Real)..x, a u) K a K' t (hA t ht) (hK t ht)) have hle : ∀ t ∈ Set.Icc (0 : Real) t1, f' t ≤ g t := by intro t ht simpa [f', g] using (gronwallIntegratingFactorDerivativeInequalityScalar (a t) (Real.exp (∫ u in (0 : Real)..t, a u)) (K t) (K' t) (b t) (le_of_lt (Real.exp_pos _)) (hineq t ht)) have hint : F t1 - F 0 ≤ ∫ t in (0 : Real)..t1, g t := by exact gronwallOrderIntegrationOfHasDerivAt F f' g t1 ht1 hF' hfInt hgInt hle have hend : Real.exp (∫ u in (0 : Real)..t1, a u) * K t1 ≤ K 0 + ∫ t in (0 : Real)..t1, g t := by exact gronwallEndpointEvaluationScalar a K t1 (∫ t in (0 : Real)..t1, g t) (by simpa [F] using hint) have hmul := gronwallEndpointMultiplyByExpNegScalar (∫ u in (0 : Real)..t1, a u) (K t1) (K 0) (∫ t in (0 : Real)..t1, g t) hend have hrewrite : Real.exp (-(∫ u in (0 : Real)..t1, a u)) * (∫ t in (0 : Real)..t1, g t) = ∫ t in (0 : Real)..t1, Real.exp (-(∫ u in t..t1, a u)) * b t := by simpa [g] using gronwallEndpointIntegralRewrite a b t1 h0t htt1 calc K t1 ≤ Real.exp (-(∫ u in (0 : Real)..t1, a u)) * K 0 + Real.exp (-(∫ u in (0 : Real)..t1, a u)) * (∫ t in (0 : Real)..t1, g t) := hmul _ = Real.exp (-(∫ u in (0 : Real)..t1, a u)) * K 0 + ∫ t in (0 : Real)..t1, Real.exp (-(∫ u in t..t1, a u)) * b t := by rw [hrewrite] /-- Endpoint-safe global Gronwall assembly with interior derivatives. This is the same displayed bound as `gronwallIntegratingFactorBoundOfDerivatives`, but the FTC/order-integration step only requires continuity of the integrating-factor product on `[0,t1]` and right derivatives on `(0,t1)`. It therefore removes the artificial need to differentiate the product at the two closed endpoints while keeping every remaining Mathlib side condition explicit. -/ theorem gronwallIntegratingFactorBoundOfInteriorDerivatives (a b K K' : Real → Real) (t1 : Real) (ht1 : 0 ≤ t1) (hFcont : ContinuousOn (fun t => Real.exp (∫ u in (0 : Real)..t, a u) * K t) (Set.Icc (0 : Real) t1)) (hA : ∀ t ∈ Set.Ioo (0 : Real) t1, HasDerivAt (fun x => ∫ u in (0 : Real)..x, a u) (a t) t) (hK : ∀ t ∈ Set.Ioo (0 : Real) t1, HasDerivAt K (K' t) t) (hineq : ∀ t ∈ Set.Icc (0 : Real) t1, K' t ≤ -a t * K t + b t) (hfInt : IntervalIntegrable (fun t => a t * Real.exp (∫ u in (0 : Real)..t, a u) * K t + Real.exp (∫ u in (0 : Real)..t, a u) * K' t) MeasureTheory.volume 0 t1) (hgInt : IntervalIntegrable (fun t => Real.exp (∫ u in (0 : Real)..t, a u) * b t) MeasureTheory.volume 0 t1) (h0t : ∀ t ∈ Set.uIcc (0 : Real) t1, IntervalIntegrable a MeasureTheory.volume 0 t) (htt1 : ∀ t ∈ Set.uIcc (0 : Real) t1, IntervalIntegrable a MeasureTheory.volume t t1) : K t1 ≤ Real.exp (-(∫ u in (0 : Real)..t1, a u)) * K 0 + ∫ t in (0 : Real)..t1, Real.exp (-(∫ u in t..t1, a u)) * b t := by let F : Real → Real := fun t => Real.exp (∫ u in (0 : Real)..t, a u) * K t let f' : Real → Real := fun t => a t * Real.exp (∫ u in (0 : Real)..t, a u) * K t + Real.exp (∫ u in (0 : Real)..t, a u) * K' t let g : Real → Real := fun t => Real.exp (∫ u in (0 : Real)..t, a u) * b t have hF' : ∀ t ∈ Set.Ioo (0 : Real) t1, HasDerivWithinAt F (f' t) (Set.Ioi t) t := by intro t ht simpa [F, f'] using (gronwallIntegratingFactorProductDerivative (fun x => ∫ u in (0 : Real)..x, a u) K a K' t (hA t ht) (hK t ht)).hasDerivWithinAt have hle : ∀ t ∈ Set.Icc (0 : Real) t1, f' t ≤ g t := by intro t ht simpa [f', g] using (gronwallIntegratingFactorDerivativeInequalityScalar (a t) (Real.exp (∫ u in (0 : Real)..t, a u)) (K t) (K' t) (b t) (le_of_lt (Real.exp_pos _)) (hineq t ht)) have hint : F t1 - F 0 ≤ ∫ t in (0 : Real)..t1, g t := by exact gronwallOrderIntegrationOfHasDerivRight F f' g t1 ht1 (by simpa [F] using hFcont) hF' hfInt hgInt hle have hend : Real.exp (∫ u in (0 : Real)..t1, a u) * K t1 ≤ K 0 + ∫ t in (0 : Real)..t1, g t := by exact gronwallEndpointEvaluationScalar a K t1 (∫ t in (0 : Real)..t1, g t) (by simpa [F] using hint) have hmul := gronwallEndpointMultiplyByExpNegScalar (∫ u in (0 : Real)..t1, a u) (K t1) (K 0) (∫ t in (0 : Real)..t1, g t) hend have hrewrite : Real.exp (-(∫ u in (0 : Real)..t1, a u)) * (∫ t in (0 : Real)..t1, g t) = ∫ t in (0 : Real)..t1, Real.exp (-(∫ u in t..t1, a u)) * b t := by simpa [g] using gronwallEndpointIntegralRewrite a b t1 h0t htt1 calc K t1 ≤ Real.exp (-(∫ u in (0 : Real)..t1, a u)) * K 0 + Real.exp (-(∫ u in (0 : Real)..t1, a u)) * (∫ t in (0 : Real)..t1, g t) := hmul _ = Real.exp (-(∫ u in (0 : Real)..t1, a u)) * K 0 + ∫ t in (0 : Real)..t1, Real.exp (-(∫ u in t..t1, a u)) * b t := by rw [hrewrite] /-- FTC-backed form of the global Gronwall assembly. This uses Mathlib's right-endpoint derivative theorem for the integral `t ↦ ∫_0^t a`, then delegates the rest of the proof to `gronwallIntegratingFactorBoundOfDerivatives`. -/ theorem gronwallIntegratingFactorBoundOfIntegral (a b K K' : Real → Real) (t1 : Real) (ht1 : 0 ≤ t1) (haInt : ∀ t ∈ Set.uIcc (0 : Real) t1, IntervalIntegrable a MeasureTheory.volume 0 t) (haMeas : ∀ t ∈ Set.uIcc (0 : Real) t1, StronglyMeasurableAtFilter a (nhds t) MeasureTheory.volume) (haCont : ∀ t ∈ Set.uIcc (0 : Real) t1, ContinuousAt a t) (hK : ∀ t ∈ Set.uIcc (0 : Real) t1, HasDerivAt K (K' t) t) (hineq : ∀ t ∈ Set.Icc (0 : Real) t1, K' t ≤ -a t * K t + b t) (hfInt : IntervalIntegrable (fun t => a t * Real.exp (∫ u in (0 : Real)..t, a u) * K t + Real.exp (∫ u in (0 : Real)..t, a u) * K' t) MeasureTheory.volume 0 t1) (hgInt : IntervalIntegrable (fun t => Real.exp (∫ u in (0 : Real)..t, a u) * b t) MeasureTheory.volume 0 t1) (htt1 : ∀ t ∈ Set.uIcc (0 : Real) t1, IntervalIntegrable a MeasureTheory.volume t t1) : K t1 ≤ Real.exp (-(∫ u in (0 : Real)..t1, a u)) * K 0 + ∫ t in (0 : Real)..t1, Real.exp (-(∫ u in t..t1, a u)) * b t := by exact gronwallIntegratingFactorBoundOfDerivatives a b K K' t1 ht1 (fun t ht => intervalIntegral.integral_hasDerivAt_right (haInt t ht) (haMeas t ht) (haCont t ht)) hK hineq hfInt hgInt haInt htt1 /-- Continuous coefficients supply the interval-integrability and local FTC side conditions needed by the appendix Gronwall assembly. This is still below the source lemma: it assumes global continuity of the coefficient `a`, while the remaining Gronwall wrapper must still choose an endpoint-safe derivative formulation for `K` on the closed source interval. -/ theorem gronwallCoefficientSideConditionsOfContinuous (a : Real → Real) (t1 : Real) (ha : Continuous a) : (∀ t ∈ Set.uIcc (0 : Real) t1, IntervalIntegrable a MeasureTheory.volume 0 t) ∧ (∀ t ∈ Set.uIcc (0 : Real) t1, StronglyMeasurableAtFilter a (nhds t) MeasureTheory.volume) ∧ (∀ t ∈ Set.uIcc (0 : Real) t1, ContinuousAt a t) ∧ (∀ t ∈ Set.uIcc (0 : Real) t1, IntervalIntegrable a MeasureTheory.volume t t1) := by exact ⟨fun t _ => ha.intervalIntegrable 0 t, fun t _ => ha.stronglyMeasurableAtFilter MeasureTheory.volume (nhds t), fun _ _ => ha.continuousAt, fun t _ => ha.intervalIntegrable t t1⟩ /-- Continuous-data wrapper for the appendix Gronwall display. Compared with `gronwallIntegratingFactorBoundOfIntegral`, this theorem proves the integrability of the derivative-side and right-hand-side integrands from global continuity of `a`, `b`, `K`, and the selected derivative witness `K'`. It preserves the source signs and constants and does not assume positivity of `a` or `b`. -/ theorem gronwallIntegratingFactorBoundOfContinuousData (a b K K' : Real → Real) (t1 : Real) (ht1 : 0 ≤ t1) (ha : Continuous a) (hb : Continuous b) (hKcont : Continuous K) (hK'cont : Continuous K') (hK : ∀ t ∈ Set.uIcc (0 : Real) t1, HasDerivAt K (K' t) t) (hineq : ∀ t ∈ Set.Icc (0 : Real) t1, K' t ≤ -a t * K t + b t) : K t1 ≤ Real.exp (-(∫ u in (0 : Real)..t1, a u)) * K 0 + ∫ t in (0 : Real)..t1, Real.exp (-(∫ u in t..t1, a u)) * b t := by have hside := gronwallCoefficientSideConditionsOfContinuous a t1 ha have hAcont : Continuous (fun t => ∫ u in (0 : Real)..t, a u) := intervalIntegral.continuous_primitive (fun x y => ha.intervalIntegrable x y) 0 have hIcont : Continuous (fun t => Real.exp (∫ u in (0 : Real)..t, a u)) := by exact Real.continuous_exp.comp hAcont have hfInt : IntervalIntegrable (fun t => a t * Real.exp (∫ u in (0 : Real)..t, a u) * K t + Real.exp (∫ u in (0 : Real)..t, a u) * K' t) MeasureTheory.volume 0 t1 := by exact (((ha.mul hIcont).mul hKcont).add (hIcont.mul hK'cont)).intervalIntegrable 0 t1 have hgInt : IntervalIntegrable (fun t => Real.exp (∫ u in (0 : Real)..t, a u) * b t) MeasureTheory.volume 0 t1 := by exact (hIcont.mul hb).intervalIntegrable 0 t1 exact gronwallIntegratingFactorBoundOfIntegral a b K K' t1 ht1 hside.1 hside.2.1 hside.2.2.1 hK hineq hfInt hgInt hside.2.2.2 /-- Source-facing derivative wrapper for the appendix Gronwall display. This version writes the paper's derivative term as `deriv K`. It keeps one explicit interval-integrability hypothesis for the product-derivative integrand, which is the remaining FTC-side condition not supplied by a bare ``K is differentiable'' reading of `appendix.tex:47-71`. -/ theorem gronwallIntegratingFactorBoundOfDifferentiable (a b K : Real → Real) (t1 : Real) (ht1 : 0 ≤ t1) (ha : Continuous a) (hb : Continuous b) (hK : Differentiable Real K) (hfInt : IntervalIntegrable (fun t => a t * Real.exp (∫ u in (0 : Real)..t, a u) * K t + Real.exp (∫ u in (0 : Real)..t, a u) * deriv K t) MeasureTheory.volume 0 t1) (hineq : ∀ t ∈ Set.Icc (0 : Real) t1, deriv K t ≤ -a t * K t + b t) : K t1 ≤ Real.exp (-(∫ u in (0 : Real)..t1, a u)) * K 0 + ∫ t in (0 : Real)..t1, Real.exp (-(∫ u in t..t1, a u)) * b t := by have hside := gronwallCoefficientSideConditionsOfContinuous a t1 ha have hAcont : Continuous (fun t => ∫ u in (0 : Real)..t, a u) := intervalIntegral.continuous_primitive (fun x y => ha.intervalIntegrable x y) 0 have hIcont : Continuous (fun t => Real.exp (∫ u in (0 : Real)..t, a u)) := by exact Real.continuous_exp.comp hAcont have hgInt : IntervalIntegrable (fun t => Real.exp (∫ u in (0 : Real)..t, a u) * b t) MeasureTheory.volume 0 t1 := by exact (hIcont.mul hb).intervalIntegrable 0 t1 exact gronwallIntegratingFactorBoundOfIntegral a b K (deriv K) t1 ht1 hside.1 hside.2.1 hside.2.2.1 (fun t _ => (hK t).hasDerivAt) hineq hfInt hgInt hside.2.2.2 /-- C1-style source-facing Gronwall wrapper. When the selected derivative witness is continuous, the interval-integrability left explicit in `gronwallIntegratingFactorBoundOfDifferentiable` is produced from continuity. This is still recorded below the paper's bare source wording, because it interprets "differentiable" as a C1-compatible backend. -/ theorem gronwallIntegratingFactorBoundOfC1 (a b K : Real → Real) (t1 : Real) (ht1 : 0 ≤ t1) (ha : Continuous a) (hb : Continuous b) (hK : Differentiable Real K) (hKderiv : Continuous (deriv K)) (hineq : ∀ t ∈ Set.Icc (0 : Real) t1, deriv K t ≤ -a t * K t + b t) : K t1 ≤ Real.exp (-(∫ u in (0 : Real)..t1, a u)) * K 0 + ∫ t in (0 : Real)..t1, Real.exp (-(∫ u in t..t1, a u)) * b t := by have hAcont : Continuous (fun t => ∫ u in (0 : Real)..t, a u) := intervalIntegral.continuous_primitive (fun x y => ha.intervalIntegrable x y) 0 have hIcont : Continuous (fun t => Real.exp (∫ u in (0 : Real)..t, a u)) := by exact Real.continuous_exp.comp hAcont have hfInt : IntervalIntegrable (fun t => a t * Real.exp (∫ u in (0 : Real)..t, a u) * K t + Real.exp (∫ u in (0 : Real)..t, a u) * deriv K t) MeasureTheory.volume 0 t1 := by exact (((ha.mul hIcont).mul hK.continuous).add (hIcont.mul hKderiv)).intervalIntegrable 0 t1 exact gronwallIntegratingFactorBoundOfDifferentiable a b K t1 ht1 ha hb hK hfInt hineq /-- Continuous-data Gronwall assembly with only interior derivatives for `K`. This discharges the interval-integrability and integral-FTC side conditions as in `gronwallIntegratingFactorBoundOfContinuousData`, but it uses the endpoint-safe interior-derivative assembly above. Thus the derivative of `K` is needed only on `(0,t1)`; the closed endpoints are handled by continuity and the integrated endpoint evaluation. -/ theorem gronwallIntegratingFactorBoundOfInteriorContinuousData (a b K K' : Real → Real) (t1 : Real) (ht1 : 0 ≤ t1) (ha : Continuous a) (hb : Continuous b) (hKcont : Continuous K) (hK'cont : Continuous K') (hK : ∀ t ∈ Set.Ioo (0 : Real) t1, HasDerivAt K (K' t) t) (hineq : ∀ t ∈ Set.Icc (0 : Real) t1, K' t ≤ -a t * K t + b t) : K t1 ≤ Real.exp (-(∫ u in (0 : Real)..t1, a u)) * K 0 + ∫ t in (0 : Real)..t1, Real.exp (-(∫ u in t..t1, a u)) * b t := by have hside := gronwallCoefficientSideConditionsOfContinuous a t1 ha have hAcont : Continuous (fun t => ∫ u in (0 : Real)..t, a u) := intervalIntegral.continuous_primitive (fun x y => ha.intervalIntegrable x y) 0 have hIcont : Continuous (fun t => Real.exp (∫ u in (0 : Real)..t, a u)) := by exact Real.continuous_exp.comp hAcont have hFcont : ContinuousOn (fun t => Real.exp (∫ u in (0 : Real)..t, a u) * K t) (Set.Icc (0 : Real) t1) := by exact (hIcont.mul hKcont).continuousOn have hfInt : IntervalIntegrable (fun t => a t * Real.exp (∫ u in (0 : Real)..t, a u) * K t + Real.exp (∫ u in (0 : Real)..t, a u) * K' t) MeasureTheory.volume 0 t1 := by exact (((ha.mul hIcont).mul hKcont).add (hIcont.mul hK'cont)).intervalIntegrable 0 t1 have hgInt : IntervalIntegrable (fun t => Real.exp (∫ u in (0 : Real)..t, a u) * b t) MeasureTheory.volume 0 t1 := by exact (hIcont.mul hb).intervalIntegrable 0 t1 exact gronwallIntegratingFactorBoundOfInteriorDerivatives a b K K' t1 ht1 hFcont (fun t _ => intervalIntegral.integral_hasDerivAt_right (ha.intervalIntegrable 0 t) (ha.stronglyMeasurableAtFilter MeasureTheory.volume (nhds t)) ha.continuousAt) hK hineq hfInt hgInt hside.1 hside.2.2.2 /-- C1-compatible source wrapper with no endpoint derivative hypothesis on `K`. This uses `deriv K` for the paper's `dK_t/dt` term, but only assumes differentiability of `K` on the open source interval. Continuity of `K` and `deriv K` supplies the endpoint and interval-integrability data. The original paper lemma still remains an obligation because its bare wording does not state this C1-compatible closed-interval interpretation explicitly. -/ theorem gronwallIntegratingFactorBoundOfInteriorC1 (a b K : Real → Real) (t1 : Real) (ht1 : 0 ≤ t1) (ha : Continuous a) (hb : Continuous b) (hKcont : Continuous K) (hKdiff : ∀ t ∈ Set.Ioo (0 : Real) t1, DifferentiableAt Real K t) (hKderiv : Continuous (deriv K)) (hineq : ∀ t ∈ Set.Icc (0 : Real) t1, deriv K t ≤ -a t * K t + b t) : K t1 ≤ Real.exp (-(∫ u in (0 : Real)..t1, a u)) * K 0 + ∫ t in (0 : Real)..t1, Real.exp (-(∫ u in t..t1, a u)) * b t := by exact gronwallIntegratingFactorBoundOfInteriorContinuousData a b K (deriv K) t1 ht1 ha hb hKcont hKderiv (fun t ht => (hKdiff t ht).hasDerivAt) hineq /-- Assemble interval-integrability of the continuous forward-KL Gronwall coefficient from its source LSI and alpha pieces. For `thm:forward-KL`, `lsiPart` is `dot{s}(t)*C_LSI(t)` and `alphaPart` is `(1/2)*dot{s}(t)^(-1)*alpha^(-1)`. This only packages Mathlib closure of `IntervalIntegrable` under subtraction; the paper-specific regularity of each piece remains a side condition. -/ theorem forwardKlGronwallCoeffIntervalIntegrable (lsiPart alphaPart : Real → Real) (a b : Real) (hlsi : IntervalIntegrable lsiPart MeasureTheory.volume a b) (halpha : IntervalIntegrable alphaPart MeasureTheory.volume a b) : IntervalIntegrable (fun t => lsiPart t - alphaPart t) MeasureTheory.volume a b := by exact hlsi.sub halpha /-- Adjacent-interval version of `forwardKlGronwallCoeffIntervalIntegrable` for the continuous forward-KL Gronwall exponent bridge. The hypotheses are exactly the theorem-specific interval-integrability data still owed for the source pieces on `[0,t]` and `[t,T]`. -/ theorem forwardKlGronwallCoeffAdjacentIntervalIntegrable (lsiPart alphaPart : Real → Real) (T : Real) (hlsi0t : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable lsiPart MeasureTheory.volume 0 t) (halpha0t : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable alphaPart MeasureTheory.volume 0 t) (hlsiT : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable lsiPart MeasureTheory.volume t T) (halphaT : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable alphaPart MeasureTheory.volume t T) : (∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable (fun u => lsiPart u - alphaPart u) MeasureTheory.volume 0 t) ∧ (∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable (fun u => lsiPart u - alphaPart u) MeasureTheory.volume t T) := by constructor · intro t ht exact forwardKlGronwallCoeffIntervalIntegrable lsiPart alphaPart 0 t (hlsi0t t ht) (halpha0t t ht) · intro t ht exact forwardKlGronwallCoeffIntervalIntegrable lsiPart alphaPart t T (hlsiT t ht) (halphaT t ht) /-- Continuous forward-KL use site for the compiled Gronwall exponent congruence. Once the LSI and alpha pieces of `a(t)=dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1)` are interval integrable on the adjacent intervals, this applies `SALD.gronwallExpProductRewriteIntegralCongr` to the assembled coefficient. It does not prove those theorem-specific regularity hypotheses, Gronwall, endpoint rewrites, or the residual exponent monotonicity. -/ theorem forwardKlGronwallExpProductRewriteIntegralCongrOfPieces (lsiPart alphaPart b : Real → Real) (T : Real) (hlsi0t : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable lsiPart MeasureTheory.volume 0 t) (halpha0t : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable alphaPart MeasureTheory.volume 0 t) (hlsiT : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable lsiPart MeasureTheory.volume t T) (halphaT : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable alphaPart MeasureTheory.volume t T) : (∫ t in (0 : Real)..T, (Real.exp (-(∫ u in (0 : Real)..T, lsiPart u - alphaPart u)) * Real.exp (∫ u in (0 : Real)..t, lsiPart u - alphaPart u)) * b t) = ∫ t in (0 : Real)..T, Real.exp (-(∫ u in t..T, lsiPart u - alphaPart u)) * b t := by have hint := forwardKlGronwallCoeffAdjacentIntervalIntegrable lsiPart alphaPart T hlsi0t halpha0t hlsiT halphaT exact gronwallExpProductRewriteIntegralCongr (fun u => lsiPart u - alphaPart u) b T hint.1 hint.2 /-- Integral subtraction for the continuous forward-KL Gronwall coefficient. This is the local interval-integral algebra behind the source split `a(t)=dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1)`. The theorem-specific regularity of the LSI and alpha pieces remains an explicit input. -/ theorem forwardKlGronwallCoeffIntegralSub (lsiPart alphaPart : Real → Real) (a b : Real) (hlsi : IntervalIntegrable lsiPart MeasureTheory.volume a b) (halpha : IntervalIntegrable alphaPart MeasureTheory.volume a b) : (∫ u in a..b, lsiPart u - alphaPart u) = (∫ u in a..b, lsiPart u) - ∫ u in a..b, alphaPart u := by exact intervalIntegral.integral_sub hlsi halpha /-- Scalar split of the initial Gronwall exponent in `thm:forward-KL`. This proves only the Real exponential algebra in `appendix.tex:249-250`; the integral identities producing the two pieces are supplied by `forwardKlGronwallCoeffIntegralSub`. -/ theorem forwardKlGronwallInitialExponentSplitScalar (lsiInt alphaInt K0 : Real) : Real.exp (-(lsiInt - alphaInt)) * K0 = Real.exp (-lsiInt) * Real.exp alphaInt * K0 := by have harg : -(lsiInt - alphaInt) = -lsiInt + alphaInt := by ring rw [harg, Real.exp_add] /-- Initial-term exponent split for the continuous forward-KL theorem display. Given interval-integrability of the LSI and alpha pieces on `[0,T]`, this matches the source's two exponential factors multiplying the initial KL term in `main_body.tex:243-245`. -/ theorem forwardKlGronwallInitialExponentSplitOfPieces (lsiPart alphaPart : Real → Real) (T K0 : Real) (hlsi : IntervalIntegrable lsiPart MeasureTheory.volume 0 T) (halpha : IntervalIntegrable alphaPart MeasureTheory.volume 0 T) : Real.exp (-(∫ u in (0 : Real)..T, lsiPart u - alphaPart u)) * K0 = Real.exp (-(∫ u in (0 : Real)..T, lsiPart u)) * Real.exp (∫ u in (0 : Real)..T, alphaPart u) * K0 := by have hsplit := forwardKlGronwallCoeffIntegralSub lsiPart alphaPart (0 : Real) T hlsi halpha rw [hsplit] exact forwardKlGronwallInitialExponentSplitScalar (∫ u in (0 : Real)..T, lsiPart u) (∫ u in (0 : Real)..T, alphaPart u) K0 /-- Pointwise residual-exponent drop for the final forward-KL display. The source drops the nonpositive LSI contribution inside `exp(-int_t^T a)`. This lemma starts after the interval integral of the LSI piece has been shown nonnegative and after the residual integrand is known nonnegative. -/ theorem forwardKlGronwallResidualExponentDropScalar (lsiInt alphaInt b : Real) (hlsi : 0 ≤ lsiInt) (hb : 0 ≤ b) : Real.exp (-(lsiInt - alphaInt)) * b ≤ Real.exp alphaInt * b := by have hexp : Real.exp (-(lsiInt - alphaInt)) ≤ Real.exp alphaInt := by exact Real.exp_le_exp.mpr (by linarith) exact mul_le_mul_of_nonneg_right hexp hb /-- Integral residual-exponent drop for `thm:forward-KL`. This packages the last display-matching inequality in `appendix.tex:248-251` under explicit side conditions: adjacent interval-integrability for the LSI and alpha coefficient pieces, nonnegativity of the LSI integral, nonnegativity of the residual `b(t)`, and interval-integrability of both outer integrands. It does not prove those theorem-specific side conditions or the Gronwall lemma. -/ theorem forwardKlGronwallResidualExponentDropIntegral (lsiPart alphaPart b : Real → Real) (T : Real) (hT : 0 ≤ T) (hleftInt : IntervalIntegrable (fun t => Real.exp (-(∫ u in t..T, lsiPart u - alphaPart u)) * b t) MeasureTheory.volume 0 T) (hrightInt : IntervalIntegrable (fun t => Real.exp (∫ u in t..T, alphaPart u) * b t) MeasureTheory.volume 0 T) (hlsiT : ∀ t ∈ Set.Icc (0 : Real) T, IntervalIntegrable lsiPart MeasureTheory.volume t T) (halphaT : ∀ t ∈ Set.Icc (0 : Real) T, IntervalIntegrable alphaPart MeasureTheory.volume t T) (hlsiNonneg : ∀ t ∈ Set.Icc (0 : Real) T, 0 ≤ ∫ u in t..T, lsiPart u) (hbNonneg : ∀ t ∈ Set.Icc (0 : Real) T, 0 ≤ b t) : (∫ t in (0 : Real)..T, Real.exp (-(∫ u in t..T, lsiPart u - alphaPart u)) * b t) ≤ ∫ t in (0 : Real)..T, Real.exp (∫ u in t..T, alphaPart u) * b t := by apply intervalIntegral.integral_mono_on hT hleftInt hrightInt intro t ht have hsplit := forwardKlGronwallCoeffIntegralSub lsiPart alphaPart t T (hlsiT t ht) (halphaT t ht) have hscalar := forwardKlGronwallResidualExponentDropScalar (∫ u in t..T, lsiPart u) (∫ u in t..T, alphaPart u) (b t) (hlsiNonneg t ht) (hbNonneg t ht) simpa [hsplit] using hscalar /-- Scalar positive-alpha division for the continuous forward-KL DV step. After the cited DV formula gives `alpha * energy <= kl + logMgf`, the paper divides by `alpha > 0` and rewrites `alpha^(-1) * logMgf` as the alpha-complexity density. This closes only that real-order algebra; the finite log-mgf, measurability, common-space, and DV formula hypotheses remain separate obligations. -/ theorem forwardKlDvPositiveAlphaScalingScalar (alpha energy kl logMgf eAlpha : Real) (halpha : 0 < alpha) (hdv : alpha * energy ≤ kl + logMgf) (heAlpha : eAlpha = alpha⁻¹ * logMgf) : energy ≤ alpha⁻¹ * kl + eAlpha := by have hscaled : alpha⁻¹ * (alpha * energy) ≤ alpha⁻¹ * (kl + logMgf) := by exact mul_le_mul_of_nonneg_left hdv (inv_nonneg.mpr halpha.le) calc energy = alpha⁻¹ * (alpha * energy) := by field_simp [ne_of_gt halpha] _ ≤ alpha⁻¹ * (kl + logMgf) := hscaled _ = alpha⁻¹ * kl + eAlpha := by rw [heAlpha] ring /-- Coefficient-preserving form of `forwardKlDvPositiveAlphaScalingScalar`. This is the scalar handoff to the Gronwall coefficient audit: once the nonnegative prefactor, later `(1/2) * dot{s}(t)^(-1)`, has been supplied, the KL coefficient is still exactly `coeff * alpha^(-1)`. -/ theorem forwardKlDvPositiveAlphaCoefficientScalar (alpha energy kl logMgf eAlpha coeff : Real) (halpha : 0 < alpha) (hcoeff : 0 ≤ coeff) (hdv : alpha * energy ≤ kl + logMgf) (heAlpha : eAlpha = alpha⁻¹ * logMgf) : coeff * energy ≤ (coeff * alpha⁻¹) * kl + coeff * eAlpha := by have hbase := forwardKlDvPositiveAlphaScalingScalar alpha energy kl logMgf eAlpha halpha hdv heAlpha have hmul : coeff * energy ≤ coeff * (alpha⁻¹ * kl + eAlpha) := by exact mul_le_mul_of_nonneg_left hbase hcoeff calc coeff * energy ≤ coeff * (alpha⁻¹ * kl + eAlpha) := hmul _ = (coeff * alpha⁻¹) * kl + coeff * eAlpha := by ring /-- Post-DV scalar handoff to the continuous forward-KL Gronwall coefficient. This is the source step in `appendix.tex:230-244`: after the pre-DV derivative inequality has the form `dK/dt <= -(dot{s}*C_LSI)*K + coeff*energy`, and DV supplies `energy <= alpha^(-1)*K + E_alpha`, the coefficient becomes exactly `dot{s}*C_LSI - coeff*alpha^(-1)`. The KL derivative, Fokker--Planck, finite-log-mgf, and selected-test DV hypotheses remain explicit inputs. -/ theorem forwardKlPostDvGronwallCoefficientScalar (dKdt kl cLSI dotS alpha energy logMgf eAlpha coeff : Real) (hpre : dKdt ≤ -(dotS * cLSI) * kl + coeff * energy) (halpha : 0 < alpha) (hcoeff : 0 ≤ coeff) (hdv : alpha * energy ≤ kl + logMgf) (heAlpha : eAlpha = alpha⁻¹ * logMgf) : dKdt ≤ -((dotS * cLSI) - coeff * alpha⁻¹) * kl + coeff * eAlpha := by have hdvCoeff := forwardKlDvPositiveAlphaCoefficientScalar alpha energy kl logMgf eAlpha coeff halpha hcoeff hdv heAlpha calc dKdt ≤ -(dotS * cLSI) * kl + coeff * energy := hpre _ ≤ -(dotS * cLSI) * kl + ((coeff * alpha⁻¹) * kl + coeff * eAlpha) := by linarith _ = -((dotS * cLSI) - coeff * alpha⁻¹) * kl + coeff * eAlpha := by ring /-- Source-shaped post-DV handoff for `thm:forward-KL`. This specializes `forwardKlPostDvGronwallCoefficientScalar` to the coefficient `coeff=(1/2)*dot{s}(t)^(-1)` that appears immediately before the Gronwall application in `appendix.tex:230-244`. -/ theorem forwardKlPostDvGronwallCoefficientOfScheduleScalar (dKdt kl cLSI dotS alpha energy logMgf eAlpha : Real) (hpre : dKdt ≤ -(dotS * cLSI) * kl + (1 / 2) * dotS⁻¹ * energy) (halpha : 0 < alpha) (hcoeff : 0 ≤ (1 / 2) * dotS⁻¹) (hdv : alpha * energy ≤ kl + logMgf) (heAlpha : eAlpha = alpha⁻¹ * logMgf) : dKdt ≤ -((dotS * cLSI) - (1 / 2) * dotS⁻¹ * alpha⁻¹) * kl + (1 / 2) * dotS⁻¹ * eAlpha := by exact forwardKlPostDvGronwallCoefficientScalar dKdt kl cLSI dotS alpha energy logMgf eAlpha ((1 / 2) * dotS⁻¹) hpre halpha hcoeff hdv heAlpha /-- Adjacent-interval coefficient package for the continuous general VA-SALD Gronwall side conditions. For `thm:general-moving-target-SALD`, `lsiPart` is `(sigma_t^2/2)*dot{s}(t)*C_LSI(t)`, `alphaPart` is `sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1)`, and `residualPart` is `b(t)=sigma_t^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t)`. This proves only the Mathlib closure/packaging step: once those source pieces are interval integrable on the adjacent intervals, the assembled `a(t)=lsiPart-alphaPart` is interval integrable there too, and the supplied residual regularity is kept available for the Gronwall call. -/ theorem generalMovingTargetGronwallCoeffAdjacentIntervalIntegrable (lsiPart alphaPart residualPart : Real → Real) (T : Real) (hlsi0t : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable lsiPart MeasureTheory.volume 0 t) (halpha0t : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable alphaPart MeasureTheory.volume 0 t) (hresidual0t : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable residualPart MeasureTheory.volume 0 t) (hlsiT : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable lsiPart MeasureTheory.volume t T) (halphaT : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable alphaPart MeasureTheory.volume t T) (hresidualT : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable residualPart MeasureTheory.volume t T) : ((∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable (fun u => lsiPart u - alphaPart u) MeasureTheory.volume 0 t) ∧ (∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable (fun u => lsiPart u - alphaPart u) MeasureTheory.volume t T)) ∧ ((∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable residualPart MeasureTheory.volume 0 t) ∧ (∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable residualPart MeasureTheory.volume t T)) := by constructor · exact forwardKlGronwallCoeffAdjacentIntervalIntegrable lsiPart alphaPart T hlsi0t halpha0t hlsiT halphaT · exact ⟨hresidual0t, hresidualT⟩ /-- Continuous general VA-SALD use site for the compiled Gronwall exponent congruence. The theorem-specific hypotheses expose the adjacent interval-integrability of the sigma/LSI coefficient, the alpha coefficient, and the residual `b(t)`. The proof reuses the local Gronwall exponent congruence for the assembled `a(t)=lsiPart-alphaPart`. It does not prove the analytic source regularity, endpoint rewrites, residual exponent drop, pure-contraction clause, or the full Gronwall lemma. -/ theorem generalMovingTargetGronwallExpProductRewriteIntegralCongrOfPieces (lsiPart alphaPart residualPart : Real → Real) (T : Real) (hlsi0t : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable lsiPart MeasureTheory.volume 0 t) (halpha0t : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable alphaPart MeasureTheory.volume 0 t) (hresidual0t : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable residualPart MeasureTheory.volume 0 t) (hlsiT : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable lsiPart MeasureTheory.volume t T) (halphaT : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable alphaPart MeasureTheory.volume t T) (hresidualT : ∀ t ∈ Set.uIcc (0 : Real) T, IntervalIntegrable residualPart MeasureTheory.volume t T) : (∫ t in (0 : Real)..T, (Real.exp (-(∫ u in (0 : Real)..T, lsiPart u - alphaPart u)) * Real.exp (∫ u in (0 : Real)..t, lsiPart u - alphaPart u)) * residualPart t) = ∫ t in (0 : Real)..T, Real.exp (-(∫ u in t..T, lsiPart u - alphaPart u)) * residualPart t := by have hint := generalMovingTargetGronwallCoeffAdjacentIntervalIntegrable lsiPart alphaPart residualPart T hlsi0t halpha0t hresidual0t hlsiT halphaT hresidualT exact gronwallExpProductRewriteIntegralCongr (fun u => lsiPart u - alphaPart u) residualPart T hint.1.1 hint.1.2 /-- Scalar order core for the discrete forward-KL residual exponent bound. In the source application, `lsiTerm` is the nonnegative LSI contribution, `alphaTerm` is the interval alpha contribution, and `gammaTerm` is the interval Gamma contribution. The interval-integral facts producing `halpha` and `hgamma` remain theorem-specific obligations. -/ theorem discreteForwardKlGronwallCoeffIntervalIntegrable (lsiPart alphaPart gammaPart : Real → Real) (a b : Real) (hlsi : IntervalIntegrable lsiPart MeasureTheory.volume a b) (halpha : IntervalIntegrable alphaPart MeasureTheory.volume a b) (hgamma : IntervalIntegrable gammaPart MeasureTheory.volume a b) : IntervalIntegrable (fun t => lsiPart t - alphaPart t - gammaPart t) MeasureTheory.volume a b := by exact (hlsi.sub halpha).sub hgamma /-- Integral subtraction for the three-piece discrete Gronwall coefficient. For `thm:forward-KL-discrete`, the coefficient is `a(t)=dot{s}(t)*C_LSI(t)-dot{s}(t)^(-1)*alpha^(-1) -2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t)`. This lemma proves only the interval-integral algebra after the three source pieces have been shown interval-integrable. -/ theorem discreteForwardKlGronwallCoeffIntegralSubSub (lsiPart alphaPart gammaPart : Real → Real) (a b : Real) (hlsi : IntervalIntegrable lsiPart MeasureTheory.volume a b) (halpha : IntervalIntegrable alphaPart MeasureTheory.volume a b) (hgamma : IntervalIntegrable gammaPart MeasureTheory.volume a b) : (∫ u in a..b, lsiPart u - alphaPart u - gammaPart u) = (∫ u in a..b, lsiPart u) - (∫ u in a..b, alphaPart u) - ∫ u in a..b, gammaPart u := by rw [intervalIntegral.integral_sub (hlsi.sub halpha) hgamma, intervalIntegral.integral_sub hlsi halpha] /-- Scalar split of the initial Gronwall exponent for discrete forward-KL. This is the real exponential algebra behind `main_body.tex:309-315`: the initial term keeps the LSI contraction as one factor and collects the positive alpha and Gamma contributions into the second factor. -/ theorem discreteForwardKlGronwallInitialExponentSplitScalar (lsiInt alphaInt gammaInt K0 : Real) : Real.exp (-(lsiInt - alphaInt - gammaInt)) * K0 = Real.exp (-lsiInt) * Real.exp (alphaInt + gammaInt) * K0 := by have harg : -(lsiInt - alphaInt - gammaInt) = -lsiInt + (alphaInt + gammaInt) := by ring rw [harg, Real.exp_add] /-- Initial-term exponent split for the discrete forward-KL theorem display. Given interval-integrability of the LSI, alpha, and Gamma coefficient pieces on `[0,T]`, this matches the source's two exponential factors multiplying `KL(rho_0||pi_0)`. The endpoint law rewrite, linear-slowdown identities, and `barGamma` identification remain separate obligations. -/ theorem discreteForwardKlGronwallInitialExponentSplitOfPieces (lsiPart alphaPart gammaPart : Real → Real) (T K0 : Real) (hlsi : IntervalIntegrable lsiPart MeasureTheory.volume 0 T) (halpha : IntervalIntegrable alphaPart MeasureTheory.volume 0 T) (hgamma : IntervalIntegrable gammaPart MeasureTheory.volume 0 T) : Real.exp (-(∫ u in (0 : Real)..T, lsiPart u - alphaPart u - gammaPart u)) * K0 = Real.exp (-(∫ u in (0 : Real)..T, lsiPart u)) * Real.exp ((∫ u in (0 : Real)..T, alphaPart u) + ∫ u in (0 : Real)..T, gammaPart u) * K0 := by have hsplit := discreteForwardKlGronwallCoeffIntegralSubSub lsiPart alphaPart gammaPart (0 : Real) T hlsi halpha hgamma rw [hsplit] exact discreteForwardKlGronwallInitialExponentSplitScalar (∫ u in (0 : Real)..T, lsiPart u) (∫ u in (0 : Real)..T, alphaPart u) (∫ u in (0 : Real)..T, gammaPart u) K0 theorem discreteForwardKlResidualExponentBoundScalar (lsiTerm alphaTerm gammaTerm alphaBound gammaBound : Real) (hlsi : 0 ≤ lsiTerm) (halpha : alphaTerm ≤ alphaBound) (hgamma : gammaTerm ≤ gammaBound) : -(lsiTerm - alphaTerm - gammaTerm) ≤ alphaBound + gammaBound := by linarith /-- Exponential form of `discreteForwardKlResidualExponentBoundScalar`. This compiles only the monotone `Real.exp` wrapper around the scalar residual exponent inequality used in the accumulated-error bridge. -/ theorem discreteForwardKlResidualExpBoundScalar (lsiTerm alphaTerm gammaTerm alphaBound gammaBound : Real) (hlsi : 0 ≤ lsiTerm) (halpha : alphaTerm ≤ alphaBound) (hgamma : gammaTerm ≤ gammaBound) : Real.exp (-(lsiTerm - alphaTerm - gammaTerm)) ≤ Real.exp (alphaBound + gammaBound) := by exact Real.exp_le_exp.mpr (discreteForwardKlResidualExponentBoundScalar lsiTerm alphaTerm gammaTerm alphaBound gammaBound hlsi halpha hgamma) /-- Constant-factor integral core for the `A_alpha` term in the discrete forward-KL accumulated-error bridge. After the linear slowdown supplies `dot{s}(t)⁻¹ = r⁻¹`, this formalizes only the interval-integral algebra turning the residual `E_alpha` contribution into `r⁻¹ * A`. The theorem-specific identification of `A` with the paper's `A_alpha(pi,v)` remains a separate source obligation. -/ theorem discreteForwardKlAlphaComplexityCollectionScalar (energy : Real → Real) (T r A : Real) (hA : (∫ t in (0 : Real)..T, energy t) = A) : (∫ t in (0 : Real)..T, r⁻¹ * energy t) = r⁻¹ * A := by rw [intervalIntegral.integral_const_mul, hA] /-- Constant-factor integral core for the `barDelta` term in the discrete forward-KL accumulated-error bridge. After the linear slowdown supplies `dot{s}(t)=r`, this formalizes only the source algebra collecting `2*r*eta*Delta(t)` into `2*r*eta*barDelta`. The identification of `barDelta` with the paper's full-interval integral is still an obligation. -/ theorem discreteForwardKlDeltaAccumulationScalar (delta : Real → Real) (T r eta barDelta : Real) (hbarDelta : (∫ t in (0 : Real)..T, delta t) = barDelta) : (∫ t in (0 : Real)..T, 2 * r * eta * delta t) = 2 * r * eta * barDelta := by rw [intervalIntegral.integral_const_mul, hbarDelta] /-- Combined scalar/integral collection for the two additive residual terms in the discrete forward-KL accumulated-error bridge. This packages the compiled lower slice for cycle 27: once the linear-slowdown coefficient identities and the source definitions of `A_alpha` and `barDelta` are supplied, the additive residual collection has exactly the paper's `r⁻¹*A_alpha + 2*r*eta*barDelta` shape. It does not prove endpoint stitching, the residual exponent bound, or the full accumulated-error bridge. -/ theorem discreteForwardKlAccumulatedErrorCollectionScalar (energy delta : Real → Real) (T r eta A barDelta : Real) (hA : (∫ t in (0 : Real)..T, energy t) = A) (hbarDelta : (∫ t in (0 : Real)..T, delta t) = barDelta) : (∫ t in (0 : Real)..T, r⁻¹ * energy t) + (∫ t in (0 : Real)..T, 2 * r * eta * delta t) = r⁻¹ * A + 2 * r * eta * barDelta := by rw [discreteForwardKlAlphaComplexityCollectionScalar energy T r A hA, discreteForwardKlDeltaAccumulationScalar delta T r eta barDelta hbarDelta] /-- Residual-integral display bridge for discrete forward-KL. This is the cycle-61 lower scalar wrapper for the last additive term in `main_body.tex:309-323`. Once the residual Gronwall kernel has already been bounded by the common positive exponential factor, this lemma plugs in the compiled `A_alpha` and `barDelta` collection algebra. It does not prove the residual exponent bound, endpoint stitching, or the source identifications of `A_alpha` and `barDelta`. -/ theorem discreteForwardKlResidualIntegralDisplayBoundScalar (residualBound commonExp : Real) (energy delta : Real → Real) (T r eta A barDelta : Real) (hresidual : residualBound ≤ commonExp * ((∫ t in (0 : Real)..T, r⁻¹ * energy t) + (∫ t in (0 : Real)..T, 2 * r * eta * delta t))) (hA : (∫ t in (0 : Real)..T, energy t) = A) (hbarDelta : (∫ t in (0 : Real)..T, delta t) = barDelta) : residualBound ≤ commonExp * (r⁻¹ * A + 2 * r * eta * barDelta) := by rw [← discreteForwardKlAccumulatedErrorCollectionScalar energy delta T r eta A barDelta hA hbarDelta] exact hresidual /-- Main-display scalar wrapper for the discrete forward-KL accumulated-error bridge. This is the cycle-66 lower proof-producing step for `main_body.tex:309-323` after the appendix Gronwall display in `appendix.tex:557-592`. It combines the compiled initial-term exponent split with the residual-integral display wrapper. All analytic inputs to Gronwall, endpoint rewrites, residual exponent monotonicity, and source identifications of `A_alpha`, `barGamma`, and `barDelta` remain explicit hypotheses or obligations. -/ theorem discreteForwardKlMainDisplayBoundScalar (terminal residualBound lsiInt alphaInt gammaInt K0 : Real) (energy delta : Real → Real) (T r eta A barDelta : Real) (hgronwall : terminal ≤ Real.exp (-(lsiInt - alphaInt - gammaInt)) * K0 + residualBound) (hresidual : residualBound ≤ Real.exp (alphaInt + gammaInt) * ((∫ t in (0 : Real)..T, r⁻¹ * energy t) + (∫ t in (0 : Real)..T, 2 * r * eta * delta t))) (hA : (∫ t in (0 : Real)..T, energy t) = A) (hbarDelta : (∫ t in (0 : Real)..T, delta t) = barDelta) : terminal ≤ Real.exp (-lsiInt) * Real.exp (alphaInt + gammaInt) * K0 + Real.exp (alphaInt + gammaInt) * (r⁻¹ * A + 2 * r * eta * barDelta) := by have hinit := discreteForwardKlGronwallInitialExponentSplitScalar lsiInt alphaInt gammaInt K0 have hdisplay := discreteForwardKlResidualIntegralDisplayBoundScalar residualBound (Real.exp (alphaInt + gammaInt)) energy delta T r eta A barDelta hresidual hA hbarDelta calc terminal ≤ Real.exp (-(lsiInt - alphaInt - gammaInt)) * K0 + residualBound := hgronwall _ = Real.exp (-lsiInt) * Real.exp (alphaInt + gammaInt) * K0 + residualBound := by rw [hinit] _ ≤ Real.exp (-lsiInt) * Real.exp (alphaInt + gammaInt) * K0 + Real.exp (alphaInt + gammaInt) * (r⁻¹ * A + 2 * r * eta * barDelta) := by simpa [add_comm, add_left_comm, add_assoc] using add_le_add_right hdisplay (Real.exp (-lsiInt) * Real.exp (alphaInt + gammaInt) * K0) /-- Scalar inverse-schedule square identity for the discrete general VA-SALD time-change coefficient. The analytic fact that `dotT` is the derivative of the inverse schedule remains part of `sald.general_moving_target_discrete.constant_schedule_stitching`; this lemma only closes the real algebra once `dotT = dotS⁻¹` is available. -/ theorem generalMovingTargetDiscreteConstantScheduleSquareScalar (dotS dotT : Real) (hdotT : dotT = dotS⁻¹) (hdotS : dotS ≠ 0) : dotS * dotT ^ 2 = dotS⁻¹ := by rw [hdotT, pow_two] rw [← mul_assoc, mul_inv_cancel₀ hdotS] simp /-- Scalar time-change coefficient rewrite for the discrete forward-KL proof. In `appendix.tex:526-553`, the source multiplies the `s`-time DV coefficient by `dot{s}(t)` and rewrites `dot{s}(t) * dot t(s(t))^2 * coeff` as `dot{s}(t)^(-1) * coeff`. This theorem closes only that real algebra after the inverse-schedule identity `dot t(s(t)) = dot{s}(t)^(-1)` and nonzero `dot{s}(t)` have been supplied. -/ theorem discreteForwardKlTimeChangeSquareCoefficientRewriteScalar (dotS dotT coeff : Real) (hdotT : dotT = dotS⁻¹) (hdotS : dotS ≠ 0) : dotS * (dotT ^ 2 * coeff) = dotS⁻¹ * coeff := by calc dotS * (dotT ^ 2 * coeff) = (dotS * dotT ^ 2) * coeff := by ring _ = dotS⁻¹ * coeff := by rw [generalMovingTargetDiscreteConstantScheduleSquareScalar dotS dotT hdotT hdotS] /-- Scalar post-DV time-change handoff for discrete forward-KL. This is the proof-producing lower core for the cycle-56 `sald.discrete_forward_kl.gronwall_accumulation` packet. It starts after the EM/KL derivative backend, LSI comparison, frozen-defect bound, and DV velocity estimate have supplied the `s`-time inequality in appendix lines 520-523. It then performs only the real-order `s` to `t` change of variables in appendix lines 526-553, preserving the exact Gronwall coefficient `dot{s}*C_LSI - dot{s}^{-1}*alpha^{-1} - 2*dot{s}*eta^2*alpha'^{-1}*Gamma` and residual `dot{s}^{-1}*E_alpha + 2*dot{s}*eta*Delta`. -/ theorem discreteForwardKlPostDvTimeChangedDerivativeScalar (dKds dKdt kl cLSI dotS dotT alphaInv eAlpha etaSq alphaPrimeInv gamma delta eta : Real) (hdotSNonneg : 0 ≤ dotS) (hdKdt : dKdt = dotS * dKds) (hsBound : dKds ≤ -((cLSI - dotT ^ 2 * alphaInv - 2 * etaSq * alphaPrimeInv * gamma) * kl) + dotT ^ 2 * eAlpha + 2 * eta * delta) (hdotT : dotT = dotS⁻¹) (hdotS : dotS ≠ 0) : dKdt ≤ -(((dotS * cLSI) - dotS⁻¹ * alphaInv - 2 * dotS * etaSq * alphaPrimeInv * gamma) * kl) + dotS⁻¹ * eAlpha + 2 * dotS * eta * delta := by rw [hdKdt] have hmul : dotS * dKds ≤ dotS * (-((cLSI - dotT ^ 2 * alphaInv - 2 * etaSq * alphaPrimeInv * gamma) * kl) + dotT ^ 2 * eAlpha + 2 * eta * delta) := by exact mul_le_mul_of_nonneg_left hsBound hdotSNonneg calc dotS * dKds ≤ dotS * (-((cLSI - dotT ^ 2 * alphaInv - 2 * etaSq * alphaPrimeInv * gamma) * kl) + dotT ^ 2 * eAlpha + 2 * eta * delta) := hmul _ = -(((dotS * cLSI) - dotS⁻¹ * alphaInv - 2 * dotS * etaSq * alphaPrimeInv * gamma) * kl) + dotS⁻¹ * eAlpha + 2 * dotS * eta * delta := by rw [hdotT] field_simp [hdotS] /-- Pointwise Gronwall-input wrapper for the discrete forward-KL time change. The scalar theorem above handles one fixed time. This wrapper is the exact lower-facing shape needed by the Gronwall accumulation obligation: once the post-DV `s`-time derivative inequality and inverse-schedule identities are available pointwise in `t`, the source coefficient `a(t)` and residual `b(t)` from appendix lines 526-553 are available pointwise in `t`. -/ theorem discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged (K dKds dKdt cLSI dotS dotT eAlpha gamma delta : Real → Real) (alphaInv etaSq alphaPrimeInv eta : Real) (hdotSNonneg : ∀ t, 0 ≤ dotS t) (hdKdt : ∀ t, dKdt t = dotS t * dKds t) (hsBound : ∀ t, dKds t ≤ -((cLSI t - dotT t ^ 2 * alphaInv - 2 * etaSq * alphaPrimeInv * gamma t) * K t) + dotT t ^ 2 * eAlpha t + 2 * eta * delta t) (hdotT : ∀ t, dotT t = (dotS t)⁻¹) (hdotS : ∀ t, dotS t ≠ 0) : ∀ t, dKdt t ≤ -(((dotS t * cLSI t) - (dotS t)⁻¹ * alphaInv - 2 * dotS t * etaSq * alphaPrimeInv * gamma t) * K t) + (dotS t)⁻¹ * eAlpha t + 2 * dotS t * eta * delta t := by intro t exact discreteForwardKlPostDvTimeChangedDerivativeScalar (dKds t) (dKdt t) (K t) (cLSI t) (dotS t) (dotT t) alphaInv (eAlpha t) etaSq alphaPrimeInv (gamma t) (delta t) eta (hdotSNonneg t) (hdKdt t) (hsBound t) (hdotT t) (hdotS t) /-- Left-endpoint algebra for the frozen EM interpolation in `appendix.tex:260-266`. This proves only the pointwise vector identity behind `\hat X_{s_k}=X_k^\eta`: at the left endpoint the time increment and Brownian increment vanish. The stochastic law statement `\hat\rho_{s_k}=\rho_k^\eta` remains `sald.discrete_forward_kl.em_endpoint_laws`. -/ theorem discreteForwardKlEmInterpolationLeftEndpointVector {E : Type*} [AddCommGroup E] [Module Real E] (x drift w : E) (sK c : Real) : x + (sK - sK) • drift + c • (w - w) = x := by simp /-- Right-endpoint algebra for the frozen EM interpolation in `appendix.tex:260-266`. Once the mesh identity `s_{k+1}-s_k=eta` and the EM update definition for `X_{k+1}^\eta` are supplied, this identifies the interpolation endpoint with the next step. It does not prove the stochastic endpoint law `\hat\rho_{s_{k+1}}=\rho_{k+1}^\eta`. -/ theorem discreteForwardKlEmInterpolationRightEndpointVector {E : Type*} [AddCommGroup E] [Module Real E] (x xNext drift noise : E) (sK sNext eta c : Real) (hstep : sNext - sK = eta) (hupdate : xNext = x + eta • drift + c • noise) : x + (sNext - sK) • drift + c • noise = xNext := by rw [hstep, hupdate] /-- Law-level handoff from pointwise equality of random variables. This is the abstract step needed to use the endpoint-vector identities in the EM interpolation proof: once two random variables are pointwise equal, any chosen law operator gives the same law. The probabilistic construction of the law operator remains the existing endpoint-law obligation. -/ theorem discreteForwardKlLawEqOfPointwise {Ω E Law : Type*} (law : (Ω → E) → Law) (X Y : Ω → E) (hXY : ∀ ω, X ω = Y ω) : law X = law Y := by exact congrArg law (funext hXY) /-- Left-endpoint law handoff for the frozen EM interpolation. Combines the pointwise identity `\hat X_{s_k}=X_k^\eta` with an abstract law operator. It does not construct `Law`, Brownian motion, or densities. -/ theorem discreteForwardKlEmInterpolationLeftEndpointLawHandoff {Ω E Law : Type*} [AddCommGroup E] [Module Real E] (law : (Ω → E) → Law) (X drift W : Ω → E) (sK c : Real) : law (fun ω => X ω + (sK - sK) • drift ω + c • (W ω - W ω)) = law X := by exact discreteForwardKlLawEqOfPointwise law _ _ (fun ω => discreteForwardKlEmInterpolationLeftEndpointVector (X ω) (drift ω) (W ω) sK c) /-- Right-endpoint law handoff for the frozen EM interpolation. After the mesh identity and pointwise EM update definition are supplied, this turns the right-endpoint vector identity into the law equality used at `appendix.tex:334-335`. Stochastic endpoint matching is still tracked by `sald.discrete_forward_kl.em_endpoint_laws`. -/ theorem discreteForwardKlEmInterpolationRightEndpointLawHandoff {Ω E Law : Type*} [AddCommGroup E] [Module Real E] (law : (Ω → E) → Law) (X XNext drift noise : Ω → E) (sK sNext eta c : Real) (hstep : sNext - sK = eta) (hupdate : ∀ ω, XNext ω = X ω + eta • drift ω + c • noise ω) : law (fun ω => X ω + (sNext - sK) • drift ω + c • noise ω) = law XNext := by exact discreteForwardKlLawEqOfPointwise law _ _ (fun ω => discreteForwardKlEmInterpolationRightEndpointVector (X ω) (XNext ω) (drift ω) (noise ω) sK sNext eta c hstep (hupdate ω)) /-- Endpoint-law pair handoff for the frozen EM interpolation. This is the lower cycle-40 instantiation layer for `sald.discrete_forward_kl.em_endpoint_laws`: once the repository supplies named law representations for `hat rho_s`, `rho_k^eta`, and `rho_{k+1}^eta`, the compiled endpoint handoffs give the two endpoint equalities used at `appendix.tex:334-335`. The theorem does not construct the law operator, Brownian path, density, or regular conditional law. -/ theorem discreteForwardKlEmEndpointLawPairHandoff {Ω E Law : Type*} [AddCommGroup E] [Module Real E] (law : (Ω → E) → Law) (hatRho : Real → Law) (rhoEta : Nat → Law) (k : Nat) (X XNext drift W noise : Ω → E) (sK sNext eta c : Real) (hhatLeft : hatRho sK = law (fun ω => X ω + (sK - sK) • drift ω + c • (W ω - W ω))) (hrhoLeft : rhoEta k = law X) (hhatRight : hatRho sNext = law (fun ω => X ω + (sNext - sK) • drift ω + c • noise ω)) (hrhoRight : rhoEta (k + 1) = law XNext) (hstep : sNext - sK = eta) (hupdate : ∀ ω, XNext ω = X ω + eta • drift ω + c • noise ω) : hatRho sK = rhoEta k ∧ hatRho sNext = rhoEta (k + 1) := by constructor · rw [hhatLeft, hrhoLeft] exact discreteForwardKlEmInterpolationLeftEndpointLawHandoff law X drift W sK c · rw [hhatRight, hrhoRight] exact discreteForwardKlEmInterpolationRightEndpointLawHandoff law X XNext drift noise sK sNext eta c hstep hupdate /-- General VA-SALD left-endpoint law handoff for `eq:general_moving_target_SALD_frozen_interp`. This is the same abstract law transport used by the discrete forward-KL EM block, specialized to the general moving-target notation where the frozen drift is `dot t_k*c_{t_k} + (sigma_eta^2/2)*nabla log pi_{t_k}` and the Brownian increment is scaled by `sigma_eta`. It proves only the pointwise-to-law handoff; the concrete stochastic law construction remains an obligation. -/ theorem generalMovingTargetDiscreteEmInterpolationLeftEndpointLawHandoff {Ω E Law : Type*} [AddCommGroup E] [Module Real E] (law : (Ω → E) → Law) (X frozenDrift W : Ω → E) (sK sigmaEta : Real) : law (fun ω => X ω + (sK - sK) • frozenDrift ω + sigmaEta • (W ω - W ω)) = law X := by exact discreteForwardKlEmInterpolationLeftEndpointLawHandoff law X frozenDrift W sK sigmaEta /-- General VA-SALD right-endpoint law handoff for the frozen interpolation. After the mesh identity and the pointwise general EM update are supplied, this turns the source endpoint identity into the law equality `\hat\rho_{s_{k+1}}=\rho_{k+1}^\eta`. It does not prove the Brownian, conditional-law, density, or Fokker--Planck backend. -/ theorem generalMovingTargetDiscreteEmInterpolationRightEndpointLawHandoff {Ω E Law : Type*} [AddCommGroup E] [Module Real E] (law : (Ω → E) → Law) (X XNext frozenDrift noise : Ω → E) (sK sNext eta sigmaEta : Real) (hstep : sNext - sK = eta) (hupdate : ∀ ω, XNext ω = X ω + eta • frozenDrift ω + sigmaEta • noise ω) : law (fun ω => X ω + (sNext - sK) • frozenDrift ω + sigmaEta • noise ω) = law XNext := by exact discreteForwardKlEmInterpolationRightEndpointLawHandoff law X XNext frozenDrift noise sK sNext eta sigmaEta hstep hupdate /-- Endpoint-law pair handoff for the discrete general moving-target VA-SALD EM interpolation. Once named law representations for `hat rho_s`, `rho_k^eta`, and `rho_{k+1}^eta` are supplied, this proves the two endpoint law equalities used at `appendix.tex:1354-1357`. It is proof-producing endpoint bookkeeping only; the regular conditional drift, densities, and conditional Fokker--Planck equation remain separate obligations. -/ theorem generalMovingTargetDiscreteEmEndpointLawPairHandoff {Ω E Law : Type*} [AddCommGroup E] [Module Real E] (law : (Ω → E) → Law) (hatRho : Real → Law) (rhoEta : Nat → Law) (k : Nat) (X XNext frozenDrift W noise : Ω → E) (sK sNext eta sigmaEta : Real) (hhatLeft : hatRho sK = law (fun ω => X ω + (sK - sK) • frozenDrift ω + sigmaEta • (W ω - W ω))) (hrhoLeft : rhoEta k = law X) (hhatRight : hatRho sNext = law (fun ω => X ω + (sNext - sK) • frozenDrift ω + sigmaEta • noise ω)) (hrhoRight : rhoEta (k + 1) = law XNext) (hstep : sNext - sK = eta) (hupdate : ∀ ω, XNext ω = X ω + eta • frozenDrift ω + sigmaEta • noise ω) : hatRho sK = rhoEta k ∧ hatRho sNext = rhoEta (k + 1) := by constructor · rw [hhatLeft, hrhoLeft] exact generalMovingTargetDiscreteEmInterpolationLeftEndpointLawHandoff law X frozenDrift W sK sigmaEta · rw [hhatRight, hrhoRight] exact generalMovingTargetDiscreteEmInterpolationRightEndpointLawHandoff law X XNext frozenDrift noise sK sNext eta sigmaEta hstep hupdate /-- Named-interpolation endpoint-law handoff for the discrete general moving-target VA-SALD EM path. This is the lower cycle-49 endpoint slice for `appendix.tex:1354-1357`. It starts from the repository's eventual named process `hatX`, law map `hatRho s = law (hatX s)`, and named EM laws `rhoEta k`, `rhoEta (k+1)`. The result still proves only endpoint-law bookkeeping from pointwise interpolation identities; it does not construct Brownian motion, conditional laws, densities, or the conditional Fokker--Planck equation. -/ theorem generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation {Ω E Law : Type*} [AddCommGroup E] [Module Real E] (law : (Ω → E) → Law) (hatRho : Real → Law) (rhoEta : Nat → Law) (hatX : Real → Ω → E) (k : Nat) (X XNext frozenDrift W noise : Ω → E) (sK sNext eta sigmaEta : Real) (hhatLaw : ∀ s, hatRho s = law (hatX s)) (hrhoLeft : rhoEta k = law X) (hrhoRight : rhoEta (k + 1) = law XNext) (hhatLeft : ∀ ω, hatX sK ω = X ω + (sK - sK) • frozenDrift ω + sigmaEta • (W ω - W ω)) (hhatRight : ∀ ω, hatX sNext ω = X ω + (sNext - sK) • frozenDrift ω + sigmaEta • noise ω) (hstep : sNext - sK = eta) (hupdate : ∀ ω, XNext ω = X ω + eta • frozenDrift ω + sigmaEta • noise ω) : hatRho sK = rhoEta k ∧ hatRho sNext = rhoEta (k + 1) := by exact generalMovingTargetDiscreteEmEndpointLawPairHandoff law hatRho rhoEta k X XNext frozenDrift W noise sK sNext eta sigmaEta (by rw [hhatLaw sK] exact discreteForwardKlLawEqOfPointwise law _ _ hhatLeft) hrhoLeft (by rw [hhatLaw sNext] exact discreteForwardKlLawEqOfPointwise law _ _ hhatRight) hrhoRight hstep hupdate /-- Measure-level endpoint-law handoff for the discrete general moving-target VA-SALD EM path. This is the first concrete measure-theory backfill below the abstract law operator handoffs. It uses `Measure.map` and almost-everywhere endpoint identities, matching the pushforward-law style used in local SLT examples. It still does not construct Brownian motion, regular conditional drift, densities, absolute continuity, or the conditional Fokker--Planck equation. -/ theorem generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation {Ω E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [AddCommGroup E] [Module Real E] (P : MeasureTheory.Measure Ω) (hatX : Real → Ω → E) (X XNext frozenDrift W noise : Ω → E) (sK sNext eta sigmaEta : Real) (hhatLeft : hatX sK =ᵐ[P] fun ω => X ω + (sK - sK) • frozenDrift ω + sigmaEta • (W ω - W ω)) (hhatRight : hatX sNext =ᵐ[P] fun ω => X ω + (sNext - sK) • frozenDrift ω + sigmaEta • noise ω) (hstep : sNext - sK = eta) (hupdate : XNext =ᵐ[P] fun ω => X ω + eta • frozenDrift ω + sigmaEta • noise ω) : MeasureTheory.Measure.map (hatX sK) P = MeasureTheory.Measure.map X P ∧ MeasureTheory.Measure.map (hatX sNext) P = MeasureTheory.Measure.map XNext P := by constructor · exact lawMapEqOfAEEq (P := P) (X := hatX sK) (Y := X) <| by filter_upwards [hhatLeft] with ω hleft rw [hleft] exact discreteForwardKlEmInterpolationLeftEndpointVector (X ω) (frozenDrift ω) (W ω) sK sigmaEta · exact lawMapEqOfAEEq (P := P) (X := hatX sNext) (Y := XNext) <| by filter_upwards [hhatRight, hupdate] with ω hright hstepUpdate rw [hright] exact discreteForwardKlEmInterpolationRightEndpointVector (X ω) (XNext ω) (frozenDrift ω) (noise ω) sK sNext eta sigmaEta hstep hstepUpdate /-- Joint endpoint-law handoff for the discrete general moving-target VA-SALD EM path. This packages the two endpoint a.e. identities into a paired pushforward law. It is still only endpoint bookkeeping below `appendix.tex:1354-1387`; regular conditional drift, density/absolute-continuity, weak Fokker--Planck, and KL differentiation remain separate obligations. -/ theorem generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation {Ω E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [AddCommGroup E] [Module Real E] (P : MeasureTheory.Measure Ω) (hatX : Real → Ω → E) (X XNext frozenDrift W noise : Ω → E) (sK sNext eta sigmaEta : Real) (hhatLeft : hatX sK =ᵐ[P] fun ω => X ω + (sK - sK) • frozenDrift ω + sigmaEta • (W ω - W ω)) (hhatRight : hatX sNext =ᵐ[P] fun ω => X ω + (sNext - sK) • frozenDrift ω + sigmaEta • noise ω) (hstep : sNext - sK = eta) (hupdate : XNext =ᵐ[P] fun ω => X ω + eta • frozenDrift ω + sigmaEta • noise ω) : MeasureTheory.Measure.map (fun ω => (hatX sK ω, hatX sNext ω)) P = MeasureTheory.Measure.map (fun ω => (X ω, XNext ω)) P := by exact lawMapProdEqOfAEEq (P := P) (X := hatX sK) (X' := X) (Y := hatX sNext) (Y' := XNext) (by filter_upwards [hhatLeft] with ω hleft rw [hleft] exact discreteForwardKlEmInterpolationLeftEndpointVector (X ω) (frozenDrift ω) (W ω) sK sigmaEta) (by filter_upwards [hhatRight, hupdate] with ω hright hstepUpdate rw [hright] exact discreteForwardKlEmInterpolationRightEndpointVector (X ω) (XNext ω) (frozenDrift ω) (noise ω) sK sNext eta sigmaEta hstep hstepUpdate) /-- Marginal endpoint-law extraction from the joint endpoint law for the discrete general moving-target VA-SALD EM path. This composes the paired endpoint-law equality with the first/second projection lemmas. It is still common-space endpoint bookkeeping only; the conditional drift, density, weak Fokker--Planck equation, and KL derivative remain obligations. -/ theorem generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation {Ω E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [AddCommGroup E] [Module Real E] (P : MeasureTheory.Measure Ω) (hatX : Real → Ω → E) (X XNext frozenDrift W noise : Ω → E) (sK sNext eta sigmaEta : Real) (hX : Measurable X) (hXNext : Measurable XNext) (hhatLeft : hatX sK =ᵐ[P] fun ω => X ω + (sK - sK) • frozenDrift ω + sigmaEta • (W ω - W ω)) (hhatRight : hatX sNext =ᵐ[P] fun ω => X ω + (sNext - sK) • frozenDrift ω + sigmaEta • noise ω) (hstep : sNext - sK = eta) (hupdate : XNext =ᵐ[P] fun ω => X ω + eta • frozenDrift ω + sigmaEta • noise ω) : MeasureTheory.Measure.map Prod.fst (MeasureTheory.Measure.map (fun ω => (hatX sK ω, hatX sNext ω)) P) = MeasureTheory.Measure.map X P ∧ MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun ω => (hatX sK ω, hatX sNext ω)) P) = MeasureTheory.Measure.map XNext P := by have hjoint := generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation P hatX X XNext frozenDrift W noise sK sNext eta sigmaEta hhatLeft hhatRight hstep hupdate constructor · rw [hjoint] exact lawMapProdFst (P := P) (X := X) (Y := XNext) hX hXNext · rw [hjoint] exact lawMapProdSnd (P := P) (X := X) (Y := XNext) hX hXNext /-- Marginal compatibility for the joint law used in the conditional drift. For a fixed EM interpolation time `s`, the source defines `\hat\rho_s=Law(\hat X_s)` and then conditions `X_k^eta` on `\hat X_s=x`. This lemma proves only the `Measure.map` bookkeeping that the second marginal of the joint pushforward law of `(X_k^eta,\hat X_s)` is the named `\hat\rho_s`, once the named-law representation is supplied. It does not construct a regular conditional probability, conditional expectation, density, or Fokker--Planck equation. -/ theorem generalMovingTargetDiscreteHatRhoMarginalOfJointMap {Ω E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] (P : MeasureTheory.Measure Ω) (Xk hatXAtS : Ω → E) (hatRhoS : MeasureTheory.Measure E) (hXk : Measurable Xk) (hhatXAtS : Measurable hatXAtS) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) : MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) = hatRhoS := by calc MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) = MeasureTheory.Measure.map hatXAtS P := by exact lawMapProdSnd (P := P) (X := Xk) (Y := hatXAtS) hXk hhatXAtS _ = hatRhoS := hhatRhoS.symm /-- First-marginal compatibility for Mathlib's conditional-distribution orientation. For `condDistrib X_k^eta hatX_s P`, Mathlib names the joint law in the order `(hatX_s, X_k^eta)`, so the conditioning law is the first marginal. This is only `Measure.map` bookkeeping; it does not construct a conditional distribution or prove conditional-expectation identities. -/ theorem generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap {Ω E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] (P : MeasureTheory.Measure Ω) (Xk hatXAtS : Ω → E) (hatRhoS : MeasureTheory.Measure E) (hXk : Measurable Xk) (hhatXAtS : Measurable hatXAtS) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) : MeasureTheory.Measure.map Prod.fst (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P) = hatRhoS := by calc MeasureTheory.Measure.map Prod.fst (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P) = MeasureTheory.Measure.map hatXAtS P := by exact lawMapProdFst (P := P) (X := hatXAtS) (Y := Xk) hhatXAtS hXk _ = hatRhoS := hhatRhoS.symm /-- Transport a supplied conditional-kernel compatibility predicate to the named `\hat\rho_s` marginal. The analytic backend must still supply the regular conditional kernel and prove that it disintegrates the joint law. This wrapper only connects that supplied compatibility predicate from the joint law's `Measure.map Prod.snd` marginal to the named `hatRhoS = Law(hat X_s)` marginal used by the weak Fokker--Planck interface. -/ theorem generalMovingTargetDiscreteConditionalKernelCompatibilityOfJointMapMarginal {Ω E Kernel : Type*} [MeasurableSpace Ω] [MeasurableSpace E] (P : MeasureTheory.Measure Ω) (Xk hatXAtS : Ω → E) (hatRhoS : MeasureTheory.Measure E) (kernel : Kernel) (KernelCompatible : MeasureTheory.Measure (E × E) → MeasureTheory.Measure E → Kernel → Prop) (hXk : Measurable Xk) (hhatXAtS : Measurable hatXAtS) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hcompat : KernelCompatible (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) (MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P)) kernel) (hcongr : ∀ {μ ν : MeasureTheory.Measure E}, μ = ν → KernelCompatible (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) μ kernel → KernelCompatible (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) ν kernel) : KernelCompatible (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) hatRhoS kernel := by exact hcongr (generalMovingTargetDiscreteHatRhoMarginalOfJointMap P Xk hatXAtS hatRhoS hXk hhatXAtS hhatRhoS) hcompat /-- Bundle the second-marginal equality with conditional-kernel compatibility. This is the lower endpoint-to-conditional wrapper used before the weak Fokker--Planck statement: if a supplied kernel compatibility predicate is proved for the joint law and its `Measure.map Prod.snd` marginal, the same predicate holds for the named `hatRhoS = Law(hat X_s)` marginal. The theorem does not construct the kernel or prove disintegration; it only packages the `Measure.map` marginal bookkeeping needed by `appendix.tex:1368-1377`. -/ theorem generalMovingTargetDiscreteEndpointConditionalCompatibilityOfJointMap {Ω E Kernel : Type*} [MeasurableSpace Ω] [MeasurableSpace E] (P : MeasureTheory.Measure Ω) (Xk hatXAtS : Ω → E) (hatRhoS : MeasureTheory.Measure E) (kernel : Kernel) (KernelCompatible : MeasureTheory.Measure (E × E) → MeasureTheory.Measure E → Kernel → Prop) (hXk : Measurable Xk) (hhatXAtS : Measurable hatXAtS) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hcompat : KernelCompatible (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) (MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P)) kernel) : MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) = hatRhoS ∧ KernelCompatible (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) hatRhoS kernel := by have hmarg : MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) = hatRhoS := generalMovingTargetDiscreteHatRhoMarginalOfJointMap P Xk hatXAtS hatRhoS hXk hhatXAtS hhatRhoS constructor · exact hmarg · rw [← hmarg] exact hcompat /-- Conditional-expectation linearity wrapper for the frozen general VA-SALD drift. The analytic backend must still provide the regular conditional law and the linearity hypotheses for the selected conditional expectation. This lemma only packages the source algebra in `appendix.tex:1368-1377`: the conditional expectation of `dot t_k*c_{t_k}(X_k^eta) + (sigma_eta^2/2)*nabla log pi_{t_k}(X_k^eta)` splits into the corresponding scaled conditional expectations once that backend is supplied. -/ theorem generalMovingTargetDiscreteConditionalDriftLinearCombination {Omega X E : Type*} [AddCommGroup E] [Module Real E] (condExp : (Omega → E) → X → E) (c score : Omega → E) (dotTk sigmaCoeff : Real) (hadd : ∀ (f g : Omega → E) (x : X), condExp (fun omega => f omega + g omega) x = condExp f x + condExp g x) (hsmul : ∀ (a : Real) (f : Omega → E) (x : X), condExp (fun omega => a • f omega) x = a • condExp f x) : ∀ x, condExp (fun omega => dotTk • c omega + sigmaCoeff • score omega) x = dotTk • condExp c x + sigmaCoeff • condExp score x := by intro x rw [hadd, hsmul, hsmul] /-- Named-field version of `generalMovingTargetDiscreteConditionalDriftLinearCombination`. If a later analytic backend supplies `barB` as the selected conditional expectation in the source definition, this theorem rewrites it into the separate conditional-drift and conditional-score summands. It is local additive/module algebra only; existence, measurability, integrability, and the weak Fokker--Planck identity remain obligations. -/ theorem generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination {Omega X E : Type*} [AddCommGroup E] [Module Real E] (condExp : (Omega → E) → X → E) (barB : X → E) (c score : Omega → E) (dotTk sigmaCoeff : Real) (hbar : ∀ x, barB x = condExp (fun omega => dotTk • c omega + sigmaCoeff • score omega) x) (hadd : ∀ (f g : Omega → E) (x : X), condExp (fun omega => f omega + g omega) x = condExp f x + condExp g x) (hsmul : ∀ (a : Real) (f : Omega → E) (x : X), condExp (fun omega => a • f omega) x = a • condExp f x) : ∀ x, barB x = dotTk • condExp c x + sigmaCoeff • condExp score x := by intro x rw [hbar x] exact generalMovingTargetDiscreteConditionalDriftLinearCombination condExp c score dotTk sigmaCoeff hadd hsmul x /-- Named conditional-drift component handoff for `bar b_{k,s}`. After the analytic backend supplies named conditional fields for the frozen guide drift and frozen score summands, this wrapper rewrites the selected source field into those components. It is only module/linearity bookkeeping: regular conditional laws, measurability, integrability, and weak FP remain separate hypotheses and obligations. -/ theorem generalMovingTargetDiscreteNamedConditionalDriftComponents {Omega X E : Type*} [AddCommGroup E] [Module Real E] (condExp : (Omega → E) → X → E) (barB condC condScore : X → E) (c score : Omega → E) (dotTk sigmaCoeff : Real) (hbar : ∀ x, barB x = condExp (fun omega => dotTk • c omega + sigmaCoeff • score omega) x) (hcondC : ∀ x, condC x = condExp c x) (hcondScore : ∀ x, condScore x = condExp score x) (hadd : ∀ (f g : Omega → E) (x : X), condExp (fun omega => f omega + g omega) x = condExp f x + condExp g x) (hsmul : ∀ (a : Real) (f : Omega → E) (x : X), condExp (fun omega => a • f omega) x = a • condExp f x) : ∀ x, barB x = dotTk • condC x + sigmaCoeff • condScore x := by intro x calc barB x = condExp (fun omega => dotTk • c omega + sigmaCoeff • score omega) x := hbar x _ = dotTk • condExp c x + sigmaCoeff • condExp score x := generalMovingTargetDiscreteConditionalDriftLinearCombination condExp c score dotTk sigmaCoeff hadd hsmul x _ = dotTk • condC x + sigmaCoeff • condScore x := by rw [← hcondC x, ← hcondScore x] /-- Abstract regularity handoff for the named frozen drift field. The analytic backend chooses the concrete meanings of `FieldMeasurable` and `FieldIntegrable` (for example, measurability and local integrability under `\hat\rho_s`). Given congruence under pointwise equality and regularity of the named component combination, the selected `barB` field inherits those regularity predicates. This does not prove the predicates themselves. -/ theorem generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff {X E : Type*} [AddCommGroup E] [Module Real E] (FieldMeasurable FieldIntegrable : (X → E) → Prop) (barB condC condScore : X → E) (dotTk sigmaCoeff : Real) (hbar : ∀ x, barB x = dotTk • condC x + sigmaCoeff • condScore x) (hmeasCombo : FieldMeasurable (fun x => dotTk • condC x + sigmaCoeff • condScore x)) (hintCombo : FieldIntegrable (fun x => dotTk • condC x + sigmaCoeff • condScore x)) (hmeasCongr : ∀ {f g : X → E}, (∀ x, f x = g x) → FieldMeasurable f → FieldMeasurable g) (hintCongr : ∀ {f g : X → E}, (∀ x, f x = g x) → FieldIntegrable f → FieldIntegrable g) : FieldMeasurable barB ∧ FieldIntegrable barB := by constructor · exact hmeasCongr (fun x => (hbar x).symm) hmeasCombo · exact hintCongr (fun x => (hbar x).symm) hintCombo /-- Component-regularity handoff for the named frozen drift field. This is the next local bookkeeping step after naming `condC_{k,s}` and `condScore_{k,s}` in `appendix.tex:1368-1377`. If the concrete backend supplies measurability/integrability of those two conditional fields, plus closure of the chosen predicates under addition and real scalar multiplication, then `barB` inherits the same predicates from the pointwise component formula. It does not construct the regular conditional law, prove the component predicates, or prove the weak Fokker--Planck identity. -/ theorem generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents {X E : Type*} [AddCommGroup E] [Module Real E] (FieldMeasurable FieldIntegrable : (X → E) → Prop) (barB condC condScore : X → E) (dotTk sigmaCoeff : Real) (hbar : ∀ x, barB x = dotTk • condC x + sigmaCoeff • condScore x) (hmeasC : FieldMeasurable condC) (hmeasScore : FieldMeasurable condScore) (hintC : FieldIntegrable condC) (hintScore : FieldIntegrable condScore) (hmeasAdd : ∀ {f g : X → E}, FieldMeasurable f → FieldMeasurable g → FieldMeasurable (fun x => f x + g x)) (hmeasSmul : ∀ (a : Real) {f : X → E}, FieldMeasurable f → FieldMeasurable (fun x => a • f x)) (hintAdd : ∀ {f g : X → E}, FieldIntegrable f → FieldIntegrable g → FieldIntegrable (fun x => f x + g x)) (hintSmul : ∀ (a : Real) {f : X → E}, FieldIntegrable f → FieldIntegrable (fun x => a • f x)) (hmeasCongr : ∀ {f g : X → E}, (∀ x, f x = g x) → FieldMeasurable f → FieldMeasurable g) (hintCongr : ∀ {f g : X → E}, (∀ x, f x = g x) → FieldIntegrable f → FieldIntegrable g) : FieldMeasurable barB ∧ FieldIntegrable barB := by have hmeasCombo : FieldMeasurable (fun x => dotTk • condC x + sigmaCoeff • condScore x) := hmeasAdd (hmeasSmul dotTk hmeasC) (hmeasSmul sigmaCoeff hmeasScore) have hintCombo : FieldIntegrable (fun x => dotTk • condC x + sigmaCoeff • condScore x) := hintAdd (hintSmul dotTk hintC) (hintSmul sigmaCoeff hintScore) exact generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff FieldMeasurable FieldIntegrable barB condC condScore dotTk sigmaCoeff hbar hmeasCombo hintCombo hmeasCongr hintCongr /-- Mathlib `condDistrib` version of the named frozen-drift regularity handoff. This is the source-specific backfill for `appendix.tex:1368-1377` after the cycle-85 law-space conditional-integral lemmas. If `condC` and `condScore` are chosen as `hatRhoS`-a.e. versions of the canonical conditional integrals against `condDistrib Xk hatXAtS P`, then Mathlib supplies their measurability/integrability under `hatRhoS = Law(hatXAtS)`, and the existing component-combination wrapper gives regularity of `barB`. The remaining analytic boundary is the SALD-specific a.e. version choice and conditional kernel compatibility, not a supplied component-regularity hypothesis. -/ theorem generalMovingTargetDiscreteCondDistribNamedDriftRegularity {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} {barB condC condScore : State → Vec} (dotTk sigmaCoeff : Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hcondC : (fun x => ∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) =ᵐ[hatRhoS] condC) (hcondScore : (fun x => ∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) =ᵐ[hatRhoS] condScore) (hbar : ∀ x, barB x = dotTk • condC x + sigmaCoeff • condScore x) : MeasureTheory.AEStronglyMeasurable barB hatRhoS ∧ MeasureTheory.Integrable barB hatRhoS := by have hregC : MeasureTheory.AEStronglyMeasurable condC hatRhoS ∧ MeasureTheory.Integrable condC hatRhoS := AutoSamplingTheory.condDistribIntegralNamedFieldRegularity (μ := P) (hatRho := hatRhoS) (X := hatXAtS) (Y := Xk) (f := guideIntegrand) (field := condC) hhatRhoS hXk hguideMeas hguideInt hcondC have hregScore : MeasureTheory.AEStronglyMeasurable condScore hatRhoS ∧ MeasureTheory.Integrable condScore hatRhoS := AutoSamplingTheory.condDistribIntegralNamedFieldRegularity (μ := P) (hatRho := hatRhoS) (X := hatXAtS) (Y := Xk) (f := scoreIntegrand) (field := condScore) hhatRhoS hXk hscoreMeas hscoreInt hcondScore exact generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents (fun f => MeasureTheory.AEStronglyMeasurable f hatRhoS) (fun f => MeasureTheory.Integrable f hatRhoS) barB condC condScore dotTk sigmaCoeff hbar hregC.1 hregScore.1 hregC.2 hregScore.2 (by intro f g hf hg exact hf.add hg) (by intro a f hf exact hf.const_smul a) (by intro f g hf hg exact hf.add hg) (by intro a f hf exact MeasureTheory.Integrable.smul a hf) (by intro f g hfg hf exact hf.congr (Filter.Eventually.of_forall hfg)) (by intro f g hfg hf exact hf.congr (Filter.Eventually.of_forall hfg)) /-- Canonical `condDistrib` version of the frozen conditional drift field. This cycle-106 theorem removes the old supplied component-field regularity premise for the canonical representative in `appendix.tex:1368-1377`. If `bar b_{k,s}` is taken to be the sum of the two Mathlib conditional integrals against `condDistrib X_k^eta hatX_s P`, then measurability and integrability under the named law `hatRhoS = Law(hat X_s)` follow directly from the local conditional-integral lemmas. A future theorem may still need to prove that a separately named paper representative is `hatRhoS`-a.e. equal to this canonical field; that version-selection boundary is smaller than the cycle-80 supplied component-regularity hypothesis. -/ theorem generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (dotTk sigmaCoeff : Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) : MeasureTheory.AEStronglyMeasurable (fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x)) hatRhoS ∧ MeasureTheory.Integrable (fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x)) hatRhoS := by have hmeasGuide : MeasureTheory.AEStronglyMeasurable (fun x => ∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) hatRhoS := AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable (μ := P) (hatRho := hatRhoS) (X := hatXAtS) (Y := Xk) (f := guideIntegrand) hhatRhoS hXk hguideMeas have hmeasScore : MeasureTheory.AEStronglyMeasurable (fun x => ∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) hatRhoS := AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable (μ := P) (hatRho := hatRhoS) (X := hatXAtS) (Y := Xk) (f := scoreIntegrand) hhatRhoS hXk hscoreMeas have hintGuide : MeasureTheory.Integrable (fun x => ∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) hatRhoS := AutoSamplingTheory.condDistribIntegralNamedLawIntegrable (μ := P) (hatRho := hatRhoS) (X := hatXAtS) (Y := Xk) (f := guideIntegrand) hhatRhoS hXk hguideInt have hintScore : MeasureTheory.Integrable (fun x => ∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) hatRhoS := AutoSamplingTheory.condDistribIntegralNamedLawIntegrable (μ := P) (hatRho := hatRhoS) (X := hatXAtS) (Y := Xk) (f := scoreIntegrand) hhatRhoS hXk hscoreInt constructor · exact (hmeasGuide.const_smul dotTk).add (hmeasScore.const_smul sigmaCoeff) · exact (MeasureTheory.Integrable.smul dotTk hintGuide).add (MeasureTheory.Integrable.smul sigmaCoeff hintScore) /-- Named-field bridge from the canonical `condDistrib` drift representative. This is the strict versioning handoff left by `generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity`: if the paper's selected `barB` field is `hatRhoS`-a.e. equal to the canonical guide-plus-score conditional integral, then `barB` inherits the canonical measurability and integrability. The theorem does not prove the a.e. representative equality; that remains the smaller source-cited boundary for `appendix.tex:1368-1377`. -/ theorem generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} {barB : State → Vec} (dotTk sigmaCoeff : Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hbarBAe : (fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x)) =ᵐ[hatRhoS] barB) : MeasureTheory.AEStronglyMeasurable barB hatRhoS ∧ MeasureTheory.Integrable barB hatRhoS := by have hcanonical : MeasureTheory.AEStronglyMeasurable (fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x)) hatRhoS ∧ MeasureTheory.Integrable (fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x)) hatRhoS := generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity (P := P) (hatRhoS := hatRhoS) (hatXAtS := hatXAtS) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) dotTk sigmaCoeff hhatRhoS hXk hguideMeas hguideInt hscoreMeas hscoreInt exact ⟨hcanonical.1.congr hbarBAe, hcanonical.2.congr hbarBAe⟩ /-- Equality-set measurability for the named `barB` source representative. Cycle 110 removes one supplied side condition from the cycle-109 `ae_map_iff` bridge: once the canonical conditional-drift representative and the paper's selected named `barB` representative are strongly measurable, the set on which they agree is measurable. This does not choose the conditional expectation representative or prove the `barB` a.e. equality; it only discharges the equality-set measurability premise used to transport that equality from the sample space to `hatRhoS = Law(hatXAtS)`. -/ theorem generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} {barB : State → Vec} (dotTk sigmaCoeff : Real) (hcanonicalMeas : MeasureTheory.StronglyMeasurable (fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x))) (hbarBMeas : MeasureTheory.StronglyMeasurable barB) : MeasurableSet {x | dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) = barB x} := by exact MeasureTheory.StronglyMeasurable.measurableSet_eq_fun hcanonicalMeas hbarBMeas /-- Conditional-expectation uniqueness bridge for the named `barB` representative. Cycle 112 narrows the remaining `hbarBCondExp` side condition from `generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef`. It is enough to show that `barB ∘ hatXAtS` is an integrable `mState.comap hatXAtS`-measurable candidate whose set integrals over every `hatXAtS`-measurable finite-measure set match the frozen guide-plus-score drift. The theorem does not prove that source set-integral characterization; that is the smaller paper-facing boundary for `appendix.tex:1368-1377`. -/ theorem generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} {barB : State → Vec} (dotTk sigmaCoeff : Real) (hhatXAtS : Measurable hatXAtS) (hXk : AEMeasurable Xk P) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hbarBMeas : MeasureTheory.AEStronglyMeasurable (m := (inferInstance : MeasurableSpace State).comap hatXAtS) (fun ω => barB (hatXAtS ω)) P) (hbarBInt : MeasureTheory.Integrable (fun ω => barB (hatXAtS ω)) P) (hbarBSetIntegral : ∀ s : Set Ω, @MeasurableSet Ω ((inferInstance : MeasurableSpace State).comap hatXAtS) s → P s < ⊤ → (∫ ω in s, barB (hatXAtS ω) ∂P) = ∫ ω in s, dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω) ∂P) : MeasureTheory.condExp ((inferInstance : MeasurableSpace State).comap hatXAtS) P (fun ω => dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω)) =ᵐ[P] fun ω => barB (hatXAtS ω) := by have hguideSample : MeasureTheory.Integrable (fun ω => guideIntegrand (hatXAtS ω, Xk ω)) P := by simpa [Function.comp_def] using hguideInt.comp_aemeasurable (hhatXAtS.aemeasurable.prodMk hXk) have hscoreSample : MeasureTheory.Integrable (fun ω => scoreIntegrand (hatXAtS ω, Xk ω)) P := by simpa [Function.comp_def] using hscoreInt.comp_aemeasurable (hhatXAtS.aemeasurable.prodMk hXk) have hfInt : MeasureTheory.Integrable (fun ω => dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω)) P := (MeasureTheory.Integrable.smul dotTk hguideSample).add (MeasureTheory.Integrable.smul sigmaCoeff hscoreSample) have hbarBIntFinite : ∀ s : Set Ω, @MeasurableSet Ω ((inferInstance : MeasurableSpace State).comap hatXAtS) s → P s < ⊤ → MeasureTheory.IntegrableOn (fun ω => barB (hatXAtS ω)) s P := by intro s _ _ exact hbarBInt.integrableOn have hselected : (fun ω => barB (hatXAtS ω)) =ᵐ[P] MeasureTheory.condExp ((inferInstance : MeasurableSpace State).comap hatXAtS) P (fun ω => dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω)) := MeasureTheory.ae_eq_condExp_of_forall_setIntegral_eq hhatXAtS.comap_le hfInt hbarBIntFinite hbarBSetIntegral hbarBMeas exact hselected.symm /-- Source-definition bridge for the named `barB` representative. This narrows the remaining `appendix.tex:1368-1377` version-selection boundary. If the source-level conditional-expectation definition of `barB` is given using `condExpKernel.map Xk`, and that kernel is a sample-space a.e. version of Mathlib's `condDistrib Xk hatXAtS P (hatXAtS omega)`, then the named field is `hatRhoS`-a.e. equal to the canonical `condDistrib` guide-plus-score drift. The remaining analytic theorem is the measure-valued kernel alignment and the source conditional-expectation representative, not another direct `hbarBAe` hypothesis. -/ theorem generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpKernelSourceDef {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [StandardBorelSpace Ω] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} {barB : State → Vec} (dotTk sigmaCoeff : Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hhatXAtS : AEMeasurable hatXAtS P) (hbarBEqMeas : MeasurableSet {x | dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) = barB x}) (hkernel : (fun ω => ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) =ᵐ[P] fun ω => (ProbabilityTheory.condExpKernel P ((inferInstance : MeasurableSpace State).comap hatXAtS)).map Xk ω) (hbarBSource : (fun ω => dotTk • (∫ y, guideIntegrand (hatXAtS ω, y) ∂(ProbabilityTheory.condExpKernel P ((inferInstance : MeasurableSpace State).comap hatXAtS)).map Xk ω) + sigmaCoeff • (∫ y, scoreIntegrand (hatXAtS ω, y) ∂(ProbabilityTheory.condExpKernel P ((inferInstance : MeasurableSpace State).comap hatXAtS)).map Xk ω)) =ᵐ[P] fun ω => barB (hatXAtS ω)) : (fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x)) =ᵐ[hatRhoS] barB := by rw [hhatRhoS] refine (MeasureTheory.ae_map_iff hhatXAtS hbarBEqMeas).2 ?_ filter_upwards [hkernel, hbarBSource] with ω hker hsource simpa [hker] using hsource /-- Source-definition bridge for the named `barB` representative using Mathlib's product conditional-expectation theorem. This is a stricter `appendix.tex:1368-1377` backend than assuming a direct `hatRhoS`-a.e. equality for `barB`. If the paper's selected representative composed with `hatXAtS` is the Mathlib conditional expectation of the frozen guide-plus-score summand given `hatXAtS`, then `condExp_prod_ae_eq_integral_condDistrib` identifies that conditional expectation with the canonical `condDistrib` integral. The named marginal `hatRhoS = Law(hatXAtS)` then transports the equality to state space. -/ theorem generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} {barB : State → Vec} (dotTk sigmaCoeff : Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hhatXAtS : Measurable hatXAtS) (hXk : AEMeasurable Xk P) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hbarBEqMeas : MeasurableSet {x | dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) = barB x}) (hbarBCondExp : MeasureTheory.condExp ((inferInstance : MeasurableSpace State).comap hatXAtS) P (fun ω => dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω)) =ᵐ[P] fun ω => barB (hatXAtS ω)) : (fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x)) =ᵐ[hatRhoS] barB := by let combined : State × State → Vec := fun z => dotTk • guideIntegrand z + sigmaCoeff • scoreIntegrand z have hcombinedInt : MeasureTheory.Integrable combined (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P) := by exact (MeasureTheory.Integrable.smul dotTk hguideInt).add (MeasureTheory.Integrable.smul sigmaCoeff hscoreInt) have hcondCombined : MeasureTheory.condExp ((inferInstance : MeasurableSpace State).comap hatXAtS) P (fun ω => dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω)) =ᵐ[P] fun ω => ∫ y, dotTk • guideIntegrand (hatXAtS ω, y) + sigmaCoeff • scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω) := by simpa [combined] using (ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib' (μ := P) (X := hatXAtS) (Y := Xk) (f := combined) hhatXAtS hXk hcombinedInt) have hsample : (fun ω => dotTk • (∫ y, guideIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) + sigmaCoeff • (∫ y, scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω))) =ᵐ[P] fun ω => barB (hatXAtS ω) := by filter_upwards [hcondCombined, hbarBCondExp, hguideInt.condDistrib_ae hhatXAtS.aemeasurable hXk, hscoreInt.condDistrib_ae hhatXAtS.aemeasurable hXk] with ω hcond hsource hguideω hscoreω have hlin : (∫ y, dotTk • guideIntegrand (hatXAtS ω, y) + sigmaCoeff • scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) = dotTk • (∫ y, guideIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) + sigmaCoeff • (∫ y, scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) := by have hadd := MeasureTheory.integral_add (μ := ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) (f := fun y => dotTk • guideIntegrand (hatXAtS ω, y)) (g := fun y => sigmaCoeff • scoreIntegrand (hatXAtS ω, y)) (MeasureTheory.Integrable.smul dotTk hguideω) (MeasureTheory.Integrable.smul sigmaCoeff hscoreω) simpa [MeasureTheory.integral_smul] using hadd calc dotTk • (∫ y, guideIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) + sigmaCoeff • (∫ y, scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) = (∫ y, dotTk • guideIntegrand (hatXAtS ω, y) + sigmaCoeff • scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) := hlin.symm _ = MeasureTheory.condExp ((inferInstance : MeasurableSpace State).comap hatXAtS) P (fun ω => dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω)) ω := hcond.symm _ = barB (hatXAtS ω) := hsource rw [hhatRhoS] exact (MeasureTheory.ae_map_iff hhatXAtS.aemeasurable hbarBEqMeas).2 hsample /-- Source-definition bridge with the `hbarBCondExp` premise replaced by a conditional-expectation uniqueness boundary. This is the cycle-112 downstream handoff for `appendix.tex:1368-1377`. Instead of assuming directly that the Mathlib conditional expectation equals `barB (hatXAtS omega)`, it derives that a.e. equality from the set-integral characterization of the selected representative and then invokes `generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef`. -/ theorem generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} {barB : State → Vec} (dotTk sigmaCoeff : Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hhatXAtS : Measurable hatXAtS) (hXk : AEMeasurable Xk P) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hbarBEqMeas : MeasurableSet {x | dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) = barB x}) (hbarBMeas : MeasureTheory.AEStronglyMeasurable (m := (inferInstance : MeasurableSpace State).comap hatXAtS) (fun ω => barB (hatXAtS ω)) P) (hbarBInt : MeasureTheory.Integrable (fun ω => barB (hatXAtS ω)) P) (hbarBSetIntegral : ∀ s : Set Ω, @MeasurableSet Ω ((inferInstance : MeasurableSpace State).comap hatXAtS) s → P s < ⊤ → (∫ ω in s, barB (hatXAtS ω) ∂P) = ∫ ω in s, dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω) ∂P) : (fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x)) =ᵐ[hatRhoS] barB := by have hbarBCondExp : MeasureTheory.condExp ((inferInstance : MeasurableSpace State).comap hatXAtS) P (fun ω => dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω)) =ᵐ[P] fun ω => barB (hatXAtS ω) := generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq (P := P) (hatXAtS := hatXAtS) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) (barB := barB) dotTk sigmaCoeff hhatXAtS hXk hguideInt hscoreInt hbarBMeas hbarBInt hbarBSetIntegral exact generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef (P := P) (hatRhoS := hatRhoS) (hatXAtS := hatXAtS) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) (barB := barB) dotTk sigmaCoeff hhatRhoS hhatXAtS hXk hguideInt hscoreInt hbarBEqMeas hbarBCondExp /-- Convert the source-facing state-event set-integral criterion into the `comap hatXAtS` criterion used by Mathlib conditional-expectation uniqueness. The paper defines `bar b_{k,s}` by conditioning on `hat X_s = x`, so its natural verification sets are events `{omega | hatXAtS omega ∈ t}` for measurable `t : Set State`. Mathlib's uniqueness theorem asks for all sets measurable in the comap sigma-algebra; this lemma proves that those are exactly preimages of state-measurable sets for this purpose. -/ theorem generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] {P : MeasureTheory.Measure Ω} {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} {barB : State → Vec} (dotTk sigmaCoeff : Real) (hbarBStateSetIntegral : ∀ t : Set State, MeasurableSet t → P (hatXAtS ⁻¹' t) < ⊤ → (∫ ω in hatXAtS ⁻¹' t, barB (hatXAtS ω) ∂P) = ∫ ω in hatXAtS ⁻¹' t, dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω) ∂P) : ∀ s : Set Ω, @MeasurableSet Ω ((inferInstance : MeasurableSpace State).comap hatXAtS) s → P s < ⊤ → (∫ ω in s, barB (hatXAtS ω) ∂P) = ∫ ω in s, dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω) ∂P := by intro s hs hsfinite rcases (MeasurableSpace.measurableSet_comap.mp hs) with ⟨t, ht, hpre⟩ rw [← hpre] at hsfinite ⊢ exact hbarBStateSetIntegral t ht hsfinite /-- Canonical `condDistrib` drift satisfies the sample-space conditional expectation identity. This factors the `hbarBCondExp` part of the cycle-115 boundary for the canonical Mathlib representative used in `appendix.tex:1368-1377`. The proof uses Mathlib's product conditional-expectation theorem for `X_k^eta | hat X_s`, then expands the guide-plus-score Bochner integral by additivity and scalar multiplication. A separate paper-selected `barB` still needs the smaller version-selection equality to this canonical field. -/ theorem generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (dotTk sigmaCoeff : Real) (hhatXAtS : Measurable hatXAtS) (hXk : AEMeasurable Xk P) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) : MeasureTheory.condExp ((inferInstance : MeasurableSpace State).comap hatXAtS) P (fun ω => dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω)) =ᵐ[P] fun ω => dotTk • (∫ y, guideIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) + sigmaCoeff • (∫ y, scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) := by let combined : State × State → Vec := fun z => dotTk • guideIntegrand z + sigmaCoeff • scoreIntegrand z have hcombinedInt : MeasureTheory.Integrable combined (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P) := by exact (MeasureTheory.Integrable.smul dotTk hguideInt).add (MeasureTheory.Integrable.smul sigmaCoeff hscoreInt) have hcondCombined : MeasureTheory.condExp ((inferInstance : MeasurableSpace State).comap hatXAtS) P (fun ω => dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω)) =ᵐ[P] fun ω => ∫ y, dotTk • guideIntegrand (hatXAtS ω, y) + sigmaCoeff • scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω) := by simpa [combined] using (ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib' (μ := P) (X := hatXAtS) (Y := Xk) (f := combined) hhatXAtS hXk hcombinedInt) filter_upwards [hcondCombined, hguideInt.condDistrib_ae hhatXAtS.aemeasurable hXk, hscoreInt.condDistrib_ae hhatXAtS.aemeasurable hXk] with ω hcond hguideω hscoreω have hlin : (∫ y, dotTk • guideIntegrand (hatXAtS ω, y) + sigmaCoeff • scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) = dotTk • (∫ y, guideIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) + sigmaCoeff • (∫ y, scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) := by have hadd := MeasureTheory.integral_add (μ := ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) (f := fun y => dotTk • guideIntegrand (hatXAtS ω, y)) (g := fun y => sigmaCoeff • scoreIntegrand (hatXAtS ω, y)) (MeasureTheory.Integrable.smul dotTk hguideω) (MeasureTheory.Integrable.smul sigmaCoeff hscoreω) simpa [MeasureTheory.integral_smul] using hadd calc MeasureTheory.condExp ((inferInstance : MeasurableSpace State).comap hatXAtS) P (fun ω => dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω)) ω = ∫ y, dotTk • guideIntegrand (hatXAtS ω, y) + sigmaCoeff • scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω) := hcond _ = dotTk • (∫ y, guideIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) + sigmaCoeff • (∫ y, scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) := hlin /-- Canonical `condDistrib` drift satisfies the state-event set-integral criterion. This is the cycle-114 narrowing of the remaining `hbarBStateSetIntegral` boundary for `appendix.tex:1368-1377`: if `barB` is chosen to be the canonical Mathlib conditional-distribution representative, its set integrals over events `{omega | hatXAtS omega ∈ t}` agree with the frozen guide-plus-score sample integral. A separately named paper version still needs the smaller version-selection theorem identifying it with this canonical field. -/ theorem generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (dotTk sigmaCoeff : Real) (hhatXAtS : Measurable hatXAtS) (hXk : AEMeasurable Xk P) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) : ∀ t : Set State, MeasurableSet t → P (hatXAtS ⁻¹' t) < ⊤ → (∫ ω in hatXAtS ⁻¹' t, dotTk • (∫ y, guideIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) + sigmaCoeff • (∫ y, scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) ∂P) = ∫ ω in hatXAtS ⁻¹' t, dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω) ∂P := by intro t ht _ let combined : State × State → Vec := fun z => dotTk • guideIntegrand z + sigmaCoeff • scoreIntegrand z have hcombinedInt : MeasureTheory.Integrable combined (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P) := by exact (MeasureTheory.Integrable.smul dotTk hguideInt).add (MeasureTheory.Integrable.smul sigmaCoeff hscoreInt) have hsampleInt : MeasureTheory.Integrable (fun ω => dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω)) P := by simpa [combined, Function.comp_def] using hcombinedInt.comp_aemeasurable (hhatXAtS.aemeasurable.prodMk hXk) have hcanonicalCondExp : (fun ω => dotTk • (∫ y, guideIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) + sigmaCoeff • (∫ y, scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω))) =ᵐ[P] MeasureTheory.condExp ((inferInstance : MeasurableSpace State).comap hatXAtS) P (fun ω => dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω)) := (generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib (P := P) (hatXAtS := hatXAtS) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) dotTk sigmaCoeff hhatXAtS hXk hguideInt hscoreInt).symm have hpreMeas : MeasurableSet (hatXAtS ⁻¹' t) := hhatXAtS ht have hpreComap : @MeasurableSet Ω ((inferInstance : MeasurableSpace State).comap hatXAtS) (hatXAtS ⁻¹' t) := MeasurableSpace.measurableSet_comap.mpr ⟨t, ht, rfl⟩ calc (∫ ω in hatXAtS ⁻¹' t, dotTk • (∫ y, guideIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) + sigmaCoeff • (∫ y, scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) ∂P) = ∫ ω in hatXAtS ⁻¹' t, MeasureTheory.condExp ((inferInstance : MeasurableSpace State).comap hatXAtS) P (fun ω => dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω)) ω ∂P := by exact MeasureTheory.setIntegral_congr_ae hpreMeas (hcanonicalCondExp.mono fun _ hω _ => hω) _ = ∫ ω in hatXAtS ⁻¹' t, dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω) ∂P := by exact MeasureTheory.setIntegral_condExp hhatXAtS.comap_le hsampleInt hpreComap /-- Selected named `barB` inherits the canonical state-event set-integral criterion. Cycle 115 narrows the post-canonical blocker for `appendix.tex:1368-1377`. Once the paper-selected `barB` is identified `hatRhoS`-a.e. with the canonical Mathlib `condDistrib` guide-plus-score drift, the old source-facing `hbarBStateSetIntegral` input follows by pulling that a.e. equality back along `hatXAtS` and using `setIntegral_congr_ae`. The same a.e. equality also supplies `Integrable barB hatRhoS` through the canonical regularity theorem, so the remaining theorem boundary is only the selected-version equality. -/ theorem generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} {barB : State → Vec} (dotTk sigmaCoeff : Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hhatXAtS : Measurable hatXAtS) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hbarBAe : (fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x)) =ᵐ[hatRhoS] barB) : MeasureTheory.Integrable barB hatRhoS ∧ ∀ t : Set State, MeasurableSet t → P (hatXAtS ⁻¹' t) < ⊤ → (∫ ω in hatXAtS ⁻¹' t, barB (hatXAtS ω) ∂P) = ∫ ω in hatXAtS ⁻¹' t, dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω) ∂P := by have hregularity : MeasureTheory.AEStronglyMeasurable barB hatRhoS ∧ MeasureTheory.Integrable barB hatRhoS := generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq (P := P) (hatRhoS := hatRhoS) (hatXAtS := hatXAtS) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) (barB := barB) dotTk sigmaCoeff hhatRhoS hXk hguideMeas hguideInt hscoreMeas hscoreInt hbarBAe have hbarBAeMap : (fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x)) =ᵐ[ MeasureTheory.Measure.map hatXAtS P] barB := by simpa [hhatRhoS] using hbarBAe have hsampleAe : (fun ω => dotTk • (∫ y, guideIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) + sigmaCoeff • (∫ y, scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω))) =ᵐ[P] fun ω => barB (hatXAtS ω) := by simpa [Function.comp_def] using (MeasureTheory.ae_eq_comp (μ := P) (f := hatXAtS) hhatXAtS.aemeasurable hbarBAeMap) refine ⟨hregularity.2, ?_⟩ intro t ht hsfinite have hcanonicalStateSet : ∀ t : Set State, MeasurableSet t → P (hatXAtS ⁻¹' t) < ⊤ → (∫ ω in hatXAtS ⁻¹' t, dotTk • (∫ y, guideIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) + sigmaCoeff • (∫ y, scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) ∂P) = ∫ ω in hatXAtS ⁻¹' t, dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω) ∂P := generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib (P := P) (hatXAtS := hatXAtS) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) dotTk sigmaCoeff hhatXAtS hXk hguideInt hscoreInt have hpreMeas : MeasurableSet (hatXAtS ⁻¹' t) := hhatXAtS ht calc (∫ ω in hatXAtS ⁻¹' t, barB (hatXAtS ω) ∂P) = ∫ ω in hatXAtS ⁻¹' t, dotTk • (∫ y, guideIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) + sigmaCoeff • (∫ y, scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) ∂P := by exact MeasureTheory.setIntegral_congr_ae hpreMeas (hsampleAe.symm.mono fun _ hω _ => hω) _ = ∫ ω in hatXAtS ⁻¹' t, dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω) ∂P := hcanonicalStateSet t ht hsfinite /-- Source conditional-expectation version of the selected `barB` bridge. This is the cycle-115 lower packet for `appendix.tex:1368-1377`. It removes the supplied selected-to-canonical `hbarBAe` input from `generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq` when the selected paper representative is given by the Mathlib conditional-expectation definition on the sample space. The remaining source boundary is the paper's conditional-expectation representative equality `hbarBCondExp`, plus the equality-set measurability side condition if it is not discharged by the existing cycle-110 measurable-representative theorem. -/ theorem generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCondExpSourceDef {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} {barB : State → Vec} (dotTk sigmaCoeff : Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hhatXAtS : Measurable hatXAtS) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hbarBEqMeas : MeasurableSet {x | dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) = barB x}) (hbarBCondExp : MeasureTheory.condExp ((inferInstance : MeasurableSpace State).comap hatXAtS) P (fun ω => dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω)) =ᵐ[P] fun ω => barB (hatXAtS ω)) : MeasureTheory.Integrable barB hatRhoS ∧ ∀ t : Set State, MeasurableSet t → P (hatXAtS ⁻¹' t) < ⊤ → (∫ ω in hatXAtS ⁻¹' t, barB (hatXAtS ω) ∂P) = ∫ ω in hatXAtS ⁻¹' t, dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω) ∂P := by have hbarBAe : (fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x)) =ᵐ[hatRhoS] barB := generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef (P := P) (hatRhoS := hatRhoS) (hatXAtS := hatXAtS) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) (barB := barB) dotTk sigmaCoeff hhatRhoS hhatXAtS hXk hguideInt hscoreInt hbarBEqMeas hbarBCondExp exact generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq (P := P) (hatRhoS := hatRhoS) (hatXAtS := hatXAtS) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) (barB := barB) dotTk sigmaCoeff hhatRhoS hhatXAtS hXk hguideMeas hguideInt hscoreMeas hscoreInt hbarBAe /-- Canonical `condDistrib` drift inherits the selected-bridge integrability and state-event set-integral conclusion. This specializes the cycle-115 selected `barB` bridge to the canonical Mathlib conditional-distribution representative. The cycle-116 canonical conditional-expectation theorem supplies the old `hbarBCondExp` input, and the equality-set measurability side condition is tautological for this representative. A separately named paper version still has only the smaller selected-version equality boundary to the canonical field. -/ theorem generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (dotTk sigmaCoeff : Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hhatXAtS : Measurable hatXAtS) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) : MeasureTheory.Integrable (fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x)) hatRhoS ∧ ∀ t : Set State, MeasurableSet t → P (hatXAtS ⁻¹' t) < ⊤ → (∫ ω in hatXAtS ⁻¹' t, dotTk • (∫ y, guideIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) + sigmaCoeff • (∫ y, scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) ∂P) = ∫ ω in hatXAtS ⁻¹' t, dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω) ∂P := by let barB : State → Vec := fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) have hbarBEqMeas : MeasurableSet {x | dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) = barB x} := by simp [barB] have hbarBCondExp : MeasureTheory.condExp ((inferInstance : MeasurableSpace State).comap hatXAtS) P (fun ω => dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω)) =ᵐ[P] fun ω => barB (hatXAtS ω) := by simpa [barB] using generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib (P := P) (hatXAtS := hatXAtS) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) dotTk sigmaCoeff hhatXAtS hXk hguideInt hscoreInt simpa [barB] using generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCondExpSourceDef (P := P) (hatRhoS := hatRhoS) (hatXAtS := hatXAtS) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) (barB := barB) dotTk sigmaCoeff hhatRhoS hhatXAtS hXk hguideMeas hguideInt hscoreMeas hscoreInt hbarBEqMeas hbarBCondExp /-- Pointwise source selection of the canonical `condDistrib` `barB` representative. This is the cycle-117 lower handoff for the selected-version boundary at `appendix.tex:1368-1377`. If the paper's named field is chosen pointwise as the canonical conditional-distribution guide-plus-score field, then the old selected-to-canonical a.e. premise is discharged without another `hbarBCondExp` wrapper, and the downstream state-event set-integral package follows from the existing selected-version bridge. -/ theorem generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalPointwiseEq {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} {barB : State → Vec} (dotTk sigmaCoeff : Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hhatXAtS : Measurable hatXAtS) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hbarBPointwise : ∀ x, dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) = barB x) : MeasureTheory.Integrable barB hatRhoS ∧ ∀ t : Set State, MeasurableSet t → P (hatXAtS ⁻¹' t) < ⊤ → (∫ ω in hatXAtS ⁻¹' t, barB (hatXAtS ω) ∂P) = ∫ ω in hatXAtS ⁻¹' t, dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω) ∂P := by have hbarBAe : (fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x)) =ᵐ[hatRhoS] barB := by filter_upwards with x exact hbarBPointwise x exact generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq (P := P) (hatRhoS := hatRhoS) (hatXAtS := hatXAtS) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) (barB := barB) dotTk sigmaCoeff hhatRhoS hhatXAtS hXk hguideMeas hguideInt hscoreMeas hscoreInt hbarBAe /-- Direct canonical `barB` state-event interface for the EM backend. Cycle 118 consumes the source-supported canonical representative choice instead of keeping a separate named `barB` version open. The theorem existentially instantiates the downstream state-event interface with the canonical `condDistrib` guide-plus-score field and reuses the compiled canonical package from cycle 116. A separate named representative only remains relevant if the route deliberately keeps one and then proves pointwise equality to this field. -/ theorem generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (dotTk sigmaCoeff : Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hhatXAtS : Measurable hatXAtS) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) : ∃ barB : State → Vec, barB = (fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x)) ∧ MeasureTheory.Integrable barB hatRhoS ∧ ∀ t : Set State, MeasurableSet t → P (hatXAtS ⁻¹' t) < ⊤ → (∫ ω in hatXAtS ⁻¹' t, barB (hatXAtS ω) ∂P) = ∫ ω in hatXAtS ⁻¹' t, dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω) ∂P := by let canonicalBarB : State → Vec := fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) refine ⟨canonicalBarB, rfl, ?_⟩ simpa [canonicalBarB] using generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib (P := P) (hatRhoS := hatRhoS) (hatXAtS := hatXAtS) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) dotTk sigmaCoeff hhatRhoS hhatXAtS hXk hguideMeas hguideInt hscoreMeas hscoreInt /-- Source-definition bridge with the conditional-expectation set-integral boundary restricted to state events. This is a narrower form of `generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef`: instead of asking for matching integrals on every `mState.comap hatXAtS` sample-space set, it asks only for matching integrals on events pulled back from measurable subsets of the state space, which is the form directly aligned with conditioning on `hat X_s = x` in `appendix.tex:1368-1377`. -/ theorem generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} {barB : State → Vec} (dotTk sigmaCoeff : Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hhatXAtS : Measurable hatXAtS) (hXk : AEMeasurable Xk P) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hbarBEqMeas : MeasurableSet {x | dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) = barB x}) (hbarBMeas : MeasureTheory.AEStronglyMeasurable (m := (inferInstance : MeasurableSpace State).comap hatXAtS) (fun ω => barB (hatXAtS ω)) P) (hbarBInt : MeasureTheory.Integrable (fun ω => barB (hatXAtS ω)) P) (hbarBStateSetIntegral : ∀ t : Set State, MeasurableSet t → P (hatXAtS ⁻¹' t) < ⊤ → (∫ ω in hatXAtS ⁻¹' t, barB (hatXAtS ω) ∂P) = ∫ ω in hatXAtS ⁻¹' t, dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω) ∂P) : (fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x)) =ᵐ[hatRhoS] barB := by have hbarBSetIntegral : ∀ s : Set Ω, @MeasurableSet Ω ((inferInstance : MeasurableSpace State).comap hatXAtS) s → P s < ⊤ → (∫ ω in s, barB (hatXAtS ω) ∂P) = ∫ ω in s, dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω) ∂P := generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents (P := P) (hatXAtS := hatXAtS) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) (barB := barB) dotTk sigmaCoeff hbarBStateSetIntegral exact generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef (P := P) (hatRhoS := hatRhoS) (hatXAtS := hatXAtS) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) (barB := barB) dotTk sigmaCoeff hhatRhoS hhatXAtS hXk hguideInt hscoreInt hbarBEqMeas hbarBMeas hbarBInt hbarBSetIntegral /-- Candidate regularity for the named `barB` representative pulled back to the sample space. Cycle 113 discharges the `hbarBMeas`/`hbarBInt` inputs of `generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef` from source-facing state-field hypotheses: the selected representative is strongly measurable on state space and integrable under the named marginal `hatRhoS = Law(hatXAtS)`. The source state-event set-integral equality remains the non-wrapper blocker for `appendix.tex:1368-1377`. -/ theorem generalMovingTargetDiscreteNamedBarBComapRegularityOfStateField {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] {P : MeasureTheory.Measure Ω} {hatRhoS : MeasureTheory.Measure State} {hatXAtS : Ω → State} {barB : State → Vec} (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hhatXAtS : Measurable hatXAtS) (hbarBStrong : MeasureTheory.StronglyMeasurable barB) (hbarBIntHat : MeasureTheory.Integrable barB hatRhoS) : MeasureTheory.AEStronglyMeasurable (m := (inferInstance : MeasurableSpace State).comap hatXAtS) (fun ω => barB (hatXAtS ω)) P ∧ MeasureTheory.Integrable (fun ω => barB (hatXAtS ω)) P := by have hhatXAtSComap : @Measurable Ω State ((inferInstance : MeasurableSpace State).comap hatXAtS) (inferInstance : MeasurableSpace State) hatXAtS := Measurable.of_comap_le le_rfl have hbarBCompStrong : @MeasureTheory.StronglyMeasurable Ω Vec _ ((inferInstance : MeasurableSpace State).comap hatXAtS) (fun ω => barB (hatXAtS ω)) := hbarBStrong.comp_measurable hhatXAtSComap refine ⟨hbarBCompStrong.aestronglyMeasurable, ?_⟩ rw [hhatRhoS] at hbarBIntHat simpa [Function.comp_def] using hbarBIntHat.comp_measurable hhatXAtS /-- Source-definition bridge with candidate regularity pulled back from the named state marginal. This is the cycle-113 lower handoff for `appendix.tex:1368-1377`. It removes the older sample-space candidate-regularity hypotheses from the cycle-112 consumer by deriving them from state-space measurability and integrability of the selected `barB` representative under `hatRhoS = Law(hatXAtS)`. The only remaining representative-specific equality input is the source state-event Bochner set-integral characterization. -/ theorem generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateFieldSetIntegralDef {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} {barB : State → Vec} (dotTk sigmaCoeff : Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hhatXAtS : Measurable hatXAtS) (hXk : AEMeasurable Xk P) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hbarBEqMeas : MeasurableSet {x | dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) = barB x}) (hbarBStrong : MeasureTheory.StronglyMeasurable barB) (hbarBIntHat : MeasureTheory.Integrable barB hatRhoS) (hbarBStateSetIntegral : ∀ t : Set State, MeasurableSet t → P (hatXAtS ⁻¹' t) < ⊤ → (∫ ω in hatXAtS ⁻¹' t, barB (hatXAtS ω) ∂P) = ∫ ω in hatXAtS ⁻¹' t, dotTk • guideIntegrand (hatXAtS ω, Xk ω) + sigmaCoeff • scoreIntegrand (hatXAtS ω, Xk ω) ∂P) : (fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x)) =ᵐ[hatRhoS] barB := by have hbarBRegularity : MeasureTheory.AEStronglyMeasurable (m := (inferInstance : MeasurableSpace State).comap hatXAtS) (fun ω => barB (hatXAtS ω)) P ∧ MeasureTheory.Integrable (fun ω => barB (hatXAtS ω)) P := generalMovingTargetDiscreteNamedBarBComapRegularityOfStateField (P := P) (hatRhoS := hatRhoS) (hatXAtS := hatXAtS) (barB := barB) hhatRhoS hhatXAtS hbarBStrong hbarBIntHat exact generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef (P := P) (hatRhoS := hatRhoS) (hatXAtS := hatXAtS) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) (barB := barB) dotTk sigmaCoeff hhatRhoS hhatXAtS hXk hguideInt hscoreInt hbarBEqMeas hbarBRegularity.1 hbarBRegularity.2 hbarBStateSetIntegral /-- Source-specific map-a.e. transfer for named conditional component fields. The Mathlib conditional-expectation facts often first produce a sample-space a.e. statement after composing with `hatXAtS`. Since the paper names `\hat\rho_s = Law(\hat X_s)`, this lemma transports such a statement to a `hatRhoS`-a.e. equality on state space. The measurable equality-set hypothesis is the exact remaining versioning side condition needed for this transfer; the lemma does not construct the conditional law or prove weak FP. -/ theorem generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {componentIntegrand : State × State → Vec} {field : State → Vec} (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hhatXAtS : AEMeasurable hatXAtS P) (hmeasEq : MeasurableSet {x | (∫ y, componentIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) = field x}) (hcomp : (fun ω => ∫ y, componentIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) =ᵐ[P] fun ω => field (hatXAtS ω)) : (fun x => ∫ y, componentIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) =ᵐ[hatRhoS] field := by rw [hhatRhoS] exact (MeasureTheory.ae_map_iff hhatXAtS hmeasEq).2 hcomp /-- Source-specific component-version bridge from `condExpKernel.map`. This cycle-103 lower theorem targets `appendix.tex:1368-1377` for one component such as `condC_{k,s}`. It no longer takes the old sample-space `hguideComp` equality as a premise: instead, it reduces that equality to a measure-valued `condDistrib`/`condExpKernel.map` kernel alignment and a selected `condExpKernel.map` version of the named field. The remaining boundary is proving those two conditional-kernel facts, plus the equality-set measurability required by `ae_map_iff`. -/ theorem generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfCondExpKernelMap {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [StandardBorelSpace Ω] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {componentIntegrand : State × State → Vec} {field : State → Vec} (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hhatXAtS : AEMeasurable hatXAtS P) (hmeasEq : MeasurableSet {x | (∫ y, componentIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) = field x}) (hkernel : (fun ω => ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) =ᵐ[P] fun ω => (ProbabilityTheory.condExpKernel P ((inferInstance : MeasurableSpace State).comap hatXAtS)).map Xk ω) (hfieldKernel : (fun ω => ∫ y, componentIntegrand (hatXAtS ω, y) ∂(ProbabilityTheory.condExpKernel P ((inferInstance : MeasurableSpace State).comap hatXAtS)).map Xk ω) =ᵐ[P] fun ω => field (hatXAtS ω)) : (fun x => ∫ y, componentIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) =ᵐ[hatRhoS] field := by have hsample : (fun ω => ∫ y, componentIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) =ᵐ[P] fun ω => field (hatXAtS ω) := AutoSamplingTheory.condDistribIntegralSampleAeEqOfCondExpKernelMap (μ := P) (X := hatXAtS) (Y := Xk) (f := componentIntegrand) (field := field) hkernel hfieldKernel exact generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample (P := P) (hatRhoS := hatRhoS) (hatXAtS := hatXAtS) (Xk := Xk) (componentIntegrand := componentIntegrand) (field := field) hhatRhoS hhatXAtS hmeasEq hsample /-- Cycle-91 lower theorem using sample-space version equalities. This removes the direct `hatRhoS`-a.e. component-version hypotheses from `generalMovingTargetDiscreteCondDistribNamedDriftRegularity`: it is enough to prove the corresponding sample-space a.e. equalities after composing with `hatXAtS`, together with measurability of the two equality sets. The theorem still leaves the actual SALD conditional-expectation/disintegration proof as the remaining analytic boundary. -/ theorem generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions {Ω State Vec : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} {barB condC condScore : State → Vec} (dotTk sigmaCoeff : Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hhatXAtS : AEMeasurable hatXAtS P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hguideEqMeas : MeasurableSet {x | (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) = condC x}) (hguideComp : (fun ω => ∫ y, guideIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) =ᵐ[P] fun ω => condC (hatXAtS ω)) (hscoreEqMeas : MeasurableSet {x | (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) = condScore x}) (hscoreComp : (fun ω => ∫ y, scoreIntegrand (hatXAtS ω, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P (hatXAtS ω)) =ᵐ[P] fun ω => condScore (hatXAtS ω)) (hbar : ∀ x, barB x = dotTk • condC x + sigmaCoeff • condScore x) : MeasureTheory.AEStronglyMeasurable barB hatRhoS ∧ MeasureTheory.Integrable barB hatRhoS := by have hcondC : (fun x => ∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) =ᵐ[hatRhoS] condC := generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample (P := P) (hatRhoS := hatRhoS) (hatXAtS := hatXAtS) (Xk := Xk) (componentIntegrand := guideIntegrand) (field := condC) hhatRhoS hhatXAtS hguideEqMeas hguideComp have hcondScore : (fun x => ∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) =ᵐ[hatRhoS] condScore := generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample (P := P) (hatRhoS := hatRhoS) (hatXAtS := hatXAtS) (Xk := Xk) (componentIntegrand := scoreIntegrand) (field := condScore) hhatRhoS hhatXAtS hscoreEqMeas hscoreComp exact generalMovingTargetDiscreteCondDistribNamedDriftRegularity (P := P) (hatRhoS := hatRhoS) (hatXAtS := hatXAtS) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) (barB := barB) (condC := condC) (condScore := condScore) dotTk sigmaCoeff hhatRhoS hXk hguideMeas hguideInt hscoreMeas hscoreInt hcondC hcondScore hbar /-- Cycle-74 lower handoff from a supplied conditional-kernel backend. Once the cited Mathlib/disintegration layer supplies kernel compatibility for the joint law, and supplies the component integral fields with their measurability and integrability consequences, this wrapper packages exactly the part needed by the paper's `bar b_{k,s}` line: transport compatibility to the named `hat rho_s` marginal and inherit regularity for the selected drift field. It does not construct `condDistrib`, `condExpKernel`, or the weak Fokker--Planck equation. -/ theorem generalMovingTargetDiscreteConditionalKernelRegularityOfComponents {Ω State Vec Kernel : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [AddCommGroup Vec] [Module Real Vec] (P : MeasureTheory.Measure Ω) (Xk hatXAtS : Ω → State) (hatRhoS : MeasureTheory.Measure State) (kernel : Kernel) (KernelCompatible : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (KernelIntegralField : Kernel → (State → Vec) → Prop) (FieldMeasurable FieldIntegrable : (State → Vec) → Prop) (barB condC condScore : State → Vec) (dotTk sigmaCoeff : Real) (hXk : Measurable Xk) (hhatXAtS : Measurable hatXAtS) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hcompat : KernelCompatible (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) (MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P)) kernel) (hcongr : ∀ {μ ν : MeasureTheory.Measure State}, μ = ν → KernelCompatible (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) μ kernel → KernelCompatible (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) ν kernel) (hcondC : KernelIntegralField kernel condC) (hcondScore : KernelIntegralField kernel condScore) (hmeasOfKernel : ∀ {f : State → Vec}, KernelIntegralField kernel f → FieldMeasurable f) (hintOfKernel : ∀ {f : State → Vec}, KernelIntegralField kernel f → FieldIntegrable f) (hbar : ∀ x, barB x = dotTk • condC x + sigmaCoeff • condScore x) (hmeasAdd : ∀ {f g : State → Vec}, FieldMeasurable f → FieldMeasurable g → FieldMeasurable (fun x => f x + g x)) (hmeasSmul : ∀ (a : Real) {f : State → Vec}, FieldMeasurable f → FieldMeasurable (fun x => a • f x)) (hintAdd : ∀ {f g : State → Vec}, FieldIntegrable f → FieldIntegrable g → FieldIntegrable (fun x => f x + g x)) (hintSmul : ∀ (a : Real) {f : State → Vec}, FieldIntegrable f → FieldIntegrable (fun x => a • f x)) (hmeasCongr : ∀ {f g : State → Vec}, (∀ x, f x = g x) → FieldMeasurable f → FieldMeasurable g) (hintCongr : ∀ {f g : State → Vec}, (∀ x, f x = g x) → FieldIntegrable f → FieldIntegrable g) : KernelCompatible (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) hatRhoS kernel ∧ FieldMeasurable barB ∧ FieldIntegrable barB := by have hcompatNamed : KernelCompatible (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) hatRhoS kernel := generalMovingTargetDiscreteConditionalKernelCompatibilityOfJointMapMarginal P Xk hatXAtS hatRhoS kernel KernelCompatible hXk hhatXAtS hhatRhoS hcompat hcongr have hreg : FieldMeasurable barB ∧ FieldIntegrable barB := generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents FieldMeasurable FieldIntegrable barB condC condScore dotTk sigmaCoeff hbar (hmeasOfKernel hcondC) (hmeasOfKernel hcondScore) (hintOfKernel hcondC) (hintOfKernel hcondScore) hmeasAdd hmeasSmul hintAdd hintSmul hmeasCongr hintCongr exact ⟨hcompatNamed, hreg⟩ /-- Cycle-75 lower handoff for the `condDistrib` orientation. Mathlib's conditional distribution for `X_k^eta | hatX_s` supplies the joint law in the order `(hatX_s, X_k^eta)`. Existing SALD endpoint compatibility uses `(X_k^eta, hatX_s)`. This wrapper proves the first-marginal and swap bookkeeping, then transfers a supplied swapped-joint kernel compatibility to the existing SALD orientation and inherits `barB` regularity from supplied component integral fields. It does not construct `condDistrib`, `condExpKernel`, conditional expectations, or the weak Fokker--Planck theorem. -/ theorem generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents {Ω State Vec Kernel : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [AddCommGroup Vec] [Module Real Vec] (P : MeasureTheory.Measure Ω) (Xk hatXAtS : Ω → State) (hatRhoS : MeasureTheory.Measure State) (kernel : Kernel) (KernelCompatibleSwapped : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (KernelCompatibleOriginal : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (KernelIntegralField : Kernel → (State → Vec) → Prop) (FieldMeasurable FieldIntegrable : (State → Vec) → Prop) (barB condC condScore : State → Vec) (dotTk sigmaCoeff : Real) (hXk : Measurable Xk) (hhatXAtS : Measurable hatXAtS) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hcompatSwapped : KernelCompatibleSwapped (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P) hatRhoS kernel) (hcompatBridge : ∀ {joint jointSwapped : MeasureTheory.Measure (State × State)} {μ : MeasureTheory.Measure State}, MeasureTheory.Measure.map Prod.swap joint = jointSwapped → KernelCompatibleSwapped jointSwapped μ kernel → KernelCompatibleOriginal joint μ kernel) (hcondC : KernelIntegralField kernel condC) (hcondScore : KernelIntegralField kernel condScore) (hmeasOfKernel : ∀ {f : State → Vec}, KernelIntegralField kernel f → FieldMeasurable f) (hintOfKernel : ∀ {f : State → Vec}, KernelIntegralField kernel f → FieldIntegrable f) (hbar : ∀ x, barB x = dotTk • condC x + sigmaCoeff • condScore x) (hmeasAdd : ∀ {f g : State → Vec}, FieldMeasurable f → FieldMeasurable g → FieldMeasurable (fun x => f x + g x)) (hmeasSmul : ∀ (a : Real) {f : State → Vec}, FieldMeasurable f → FieldMeasurable (fun x => a • f x)) (hintAdd : ∀ {f g : State → Vec}, FieldIntegrable f → FieldIntegrable g → FieldIntegrable (fun x => f x + g x)) (hintSmul : ∀ (a : Real) {f : State → Vec}, FieldIntegrable f → FieldIntegrable (fun x => a • f x)) (hmeasCongr : ∀ {f g : State → Vec}, (∀ x, f x = g x) → FieldMeasurable f → FieldMeasurable g) (hintCongr : ∀ {f g : State → Vec}, (∀ x, f x = g x) → FieldIntegrable f → FieldIntegrable g) : MeasureTheory.Measure.map Prod.fst (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P) = hatRhoS ∧ MeasureTheory.Measure.map Prod.swap (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) = MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P ∧ KernelCompatibleOriginal (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) hatRhoS kernel ∧ FieldMeasurable barB ∧ FieldIntegrable barB := by have hfirst : MeasureTheory.Measure.map Prod.fst (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P) = hatRhoS := generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap P Xk hatXAtS hatRhoS hXk hhatXAtS hhatRhoS have hswap : MeasureTheory.Measure.map Prod.swap (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) = MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P := lawMapProdSwap (P := P) (X := Xk) (Y := hatXAtS) hXk hhatXAtS have hcompatOriginal : KernelCompatibleOriginal (MeasureTheory.Measure.map (fun ω => (Xk ω, hatXAtS ω)) P) hatRhoS kernel := hcompatBridge hswap hcompatSwapped have hreg : FieldMeasurable barB ∧ FieldIntegrable barB := generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents FieldMeasurable FieldIntegrable barB condC condScore dotTk sigmaCoeff hbar (hmeasOfKernel hcondC) (hmeasOfKernel hcondScore) (hintOfKernel hcondC) (hintOfKernel hcondScore) hmeasAdd hmeasSmul hintAdd hintSmul hmeasCongr hintCongr exact ⟨hfirst, hswap, hcompatOriginal, hreg⟩ /-- Cycle-76 endpoint-law to swapped conditional-kernel compatibility. This is the endpoint-to-conditional bridge for the active EM backend. It packages the already compiled endpoint `Measure.map` handoff for `\hat X_{s_k}` and `\hat X_{s_{k+1}}` together with the Mathlib-style swapped conditional-distribution orientation `(hat X_s, X_k^eta)`. Under an explicit supplied kernel compatibility theorem for that swapped joint law, the wrapper bridges back to the paper's `(X_k^eta, hat X_s)` orientation consumed by the weak Fokker--Planck interface. It does not construct a regular conditional law, conditional expectation, density, weak Fokker--Planck equation, or KL derivative. -/ theorem generalMovingTargetDiscreteEndpointMeasureMapToSwappedConditionalCompatibility {Ω State Kernel : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [AddCommGroup State] [Module Real State] (P : MeasureTheory.Measure Ω) (hatX : Real → Ω → State) (Xk XNext frozenDrift W noise : Ω → State) (rhoK rhoNext hatRhoS : MeasureTheory.Measure State) (kernel : Kernel) (KernelCompatibleSwapped : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (KernelCompatibleOriginal : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (s sK sNext eta sigmaEta : Real) (hXk : Measurable Xk) (hhatAtS : Measurable (hatX s)) (hhatLeft : hatX sK =ᵐ[P] fun ω => Xk ω + (sK - sK) • frozenDrift ω + sigmaEta • (W ω - W ω)) (hhatRight : hatX sNext =ᵐ[P] fun ω => Xk ω + (sNext - sK) • frozenDrift ω + sigmaEta • noise ω) (hstep : sNext - sK = eta) (hupdate : XNext =ᵐ[P] fun ω => Xk ω + eta • frozenDrift ω + sigmaEta • noise ω) (hrhoK : rhoK = MeasureTheory.Measure.map Xk P) (hrhoNext : rhoNext = MeasureTheory.Measure.map XNext P) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map (hatX s) P) (hcompatSwapped : KernelCompatibleSwapped (MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P) hatRhoS kernel) (hcompatBridge : ∀ {joint jointSwapped : MeasureTheory.Measure (State × State)} {μ : MeasureTheory.Measure State}, MeasureTheory.Measure.map Prod.swap joint = jointSwapped → KernelCompatibleSwapped jointSwapped μ kernel → KernelCompatibleOriginal joint μ kernel) : MeasureTheory.Measure.map (hatX sK) P = rhoK ∧ MeasureTheory.Measure.map (hatX sNext) P = rhoNext ∧ MeasureTheory.Measure.map Prod.fst (MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P) = hatRhoS ∧ MeasureTheory.Measure.map Prod.swap (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) = MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P ∧ KernelCompatibleOriginal (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) hatRhoS kernel := by have hendpoints : MeasureTheory.Measure.map (hatX sK) P = MeasureTheory.Measure.map Xk P ∧ MeasureTheory.Measure.map (hatX sNext) P = MeasureTheory.Measure.map XNext P := generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation P hatX Xk XNext frozenDrift W noise sK sNext eta sigmaEta hhatLeft hhatRight hstep hupdate have hfirst : MeasureTheory.Measure.map Prod.fst (MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P) = hatRhoS := generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap P Xk (hatX s) hatRhoS hXk hhatAtS hhatRhoS have hswap : MeasureTheory.Measure.map Prod.swap (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) = MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P := lawMapProdSwap (P := P) (X := Xk) (Y := hatX s) hXk hhatAtS have hcompatOriginal : KernelCompatibleOriginal (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) hatRhoS kernel := hcompatBridge hswap hcompatSwapped refine ⟨?_, ?_, hfirst, hswap, hcompatOriginal⟩ · calc MeasureTheory.Measure.map (hatX sK) P = MeasureTheory.Measure.map Xk P := hendpoints.1 _ = rhoK := hrhoK.symm · calc MeasureTheory.Measure.map (hatX sNext) P = MeasureTheory.Measure.map XNext P := hendpoints.2 _ = rhoNext := hrhoNext.symm /-- Cycle-76 lower handoff with the original joint-law marginal exposed. The previous endpoint-to-swapped wrapper returns the first marginal in the Mathlib-style `(hat X_s, X_k^eta)` orientation. The weak Fokker--Planck interface also names the paper orientation `(X_k^eta, hat X_s)`, whose second marginal is the same `hatRhoS`. This theorem packages both marginal views with the endpoint law equalities and the supplied bridge to original-orientation kernel compatibility. It is still only `Measure.map` bookkeeping under explicit hypotheses; it does not construct `condDistrib`, conditional expectations, density, a weak Fokker--Planck equation, or the KL derivative. -/ theorem generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility {Ω State Kernel : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [AddCommGroup State] [Module Real State] (P : MeasureTheory.Measure Ω) (hatX : Real → Ω → State) (Xk XNext frozenDrift W noise : Ω → State) (rhoK rhoNext hatRhoS : MeasureTheory.Measure State) (kernel : Kernel) (KernelCompatibleSwapped : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (KernelCompatibleOriginal : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (s sK sNext eta sigmaEta : Real) (hXk : Measurable Xk) (hhatAtS : Measurable (hatX s)) (hhatLeft : hatX sK =ᵐ[P] fun ω => Xk ω + (sK - sK) • frozenDrift ω + sigmaEta • (W ω - W ω)) (hhatRight : hatX sNext =ᵐ[P] fun ω => Xk ω + (sNext - sK) • frozenDrift ω + sigmaEta • noise ω) (hstep : sNext - sK = eta) (hupdate : XNext =ᵐ[P] fun ω => Xk ω + eta • frozenDrift ω + sigmaEta • noise ω) (hrhoK : rhoK = MeasureTheory.Measure.map Xk P) (hrhoNext : rhoNext = MeasureTheory.Measure.map XNext P) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map (hatX s) P) (hcompatSwapped : KernelCompatibleSwapped (MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P) hatRhoS kernel) (hcompatBridge : ∀ {joint jointSwapped : MeasureTheory.Measure (State × State)} {μ : MeasureTheory.Measure State}, MeasureTheory.Measure.map Prod.swap joint = jointSwapped → KernelCompatibleSwapped jointSwapped μ kernel → KernelCompatibleOriginal joint μ kernel) : MeasureTheory.Measure.map (hatX sK) P = rhoK ∧ MeasureTheory.Measure.map (hatX sNext) P = rhoNext ∧ MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) = hatRhoS ∧ MeasureTheory.Measure.map Prod.fst (MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P) = hatRhoS ∧ MeasureTheory.Measure.map Prod.swap (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) = MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P ∧ KernelCompatibleOriginal (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) hatRhoS kernel := by have hswapped := generalMovingTargetDiscreteEndpointMeasureMapToSwappedConditionalCompatibility P hatX Xk XNext frozenDrift W noise rhoK rhoNext hatRhoS kernel KernelCompatibleSwapped KernelCompatibleOriginal s sK sNext eta sigmaEta hXk hhatAtS hhatLeft hhatRight hstep hupdate hrhoK hrhoNext hhatRhoS hcompatSwapped hcompatBridge have horiginalMarginal : MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) = hatRhoS := generalMovingTargetDiscreteHatRhoMarginalOfJointMap P Xk (hatX s) hatRhoS hXk hhatAtS hhatRhoS exact ⟨hswapped.1, hswapped.2.1, horiginalMarginal, hswapped.2.2.1, hswapped.2.2.2.1, hswapped.2.2.2.2⟩ /-- Cycle-81 lower handoff from endpoint `Measure.map` compatibility to the weak-FP prerequisite layer. This is the endpoint-only part of the cycle-81 backend: once the named endpoint laws, the original/swapped `hatRhoS` marginal views, the swap bridge, and original-orientation kernel compatibility are available, already-supplied regularity of `barB` is enough to feed the abstract weak-FP prerequisite predicate. It deliberately does not reconstruct the component conditional fields; that remains the cycle-80 drift-regularity handoff. -/ theorem generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff {Ω State Vec Kernel : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [AddCommGroup State] [Module Real State] (P : MeasureTheory.Measure Ω) (hatX : Real → Ω → State) (Xk XNext frozenDrift W noise : Ω → State) (rhoK rhoNext hatRhoS : MeasureTheory.Measure State) (kernel : Kernel) (KernelCompatibleSwapped : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (KernelCompatibleOriginal : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (FieldMeasurable FieldIntegrable : (State → Vec) → Prop) (WeakFpPrereq : MeasureTheory.Measure State → Kernel → (State → Vec) → Prop) (barB : State → Vec) (s sK sNext eta sigmaEta : Real) (hXk : Measurable Xk) (hhatAtS : Measurable (hatX s)) (hhatLeft : hatX sK =ᵐ[P] fun ω => Xk ω + (sK - sK) • frozenDrift ω + sigmaEta • (W ω - W ω)) (hhatRight : hatX sNext =ᵐ[P] fun ω => Xk ω + (sNext - sK) • frozenDrift ω + sigmaEta • noise ω) (hstep : sNext - sK = eta) (hupdate : XNext =ᵐ[P] fun ω => Xk ω + eta • frozenDrift ω + sigmaEta • noise ω) (hrhoK : rhoK = MeasureTheory.Measure.map Xk P) (hrhoNext : rhoNext = MeasureTheory.Measure.map XNext P) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map (hatX s) P) (hcompatSwapped : KernelCompatibleSwapped (MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P) hatRhoS kernel) (hcompatBridge : ∀ {joint jointSwapped : MeasureTheory.Measure (State × State)} {μ : MeasureTheory.Measure State}, MeasureTheory.Measure.map Prod.swap joint = jointSwapped → KernelCompatibleSwapped jointSwapped μ kernel → KernelCompatibleOriginal joint μ kernel) (hbarMeas : FieldMeasurable barB) (hbarInt : FieldIntegrable barB) (hweakReady : MeasureTheory.Measure.map (hatX sK) P = rhoK → MeasureTheory.Measure.map (hatX sNext) P = rhoNext → MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) = hatRhoS → MeasureTheory.Measure.map Prod.fst (MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P) = hatRhoS → MeasureTheory.Measure.map Prod.swap (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) = MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P → KernelCompatibleOriginal (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) hatRhoS kernel → FieldMeasurable barB → FieldIntegrable barB → WeakFpPrereq hatRhoS kernel barB) : MeasureTheory.Measure.map (hatX sK) P = rhoK ∧ MeasureTheory.Measure.map (hatX sNext) P = rhoNext ∧ MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) = hatRhoS ∧ MeasureTheory.Measure.map Prod.fst (MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P) = hatRhoS ∧ MeasureTheory.Measure.map Prod.swap (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) = MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P ∧ KernelCompatibleOriginal (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) hatRhoS kernel ∧ FieldMeasurable barB ∧ FieldIntegrable barB ∧ WeakFpPrereq hatRhoS kernel barB := by have hcompatPack := generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility P hatX Xk XNext frozenDrift W noise rhoK rhoNext hatRhoS kernel KernelCompatibleSwapped KernelCompatibleOriginal s sK sNext eta sigmaEta hXk hhatAtS hhatLeft hhatRight hstep hupdate hrhoK hrhoNext hhatRhoS hcompatSwapped hcompatBridge exact ⟨hcompatPack.1, hcompatPack.2.1, hcompatPack.2.2.1, hcompatPack.2.2.2.1, hcompatPack.2.2.2.2.1, hcompatPack.2.2.2.2.2, hbarMeas, hbarInt, hweakReady hcompatPack.1 hcompatPack.2.1 hcompatPack.2.2.1 hcompatPack.2.2.2.1 hcompatPack.2.2.2.2.1 hcompatPack.2.2.2.2.2 hbarMeas hbarInt⟩ /-- Cycle-80 endpoint/conditional drift-regularity handoff. This composes the cycle-76 endpoint-to-conditional compatibility wrapper with the named component-field regularity wrapper from the conditional-drift layer. It is still a supplied-hypothesis theorem: the regular conditional kernel, component conditional-integral fields, and their measurability/integrability consequences are not constructed here. -/ theorem generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff {Ω State Vec Kernel : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [AddCommGroup State] [Module Real State] [AddCommGroup Vec] [Module Real Vec] (P : MeasureTheory.Measure Ω) (hatX : Real → Ω → State) (Xk XNext frozenDrift W noise : Ω → State) (rhoK rhoNext hatRhoS : MeasureTheory.Measure State) (kernel : Kernel) (KernelCompatibleSwapped : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (KernelCompatibleOriginal : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (KernelIntegralField : Kernel → (State → Vec) → Prop) (FieldMeasurable FieldIntegrable : (State → Vec) → Prop) (barB condC condScore : State → Vec) (s sK sNext eta sigmaEta dotTk sigmaCoeff : Real) (hXk : Measurable Xk) (hhatAtS : Measurable (hatX s)) (hhatLeft : hatX sK =ᵐ[P] fun ω => Xk ω + (sK - sK) • frozenDrift ω + sigmaEta • (W ω - W ω)) (hhatRight : hatX sNext =ᵐ[P] fun ω => Xk ω + (sNext - sK) • frozenDrift ω + sigmaEta • noise ω) (hstep : sNext - sK = eta) (hupdate : XNext =ᵐ[P] fun ω => Xk ω + eta • frozenDrift ω + sigmaEta • noise ω) (hrhoK : rhoK = MeasureTheory.Measure.map Xk P) (hrhoNext : rhoNext = MeasureTheory.Measure.map XNext P) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map (hatX s) P) (hcompatSwapped : KernelCompatibleSwapped (MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P) hatRhoS kernel) (hcompatBridge : ∀ {joint jointSwapped : MeasureTheory.Measure (State × State)} {μ : MeasureTheory.Measure State}, MeasureTheory.Measure.map Prod.swap joint = jointSwapped → KernelCompatibleSwapped jointSwapped μ kernel → KernelCompatibleOriginal joint μ kernel) (hcondC : KernelIntegralField kernel condC) (hcondScore : KernelIntegralField kernel condScore) (hmeasOfKernel : ∀ {f : State → Vec}, KernelIntegralField kernel f → FieldMeasurable f) (hintOfKernel : ∀ {f : State → Vec}, KernelIntegralField kernel f → FieldIntegrable f) (hbar : ∀ x, barB x = dotTk • condC x + sigmaCoeff • condScore x) (hmeasAdd : ∀ {f g : State → Vec}, FieldMeasurable f → FieldMeasurable g → FieldMeasurable (fun x => f x + g x)) (hmeasSmul : ∀ (a : Real) {f : State → Vec}, FieldMeasurable f → FieldMeasurable (fun x => a • f x)) (hintAdd : ∀ {f g : State → Vec}, FieldIntegrable f → FieldIntegrable g → FieldIntegrable (fun x => f x + g x)) (hintSmul : ∀ (a : Real) {f : State → Vec}, FieldIntegrable f → FieldIntegrable (fun x => a • f x)) (hmeasCongr : ∀ {f g : State → Vec}, (∀ x, f x = g x) → FieldMeasurable f → FieldMeasurable g) (hintCongr : ∀ {f g : State → Vec}, (∀ x, f x = g x) → FieldIntegrable f → FieldIntegrable g) : MeasureTheory.Measure.map (hatX sK) P = rhoK ∧ MeasureTheory.Measure.map (hatX sNext) P = rhoNext ∧ MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) = hatRhoS ∧ MeasureTheory.Measure.map Prod.fst (MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P) = hatRhoS ∧ MeasureTheory.Measure.map Prod.swap (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) = MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P ∧ KernelCompatibleOriginal (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) hatRhoS kernel ∧ FieldMeasurable barB ∧ FieldIntegrable barB := by have hcompatPack := generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility P hatX Xk XNext frozenDrift W noise rhoK rhoNext hatRhoS kernel KernelCompatibleSwapped KernelCompatibleOriginal s sK sNext eta sigmaEta hXk hhatAtS hhatLeft hhatRight hstep hupdate hrhoK hrhoNext hhatRhoS hcompatSwapped hcompatBridge have hreg : FieldMeasurable barB ∧ FieldIntegrable barB := generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents FieldMeasurable FieldIntegrable barB condC condScore dotTk sigmaCoeff hbar (hmeasOfKernel hcondC) (hmeasOfKernel hcondScore) (hintOfKernel hcondC) (hintOfKernel hcondScore) hmeasAdd hmeasSmul hintAdd hintSmul hmeasCongr hintCongr exact ⟨hcompatPack.1, hcompatPack.2.1, hcompatPack.2.2.1, hcompatPack.2.2.2.1, hcompatPack.2.2.2.2.1, hcompatPack.2.2.2.2.2, hreg⟩ /-- Cycle-81 middle handoff from endpoint/conditional bookkeeping to weak-FP readiness. This wrapper does not prove the weak conditional Fokker--Planck theorem. It only packages the endpoint `Measure.map` law equalities, the named `hat rho_s` marginal in both joint-law orientations, the swap bridge, original-orientation kernel compatibility, and regularity of `barB` into an abstract predicate representing the prerequisites a future weak-FP theorem will consume. -/ theorem generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff {Ω State Vec Kernel : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [AddCommGroup State] [Module Real State] [AddCommGroup Vec] [Module Real Vec] (P : MeasureTheory.Measure Ω) (hatX : Real → Ω → State) (Xk XNext frozenDrift W noise : Ω → State) (rhoK rhoNext hatRhoS : MeasureTheory.Measure State) (kernel : Kernel) (KernelCompatibleSwapped : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (KernelCompatibleOriginal : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (KernelIntegralField : Kernel → (State → Vec) → Prop) (FieldMeasurable FieldIntegrable : (State → Vec) → Prop) (WeakFpPrereq : MeasureTheory.Measure State → Kernel → (State → Vec) → Prop) (barB condC condScore : State → Vec) (s sK sNext eta sigmaEta dotTk sigmaCoeff : Real) (hXk : Measurable Xk) (hhatAtS : Measurable (hatX s)) (hhatLeft : hatX sK =ᵐ[P] fun ω => Xk ω + (sK - sK) • frozenDrift ω + sigmaEta • (W ω - W ω)) (hhatRight : hatX sNext =ᵐ[P] fun ω => Xk ω + (sNext - sK) • frozenDrift ω + sigmaEta • noise ω) (hstep : sNext - sK = eta) (hupdate : XNext =ᵐ[P] fun ω => Xk ω + eta • frozenDrift ω + sigmaEta • noise ω) (hrhoK : rhoK = MeasureTheory.Measure.map Xk P) (hrhoNext : rhoNext = MeasureTheory.Measure.map XNext P) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map (hatX s) P) (hcompatSwapped : KernelCompatibleSwapped (MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P) hatRhoS kernel) (hcompatBridge : ∀ {joint jointSwapped : MeasureTheory.Measure (State × State)} {μ : MeasureTheory.Measure State}, MeasureTheory.Measure.map Prod.swap joint = jointSwapped → KernelCompatibleSwapped jointSwapped μ kernel → KernelCompatibleOriginal joint μ kernel) (hcondC : KernelIntegralField kernel condC) (hcondScore : KernelIntegralField kernel condScore) (hmeasOfKernel : ∀ {f : State → Vec}, KernelIntegralField kernel f → FieldMeasurable f) (hintOfKernel : ∀ {f : State → Vec}, KernelIntegralField kernel f → FieldIntegrable f) (hbar : ∀ x, barB x = dotTk • condC x + sigmaCoeff • condScore x) (hmeasAdd : ∀ {f g : State → Vec}, FieldMeasurable f → FieldMeasurable g → FieldMeasurable (fun x => f x + g x)) (hmeasSmul : ∀ (a : Real) {f : State → Vec}, FieldMeasurable f → FieldMeasurable (fun x => a • f x)) (hintAdd : ∀ {f g : State → Vec}, FieldIntegrable f → FieldIntegrable g → FieldIntegrable (fun x => f x + g x)) (hintSmul : ∀ (a : Real) {f : State → Vec}, FieldIntegrable f → FieldIntegrable (fun x => a • f x)) (hmeasCongr : ∀ {f g : State → Vec}, (∀ x, f x = g x) → FieldMeasurable f → FieldMeasurable g) (hintCongr : ∀ {f g : State → Vec}, (∀ x, f x = g x) → FieldIntegrable f → FieldIntegrable g) (hweakReady : MeasureTheory.Measure.map (hatX sK) P = rhoK → MeasureTheory.Measure.map (hatX sNext) P = rhoNext → MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) = hatRhoS → MeasureTheory.Measure.map Prod.fst (MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P) = hatRhoS → MeasureTheory.Measure.map Prod.swap (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) = MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P → KernelCompatibleOriginal (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) hatRhoS kernel → FieldMeasurable barB → FieldIntegrable barB → WeakFpPrereq hatRhoS kernel barB) : MeasureTheory.Measure.map (hatX sK) P = rhoK ∧ MeasureTheory.Measure.map (hatX sNext) P = rhoNext ∧ MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) = hatRhoS ∧ MeasureTheory.Measure.map Prod.fst (MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P) = hatRhoS ∧ MeasureTheory.Measure.map Prod.swap (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) = MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P ∧ KernelCompatibleOriginal (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) hatRhoS kernel ∧ FieldMeasurable barB ∧ FieldIntegrable barB ∧ WeakFpPrereq hatRhoS kernel barB := by have hpack := generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff P hatX Xk XNext frozenDrift W noise rhoK rhoNext hatRhoS kernel KernelCompatibleSwapped KernelCompatibleOriginal KernelIntegralField FieldMeasurable FieldIntegrable barB condC condScore s sK sNext eta sigmaEta dotTk sigmaCoeff hXk hhatAtS hhatLeft hhatRight hstep hupdate hrhoK hrhoNext hhatRhoS hcompatSwapped hcompatBridge hcondC hcondScore hmeasOfKernel hintOfKernel hbar hmeasAdd hmeasSmul hintAdd hintSmul hmeasCongr hintCongr exact ⟨hpack.1, hpack.2.1, hpack.2.2.1, hpack.2.2.2.1, hpack.2.2.2.2.1, hpack.2.2.2.2.2.1, hpack.2.2.2.2.2.2.1, hpack.2.2.2.2.2.2.2, hweakReady hpack.1 hpack.2.1 hpack.2.2.1 hpack.2.2.2.1 hpack.2.2.2.2.1 hpack.2.2.2.2.2.1 hpack.2.2.2.2.2.2.1 hpack.2.2.2.2.2.2.2⟩ /-- Divergence-linearity algebra for the conditional-drift FP regrouping. In `appendix.tex:377-385`, after the analytic Laplacian split and linearity of the divergence operator have been supplied, the source regroups `-div(rho*bbar)+div(rho*A)+div(rho*grad log tilde pi)` as `div(rho*A)+div(rho*(grad log tilde pi-bbar))`. This lemma formalizes only that additive regrouping through an abstract additive operator `D`. -/ theorem discreteForwardKlConditionalFpDivergenceDriftSplit {E F : Type*} [AddCommGroup E] [AddCommGroup F] (D : E → F) (hadd : ∀ x y, D (x + y) = D x + D y) (hneg : ∀ x, D (-x) = -D x) (frozen rel target : E) : -D frozen + D rel + D target = D rel + D (target - frozen) := by rw [sub_eq_add_neg, hadd, hneg] abel /-- Lower handoff algebra for the EM conditional Fokker--Planck split. This composes the two analytic inputs used in `appendix.tex:357-385`: the conditional-drift Fokker--Planck equation and the Laplacian split relative to `\tilde\pi_s`. The hypotheses `hfp` and `hlap` remain analytic obligations; the theorem only verifies that, once they are supplied, the source's regrouped right-hand side follows by additive algebra and divergence linearity. -/ theorem discreteForwardKlConditionalFpLaplacianSplitHandoff {E F : Type*} [AddCommGroup E] [AddCommGroup F] (D : E → F) (hadd : ∀ x y, D (x + y) = D x + D y) (hneg : ∀ x, D (-x) = -D x) (partialS laplacian : F) (frozen rel target : E) (hfp : partialS = -D frozen + laplacian) (hlap : laplacian = D rel + D target) : partialS = D rel + D (target - frozen) := by calc partialS = -D frozen + (D rel + D target) := by rw [hfp, hlap] _ = -D frozen + D rel + D target := by abel _ = D rel + D (target - frozen) := by exact discreteForwardKlConditionalFpDivergenceDriftSplit D hadd hneg frozen rel target /-- Weak-test source-sign handoff for the discrete general EM Fokker--Planck equation. For each admissible weak test, the analytic backend should supply the conditional Fokker--Planck identity with the drift contribution already written as `-div(hat rho_s*bar b_{k,s})` and the diffusion contribution with a named coefficient. This wrapper rewrites that coefficient to the paper's `sigma_eta^2/2` while preserving the negative drift-divergence and positive Laplacian signs from `appendix.tex:1379-1387`. -/ theorem generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff {Test F : Type*} [AddCommGroup F] [Module Real F] (partialS driftDiv laplacian : Test → F) (sigmaEta sigmaCoeff : Real) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hweak : ∀ φ, partialS φ = -(driftDiv φ) + sigmaCoeff • laplacian φ) : ∀ φ, partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := by intro φ rw [hweak φ, hcoeff] /-- Admissible-test version of the weak conditional Fokker--Planck source-sign handoff. The paper's weak form is only meant for an admissible test class. This local wrapper keeps that predicate explicit while doing the same coefficient rewrite as `generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff`: the analytic backend must still supply the weak FP identity on admissible tests. -/ theorem generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff {Test F : Type*} [AddCommGroup F] [Module Real F] (Admissible : Test → Prop) (partialS driftDiv laplacian : Test → F) (sigmaEta sigmaCoeff : Real) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hweak : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + sigmaCoeff • laplacian φ) : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := by intro φ hφ rw [hweak φ hφ, hcoeff] /-- Generator-level handoff for the weak conditional Fokker--Planck source signs. This is the cycle-77 refinement of the cycle-72 wrapper. It separates the analytic input into two supplied facts: first, the EM interpolation generator represents the time derivative of `hat rho_s` on admissible tests; second, that generator expands with the paper's drift and diffusion orientation. The local proof only composes those supplied facts and rewrites the coefficient to `sigma_eta^2/2`. -/ theorem generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff {Test F : Type*} [AddCommGroup F] [Module Real F] (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (partialS generatorAction driftDiv laplacian : Test → F) (sigmaEta sigmaCoeff : Real) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hgenerator : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → partialS φ = generatorAction φ) (hsource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → generatorAction φ = -(driftDiv φ) + sigmaCoeff • laplacian φ) : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := by intro φ hφ rw [ hgenerator φ hφ hcommon hkernel hdrift hdensity htests hboundary, hsource φ hφ hcommon hkernel hdrift hdensity htests hboundary, hcoeff ] /-- Component-split generator handoff for the weak conditional Fokker--Planck source signs. This lower cycle-77 wrapper is one step closer to the source invocation than `generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff`: the supplied generator expansion is split into a drift action and a diffusion action. Once those two pieces are identified with the paper's `-div(hat rho_s*bar b_{k,s})` and positive Laplacian contributions, the theorem composes them into the source-signed weak FP identity. It still treats the Brownian/EM generator theorem, conditional law, densities, and boundary integration by parts as analytic hypotheses. -/ theorem generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff {Test F : Type*} [AddCommGroup F] [Module Real F] (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (partialS generatorAction driftAction diffusionAction driftDiv laplacian : Test → F) (sigmaEta sigmaCoeff : Real) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hgenerator : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → partialS φ = generatorAction φ) (hgeneratorSplit : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → generatorAction φ = driftAction φ + diffusionAction φ) (hdriftSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ)) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := by intro φ hφ calc partialS φ = generatorAction φ := by exact hgenerator φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = driftAction φ + diffusionAction φ := by exact hgeneratorSplit φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = -(driftDiv φ) + sigmaCoeff • laplacian φ := by rw [ hdriftSource φ hφ hcommon hkernel hdrift hdensity htests hboundary, hdiffusionSource φ hφ hcommon hkernel hdrift hdensity htests hboundary ] _ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := by rw [hcoeff] /-- Source-facing diffusion action handoff for the weak conditional Fokker--Planck identity. The direct downstream hypothesis `diffusionAction phi = sigmaCoeff • laplacian phi` is narrowed into two source steps: the EM/Brownian diffusion part of the generator equals a named weak diffusion action, and the weak Laplacian integration-by-parts theorem identifies that action with the positive Laplacian contribution. This theorem only composes those two source-facing facts; it does not prove the analytic Laplacian theorem. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianAction {Test : Type*} (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (diffusionAction weakDiffusionAction laplacian : Test → Real) (sigmaCoeff : Real) (hdiffusionAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = weakDiffusionAction φ) (hlaplacianAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • laplacian φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary calc diffusionAction φ = weakDiffusionAction φ := by exact hdiffusionAction φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = sigmaCoeff • laplacian φ := by exact hlaplacianAction φ hφ hcommon hkernel hdrift hdensity htests hboundary /-- Weak Laplacian integration-by-parts interface for the diffusion action. This narrows the cycle-129 `hlaplacianAction` boundary. The source Fokker--Planck display in `appendix.tex:1379-1387` contributes the positive `sigmaCoeff`-scaled density-Laplacian weak action; the remaining analytic theorem is the weak Laplacian integration-by-parts identity that identifies that density action with the test-Laplacian action. This theorem only composes those two source-facing facts. -/ theorem generalMovingTargetDiscreteWeakConditionalFpLaplacianActionOfIntegrationByParts {Test : Type*} (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (weakDiffusionAction densityLaplacianAction laplacian : Test → Real) (sigmaCoeff : Real) (hdiffusionLaplacianTerm : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • densityLaplacianAction φ) (hweakLaplacianIbP : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ = laplacian φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • laplacian φ := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary calc weakDiffusionAction φ = sigmaCoeff • densityLaplacianAction φ := by exact hdiffusionLaplacianTerm φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = sigmaCoeff • laplacian φ := by rw [ hweakLaplacianIbP φ hφ hcommon hkernel hdrift hdensity htests hboundary ] /-- Diffusion-source handoff with the weak Laplacian integration-by-parts boundary exposed. This keeps the EM/Brownian generator action `hdiffusionAction` separate and replaces the broader cycle-129 `hlaplacianAction` premise by the two source-facing Laplacian facts: the positive density-Laplacian weak term from `appendix.tex:1379-1387` and the weak Laplacian integration-by-parts theorem. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianIntegrationByParts {Test : Type*} (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (diffusionAction weakDiffusionAction densityLaplacianAction laplacian : Test → Real) (sigmaCoeff : Real) (hdiffusionAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = weakDiffusionAction φ) (hdiffusionLaplacianTerm : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • densityLaplacianAction φ) (hweakLaplacianIbP : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ = laplacian φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ := by have hlaplacianAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • laplacian φ := generalMovingTargetDiscreteWeakConditionalFpLaplacianActionOfIntegrationByParts Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior weakDiffusionAction densityLaplacianAction laplacian sigmaCoeff hdiffusionLaplacianTerm hweakLaplacianIbP exact generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianAction Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior diffusionAction weakDiffusionAction laplacian sigmaCoeff hdiffusionAction hlaplacianAction /-- Green-identity scout route for the weak Laplacian integration-by-parts boundary. This narrows the direct `hweakLaplacianIbP` input exposed in cycle 130 to the two no-boundary Green steps used by the source proof: move the density Laplacian onto the density-test gradient pairing, then move that pairing onto the test-Laplacian action. The final normalization identifies the local test-Laplacian action with the abstract `laplacian` consumed by the weak-FP helper. -/ theorem generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity {Test : Type*} (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian : Test → Real) (hfirstGreen : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ = negativeGradientPairAction φ) (hsecondGreen : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → negativeGradientPairAction φ = testLaplacianAction φ) (htestLaplacian : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → testLaplacianAction φ = laplacian φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ = laplacian φ := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary calc densityLaplacianAction φ = negativeGradientPairAction φ := by exact hfirstGreen φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = testLaplacianAction φ := by exact hsecondGreen φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = laplacian φ := by exact htestLaplacian φ hφ hcommon hkernel hdrift hdensity htests hboundary /-- Diffusion-source handoff with the weak Laplacian IBP route exposed as two Green identities. This removes the direct `hweakLaplacianIbP` premise from the cycle-130 diffusion-source helper. The remaining analytic leaves are the first and second Green identities, plus the local normalization of the test-Laplacian action. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfGreenLaplacianIbP {Test : Type*} (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian : Test → Real) (sigmaCoeff : Real) (hdiffusionAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = weakDiffusionAction φ) (hdiffusionLaplacianTerm : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • densityLaplacianAction φ) (hfirstGreen : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ = negativeGradientPairAction φ) (hsecondGreen : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → negativeGradientPairAction φ = testLaplacianAction φ) (htestLaplacian : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → testLaplacianAction φ = laplacian φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ := by have hweakLaplacianIbP : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ = laplacian φ := generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian hfirstGreen hsecondGreen htestLaplacian exact generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianIntegrationByParts Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior diffusionAction weakDiffusionAction densityLaplacianAction laplacian sigmaCoeff hdiffusionAction hdiffusionLaplacianTerm hweakLaplacianIbP /-- Diffusion-source handoff with the first Green identity factored through no-boundary flux algebra. This lower helper narrows the direct `hfirstGreen` premise from the lower_1 Green route. The first Green identity is reconstructed from the density-Laplacian residual identity, a divergence-theorem boundary-flux identity, and the zero boundary-flux condition. The second Green identity and test-Laplacian normalization remain explicit analytic leaves. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfFirstGreenNoBoundaryFlux {Test : Type*} (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux : Test → Real) (sigmaCoeff : Real) (hdiffusionAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = weakDiffusionAction φ) (hdiffusionLaplacianTerm : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • densityLaplacianAction φ) (hfirstGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ - negativeGradientPairAction φ = firstGreenTotal φ) (hfirstGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenTotal φ = firstGreenBoundaryFlux φ) (hfirstGreenZeroBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenBoundaryFlux φ = 0) (hsecondGreen : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → negativeGradientPairAction φ = testLaplacianAction φ) (htestLaplacian : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → testLaplacianAction φ = laplacian φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ := by have hfirstGreen : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ = negativeGradientPairAction φ := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary have hresidualZero : densityLaplacianAction φ - negativeGradientPairAction φ = 0 := by calc densityLaplacianAction φ - negativeGradientPairAction φ = firstGreenTotal φ := hfirstGreenResidual φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = firstGreenBoundaryFlux φ := hfirstGreenDivergence φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = 0 := hfirstGreenZeroBoundary φ hφ hcommon hkernel hdrift hdensity htests hboundary linarith exact generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfGreenLaplacianIbP Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian sigmaCoeff hdiffusionAction hdiffusionLaplacianTerm hfirstGreen hsecondGreen htestLaplacian /-- Diffusion-source handoff with both Green identities factored through no-boundary flux algebra. This narrows the remaining direct `hsecondGreen` premise left by `generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfFirstGreenNoBoundaryFlux`. The second Green identity is reconstructed from its residual identity, a divergence-theorem boundary-flux identity, and the zero boundary-flux condition. The first-Green residual facts and the test-Laplacian normalization remain separate analytic leaves. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenNoBoundaryFlux {Test : Type*} (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux : Test → Real) (sigmaCoeff : Real) (hdiffusionAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = weakDiffusionAction φ) (hdiffusionLaplacianTerm : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • densityLaplacianAction φ) (hfirstGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ - negativeGradientPairAction φ = firstGreenTotal φ) (hfirstGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenTotal φ = firstGreenBoundaryFlux φ) (hfirstGreenZeroBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenBoundaryFlux φ = 0) (hsecondGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → negativeGradientPairAction φ - testLaplacianAction φ = secondGreenTotal φ) (hsecondGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenTotal φ = secondGreenBoundaryFlux φ) (hsecondGreenZeroBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenBoundaryFlux φ = 0) (htestLaplacian : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → testLaplacianAction φ = laplacian φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ := by have hsecondGreen : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → negativeGradientPairAction φ = testLaplacianAction φ := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary have hresidualZero : negativeGradientPairAction φ - testLaplacianAction φ = 0 := by calc negativeGradientPairAction φ - testLaplacianAction φ = secondGreenTotal φ := hsecondGreenResidual φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = secondGreenBoundaryFlux φ := hsecondGreenDivergence φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = 0 := hsecondGreenZeroBoundary φ hφ hcommon hkernel hdrift hdensity htests hboundary linarith exact generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfFirstGreenNoBoundaryFlux Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux sigmaCoeff hdiffusionAction hdiffusionLaplacianTerm hfirstGreenResidual hfirstGreenDivergence hfirstGreenZeroBoundary hsecondGreen htestLaplacian /-- Second-Green diffusion-source handoff with zero boundary flux narrowed to a boundary trace-product condition. This keeps the cycle-131 residual and divergence facts explicit, but replaces the direct `hsecondGreenZeroBoundary` input by the same no-boundary trace interface used elsewhere in the EM backend: a boundary integral representation for the second-Green flux and an a.e. zero trace product on that boundary. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenTraceBoundary {Boundary Test : Type*} [MeasurableSpace Boundary] (boundaryMeasure : MeasureTheory.Measure Boundary) (secondGreenTestTrace : Test → Boundary → Real) (secondGreenNormalFluxTrace : Boundary → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux : Test → Real) (sigmaCoeff : Real) (hdiffusionAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = weakDiffusionAction φ) (hdiffusionLaplacianTerm : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • densityLaplacianAction φ) (hfirstGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ - negativeGradientPairAction φ = firstGreenTotal φ) (hfirstGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenTotal φ = firstGreenBoundaryFlux φ) (hfirstGreenZeroBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenBoundaryFlux φ = 0) (hsecondGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → negativeGradientPairAction φ - testLaplacianAction φ = secondGreenTotal φ) (hsecondGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenTotal φ = secondGreenBoundaryFlux φ) (hsecondGreenBoundaryFluxIntegral : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenBoundaryFlux φ = ∫ y, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y ∂boundaryMeasure) (hsecondGreenTraceProductZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y = 0) (htestLaplacian : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → testLaplacianAction φ = laplacian φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ := by have hsecondGreenZeroBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenBoundaryFlux φ = 0 := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary calc secondGreenBoundaryFlux φ = ∫ y, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y ∂boundaryMeasure := hsecondGreenBoundaryFluxIntegral φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = ∫ _y, (0 : Real) ∂boundaryMeasure := MeasureTheory.integral_congr_ae (hsecondGreenTraceProductZero φ hφ hcommon hkernel hdrift hdensity htests hboundary) _ = 0 := by simp exact generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenNoBoundaryFlux Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux sigmaCoeff hdiffusionAction hdiffusionLaplacianTerm hfirstGreenResidual hfirstGreenDivergence hfirstGreenZeroBoundary hsecondGreenResidual hsecondGreenDivergence hsecondGreenZeroBoundary htestLaplacian /-- Sample-space generator derivative handoff for the weak conditional Fokker--Planck source signs. This cycle-86 refinement removes the coarse supplied generator/time-derivative equality from the cycle-77 source-sign wrapper. For each admissible weak test, the sample-space EM interpolation derivative is transported to the law integral by `lawMapIntegralHasDerivAtOfSample`; uniqueness of `HasDerivAt` then identifies the law weak derivative `partialS` with the generator action. The remaining sample-path derivative, weak law derivative, generator split, source-action, and boundary hypotheses are the smaller analytic backend below `appendix.tex` lines 1379-1387. -/ theorem generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff {Ω State Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) (hatX : Real → Ω → State) (testEval : Test → State → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (partialS generatorAction driftAction diffusionAction driftDiv laplacian : Test → Real) (s0 sigmaEta sigmaCoeff : Real) (hhatX : ∀ s, AEMeasurable (hatX s) P) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (MeasureTheory.Measure.map (hatX s) P)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hsampleGenerator : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → HasDerivAt (fun s => ∫ ω, testEval φ (hatX s ω) ∂P) (generatorAction φ) s0) (hlawDerivative : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → HasDerivAt (fun s => ∫ x, testEval φ x ∂MeasureTheory.Measure.map (hatX s) P) (partialS φ) s0) (hgeneratorSplit : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → generatorAction φ = driftAction φ + diffusionAction φ) (hdriftSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ)) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := by have hgenerator : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → partialS φ = generatorAction φ := by intro φ hφ hcommon' hkernel' hdrift' hdensity' htests' hboundary' have hlawGenerator : HasDerivAt (fun s => ∫ x, testEval φ x ∂MeasureTheory.Measure.map (hatX s) P) (generatorAction φ) s0 := lawMapIntegralHasDerivAtOfSample (P := P) (X := hatX) (φ := testEval φ) hhatX (fun s => htestMeas φ s hφ) (hsampleGenerator φ hφ hcommon' hkernel' hdrift' hdensity' htests' hboundary') exact (hlawDerivative φ hφ hcommon' hkernel' hdrift' hdensity' htests' hboundary').unique hlawGenerator exact generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior partialS generatorAction driftAction diffusionAction driftDiv laplacian sigmaEta sigmaCoeff hcommon hkernel hdrift hdensity htests hboundary hcoeff hgenerator hgeneratorSplit hdriftSource hdiffusionSource /-- Sample-space generator handoff with a definitionally split generator action. This cycle-92 refinement removes the explicit `hgeneratorSplit` input from the cycle-86 theorem. The generator action consumed by the law-transport step is the paper-shaped sum of a drift action and a diffusion action. The real analytic work is still visible in the remaining hypotheses: sample-space parametric-integral differentiation for that sum, the law weak derivative, and the two source-action identifications for `-div(hat rho_s * bar b_{k,s})` and `+(sigma_eta^2/2) Delta hat rho_s`. -/ theorem generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff {Ω State Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) (hatX : Real → Ω → State) (testEval : Test → State → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (partialS driftAction diffusionAction driftDiv laplacian : Test → Real) (s0 sigmaEta sigmaCoeff : Real) (hhatX : ∀ s, AEMeasurable (hatX s) P) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (MeasureTheory.Measure.map (hatX s) P)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hsampleGenerator : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → HasDerivAt (fun s => ∫ ω, testEval φ (hatX s ω) ∂P) (driftAction φ + diffusionAction φ) s0) (hlawDerivative : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → HasDerivAt (fun s => ∫ x, testEval φ x ∂MeasureTheory.Measure.map (hatX s) P) (partialS φ) s0) (hdriftSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ)) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := by refine generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff P hatX testEval Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior partialS (fun φ => driftAction φ + diffusionAction φ) driftAction diffusionAction driftDiv laplacian s0 sigmaEta sigmaCoeff hhatX htestMeas hcommon hkernel hdrift hdensity htests hboundary hcoeff hsampleGenerator hlawDerivative ?_ hdriftSource hdiffusionSource intro φ hφ hcommon' hkernel' hdrift' hdensity' htests' hboundary' rfl /-- Mapped-law weak derivative from a sample-space split generator. This cycle-92 companion removes the separate supplied `hlawDerivative` input when the immediate goal is the weak-test Fokker--Planck derivative itself. A sample-space derivative for the drift-plus-diffusion generator is transported to the mapped law by `lawMapIntegralHasDerivAtOfSample`; the drift and diffusion source-action hypotheses then rewrite the derivative value to the paper signs `-div(hat rho_s * bar b_{k,s})` and `+(sigma_eta^2/2) Delta hat rho_s`. The sample-path/Bochner generator derivative and the two source-action identifications remain analytic inputs. -/ theorem generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff {Ω State Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) (hatX : Real → Ω → State) (testEval : Test → State → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction driftDiv laplacian : Test → Real) (s0 sigmaEta sigmaCoeff : Real) (hhatX : ∀ s, AEMeasurable (hatX s) P) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (MeasureTheory.Measure.map (hatX s) P)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hsampleGenerator : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → HasDerivAt (fun s => ∫ ω, testEval φ (hatX s ω) ∂P) (driftAction φ + diffusionAction φ) s0) (hdriftSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ)) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂MeasureTheory.Measure.map (hatX s) P) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by intro φ hφ have hlaw : HasDerivAt (fun s => ∫ x, testEval φ x ∂MeasureTheory.Measure.map (hatX s) P) (driftAction φ + diffusionAction φ) s0 := lawMapIntegralHasDerivAtOfSample (P := P) (X := hatX) (φ := testEval φ) hhatX (fun s => htestMeas φ s hφ) (hsampleGenerator φ hφ hcommon hkernel hdrift hdensity htests hboundary) have hvalue : driftAction φ + diffusionAction φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := by calc driftAction φ + diffusionAction φ = -(driftDiv φ) + sigmaCoeff • laplacian φ := by rw [ hdriftSource φ hφ hcommon hkernel hdrift hdensity htests hboundary, hdiffusionSource φ hφ hcommon hkernel hdrift hdensity htests hboundary ] _ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := by rw [hcoeff] rw [hvalue] at hlaw exact hlaw /-- Named-law weak derivative from a sample-space split generator. Cycle 104 removes the remaining bookkeeping gap between the paper notation `\hat\rho_s = Law(\hat X_s)` and the cycle-92 `Measure.map` derivative route. If `hatRhoS s` is the named law of `hatX s`, the sample-space split-generator derivative is transported directly to the named weak-test law integral. The sample-path derivative and the drift/diffusion source-action identifications remain the analytic inputs; no Fokker--Planck theorem is closed here. -/ theorem generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfSampleSplitGeneratorHandoff {Ω State Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) (hatX : Real → Ω → State) (hatRhoS : Real → MeasureTheory.Measure State) (testEval : Test → State → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction driftDiv laplacian : Test → Real) (s0 sigmaEta sigmaCoeff : Real) (hhatRhoS : ∀ s, hatRhoS s = MeasureTheory.Measure.map (hatX s) P) (hhatX : ∀ s, AEMeasurable (hatX s) P) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (hatRhoS s)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hsampleGenerator : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → HasDerivAt (fun s => ∫ ω, testEval φ (hatX s ω) ∂P) (driftAction φ + diffusionAction φ) s0) (hdriftSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ)) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by intro φ hφ have hlaw : HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (driftAction φ + diffusionAction φ) s0 := AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndSample (P := P) (X := hatX) (ρ := hatRhoS) (φ := testEval φ) hhatRhoS hhatX (fun s => htestMeas φ s hφ) (hsampleGenerator φ hφ hcommon hkernel hdrift hdensity htests hboundary) have hvalue : driftAction φ + diffusionAction φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := by calc driftAction φ + diffusionAction φ = -(driftDiv φ) + sigmaCoeff • laplacian φ := by rw [ hdriftSource φ hφ hcommon hkernel hdrift hdensity htests hboundary, hdiffusionSource φ hφ hcommon hkernel hdrift hdensity htests hboundary ] _ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := by rw [hcoeff] rw [hvalue] at hlaw exact hlaw /-- Named-law weak derivative from dominated pointwise sample-path derivatives. This cycle-110 generator-to-law refinement removes the integral-level `hsampleGenerator` premise from `generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfSampleSplitGeneratorHandoff`. For every admissible weak test, Mathlib's dominated parametric-integral theorem derives the sample-space derivative from pointwise path derivatives and an integrable bound; the existing law-map transport then gives the named-law weak Fokker--Planck derivative with the paper source signs. The remaining analytic work is the source-cited EM path derivative and domination package, plus the drift and diffusion source-action identifications. -/ theorem generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff {Ω State Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) (hatX : Real → Ω → State) (hatRhoS : Real → MeasureTheory.Measure State) (testEval : Test → State → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction driftDiv laplacian : Test → Real) (s0 sigmaEta sigmaCoeff : Real) (sampleDeriv : Test → Real → Ω → Real) (neighborhood : Test → Set Real) (bound : Test → Ω → Real) (hhatRhoS : ∀ s, hatRhoS s = MeasureTheory.Measure.map (hatX s) P) (hhatX : ∀ s, AEMeasurable (hatX s) P) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (hatRhoS s)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hsampleNeighborhood : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → neighborhood φ ∈ 𝓝 s0) (hsampleMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᶠ s in 𝓝 s0, MeasureTheory.AEStronglyMeasurable (fun ω => testEval φ (hatX s ω)) P) (hsampleInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (fun ω => testEval φ (hatX s0 ω)) P) (hsampleDerivMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (sampleDeriv φ s0) P) (hsampleDerivBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ neighborhood φ, ‖sampleDeriv φ s ω‖ ≤ bound φ ω) (hboundInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (bound φ) P) (hpathDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ neighborhood φ, HasDerivAt (fun t => testEval φ (hatX t ω)) (sampleDeriv φ s ω) s) (hderivValue : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∫ ω, sampleDeriv φ s0 ω ∂P) = driftAction φ + diffusionAction φ) (hdriftSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ)) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by intro φ hφ have hlaw : HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (driftAction φ + diffusionAction φ) s0 := AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndDominated (P := P) (X := hatX) (ρ := hatRhoS) (φ := testEval φ) (sampleDeriv := sampleDeriv φ) (neighborhood := neighborhood φ) (bound := bound φ) hhatRhoS hhatX (fun s => htestMeas φ s hφ) (hsampleNeighborhood φ hφ hcommon hkernel hdrift hdensity htests hboundary) (hsampleMeas φ hφ hcommon hkernel hdrift hdensity htests hboundary) (hsampleInt φ hφ hcommon hkernel hdrift hdensity htests hboundary) (hsampleDerivMeas φ hφ hcommon hkernel hdrift hdensity htests hboundary) (hsampleDerivBound φ hφ hcommon hkernel hdrift hdensity htests hboundary) (hboundInt φ hφ hcommon hkernel hdrift hdensity htests hboundary) (hpathDeriv φ hφ hcommon hkernel hdrift hdensity htests hboundary) (hderivValue φ hφ hcommon hkernel hdrift hdensity htests hboundary) have hvalue : driftAction φ + diffusionAction φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := by calc driftAction φ + diffusionAction φ = -(driftDiv φ) + sigmaCoeff • laplacian φ := by rw [ hdriftSource φ hφ hcommon hkernel hdrift hdensity htests hboundary, hdiffusionSource φ hφ hcommon hkernel hdrift hdensity htests hboundary ] _ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := by rw [hcoeff] rw [hvalue] at hlaw exact hlaw /-- Drift source action from the named conditional drift weak pairing. This cycle-94 helper replaces the primitive `hdriftSource` shape used by the weak conditional Fokker--Planck handoffs. The new inputs expose the two paper substeps behind the drift source sign: the frozen EM drift action is the weak test-gradient pairing against the conditional drift `barB`, and the remaining source-cited integration-by-parts/no-boundary theorem identifies that pairing with `-div(hat rho_s * barB)`. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction {State Vec Test F : Type*} [AddCommGroup F] (barB : State → Vec) (weakGradPairing : (State → Vec) → Test → F) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction driftDiv : Test → F) (hdriftBarBAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = weakGradPairing barB φ) (hbarBWeakDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing barB φ = -(driftDiv φ)) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ) := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary calc driftAction φ = weakGradPairing barB φ := by exact hdriftBarBAction φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = -(driftDiv φ) := by exact hbarBWeakDivergence φ hφ hcommon hkernel hdrift hdensity htests hboundary /-- Integrability of the `barB` weak-test contraction from integrability of the conditional drift field. This cycle-98 lower theorem removes the primitive paired-integrability input from the divergence/no-boundary route when the weak-test contraction is controlled by a scalar multiple of `‖barB‖`. For the paper line `-div(hat rho_s * bar b_{k,s})`, the remaining analytic work is to prove the test-gradient bound and the no-boundary integration-by-parts identity; the Bochner integrability step is now local Mathlib bookkeeping. -/ theorem generalMovingTargetDiscreteBarBPairIntegrableOfNormBound {State Vec Test : Type*} [MeasurableSpace State] [NormedAddCommGroup Vec] (hatRhoS : MeasureTheory.Measure State) (barB : State → Vec) (testGradPairing : Test → State → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (pairBound : Test → Real) (hbarBIntegrable : MeasureTheory.Integrable barB hatRhoS) (hpairMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (testGradPairing φ) hatRhoS) (hpairNormBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂hatRhoS, ‖testGradPairing φ x‖ ≤ pairBound φ * ‖barB x‖) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (testGradPairing φ) hatRhoS := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary exact MeasureTheory.Integrable.mono' ((hbarBIntegrable.norm).const_mul (pairBound φ)) (hpairMeas φ hφ hcommon hkernel hdrift hdensity htests hboundary) (hpairNormBound φ hφ hcommon hkernel hdrift hdensity htests hboundary) /-- `barB` weak divergence from the integral no-boundary identity. This cycle-98 handoff narrows the remaining divergence half of `ASTIS.SALD.cycle94.remaining_barB_divergence_boundary`. Instead of assuming directly that the weak gradient pairing against `barB` is `-(driftDiv phi)`, it asks for the two paper-facing definitions at `appendix.tex:1379-1387`: the weak gradient pairing is the law integral of the test-gradient contraction with `barB`, and the divergence term is the negative of that same integral after the no-boundary integration-by-parts theorem. The actual divergence theorem and boundary regularity remain source-cited analytic inputs. -/ theorem generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral {State Vec Test : Type*} [MeasurableSpace State] (hatRhoS : MeasureTheory.Measure State) (barB : State → Vec) (testGradPairing : Test → State → Real) (weakGradPairing : (State → Vec) → Test → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftDiv : Test → Real) (hpairIntegrable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (testGradPairing φ) hatRhoS) (hweakGradIntegral : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing barB φ = ∫ x, testGradPairing φ x ∂hatRhoS) (hdivNoBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, testGradPairing φ x ∂hatRhoS)) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing barB φ = -(driftDiv φ) := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary have _ := hpairIntegrable φ hφ hcommon hkernel hdrift hdensity htests hboundary calc weakGradPairing barB φ = ∫ x, testGradPairing φ x ∂hatRhoS := by exact hweakGradIntegral φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = -(driftDiv φ) := by rw [hdivNoBoundary φ hφ hcommon hkernel hdrift hdensity htests hboundary] simp /-- `barB` weak divergence with paired integrability discharged by a norm bound against the integrable conditional drift field. This composes the local integrability theorem `generalMovingTargetDiscreteBarBPairIntegrableOfNormBound` with the cycle-98 integral no-boundary handoff. It strictly narrows the lower boundary: the route no longer takes `hpairIntegrable` as a supplied hypothesis, and instead asks for `barB` integrability plus the concrete weak-test contraction bound. -/ theorem generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing {State Vec Test : Type*} [MeasurableSpace State] [NormedAddCommGroup Vec] (hatRhoS : MeasureTheory.Measure State) (barB : State → Vec) (testGradPairing : Test → State → Real) (weakGradPairing : (State → Vec) → Test → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftDiv : Test → Real) (pairBound : Test → Real) (hbarBIntegrable : MeasureTheory.Integrable barB hatRhoS) (hpairMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (testGradPairing φ) hatRhoS) (hpairNormBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂hatRhoS, ‖testGradPairing φ x‖ ≤ pairBound φ * ‖barB x‖) (hweakGradIntegral : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing barB φ = ∫ x, testGradPairing φ x ∂hatRhoS) (hdivNoBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, testGradPairing φ x ∂hatRhoS)) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing barB φ = -(driftDiv φ) := by have hpairIntegrable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (testGradPairing φ) hatRhoS := generalMovingTargetDiscreteBarBPairIntegrableOfNormBound hatRhoS barB testGradPairing Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior pairBound hbarBIntegrable hpairMeas hpairNormBound exact generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral hatRhoS barB testGradPairing weakGradPairing Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftDiv hpairIntegrable hweakGradIntegral hdivNoBoundary /-- Drift source action with the `barB` divergence fact reduced to an integral no-boundary theorem. This composes the cycle-94 `barB` weak-action handoff with the cycle-98 integral no-boundary handoff. The downstream weak-FP routes no longer need a primitive `hbarBWeakDivergence`; lower work can target the explicit law-integral definition of the weak gradient pairing and the Mathlib/local integration-by-parts theorem that makes `driftDiv` the negative of that integral. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral {State Vec Test : Type*} [MeasurableSpace State] (hatRhoS : MeasureTheory.Measure State) (barB : State → Vec) (testGradPairing : Test → State → Real) (weakGradPairing : (State → Vec) → Test → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction driftDiv : Test → Real) (hdriftBarBAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = weakGradPairing barB φ) (hpairIntegrable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (testGradPairing φ) hatRhoS) (hweakGradIntegral : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing barB φ = ∫ x, testGradPairing φ x ∂hatRhoS) (hdivNoBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, testGradPairing φ x ∂hatRhoS)) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ) := by have hbarBWeakDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing barB φ = -(driftDiv φ) := generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral hatRhoS barB testGradPairing weakGradPairing Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftDiv hpairIntegrable hweakGradIntegral hdivNoBoundary exact generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction barB weakGradPairing Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction driftDiv hdriftBarBAction hbarBWeakDivergence /-- Drift source action with the `barB` divergence fact reduced to a bounded law-integral no-boundary theorem. Compared with `generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral`, this version discharges the supplied `hpairIntegrable` premise from `Integrable barB hatRhoS` and an a.e. bound on the test-gradient contraction. The source signs and coefficient conventions are unchanged; the still-open analytic fact is the no-boundary identity for `hatRhoS * barB`. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral {State Vec Test : Type*} [MeasurableSpace State] [NormedAddCommGroup Vec] (hatRhoS : MeasureTheory.Measure State) (barB : State → Vec) (testGradPairing : Test → State → Real) (weakGradPairing : (State → Vec) → Test → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction driftDiv : Test → Real) (pairBound : Test → Real) (hdriftBarBAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = weakGradPairing barB φ) (hbarBIntegrable : MeasureTheory.Integrable barB hatRhoS) (hpairMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (testGradPairing φ) hatRhoS) (hpairNormBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂hatRhoS, ‖testGradPairing φ x‖ ≤ pairBound φ * ‖barB x‖) (hweakGradIntegral : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing barB φ = ∫ x, testGradPairing φ x ∂hatRhoS) (hdivNoBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, testGradPairing φ x ∂hatRhoS)) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ) := by have hbarBWeakDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing barB φ = -(driftDiv φ) := generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing hatRhoS barB testGradPairing weakGradPairing Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftDiv pairBound hbarBIntegrable hpairMeas hpairNormBound hweakGradIntegral hdivNoBoundary exact generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction barB weakGradPairing Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction driftDiv hdriftBarBAction hbarBWeakDivergence /-- `barB` weak divergence when the weak pairing is the law-integral definition. This cycle-100 helper removes the separate `hweakGradIntegral` supplied hypothesis from the bounded no-boundary route. It specializes `weakGradPairing` to the paper-facing law integral `fun f phi => ∫ x, fieldPairing phi f x d hatRhoS`, so only the concrete contraction bound and the no-boundary `driftDiv` identity remain as analytic inputs. -/ theorem generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef {State Vec Test : Type*} [MeasurableSpace State] [NormedAddCommGroup Vec] (hatRhoS : MeasureTheory.Measure State) (barB : State → Vec) (fieldPairing : Test → (State → Vec) → State → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftDiv : Test → Real) (pairBound : Test → Real) (hbarBIntegrable : MeasureTheory.Integrable barB hatRhoS) (hpairMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fieldPairing φ barB) hatRhoS) (hpairNormBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂hatRhoS, ‖fieldPairing φ barB x‖ ≤ pairBound φ * ‖barB x‖) (hdivNoBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, fieldPairing φ barB x ∂hatRhoS)) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (fun f ψ => ∫ x, fieldPairing ψ f x ∂hatRhoS) barB φ = -(driftDiv φ) := by exact generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing hatRhoS barB (fun φ x => fieldPairing φ barB x) (fun f φ => ∫ x, fieldPairing φ f x ∂hatRhoS) Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftDiv pairBound hbarBIntegrable hpairMeas hpairNormBound (by intro φ _ _ _ _ _ _ _ rfl) hdivNoBoundary /-- Drift source action with the weak-gradient pairing definition aligned to the law integral. This is the cycle-100 lower-ready version of `generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral`. It removes `hweakGradIntegral` as a primitive premise by making the weak pairing definition explicit. The remaining source-cited boundary is the concrete a.e. contraction bound plus the no-boundary divergence theorem giving `driftDiv phi = -∫ fieldPairing phi barB`. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryWeakGradDef {State Vec Test : Type*} [MeasurableSpace State] [NormedAddCommGroup Vec] (hatRhoS : MeasureTheory.Measure State) (barB : State → Vec) (fieldPairing : Test → (State → Vec) → State → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction driftDiv : Test → Real) (pairBound : Test → Real) (hdriftBarBAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = (fun f ψ => ∫ x, fieldPairing ψ f x ∂hatRhoS) barB φ) (hbarBIntegrable : MeasureTheory.Integrable barB hatRhoS) (hpairMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fieldPairing φ barB) hatRhoS) (hpairNormBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂hatRhoS, ‖fieldPairing φ barB x‖ ≤ pairBound φ * ‖barB x‖) (hdivNoBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, fieldPairing φ barB x ∂hatRhoS)) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ) := by have hbarBWeakDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (fun f ψ => ∫ x, fieldPairing ψ f x ∂hatRhoS) barB φ = -(driftDiv φ) := generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef hatRhoS barB fieldPairing Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftDiv pairBound hbarBIntegrable hpairMeas hpairNormBound hdivNoBoundary exact generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction barB (fun f φ => ∫ x, fieldPairing φ f x ∂hatRhoS) Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction driftDiv hdriftBarBAction hbarBWeakDivergence /-- Inner-product weak-test contraction from a gradient norm bound. This cycle-100 lower theorem removes the remaining `hpairNormBound` supplied hypothesis when the paper-facing weak pairing is the real inner product of the test gradient with the conditional drift `barB`. The only analytic bound left on this side is the smaller weak-test gradient estimate `‖testGrad phi x‖ <= pairBound phi`; Cauchy--Schwarz supplies the contraction against `barB`. -/ theorem generalMovingTargetDiscreteBarBPairNormBoundOfInnerGradientBound {State Vec Test : Type*} [MeasurableSpace State] [NormedAddCommGroup Vec] [InnerProductSpace Real Vec] (hatRhoS : MeasureTheory.Measure State) (barB : State → Vec) (testGrad : Test → State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (pairBound : Test → Real) (hgradNormBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂hatRhoS, ‖testGrad φ x‖ ≤ pairBound φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂hatRhoS, ‖inner Real (testGrad φ x) (barB x)‖ ≤ pairBound φ * ‖barB x‖ := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary filter_upwards [hgradNormBound φ hφ hcommon hkernel hdrift hdensity htests hboundary] with x hx calc ‖inner Real (testGrad φ x) (barB x)‖ ≤ ‖testGrad φ x‖ * ‖barB x‖ := by exact norm_inner_le_norm (𝕜 := Real) (testGrad φ x) (barB x) _ ≤ pairBound φ * ‖barB x‖ := by exact mul_le_mul_of_nonneg_right hx (norm_nonneg (barB x)) /-- `barB` weak divergence with inner-product contraction supplied by Cauchy--Schwarz. Compared with `generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef`, this version no longer takes `hpairNormBound`. It asks only for the weak-test gradient norm estimate and the existing no-boundary divergence identity. -/ theorem generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryInnerGradientBound {State Vec Test : Type*} [MeasurableSpace State] [NormedAddCommGroup Vec] [InnerProductSpace Real Vec] (hatRhoS : MeasureTheory.Measure State) (barB : State → Vec) (testGrad : Test → State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftDiv : Test → Real) (pairBound : Test → Real) (hbarBIntegrable : MeasureTheory.Integrable barB hatRhoS) (hpairMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun x => inner Real (testGrad φ x) (barB x)) hatRhoS) (hgradNormBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂hatRhoS, ‖testGrad φ x‖ ≤ pairBound φ) (hdivNoBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, inner Real (testGrad φ x) (barB x) ∂hatRhoS)) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂hatRhoS) barB φ = -(driftDiv φ) := by have hpairNormBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂hatRhoS, ‖inner Real (testGrad φ x) (barB x)‖ ≤ pairBound φ * ‖barB x‖ := generalMovingTargetDiscreteBarBPairNormBoundOfInnerGradientBound hatRhoS barB testGrad Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior pairBound hgradNormBound exact generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef hatRhoS barB (fun φ f x => inner Real (testGrad φ x) (f x)) Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftDiv pairBound hbarBIntegrable hpairMeas hpairNormBound hdivNoBoundary /-- Drift source action with both weak-pairing definition alignment and the inner-product contraction bound discharged locally. The remaining analytic boundary is now the weak-test gradient norm estimate plus the no-boundary identity `driftDiv phi = -int inner (testGrad phi) barB`. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound {State Vec Test : Type*} [MeasurableSpace State] [NormedAddCommGroup Vec] [InnerProductSpace Real Vec] (hatRhoS : MeasureTheory.Measure State) (barB : State → Vec) (testGrad : Test → State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction driftDiv : Test → Real) (pairBound : Test → Real) (hdriftBarBAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = (fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂hatRhoS) barB φ) (hbarBIntegrable : MeasureTheory.Integrable barB hatRhoS) (hpairMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun x => inner Real (testGrad φ x) (barB x)) hatRhoS) (hgradNormBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂hatRhoS, ‖testGrad φ x‖ ≤ pairBound φ) (hdivNoBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, inner Real (testGrad φ x) (barB x) ∂hatRhoS)) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ) := by have hbarBWeakDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂hatRhoS) barB φ = -(driftDiv φ) := generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryInnerGradientBound hatRhoS barB testGrad Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftDiv pairBound hbarBIntegrable hpairMeas hgradNormBound hdivNoBoundary exact generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction barB (fun f φ => ∫ x, inner Real (testGrad φ x) (f x) ∂hatRhoS) Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction driftDiv hdriftBarBAction hbarBWeakDivergence /-- Product-rule/no-boundary algebra for the `barB` drift-divergence term. This lower helper narrows the monolithic `hdivNoBoundary` premise to the source-facing pieces expected from the no-boundary divergence theorem: the product-rule total divergence identity, the Mathlib divergence-theorem boundary-flux identity, and the zero boundary-flux condition. -/ theorem generalMovingTargetDiscreteDriftDivNoBoundaryOfProductRule {Test : Type*} (driftDiv pairing divTotal boundaryFlux : Test → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (hproductRule : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ + pairing φ = divTotal φ) (hdivergenceTheorem : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → divTotal φ = boundaryFlux φ) (hzeroBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → boundaryFlux φ = 0) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(pairing φ) := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary have hsum : driftDiv φ + pairing φ = 0 := by calc driftDiv φ + pairing φ = divTotal φ := hproductRule φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = boundaryFlux φ := hdivergenceTheorem φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = 0 := hzeroBoundary φ hφ hcommon hkernel hdrift hdensity htests hboundary linarith /-- Drift source action with the no-boundary premise factored through the product-rule and boundary-flux identities. Compared with `generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound`, this version no longer takes `hdivNoBoundary` directly. It asks for the source-local product-rule identity, the divergence-theorem boundary-flux identity, and zero boundary flux for `hatRhoS * barB`; `hgradNormBound` remains explicit. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientProductRuleBoundary {State Vec Test : Type*} [MeasurableSpace State] [NormedAddCommGroup Vec] [InnerProductSpace Real Vec] (hatRhoS : MeasureTheory.Measure State) (barB : State → Vec) (testGrad : Test → State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction driftDiv : Test → Real) (pairBound : Test → Real) (divTotal boundaryFlux : Test → Real) (hdriftBarBAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = (fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂hatRhoS) barB φ) (hbarBIntegrable : MeasureTheory.Integrable barB hatRhoS) (hpairMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun x => inner Real (testGrad φ x) (barB x)) hatRhoS) (hgradNormBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂hatRhoS, ‖testGrad φ x‖ ≤ pairBound φ) (hproductRule : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ + (∫ x, inner Real (testGrad φ x) (barB x) ∂hatRhoS) = divTotal φ) (hdivergenceTheorem : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → divTotal φ = boundaryFlux φ) (hzeroBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → boundaryFlux φ = 0) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ) := by have hdivNoBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, inner Real (testGrad φ x) (barB x) ∂hatRhoS) := generalMovingTargetDiscreteDriftDivNoBoundaryOfProductRule driftDiv (fun φ => ∫ x, inner Real (testGrad φ x) (barB x) ∂hatRhoS) divTotal boundaryFlux Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior hproductRule hdivergenceTheorem hzeroBoundary exact generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound hatRhoS barB testGrad Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction driftDiv pairBound hdriftBarBAction hbarBIntegrable hpairMeas hgradNormBound hdivNoBoundary /-- Dominated named-law weak derivative with canonical `barB` integrability supplied by the conditional-drift state-event interface. This cycle-119 lower theorem consumes `generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface` in the weak-FP generator route. The theorem fixes `barB` to the source-supported canonical `condDistrib` guide-plus-score field from `appendix.tex:1368-1377`, derives its `hatRhoS s0`-integrability locally, feeds that into the existing inner-gradient/no-boundary drift-source handoff, and then applies the cycle-110 dominated generator-to-law theorem. The remaining analytic boundary is precisely the EM pointwise derivative/domination package, the canonical drift weak-action identity, the concrete no-boundary divergence identity, and the diffusion source-action identity. -/ theorem generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated {Ω State Vec Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [InnerProductSpace Real Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatX : Real → Ω → State} {hatRhoS : Real → MeasureTheory.Measure State} {Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (testEval : Test → State → Real) (testGrad : Test → State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction driftDiv laplacian : Test → Real) (s0 dotTk sigmaEta sigmaCoeff : Real) (pairBound : Test → Real) (sampleDeriv : Test → Real → Ω → Real) (neighborhood : Test → Set Real) (bound : Test → Ω → Real) (hhatRhoS : ∀ s, hatRhoS s = MeasureTheory.Measure.map (hatX s) P) (hhatXAtS : Measurable (hatX s0)) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (hatRhoS s)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hsampleNeighborhood : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → neighborhood φ ∈ 𝓝 s0) (hsampleMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᶠ s in 𝓝 s0, MeasureTheory.AEStronglyMeasurable (fun ω => testEval φ (hatX s ω)) P) (hsampleInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (fun ω => testEval φ (hatX s0 ω)) P) (hsampleDerivMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (sampleDeriv φ s0) P) (hsampleDerivBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ neighborhood φ, ‖sampleDeriv φ s ω‖ ≤ bound φ ω) (hboundInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (bound φ) P) (hpathDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ neighborhood φ, HasDerivAt (fun t => testEval φ (hatX t ω)) (sampleDeriv φ s ω) s) (hderivValue : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∫ ω, sampleDeriv φ s0 ω ∂P) = driftAction φ + diffusionAction φ) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : let canonicalBarB : State → Vec := fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = (fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂(hatRhoS s0)) canonicalBarB φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun x => inner Real (testGrad φ x) (canonicalBarB x)) (hatRhoS s0)) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂(hatRhoS s0), ‖testGrad φ x‖ ≤ pairBound φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, inner Real (testGrad φ x) (canonicalBarB x) ∂(hatRhoS s0))) → ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by dsimp only intro hdriftBarBAction hpairMeas hgradNormBound hdivNoBoundary let canonicalBarB : State → Vec := fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) obtain ⟨barB, hbarBDef, hbarBInt, _hbarBStateSetIntegral⟩ := generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface (P := P) (hatRhoS := hatRhoS s0) (hatXAtS := hatX s0) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) dotTk sigmaCoeff (hhatRhoS s0) hhatXAtS hXk hguideMeas hguideInt hscoreMeas hscoreInt have hcanonicalIntegrable : MeasureTheory.Integrable canonicalBarB (hatRhoS s0) := by simpa [canonicalBarB, ← hbarBDef] using hbarBInt have hdriftSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ) := generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound (hatRhoS s0) canonicalBarB testGrad Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction driftDiv pairBound (by simpa [canonicalBarB] using hdriftBarBAction) hcanonicalIntegrable (by simpa [canonicalBarB] using hpairMeas) hgradNormBound (by simpa [canonicalBarB] using hdivNoBoundary) exact generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff P hatX hatRhoS testEval Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction diffusionAction driftDiv laplacian s0 sigmaEta sigmaCoeff sampleDeriv neighborhood bound hhatRhoS hhatX htestMeas hcommon hkernel hdrift hdensity htests hboundary hcoeff hsampleNeighborhood hsampleMeas hsampleInt hsampleDerivMeas hsampleDerivBound hboundInt hpathDeriv hderivValue hdriftSource hdiffusionSource /-- Canonical `barB` dominated weak derivative on the source EM interval. This cycle-120 lower theorem removes the supplied `hsampleNeighborhood` premise from the canonical `barB` weak-FP consumer by specializing the local dominated derivative theorem to the open EM interval containing `s0`. The source-cited stochastic/calculus work remains exactly the six Mathlib-facing sample inputs on that interval, together with the downstream derivative-value and source-action identities. -/ theorem generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalDominated {Ω State Vec Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [InnerProductSpace Real Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatX : Real → Ω → State} {hatRhoS : Real → MeasureTheory.Measure State} {Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (testEval : Test → State → Real) (testGrad : Test → State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction driftDiv laplacian : Test → Real) (s0 sLeft sRight dotTk sigmaEta sigmaCoeff : Real) (pairBound : Test → Real) (sampleDeriv : Test → Real → Ω → Real) (bound : Test → Ω → Real) (hhatRhoS : ∀ s, hatRhoS s = MeasureTheory.Measure.map (hatX s) P) (hhatXAtS : Measurable (hatX s0)) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (hatRhoS s)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hs0Interval : s0 ∈ Set.Ioo sLeft sRight) (hsampleMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᶠ s in 𝓝 s0, MeasureTheory.AEStronglyMeasurable (fun ω => testEval φ (hatX s ω)) P) (hsampleInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (fun ω => testEval φ (hatX s0 ω)) P) (hsampleDerivMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (sampleDeriv φ s0) P) (hsampleDerivBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, ‖sampleDeriv φ s ω‖ ≤ bound φ ω) (hboundInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (bound φ) P) (hpathDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, HasDerivAt (fun t => testEval φ (hatX t ω)) (sampleDeriv φ s ω) s) (hderivValue : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∫ ω, sampleDeriv φ s0 ω ∂P) = driftAction φ + diffusionAction φ) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : let canonicalBarB : State → Vec := fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = (fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂(hatRhoS s0)) canonicalBarB φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun x => inner Real (testGrad φ x) (canonicalBarB x)) (hatRhoS s0)) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂(hatRhoS s0), ‖testGrad φ x‖ ≤ pairBound φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, inner Real (testGrad φ x) (canonicalBarB x) ∂(hatRhoS s0))) → ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by exact generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated testEval testGrad Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction diffusionAction driftDiv laplacian s0 dotTk sigmaEta sigmaCoeff pairBound sampleDeriv (fun _ => Set.Ioo sLeft sRight) bound hhatRhoS hhatXAtS hhatX hXk hguideMeas hguideInt hscoreMeas hscoreInt htestMeas hcommon hkernel hdrift hdensity htests hboundary hcoeff (by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary exact isOpen_Ioo.mem_nhds hs0Interval) hsampleMeas hsampleInt hsampleDerivMeas hsampleDerivBound hboundInt hpathDeriv hderivValue hdiffusionSource /-- Canonical `barB` dominated weak derivative on the source EM interval, with sample measurability derived from law-space test measurability. This cycle-121 lower theorem removes the supplied `hsampleMeas` premise from the cycle-120 interval packet. For each time `s`, `htestMeas` gives measurability of the weak test on the named law `hatRhoS s`; rewriting `hatRhoS s` as the mapped law of `hatX s` and composing with the a.e. measurable sample path gives the sample-space measurability required by the dominated parametric-integral handoff. The remaining source-cited EM calculus facts are integrability, derivative measurability, local domination, pointwise `HasDerivAt`, derivative-value splitting, and source actions. -/ theorem generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasDominated {Ω State Vec Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [InnerProductSpace Real Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatX : Real → Ω → State} {hatRhoS : Real → MeasureTheory.Measure State} {Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (testEval : Test → State → Real) (testGrad : Test → State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction driftDiv laplacian : Test → Real) (s0 sLeft sRight dotTk sigmaEta sigmaCoeff : Real) (pairBound : Test → Real) (sampleDeriv : Test → Real → Ω → Real) (bound : Test → Ω → Real) (hhatRhoS : ∀ s, hatRhoS s = MeasureTheory.Measure.map (hatX s) P) (hhatXAtS : Measurable (hatX s0)) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (hatRhoS s)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hs0Interval : s0 ∈ Set.Ioo sLeft sRight) (hsampleInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (fun ω => testEval φ (hatX s0 ω)) P) (hsampleDerivMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (sampleDeriv φ s0) P) (hsampleDerivBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, ‖sampleDeriv φ s ω‖ ≤ bound φ ω) (hboundInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (bound φ) P) (hpathDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, HasDerivAt (fun t => testEval φ (hatX t ω)) (sampleDeriv φ s ω) s) (hderivValue : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∫ ω, sampleDeriv φ s0 ω ∂P) = driftAction φ + diffusionAction φ) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : let canonicalBarB : State → Vec := fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = (fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂(hatRhoS s0)) canonicalBarB φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun x => inner Real (testGrad φ x) (canonicalBarB x)) (hatRhoS s0)) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂(hatRhoS s0), ‖testGrad φ x‖ ≤ pairBound φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, inner Real (testGrad φ x) (canonicalBarB x) ∂(hatRhoS s0))) → ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by exact generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalDominated testEval testGrad Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction diffusionAction driftDiv laplacian s0 sLeft sRight dotTk sigmaEta sigmaCoeff pairBound sampleDeriv bound hhatRhoS hhatXAtS hhatX hXk hguideMeas hguideInt hscoreMeas hscoreInt htestMeas hcommon hkernel hdrift hdensity htests hboundary hcoeff hs0Interval (by intro φ hφ _hcommon _hkernel _hdrift _hdensity _htests _hboundary filter_upwards with s have htestMap : MeasureTheory.AEStronglyMeasurable (testEval φ) (MeasureTheory.Measure.map (hatX s) P) := by simpa [hhatRhoS s] using htestMeas φ s hφ exact htestMap.comp_aemeasurable (hhatX s)) hsampleInt hsampleDerivMeas hsampleDerivBound hboundInt hpathDeriv hderivValue hdiffusionSource /-- Canonical `barB` dominated weak derivative on the source EM interval, with sample measurability and sample integrability transported from the named law. This cycle-122 lower theorem removes the supplied `hsampleInt` premise from the cycle-121 packet. For the frozen time `s0`, law-space integrability of the weak test under `hatRhoS s0`, together with `hatRhoS s0 = Measure.map (hatX s0) P` and a.e. measurability of `hatX s0`, gives the sample-space integrability required by the dominated parametric-integral handoff. The remaining source-cited EM calculus facts are derivative measurability, local domination, pointwise `HasDerivAt`, derivative-value splitting, and source actions. -/ theorem generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDominated {Ω State Vec Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [InnerProductSpace Real Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatX : Real → Ω → State} {hatRhoS : Real → MeasureTheory.Measure State} {Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (testEval : Test → State → Real) (testGrad : Test → State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction driftDiv laplacian : Test → Real) (s0 sLeft sRight dotTk sigmaEta sigmaCoeff : Real) (pairBound : Test → Real) (sampleDeriv : Test → Real → Ω → Real) (bound : Test → Ω → Real) (hhatRhoS : ∀ s, hatRhoS s = MeasureTheory.Measure.map (hatX s) P) (hhatXAtS : Measurable (hatX s0)) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (hatRhoS s)) (htestInt : ∀ φ s, Admissible φ → MeasureTheory.Integrable (testEval φ) (hatRhoS s)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hs0Interval : s0 ∈ Set.Ioo sLeft sRight) (hsampleDerivMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (sampleDeriv φ s0) P) (hsampleDerivBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, ‖sampleDeriv φ s ω‖ ≤ bound φ ω) (hboundInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (bound φ) P) (hpathDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, HasDerivAt (fun t => testEval φ (hatX t ω)) (sampleDeriv φ s ω) s) (hderivValue : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∫ ω, sampleDeriv φ s0 ω ∂P) = driftAction φ + diffusionAction φ) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : let canonicalBarB : State → Vec := fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = (fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂(hatRhoS s0)) canonicalBarB φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun x => inner Real (testGrad φ x) (canonicalBarB x)) (hatRhoS s0)) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂(hatRhoS s0), ‖testGrad φ x‖ ≤ pairBound φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, inner Real (testGrad φ x) (canonicalBarB x) ∂(hatRhoS s0))) → ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by exact generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasDominated testEval testGrad Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction diffusionAction driftDiv laplacian s0 sLeft sRight dotTk sigmaEta sigmaCoeff pairBound sampleDeriv bound hhatRhoS hhatXAtS hhatX hXk hguideMeas hguideInt hscoreMeas hscoreInt htestMeas hcommon hkernel hdrift hdensity htests hboundary hcoeff hs0Interval (by intro φ hφ _hcommon _hkernel _hdrift _hdensity _htests _hboundary have htestIntMap : MeasureTheory.Integrable (testEval φ) (MeasureTheory.Measure.map (hatX s0) P) := by simpa [hhatRhoS s0] using htestInt φ s0 hφ simpa [Function.comp_def] using htestIntMap.comp_aemeasurable (hhatX s0)) hsampleDerivMeas hsampleDerivBound hboundInt hpathDeriv hderivValue hdiffusionSource /-- Canonical `barB` dominated weak derivative on the source EM interval, with sample derivative measurability derived from a concrete EM derivative representative. This cycle-123 lower theorem removes the supplied `hsampleDerivMeas` premise from the cycle-122 packet. The source-facing input is now that the concrete sample derivative `fun omega => deriv (fun t => testEval phi (hatX t omega)) s0` is a.e. strongly measurable and agrees a.e. with the chosen `sampleDeriv phi s0` representative. The remaining EM calculus facts are local domination, bound integrability, pointwise `HasDerivAt`, derivative-value splitting, and source actions. -/ theorem generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasDominated {Ω State Vec Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [InnerProductSpace Real Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatX : Real → Ω → State} {hatRhoS : Real → MeasureTheory.Measure State} {Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (testEval : Test → State → Real) (testGrad : Test → State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction driftDiv laplacian : Test → Real) (s0 sLeft sRight dotTk sigmaEta sigmaCoeff : Real) (pairBound : Test → Real) (sampleDeriv : Test → Real → Ω → Real) (bound : Test → Ω → Real) (hhatRhoS : ∀ s, hatRhoS s = MeasureTheory.Measure.map (hatX s) P) (hhatXAtS : Measurable (hatX s0)) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (hatRhoS s)) (htestInt : ∀ φ s, Admissible φ → MeasureTheory.Integrable (testEval φ) (hatRhoS s)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hs0Interval : s0 ∈ Set.Ioo sLeft sRight) (hconcreteSampleDerivMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun ω => deriv (fun t : Real => testEval φ (hatX t ω)) s0) P) (hsampleDerivConcreteAeEq : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → sampleDeriv φ s0 =ᵐ[P] (fun ω => deriv (fun t : Real => testEval φ (hatX t ω)) s0)) (hsampleDerivBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, ‖sampleDeriv φ s ω‖ ≤ bound φ ω) (hboundInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (bound φ) P) (hpathDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, HasDerivAt (fun t => testEval φ (hatX t ω)) (sampleDeriv φ s ω) s) (hderivValue : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∫ ω, sampleDeriv φ s0 ω ∂P) = driftAction φ + diffusionAction φ) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : let canonicalBarB : State → Vec := fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = (fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂(hatRhoS s0)) canonicalBarB φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun x => inner Real (testGrad φ x) (canonicalBarB x)) (hatRhoS s0)) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂(hatRhoS s0), ‖testGrad φ x‖ ≤ pairBound φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, inner Real (testGrad φ x) (canonicalBarB x) ∂(hatRhoS s0))) → ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by exact generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDominated testEval testGrad Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction diffusionAction driftDiv laplacian s0 sLeft sRight dotTk sigmaEta sigmaCoeff pairBound sampleDeriv bound hhatRhoS hhatXAtS hhatX hXk hguideMeas hguideInt hscoreMeas hscoreInt htestMeas htestInt hcommon hkernel hdrift hdensity htests hboundary hcoeff hs0Interval (by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary exact (hconcreteSampleDerivMeas φ hφ hcommon hkernel hdrift hdensity htests hboundary).congr (hsampleDerivConcreteAeEq φ hφ hcommon hkernel hdrift hdensity htests hboundary).symm) hsampleDerivBound hboundInt hpathDeriv hderivValue hdiffusionSource /-- Canonical `barB` dominated weak derivative on the source EM interval, with the local sample-derivative bound transported from the concrete EM derivative representative. This cycle-124 lower theorem removes the supplied `hsampleDerivBound` premise from the cycle-123 packet. The source-facing input is now a concrete bound for `deriv (fun t => testEval phi (hatX t omega)) s` on the EM interval plus an a.e. interval equality between that concrete derivative and the chosen `sampleDeriv phi s` representative. The theorem keeps bound integrability, pointwise `HasDerivAt`, derivative-value splitting, and source actions explicit. -/ theorem generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundDominated {Ω State Vec Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [InnerProductSpace Real Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatX : Real → Ω → State} {hatRhoS : Real → MeasureTheory.Measure State} {Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (testEval : Test → State → Real) (testGrad : Test → State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction driftDiv laplacian : Test → Real) (s0 sLeft sRight dotTk sigmaEta sigmaCoeff : Real) (pairBound : Test → Real) (sampleDeriv : Test → Real → Ω → Real) (bound : Test → Ω → Real) (hhatRhoS : ∀ s, hatRhoS s = MeasureTheory.Measure.map (hatX s) P) (hhatXAtS : Measurable (hatX s0)) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (hatRhoS s)) (htestInt : ∀ φ s, Admissible φ → MeasureTheory.Integrable (testEval φ) (hatRhoS s)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hs0Interval : s0 ∈ Set.Ioo sLeft sRight) (hconcreteSampleDerivMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun ω => deriv (fun t : Real => testEval φ (hatX t ω)) s0) P) (hsampleDerivConcreteAeEqInterval : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, sampleDeriv φ s ω = deriv (fun t : Real => testEval φ (hatX t ω)) s) (hconcreteSampleDerivBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, ‖deriv (fun t : Real => testEval φ (hatX t ω)) s‖ ≤ bound φ ω) (hboundInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (bound φ) P) (hpathDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, HasDerivAt (fun t => testEval φ (hatX t ω)) (sampleDeriv φ s ω) s) (hderivValue : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∫ ω, sampleDeriv φ s0 ω ∂P) = driftAction φ + diffusionAction φ) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : let canonicalBarB : State → Vec := fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = (fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂(hatRhoS s0)) canonicalBarB φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun x => inner Real (testGrad φ x) (canonicalBarB x)) (hatRhoS s0)) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂(hatRhoS s0), ‖testGrad φ x‖ ≤ pairBound φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, inner Real (testGrad φ x) (canonicalBarB x) ∂(hatRhoS s0))) → ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by exact generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasDominated testEval testGrad Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction diffusionAction driftDiv laplacian s0 sLeft sRight dotTk sigmaEta sigmaCoeff pairBound sampleDeriv bound hhatRhoS hhatXAtS hhatX hXk hguideMeas hguideInt hscoreMeas hscoreInt htestMeas htestInt hcommon hkernel hdrift hdensity htests hboundary hcoeff hs0Interval hconcreteSampleDerivMeas (by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary filter_upwards [hsampleDerivConcreteAeEqInterval φ hφ hcommon hkernel hdrift hdensity htests hboundary] with ω hω exact hω s0 hs0Interval) (by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary filter_upwards [hconcreteSampleDerivBound φ hφ hcommon hkernel hdrift hdensity htests hboundary, hsampleDerivConcreteAeEqInterval φ hφ hcommon hkernel hdrift hdensity htests hboundary] with ω hbound heq s hs calc ‖sampleDeriv φ s ω‖ = ‖deriv (fun t : Real => testEval φ (hatX t ω)) s‖ := by rw [heq s hs] _ ≤ bound φ ω := hbound s hs) hboundInt hpathDeriv hderivValue hdiffusionSource /-- Canonical `barB` dominated weak derivative on the source EM interval, with the dominating bound integrability transported from the joint EM law. This lower theorem removes the supplied `hboundInt` premise from the cycle-124 interval packet. The source-facing input is now an integrable bound on the joint law of `(hatX s0, Xk)` together with an a.e. equality identifying the selected sample-space bound with that joint-law representative. Pointwise `HasDerivAt`, derivative-value splitting, and source actions remain explicit. -/ theorem generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntDominated {Ω State Vec Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [InnerProductSpace Real Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatX : Real → Ω → State} {hatRhoS : Real → MeasureTheory.Measure State} {Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (testEval : Test → State → Real) (testGrad : Test → State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction driftDiv laplacian : Test → Real) (s0 sLeft sRight dotTk sigmaEta sigmaCoeff : Real) (pairBound : Test → Real) (sampleDeriv : Test → Real → Ω → Real) (bound : Test → Ω → Real) (jointBound : Test → State × State → Real) (hhatRhoS : ∀ s, hatRhoS s = MeasureTheory.Measure.map (hatX s) P) (hhatXAtS : Measurable (hatX s0)) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (hatRhoS s)) (htestInt : ∀ φ s, Admissible φ → MeasureTheory.Integrable (testEval φ) (hatRhoS s)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hs0Interval : s0 ∈ Set.Ioo sLeft sRight) (hconcreteSampleDerivMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun ω => deriv (fun t : Real => testEval φ (hatX t ω)) s0) P) (hsampleDerivConcreteAeEqInterval : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, sampleDeriv φ s ω = deriv (fun t : Real => testEval φ (hatX t ω)) s) (hconcreteSampleDerivBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, ‖deriv (fun t : Real => testEval φ (hatX t ω)) s‖ ≤ bound φ ω) (hjointBoundInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (jointBound φ) (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hboundAeEqJoint : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → bound φ =ᵐ[P] (fun ω => jointBound φ (hatX s0 ω, Xk ω))) (hpathDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, HasDerivAt (fun t => testEval φ (hatX t ω)) (sampleDeriv φ s ω) s) (hderivValue : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∫ ω, sampleDeriv φ s0 ω ∂P) = driftAction φ + diffusionAction φ) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : let canonicalBarB : State → Vec := fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = (fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂(hatRhoS s0)) canonicalBarB φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun x => inner Real (testGrad φ x) (canonicalBarB x)) (hatRhoS s0)) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂(hatRhoS s0), ‖testGrad φ x‖ ≤ pairBound φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, inner Real (testGrad φ x) (canonicalBarB x) ∂(hatRhoS s0))) → ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by exact generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundDominated testEval testGrad Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction diffusionAction driftDiv laplacian s0 sLeft sRight dotTk sigmaEta sigmaCoeff pairBound sampleDeriv bound hhatRhoS hhatXAtS hhatX hXk hguideMeas hguideInt hscoreMeas hscoreInt htestMeas htestInt hcommon hkernel hdrift hdensity htests hboundary hcoeff hs0Interval hconcreteSampleDerivMeas hsampleDerivConcreteAeEqInterval hconcreteSampleDerivBound (by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary have hpair : AEMeasurable (fun ω => (hatX s0 ω, Xk ω)) P := (hhatX s0).prodMk hXk have hjoint : MeasureTheory.Integrable ((jointBound φ) ∘ (fun ω => (hatX s0 ω, Xk ω))) P := (hjointBoundInt φ hφ hcommon hkernel hdrift hdensity htests hboundary).comp_aemeasurable hpair simpa [Function.comp_def] using hjoint.congr (hboundAeEqJoint φ hφ hcommon hkernel hdrift hdensity htests hboundary).symm) hpathDeriv hderivValue hdiffusionSource /-- Canonical `barB` dominated weak derivative on the source EM interval, with the path derivative obtained from differentiability of the concrete EM weak-test path. This cycle-125 lower theorem removes the supplied `hpathDeriv` premise from the cycle-124 bound-integrability packet. The source-facing input is now a.e. differentiability on the EM interval of `fun t => testEval phi (hatX t omega)`. Mathlib's `DifferentiableAt.hasDerivAt` gives the concrete derivative representative `deriv (fun t => testEval phi (hatX t omega)) s`; the existing interval a.e. equality then transports it to the selected `sampleDeriv` representative. Derivative-value splitting and source actions remain explicit. -/ theorem generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated {Ω State Vec Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [InnerProductSpace Real Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatX : Real → Ω → State} {hatRhoS : Real → MeasureTheory.Measure State} {Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (testEval : Test → State → Real) (testGrad : Test → State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction driftDiv laplacian : Test → Real) (s0 sLeft sRight dotTk sigmaEta sigmaCoeff : Real) (pairBound : Test → Real) (sampleDeriv : Test → Real → Ω → Real) (bound : Test → Ω → Real) (jointBound : Test → State × State → Real) (hhatRhoS : ∀ s, hatRhoS s = MeasureTheory.Measure.map (hatX s) P) (hhatXAtS : Measurable (hatX s0)) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (hatRhoS s)) (htestInt : ∀ φ s, Admissible φ → MeasureTheory.Integrable (testEval φ) (hatRhoS s)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hs0Interval : s0 ∈ Set.Ioo sLeft sRight) (hconcreteSampleDerivMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun ω => deriv (fun t : Real => testEval φ (hatX t ω)) s0) P) (hsampleDerivConcreteAeEqInterval : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, sampleDeriv φ s ω = deriv (fun t : Real => testEval φ (hatX t ω)) s) (hconcreteSampleDerivBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, ‖deriv (fun t : Real => testEval φ (hatX t ω)) s‖ ≤ bound φ ω) (hjointBoundInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (jointBound φ) (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hboundAeEqJoint : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → bound φ =ᵐ[P] (fun ω => jointBound φ (hatX s0 ω, Xk ω))) (hconcretePathDifferentiable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, DifferentiableAt Real (fun t : Real => testEval φ (hatX t ω)) s) (hderivValue : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∫ ω, sampleDeriv φ s0 ω ∂P) = driftAction φ + diffusionAction φ) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : let canonicalBarB : State → Vec := fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = (fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂(hatRhoS s0)) canonicalBarB φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun x => inner Real (testGrad φ x) (canonicalBarB x)) (hatRhoS s0)) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂(hatRhoS s0), ‖testGrad φ x‖ ≤ pairBound φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, inner Real (testGrad φ x) (canonicalBarB x) ∂(hatRhoS s0))) → ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by exact generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntDominated testEval testGrad Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction diffusionAction driftDiv laplacian s0 sLeft sRight dotTk sigmaEta sigmaCoeff pairBound sampleDeriv bound jointBound hhatRhoS hhatXAtS hhatX hXk hguideMeas hguideInt hscoreMeas hscoreInt htestMeas htestInt hcommon hkernel hdrift hdensity htests hboundary hcoeff hs0Interval hconcreteSampleDerivMeas hsampleDerivConcreteAeEqInterval hconcreteSampleDerivBound hjointBoundInt hboundAeEqJoint (by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary filter_upwards [hconcretePathDifferentiable φ hφ hcommon hkernel hdrift hdensity htests hboundary, hsampleDerivConcreteAeEqInterval φ hφ hcommon hkernel hdrift hdensity htests hboundary] with ω hdiff heq s hs simpa [heq s hs] using (hdiff s hs).hasDerivAt) hderivValue hdiffusionSource /-- Canonical `barB` dominated weak derivative on the source EM interval, with the derivative-value identity transported from the concrete EM derivative. This cycle-126 lower theorem removes the supplied `hderivValue` premise from the cycle-125 path-derivative packet. The source-facing input is the smaller concrete integral split for `deriv (fun t => testEval phi (hatX t omega)) s0`. The existing interval a.e. equality between that concrete representative and `sampleDeriv` is specialized at `s0` and transported through `MeasureTheory.integral_congr_ae`. The canonical `barB` weak action, no-boundary divergence, and diffusion source action remain explicit. -/ theorem generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDominated {Ω State Vec Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [InnerProductSpace Real Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatX : Real → Ω → State} {hatRhoS : Real → MeasureTheory.Measure State} {Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (testEval : Test → State → Real) (testGrad : Test → State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction driftDiv laplacian : Test → Real) (s0 sLeft sRight dotTk sigmaEta sigmaCoeff : Real) (pairBound : Test → Real) (sampleDeriv : Test → Real → Ω → Real) (bound : Test → Ω → Real) (jointBound : Test → State × State → Real) (hhatRhoS : ∀ s, hatRhoS s = MeasureTheory.Measure.map (hatX s) P) (hhatXAtS : Measurable (hatX s0)) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (hatRhoS s)) (htestInt : ∀ φ s, Admissible φ → MeasureTheory.Integrable (testEval φ) (hatRhoS s)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hs0Interval : s0 ∈ Set.Ioo sLeft sRight) (hconcreteSampleDerivMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun ω => deriv (fun t : Real => testEval φ (hatX t ω)) s0) P) (hsampleDerivConcreteAeEqInterval : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, sampleDeriv φ s ω = deriv (fun t : Real => testEval φ (hatX t ω)) s) (hconcreteSampleDerivBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, ‖deriv (fun t : Real => testEval φ (hatX t ω)) s‖ ≤ bound φ ω) (hjointBoundInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (jointBound φ) (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hboundAeEqJoint : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → bound φ =ᵐ[P] (fun ω => jointBound φ (hatX s0 ω, Xk ω))) (hconcretePathDifferentiable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, DifferentiableAt Real (fun t : Real => testEval φ (hatX t ω)) s) (hconcreteDerivativeIntegralSplit : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∫ ω, deriv (fun t : Real => testEval φ (hatX t ω)) s0 ∂P) = driftAction φ + diffusionAction φ) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : let canonicalBarB : State → Vec := fun x => dotTk • (∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) + sigmaCoeff • (∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x) (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = (fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂(hatRhoS s0)) canonicalBarB φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun x => inner Real (testGrad φ x) (canonicalBarB x)) (hatRhoS s0)) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂(hatRhoS s0), ‖testGrad φ x‖ ≤ pairBound φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, inner Real (testGrad φ x) (canonicalBarB x) ∂(hatRhoS s0))) → ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by exact generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated testEval testGrad Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction diffusionAction driftDiv laplacian s0 sLeft sRight dotTk sigmaEta sigmaCoeff pairBound sampleDeriv bound jointBound hhatRhoS hhatXAtS hhatX hXk hguideMeas hguideInt hscoreMeas hscoreInt htestMeas htestInt hcommon hkernel hdrift hdensity htests hboundary hcoeff hs0Interval hconcreteSampleDerivMeas hsampleDerivConcreteAeEqInterval hconcreteSampleDerivBound hjointBoundInt hboundAeEqJoint hconcretePathDifferentiable (by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary have hae : sampleDeriv φ s0 =ᵐ[P] (fun ω => deriv (fun t : Real => testEval φ (hatX t ω)) s0) := by filter_upwards [hsampleDerivConcreteAeEqInterval φ hφ hcommon hkernel hdrift hdensity htests hboundary] with ω hω exact hω s0 hs0Interval calc (∫ ω, sampleDeriv φ s0 ω ∂P) = ∫ ω, deriv (fun t : Real => testEval φ (hatX t ω)) s0 ∂P := by exact MeasureTheory.integral_congr_ae hae _ = driftAction φ + diffusionAction φ := hconcreteDerivativeIntegralSplit φ hφ hcommon hkernel hdrift hdensity htests hboundary) hdiffusionSource /-- Canonical `barB` dominated weak derivative on the source EM interval, with the drift weak-action identity derived from guide/score component pairings. This cycle-127 dynamic-leaf theorem removes the first post-`canonicalBarB` continuation from the cycle-126 packet. Instead of taking the supplied `hdriftBarBAction` identity directly, it derives the canonical drift action from the paper's guide and score component actions and the concrete Bochner integral linearity of the weak test-gradient pairing. Pairing measurability, the gradient bound, no-boundary divergence, and diffusion source action remain explicit. -/ theorem generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionDominated {Ω State Vec Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [InnerProductSpace Real Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatX : Real → Ω → State} {hatRhoS : Real → MeasureTheory.Measure State} {Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (testEval : Test → State → Real) (testGrad : Test → State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction driftDiv laplacian : Test → Real) (guideAction scoreAction : Test → Real) (s0 sLeft sRight dotTk sigmaEta sigmaCoeff : Real) (pairBound : Test → Real) (sampleDeriv : Test → Real → Ω → Real) (bound : Test → Ω → Real) (jointBound : Test → State × State → Real) (hhatRhoS : ∀ s, hatRhoS s = MeasureTheory.Measure.map (hatX s) P) (hhatXAtS : Measurable (hatX s0)) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (hatRhoS s)) (htestInt : ∀ φ s, Admissible φ → MeasureTheory.Integrable (testEval φ) (hatRhoS s)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hs0Interval : s0 ∈ Set.Ioo sLeft sRight) (hconcreteSampleDerivMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun ω => deriv (fun t : Real => testEval φ (hatX t ω)) s0) P) (hsampleDerivConcreteAeEqInterval : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, sampleDeriv φ s ω = deriv (fun t : Real => testEval φ (hatX t ω)) s) (hconcreteSampleDerivBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, ‖deriv (fun t : Real => testEval φ (hatX t ω)) s‖ ≤ bound φ ω) (hjointBoundInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (jointBound φ) (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hboundAeEqJoint : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → bound φ =ᵐ[P] (fun ω => jointBound φ (hatX s0 ω, Xk ω))) (hconcretePathDifferentiable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, DifferentiableAt Real (fun t : Real => testEval φ (hatX t ω)) s) (hconcreteDerivativeIntegralSplit : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∫ ω, deriv (fun t : Real => testEval φ (hatX t ω)) s0 ∂P) = driftAction φ + diffusionAction φ) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : let canonicalGuide : State → Vec := fun x => ∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalScore : State → Vec := fun x => ∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalBarB : State → Vec := fun x => dotTk • canonicalGuide x + sigmaCoeff • canonicalScore x let weakGradPairing : (State → Vec) → Test → Real := fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂(hatRhoS s0) (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = dotTk • guideAction φ + sigmaCoeff • scoreAction φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → guideAction φ = weakGradPairing canonicalGuide φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → scoreAction φ = weakGradPairing canonicalScore φ) → (∀ {f g : State → Vec}, (∀ x, f x = g x) → ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing f φ = weakGradPairing g φ) → (∀ (f g : State → Vec) (φ : Test), Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing (fun x => f x + g x) φ = weakGradPairing f φ + weakGradPairing g φ) → (∀ (a : Real) (f : State → Vec) (φ : Test), Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing (fun x => a • f x) φ = a • weakGradPairing f φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun x => inner Real (testGrad φ x) (canonicalBarB x)) (hatRhoS s0)) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂(hatRhoS s0), ‖testGrad φ x‖ ≤ pairBound φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, inner Real (testGrad φ x) (canonicalBarB x) ∂(hatRhoS s0))) → ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by dsimp only intro hdriftActionComponents hguideActionPairing hscoreActionPairing hweakCongr hweakAdd hweakSmul hpairMeas hgradNormBound hdivNoBoundary let canonicalGuide : State → Vec := fun x => ∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalScore : State → Vec := fun x => ∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalBarB : State → Vec := fun x => dotTk • canonicalGuide x + sigmaCoeff • canonicalScore x let weakGradPairing : (State → Vec) → Test → Real := fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂(hatRhoS s0) have hdriftComponents : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = dotTk • weakGradPairing canonicalGuide φ + sigmaCoeff • weakGradPairing canonicalScore φ := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary calc driftAction φ = dotTk • guideAction φ + sigmaCoeff • scoreAction φ := by exact hdriftActionComponents φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = dotTk • weakGradPairing canonicalGuide φ + sigmaCoeff • weakGradPairing canonicalScore φ := by rw [ hguideActionPairing φ hφ hcommon hkernel hdrift hdensity htests hboundary, hscoreActionPairing φ hφ hcommon hkernel hdrift hdensity htests hboundary ] have hdriftBarBAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = weakGradPairing canonicalBarB φ := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary have hcombo : weakGradPairing canonicalBarB φ = dotTk • weakGradPairing canonicalGuide φ + sigmaCoeff • weakGradPairing canonicalScore φ := by calc weakGradPairing canonicalBarB φ = weakGradPairing (fun x => dotTk • canonicalGuide x + sigmaCoeff • canonicalScore x) φ := by exact hweakCongr (by intro x; simp [canonicalBarB]) φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = weakGradPairing (fun x => dotTk • canonicalGuide x) φ + weakGradPairing (fun x => sigmaCoeff • canonicalScore x) φ := by exact hweakAdd (fun x => dotTk • canonicalGuide x) (fun x => sigmaCoeff • canonicalScore x) φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = dotTk • weakGradPairing canonicalGuide φ + sigmaCoeff • weakGradPairing canonicalScore φ := by have hguideSmul : weakGradPairing (fun x => dotTk • canonicalGuide x) φ = dotTk • weakGradPairing canonicalGuide φ := by simpa [weakGradPairing] using hweakSmul dotTk canonicalGuide φ hφ hcommon hkernel hdrift hdensity htests hboundary have hscoreSmul : weakGradPairing (fun x => sigmaCoeff • canonicalScore x) φ = sigmaCoeff • weakGradPairing canonicalScore φ := by simpa [weakGradPairing] using hweakSmul sigmaCoeff canonicalScore φ hφ hcommon hkernel hdrift hdensity htests hboundary rw [ hguideSmul, hscoreSmul ] calc driftAction φ = dotTk • weakGradPairing canonicalGuide φ + sigmaCoeff • weakGradPairing canonicalScore φ := by exact hdriftComponents φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = weakGradPairing canonicalBarB φ := hcombo.symm exact generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDominated testEval testGrad Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction diffusionAction driftDiv laplacian s0 sLeft sRight dotTk sigmaEta sigmaCoeff pairBound sampleDeriv bound jointBound hhatRhoS hhatXAtS hhatX hXk hguideMeas hguideInt hscoreMeas hscoreInt htestMeas htestInt hcommon hkernel hdrift hdensity htests hboundary hcoeff hs0Interval hconcreteSampleDerivMeas hsampleDerivConcreteAeEqInterval hconcreteSampleDerivBound hjointBoundInt hboundAeEqJoint hconcretePathDifferentiable hconcreteDerivativeIntegralSplit hdiffusionSource (by simpa [weakGradPairing, canonicalBarB] using hdriftBarBAction) (by simpa [canonicalBarB] using hpairMeas) hgradNormBound (by simpa [canonicalBarB] using hdivNoBoundary) /-- Canonical `barB` dominated weak derivative on the source EM interval, with the drift weak-action identity and pairing measurability derived from separate field measurability facts. This lower refinement removes the raw `hpairMeas` continuation from the cycle-127 drift-action packet. The replacement inputs are the source-facing measurability of the admissible weak-test gradient and of the canonical conditional drift field. Mathlib's `AEStronglyMeasurable.inner` supplies the measurability of their real inner-product contraction; the gradient bound, no-boundary divergence, and diffusion source action remain explicit. -/ theorem generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasDominated {Ω State Vec Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [InnerProductSpace Real Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatX : Real → Ω → State} {hatRhoS : Real → MeasureTheory.Measure State} {Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (testEval : Test → State → Real) (testGrad : Test → State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction driftDiv laplacian : Test → Real) (guideAction scoreAction : Test → Real) (s0 sLeft sRight dotTk sigmaEta sigmaCoeff : Real) (pairBound : Test → Real) (sampleDeriv : Test → Real → Ω → Real) (bound : Test → Ω → Real) (jointBound : Test → State × State → Real) (hhatRhoS : ∀ s, hatRhoS s = MeasureTheory.Measure.map (hatX s) P) (hhatXAtS : Measurable (hatX s0)) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (hatRhoS s)) (htestInt : ∀ φ s, Admissible φ → MeasureTheory.Integrable (testEval φ) (hatRhoS s)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hs0Interval : s0 ∈ Set.Ioo sLeft sRight) (hconcreteSampleDerivMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun ω => deriv (fun t : Real => testEval φ (hatX t ω)) s0) P) (hsampleDerivConcreteAeEqInterval : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, sampleDeriv φ s ω = deriv (fun t : Real => testEval φ (hatX t ω)) s) (hconcreteSampleDerivBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, ‖deriv (fun t : Real => testEval φ (hatX t ω)) s‖ ≤ bound φ ω) (hjointBoundInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (jointBound φ) (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hboundAeEqJoint : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → bound φ =ᵐ[P] (fun ω => jointBound φ (hatX s0 ω, Xk ω))) (hconcretePathDifferentiable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, DifferentiableAt Real (fun t : Real => testEval φ (hatX t ω)) s) (hconcreteDerivativeIntegralSplit : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∫ ω, deriv (fun t : Real => testEval φ (hatX t ω)) s0 ∂P) = driftAction φ + diffusionAction φ) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : let canonicalGuide : State → Vec := fun x => ∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalScore : State → Vec := fun x => ∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalBarB : State → Vec := fun x => dotTk • canonicalGuide x + sigmaCoeff • canonicalScore x let weakGradPairing : (State → Vec) → Test → Real := fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂(hatRhoS s0) (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = dotTk • guideAction φ + sigmaCoeff • scoreAction φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → guideAction φ = weakGradPairing canonicalGuide φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → scoreAction φ = weakGradPairing canonicalScore φ) → (∀ {f g : State → Vec}, (∀ x, f x = g x) → ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing f φ = weakGradPairing g φ) → (∀ (f g : State → Vec) (φ : Test), Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing (fun x => f x + g x) φ = weakGradPairing f φ + weakGradPairing g φ) → (∀ (a : Real) (f : State → Vec) (φ : Test), Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing (fun x => a • f x) φ = a • weakGradPairing f φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (testGrad φ) (hatRhoS s0)) → (commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable canonicalBarB (hatRhoS s0)) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂(hatRhoS s0), ‖testGrad φ x‖ ≤ pairBound φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, inner Real (testGrad φ x) (canonicalBarB x) ∂(hatRhoS s0))) → ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by dsimp only intro hdriftActionComponents hguideActionPairing hscoreActionPairing hweakCongr hweakAdd hweakSmul htestGradMeas hcanonicalBarBMeas hgradNormBound hdivNoBoundary let canonicalGuide : State → Vec := fun x => ∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalScore : State → Vec := fun x => ∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalBarB : State → Vec := fun x => dotTk • canonicalGuide x + sigmaCoeff • canonicalScore x have hpairMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun x => inner Real (testGrad φ x) (canonicalBarB x)) (hatRhoS s0) := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary have hbarBMeas : MeasureTheory.AEStronglyMeasurable canonicalBarB (hatRhoS s0) := by simpa [canonicalGuide, canonicalScore, canonicalBarB] using hcanonicalBarBMeas hcommon hkernel hdrift hdensity htests hboundary exact (htestGradMeas φ hφ hcommon hkernel hdrift hdensity htests hboundary).inner hbarBMeas exact generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionDominated testEval testGrad Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction diffusionAction driftDiv laplacian guideAction scoreAction s0 sLeft sRight dotTk sigmaEta sigmaCoeff pairBound sampleDeriv bound jointBound hhatRhoS hhatXAtS hhatX hXk hguideMeas hguideInt hscoreMeas hscoreInt htestMeas htestInt hcommon hkernel hdrift hdensity htests hboundary hcoeff hs0Interval hconcreteSampleDerivMeas hsampleDerivConcreteAeEqInterval hconcreteSampleDerivBound hjointBoundInt hboundAeEqJoint hconcretePathDifferentiable hconcreteDerivativeIntegralSplit hdiffusionSource hdriftActionComponents hguideActionPairing hscoreActionPairing hweakCongr hweakAdd hweakSmul (by simpa [canonicalGuide, canonicalScore, canonicalBarB] using hpairMeas) hgradNormBound (by simpa [canonicalGuide, canonicalScore, canonicalBarB] using hdivNoBoundary) /-- Canonical `barB` dominated weak derivative with no-boundary narrowed to trace/product-rule facts. This cycle-128 refinement removes the direct `hdivNoBoundary` continuation from the post-cycle-127 pair-measurability theorem. The replacement inputs are the source-facing product rule for `hatRhoS s0 * canonicalBarB`, the divergence-theorem boundary-flux identity, the boundary trace integral, and zero admissible-test trace. The separate weak-test-gradient/canonical-field measurability inputs, `hgradNormBound`, and `hdiffusionSource` remain explicit. -/ theorem generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceDominated {Ω State Vec Test Boundary : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [MeasurableSpace Boundary] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [InnerProductSpace Real Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatX : Real → Ω → State} {hatRhoS : Real → MeasureTheory.Measure State} {Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (testEval : Test → State → Real) (testGrad : Test → State → Vec) (boundaryMeasure : MeasureTheory.Measure Boundary) (testTrace : Test → Boundary → Real) (normalFluxTrace : Boundary → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction driftDiv laplacian : Test → Real) (guideAction scoreAction : Test → Real) (divTotal boundaryFlux : Test → Real) (s0 sLeft sRight dotTk sigmaEta sigmaCoeff : Real) (pairBound : Test → Real) (sampleDeriv : Test → Real → Ω → Real) (bound : Test → Ω → Real) (jointBound : Test → State × State → Real) (hhatRhoS : ∀ s, hatRhoS s = MeasureTheory.Measure.map (hatX s) P) (hhatXAtS : Measurable (hatX s0)) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (hatRhoS s)) (htestInt : ∀ φ s, Admissible φ → MeasureTheory.Integrable (testEval φ) (hatRhoS s)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hs0Interval : s0 ∈ Set.Ioo sLeft sRight) (hconcreteSampleDerivMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun ω => deriv (fun t : Real => testEval φ (hatX t ω)) s0) P) (hsampleDerivConcreteAeEqInterval : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, sampleDeriv φ s ω = deriv (fun t : Real => testEval φ (hatX t ω)) s) (hconcreteSampleDerivBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, ‖deriv (fun t : Real => testEval φ (hatX t ω)) s‖ ≤ bound φ ω) (hjointBoundInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (jointBound φ) (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hboundAeEqJoint : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → bound φ =ᵐ[P] (fun ω => jointBound φ (hatX s0 ω, Xk ω))) (hconcretePathDifferentiable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, DifferentiableAt Real (fun t : Real => testEval φ (hatX t ω)) s) (hconcreteDerivativeIntegralSplit : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∫ ω, deriv (fun t : Real => testEval φ (hatX t ω)) s0 ∂P) = driftAction φ + diffusionAction φ) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : let canonicalGuide : State → Vec := fun x => ∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalScore : State → Vec := fun x => ∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalBarB : State → Vec := fun x => dotTk • canonicalGuide x + sigmaCoeff • canonicalScore x let weakGradPairing : (State → Vec) → Test → Real := fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂(hatRhoS s0) (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = dotTk • guideAction φ + sigmaCoeff • scoreAction φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → guideAction φ = weakGradPairing canonicalGuide φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → scoreAction φ = weakGradPairing canonicalScore φ) → (∀ {f g : State → Vec}, (∀ x, f x = g x) → ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing f φ = weakGradPairing g φ) → (∀ (f g : State → Vec) (φ : Test), Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing (fun x => f x + g x) φ = weakGradPairing f φ + weakGradPairing g φ) → (∀ (a : Real) (f : State → Vec) (φ : Test), Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing (fun x => a • f x) φ = a • weakGradPairing f φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (testGrad φ) (hatRhoS s0)) → (commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable canonicalBarB (hatRhoS s0)) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂(hatRhoS s0), ‖testGrad φ x‖ ≤ pairBound φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ + (∫ x, inner Real (testGrad φ x) (canonicalBarB x) ∂(hatRhoS s0)) = divTotal φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → divTotal φ = boundaryFlux φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → boundaryFlux φ = ∫ y, testTrace φ y * normalFluxTrace y ∂boundaryMeasure) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, testTrace φ y = 0) → ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by dsimp only intro hdriftActionComponents hguideActionPairing hscoreActionPairing hweakCongr hweakAdd hweakSmul htestGradMeas hcanonicalBarBMeas hgradNormBound hproductRule hdivergenceTheorem hboundaryFluxIntegral htestTraceZero let canonicalGuide : State → Vec := fun x => ∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalScore : State → Vec := fun x => ∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalBarB : State → Vec := fun x => dotTk • canonicalGuide x + sigmaCoeff • canonicalScore x have htraceProductZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, testTrace φ y * normalFluxTrace y = 0 := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary exact (htestTraceZero φ hφ hcommon hkernel hdrift hdensity htests hboundary).mono (by intro y hy simp [hy]) have hzeroBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → boundaryFlux φ = 0 := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary calc boundaryFlux φ = ∫ y, testTrace φ y * normalFluxTrace y ∂boundaryMeasure := hboundaryFluxIntegral φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = ∫ _y, (0 : Real) ∂boundaryMeasure := MeasureTheory.integral_congr_ae (htraceProductZero φ hφ hcommon hkernel hdrift hdensity htests hboundary) _ = 0 := by simp have hdivNoBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ = -(∫ x, inner Real (testGrad φ x) (canonicalBarB x) ∂(hatRhoS s0)) := generalMovingTargetDiscreteDriftDivNoBoundaryOfProductRule driftDiv (fun φ => ∫ x, inner Real (testGrad φ x) (canonicalBarB x) ∂(hatRhoS s0)) divTotal boundaryFlux Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior hproductRule hdivergenceTheorem hzeroBoundary exact generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasDominated testEval testGrad Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction diffusionAction driftDiv laplacian guideAction scoreAction s0 sLeft sRight dotTk sigmaEta sigmaCoeff pairBound sampleDeriv bound jointBound hhatRhoS hhatXAtS hhatX hXk hguideMeas hguideInt hscoreMeas hscoreInt htestMeas htestInt hcommon hkernel hdrift hdensity htests hboundary hcoeff hs0Interval hconcreteSampleDerivMeas hsampleDerivConcreteAeEqInterval hconcreteSampleDerivBound hjointBoundInt hboundAeEqJoint hconcretePathDifferentiable hconcreteDerivativeIntegralSplit hdiffusionSource hdriftActionComponents hguideActionPairing hscoreActionPairing hweakCongr hweakAdd hweakSmul htestGradMeas hcanonicalBarBMeas hgradNormBound (by simpa [canonicalGuide, canonicalScore, canonicalBarB] using hdivNoBoundary) /-- Canonical `barB` dominated weak derivative with no-boundary narrowed and canonical-field measurability derived from the condDistrib regularity theorem. This follow-on removes the separate `hcanonicalBarBMeas` premise from the cycle-128 no-boundary trace refiner. The canonical guide-plus-score field is measurable under `hatRhoS s0` by `generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity`; weak-test gradient measurability, the gradient bound, and the diffusion source action remain explicit. -/ theorem generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDominated {Ω State Vec Test Boundary : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [MeasurableSpace Boundary] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [InnerProductSpace Real Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatX : Real → Ω → State} {hatRhoS : Real → MeasureTheory.Measure State} {Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (testEval : Test → State → Real) (testGrad : Test → State → Vec) (boundaryMeasure : MeasureTheory.Measure Boundary) (testTrace : Test → Boundary → Real) (normalFluxTrace : Boundary → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction driftDiv laplacian : Test → Real) (guideAction scoreAction : Test → Real) (divTotal boundaryFlux : Test → Real) (s0 sLeft sRight dotTk sigmaEta sigmaCoeff : Real) (pairBound : Test → Real) (sampleDeriv : Test → Real → Ω → Real) (bound : Test → Ω → Real) (jointBound : Test → State × State → Real) (hhatRhoS : ∀ s, hatRhoS s = MeasureTheory.Measure.map (hatX s) P) (hhatXAtS : Measurable (hatX s0)) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (hatRhoS s)) (htestInt : ∀ φ s, Admissible φ → MeasureTheory.Integrable (testEval φ) (hatRhoS s)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hs0Interval : s0 ∈ Set.Ioo sLeft sRight) (hconcreteSampleDerivMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun ω => deriv (fun t : Real => testEval φ (hatX t ω)) s0) P) (hsampleDerivConcreteAeEqInterval : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, sampleDeriv φ s ω = deriv (fun t : Real => testEval φ (hatX t ω)) s) (hconcreteSampleDerivBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, ‖deriv (fun t : Real => testEval φ (hatX t ω)) s‖ ≤ bound φ ω) (hjointBoundInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (jointBound φ) (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hboundAeEqJoint : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → bound φ =ᵐ[P] (fun ω => jointBound φ (hatX s0 ω, Xk ω))) (hconcretePathDifferentiable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, DifferentiableAt Real (fun t : Real => testEval φ (hatX t ω)) s) (hconcreteDerivativeIntegralSplit : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∫ ω, deriv (fun t : Real => testEval φ (hatX t ω)) s0 ∂P) = driftAction φ + diffusionAction φ) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : let canonicalGuide : State → Vec := fun x => ∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalScore : State → Vec := fun x => ∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalBarB : State → Vec := fun x => dotTk • canonicalGuide x + sigmaCoeff • canonicalScore x let weakGradPairing : (State → Vec) → Test → Real := fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂(hatRhoS s0) (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = dotTk • guideAction φ + sigmaCoeff • scoreAction φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → guideAction φ = weakGradPairing canonicalGuide φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → scoreAction φ = weakGradPairing canonicalScore φ) → (∀ {f g : State → Vec}, (∀ x, f x = g x) → ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing f φ = weakGradPairing g φ) → (∀ (f g : State → Vec) (φ : Test), Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing (fun x => f x + g x) φ = weakGradPairing f φ + weakGradPairing g φ) → (∀ (a : Real) (f : State → Vec) (φ : Test), Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing (fun x => a • f x) φ = a • weakGradPairing f φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (testGrad φ) (hatRhoS s0)) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂(hatRhoS s0), ‖testGrad φ x‖ ≤ pairBound φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ + (∫ x, inner Real (testGrad φ x) (canonicalBarB x) ∂(hatRhoS s0)) = divTotal φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → divTotal φ = boundaryFlux φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → boundaryFlux φ = ∫ y, testTrace φ y * normalFluxTrace y ∂boundaryMeasure) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, testTrace φ y = 0) → ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by dsimp only intro hdriftActionComponents hguideActionPairing hscoreActionPairing hweakCongr hweakAdd hweakSmul htestGradMeas hgradNormBound hproductRule hdivergenceTheorem hboundaryFluxIntegral htestTraceZero let canonicalGuide : State → Vec := fun x => ∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalScore : State → Vec := fun x => ∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalBarB : State → Vec := fun x => dotTk • canonicalGuide x + sigmaCoeff • canonicalScore x have hcanonicalBarBMeas : commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable canonicalBarB (hatRhoS s0) := by intro _ _ _ _ _ _ have hcanonicalRegularity : MeasureTheory.AEStronglyMeasurable canonicalBarB (hatRhoS s0) ∧ MeasureTheory.Integrable canonicalBarB (hatRhoS s0) := by simpa [canonicalGuide, canonicalScore, canonicalBarB] using (generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity (P := P) (hatRhoS := hatRhoS s0) (hatXAtS := hatX s0) (Xk := Xk) (guideIntegrand := guideIntegrand) (scoreIntegrand := scoreIntegrand) dotTk sigmaCoeff (hhatRhoS s0) hXk hguideMeas hguideInt hscoreMeas hscoreInt) exact hcanonicalRegularity.1 exact generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceDominated testEval testGrad boundaryMeasure testTrace normalFluxTrace Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction diffusionAction driftDiv laplacian guideAction scoreAction divTotal boundaryFlux s0 sLeft sRight dotTk sigmaEta sigmaCoeff pairBound sampleDeriv bound jointBound hhatRhoS hhatXAtS hhatX hXk hguideMeas hguideInt hscoreMeas hscoreInt htestMeas htestInt hcommon hkernel hdrift hdensity htests hboundary hcoeff hs0Interval hconcreteSampleDerivMeas hsampleDerivConcreteAeEqInterval hconcreteSampleDerivBound hjointBoundInt hboundAeEqJoint hconcretePathDifferentiable hconcreteDerivativeIntegralSplit hdiffusionSource hdriftActionComponents hguideActionPairing hscoreActionPairing hweakCongr hweakAdd hweakSmul htestGradMeas (by simpa [canonicalGuide, canonicalScore, canonicalBarB] using hcanonicalBarBMeas) hgradNormBound hproductRule hdivergenceTheorem hboundaryFluxIntegral htestTraceZero /-- Canonical `barB` dominated weak derivative with the diffusion source action split into the EM/Brownian weak diffusion action and weak Laplacian action. This cycle-129 lower continuation removes the direct `hdiffusionSource` premise from the canonical EM weak-FP consumer. The remaining analytic boundary is the source-facing pair from `appendix.tex:1379-1387`: identify the frozen EM/Brownian diffusion generator contribution with a named weak diffusion action, then identify that action with the positive `sigmaCoeff • laplacian phi` term by the weak Laplacian integration-by-parts theorem. -/ theorem generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDiffusionSourceDominated {Ω State Vec Test Boundary : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [MeasurableSpace Boundary] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [CompleteSpace Vec] [InnerProductSpace Real Vec] [StandardBorelSpace State] [Nonempty State] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatX : Real → Ω → State} {hatRhoS : Real → MeasureTheory.Measure State} {Xk : Ω → State} {guideIntegrand scoreIntegrand : State × State → Vec} (testEval : Test → State → Real) (testGrad : Test → State → Vec) (boundaryMeasure : MeasureTheory.Measure Boundary) (testTrace : Test → Boundary → Real) (normalFluxTrace : Boundary → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction weakDiffusionAction driftDiv laplacian : Test → Real) (guideAction scoreAction : Test → Real) (divTotal boundaryFlux : Test → Real) (s0 sLeft sRight dotTk sigmaEta sigmaCoeff : Real) (pairBound : Test → Real) (sampleDeriv : Test → Real → Ω → Real) (bound : Test → Ω → Real) (jointBound : Test → State × State → Real) (hhatRhoS : ∀ s, hatRhoS s = MeasureTheory.Measure.map (hatX s) P) (hhatXAtS : Measurable (hatX s0)) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hXk : AEMeasurable Xk P) (hguideMeas : MeasureTheory.AEStronglyMeasurable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hguideInt : MeasureTheory.Integrable guideIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreMeas : MeasureTheory.AEStronglyMeasurable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hscoreInt : MeasureTheory.Integrable scoreIntegrand (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (hatRhoS s)) (htestInt : ∀ φ s, Admissible φ → MeasureTheory.Integrable (testEval φ) (hatRhoS s)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hs0Interval : s0 ∈ Set.Ioo sLeft sRight) (hconcreteSampleDerivMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun ω => deriv (fun t : Real => testEval φ (hatX t ω)) s0) P) (hsampleDerivConcreteAeEqInterval : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, sampleDeriv φ s ω = deriv (fun t : Real => testEval φ (hatX t ω)) s) (hconcreteSampleDerivBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, ‖deriv (fun t : Real => testEval φ (hatX t ω)) s‖ ≤ bound φ ω) (hjointBoundInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (jointBound φ) (MeasureTheory.Measure.map (fun ω => (hatX s0 ω, Xk ω)) P)) (hboundAeEqJoint : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → bound φ =ᵐ[P] (fun ω => jointBound φ (hatX s0 ω, Xk ω))) (hconcretePathDifferentiable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ ω ∂P, ∀ s ∈ Set.Ioo sLeft sRight, DifferentiableAt Real (fun t : Real => testEval φ (hatX t ω)) s) (hconcreteDerivativeIntegralSplit : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∫ ω, deriv (fun t : Real => testEval φ (hatX t ω)) s0 ∂P) = driftAction φ + diffusionAction φ) (hdiffusionAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = weakDiffusionAction φ) (hlaplacianAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • laplacian φ) : let canonicalGuide : State → Vec := fun x => ∫ y, guideIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalScore : State → Vec := fun x => ∫ y, scoreIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk (hatX s0) P x let canonicalBarB : State → Vec := fun x => dotTk • canonicalGuide x + sigmaCoeff • canonicalScore x let weakGradPairing : (State → Vec) → Test → Real := fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂(hatRhoS s0) (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = dotTk • guideAction φ + sigmaCoeff • scoreAction φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → guideAction φ = weakGradPairing canonicalGuide φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → scoreAction φ = weakGradPairing canonicalScore φ) → (∀ {f g : State → Vec}, (∀ x, f x = g x) → ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing f φ = weakGradPairing g φ) → (∀ (f g : State → Vec) (φ : Test), Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing (fun x => f x + g x) φ = weakGradPairing f φ + weakGradPairing g φ) → (∀ (a : Real) (f : State → Vec) (φ : Test), Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing (fun x => a • f x) φ = a • weakGradPairing f φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (testGrad φ) (hatRhoS s0)) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂(hatRhoS s0), ‖testGrad φ x‖ ≤ pairBound φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ + (∫ x, inner Real (testGrad φ x) (canonicalBarB x) ∂(hatRhoS s0)) = divTotal φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → divTotal φ = boundaryFlux φ) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → boundaryFlux φ = ∫ y, testTrace φ y * normalFluxTrace y ∂boundaryMeasure) → (∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, testTrace φ y = 0) → ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂hatRhoS s) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by have hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ := generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianAction Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior diffusionAction weakDiffusionAction laplacian sigmaCoeff hdiffusionAction hlaplacianAction exact generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDominated testEval testGrad boundaryMeasure testTrace normalFluxTrace Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction diffusionAction driftDiv laplacian guideAction scoreAction divTotal boundaryFlux s0 sLeft sRight dotTk sigmaEta sigmaCoeff pairBound sampleDeriv bound jointBound hhatRhoS hhatXAtS hhatX hXk hguideMeas hguideInt hscoreMeas hscoreInt htestMeas htestInt hcommon hkernel hdrift hdensity htests hboundary hcoeff hs0Interval hconcreteSampleDerivMeas hsampleDerivConcreteAeEqInterval hconcreteSampleDerivBound hjointBoundInt hboundAeEqJoint hconcretePathDifferentiable hconcreteDerivativeIntegralSplit hdiffusionSource /-- Zero boundary flux from a boundary-integral trace product that vanishes almost everywhere. This is the source-facing no-boundary specialization used for the `hatRhoS * barB` drift term: after the divergence theorem has identified the boundary flux as an integral of the test trace times the normal trace of the weighted field, it is enough to prove that trace product is zero a.e. on the boundary. The trace-product condition can be supplied by compact support, decay at infinity, or an explicit zero normal trace hypothesis. -/ theorem generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero {Boundary Test : Type*} [MeasurableSpace Boundary] (boundaryMeasure : MeasureTheory.Measure Boundary) (boundaryFlux : Test → Real) (testTrace : Test → Boundary → Real) (normalFluxTrace : Boundary → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (hboundaryFluxIntegral : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → boundaryFlux φ = ∫ y, testTrace φ y * normalFluxTrace y ∂boundaryMeasure) (htraceProductZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, testTrace φ y * normalFluxTrace y = 0) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → boundaryFlux φ = 0 := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary calc boundaryFlux φ = ∫ y, testTrace φ y * normalFluxTrace y ∂boundaryMeasure := hboundaryFluxIntegral φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = ∫ _y, (0 : Real) ∂boundaryMeasure := MeasureTheory.integral_congr_ae (htraceProductZero φ hφ hcommon hkernel hdrift hdensity htests hboundary) _ = 0 := by simp /-- Boundary trace-product vanishing from zero admissible-test trace. This is the compact-support/zero-trace lower handoff for the cycle-102 boundary packet. It does not prove the analytic trace theorem; it only removes the product-zero premise once admissible tests have zero boundary trace a.e. -/ theorem generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero {Boundary Test : Type*} [MeasurableSpace Boundary] (boundaryMeasure : MeasureTheory.Measure Boundary) (testTrace : Test → Boundary → Real) (normalFluxTrace : Boundary → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (htestTraceZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, testTrace φ y = 0) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, testTrace φ y * normalFluxTrace y = 0 := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary exact (htestTraceZero φ hφ hcommon hkernel hdrift hdensity htests hboundary).mono (by intro y hy simp [hy]) /-- Boundary-flux integral representation from Mathlib's box divergence theorem. This is the lower-ready Mathlib specialization for the `appendix.tex:1379-1387` no-boundary drift packet. It does not prove the weighted-field product rule, the identification of the paper boundary flux with the box divergence integral, or the trace parametrization of the signed faces. Instead, it exposes exactly those smaller inputs and uses `MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable` to connect the box divergence integral to the signed face sum. -/ theorem generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox {n : Nat} {Test Boundary : Type*} [MeasurableSpace Boundary] (a b : Fin (n + 1) → Real) (weightedField : Test → (Fin (n + 1) → Real) → Fin (n + 1) → Real) (weightedFieldDeriv : Test → (Fin (n + 1) → Real) → (Fin (n + 1) → Real) →L[Real] Fin (n + 1) → Real) (exceptionSet : Test → Set (Fin (n + 1) → Real)) (boundaryMeasure : MeasureTheory.Measure Boundary) (boundaryFlux : Test → Real) (testTrace : Test → Boundary → Real) (normalFluxTrace : Boundary → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (hle : a ≤ b) (hcontinuous : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ContinuousOn (weightedField φ) (Set.Icc a b)) (hexceptionCountable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (exceptionSet φ).Countable) (hderiv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ exceptionSet φ, HasFDerivAt (weightedField φ) (weightedFieldDeriv φ x) x) (hdivIntegrable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.IntegrableOn (fun x => ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (Set.Icc a b) MeasureTheory.volume) (hboundaryFluxDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → boundaryFlux φ = ∫ x in Set.Icc a b, ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (hfaceTrace : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (b i) x) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (a i) x) i)) = ∫ y, testTrace φ y * normalFluxTrace y ∂boundaryMeasure) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → boundaryFlux φ = ∫ y, testTrace φ y * normalFluxTrace y ∂boundaryMeasure := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary have hbox : (∫ x in Set.Icc a b, ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) = ∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (b i) x) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (a i) x) i) := MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable a b hle (weightedField φ) (weightedFieldDeriv φ) (exceptionSet φ) (hexceptionCountable φ hφ hcommon hkernel hdrift hdensity htests hboundary) (hcontinuous φ hφ hcommon hkernel hdrift hdensity htests hboundary) (hderiv φ hφ hcommon hkernel hdrift hdensity htests hboundary) (hdivIntegrable φ hφ hcommon hkernel hdrift hdensity htests hboundary) calc boundaryFlux φ = ∫ x in Set.Icc a b, ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i := hboundaryFluxDivergence φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = ∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (b i) x) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (a i) x) i) := hbox _ = ∫ y, testTrace φ y * normalFluxTrace y ∂boundaryMeasure := hfaceTrace φ hφ hcommon hkernel hdrift hdensity htests hboundary /-- Second-Green diffusion-source handoff with the boundary-flux integral represented by the local box divergence theorem interface. This lower helper removes the direct `hsecondGreenBoundaryFluxIntegral` premise from `generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenTraceBoundary`. It derives that premise from the same source-facing Mathlib box divergence package used for the drift no-boundary backend, while keeping the trace-product zero condition, second-Green residual/divergence facts, and test-Laplacian normalization explicit. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFlux {n : Nat} {Boundary Test : Type*} [MeasurableSpace Boundary] (a b : Fin (n + 1) → Real) (weightedField : Test → (Fin (n + 1) → Real) → Fin (n + 1) → Real) (weightedFieldDeriv : Test → (Fin (n + 1) → Real) → (Fin (n + 1) → Real) →L[Real] Fin (n + 1) → Real) (exceptionSet : Test → Set (Fin (n + 1) → Real)) (boundaryMeasure : MeasureTheory.Measure Boundary) (secondGreenTestTrace : Test → Boundary → Real) (secondGreenNormalFluxTrace : Boundary → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux : Test → Real) (sigmaCoeff : Real) (hdiffusionAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = weakDiffusionAction φ) (hdiffusionLaplacianTerm : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • densityLaplacianAction φ) (hfirstGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ - negativeGradientPairAction φ = firstGreenTotal φ) (hfirstGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenTotal φ = firstGreenBoundaryFlux φ) (hfirstGreenZeroBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenBoundaryFlux φ = 0) (hsecondGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → negativeGradientPairAction φ - testLaplacianAction φ = secondGreenTotal φ) (hsecondGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenTotal φ = secondGreenBoundaryFlux φ) (hle : a ≤ b) (hsecondGreenWeightedFieldContinuous : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ContinuousOn (weightedField φ) (Set.Icc a b)) (hsecondGreenExceptionCountable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (exceptionSet φ).Countable) (hsecondGreenDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ exceptionSet φ, HasFDerivAt (weightedField φ) (weightedFieldDeriv φ x) x) (hsecondGreenDivIntegrable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.IntegrableOn (fun x => ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (Set.Icc a b) MeasureTheory.volume) (hsecondGreenBoundaryFluxDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenBoundaryFlux φ = ∫ x in Set.Icc a b, ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (hsecondGreenFaceTrace : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (b i) x) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (a i) x) i)) = ∫ y, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y ∂boundaryMeasure) (hsecondGreenTraceProductZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y = 0) (htestLaplacian : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → testLaplacianAction φ = laplacian φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ := by have hsecondGreenBoundaryFluxIntegral : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenBoundaryFlux φ = ∫ y, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y ∂boundaryMeasure := generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox a b weightedField weightedFieldDeriv exceptionSet boundaryMeasure secondGreenBoundaryFlux secondGreenTestTrace secondGreenNormalFluxTrace Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior hle hsecondGreenWeightedFieldContinuous hsecondGreenExceptionCountable hsecondGreenDeriv hsecondGreenDivIntegrable hsecondGreenBoundaryFluxDivergence hsecondGreenFaceTrace exact generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenTraceBoundary boundaryMeasure secondGreenTestTrace secondGreenNormalFluxTrace Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux sigmaCoeff hdiffusionAction hdiffusionLaplacianTerm hfirstGreenResidual hfirstGreenDivergence hfirstGreenZeroBoundary hsecondGreenResidual hsecondGreenDivergence hsecondGreenBoundaryFluxIntegral hsecondGreenTraceProductZero htestLaplacian /-- Second-Green diffusion-source handoff with the trace-product condition narrowed to zero admissible-test trace. This is the cycle-132 source-facing continuation of the cycle-131 box-boundary-flux packet. It keeps the Mathlib box-divergence inputs, second-Green residual/divergence facts, first-Green subfacts, and test-Laplacian normalization explicit, but removes the supplied `hsecondGreenTraceProductZero` input by deriving it from zero trace of the admissible test on the boundary. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTestTraceZero {n : Nat} {Boundary Test : Type*} [MeasurableSpace Boundary] (a b : Fin (n + 1) → Real) (weightedField : Test → (Fin (n + 1) → Real) → Fin (n + 1) → Real) (weightedFieldDeriv : Test → (Fin (n + 1) → Real) → (Fin (n + 1) → Real) →L[Real] Fin (n + 1) → Real) (exceptionSet : Test → Set (Fin (n + 1) → Real)) (boundaryMeasure : MeasureTheory.Measure Boundary) (secondGreenTestTrace : Test → Boundary → Real) (secondGreenNormalFluxTrace : Boundary → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux : Test → Real) (sigmaCoeff : Real) (hdiffusionAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = weakDiffusionAction φ) (hdiffusionLaplacianTerm : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • densityLaplacianAction φ) (hfirstGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ - negativeGradientPairAction φ = firstGreenTotal φ) (hfirstGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenTotal φ = firstGreenBoundaryFlux φ) (hfirstGreenZeroBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenBoundaryFlux φ = 0) (hsecondGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → negativeGradientPairAction φ - testLaplacianAction φ = secondGreenTotal φ) (hsecondGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenTotal φ = secondGreenBoundaryFlux φ) (hle : a ≤ b) (hsecondGreenWeightedFieldContinuous : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ContinuousOn (weightedField φ) (Set.Icc a b)) (hsecondGreenExceptionCountable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (exceptionSet φ).Countable) (hsecondGreenDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ exceptionSet φ, HasFDerivAt (weightedField φ) (weightedFieldDeriv φ x) x) (hsecondGreenDivIntegrable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.IntegrableOn (fun x => ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (Set.Icc a b) MeasureTheory.volume) (hsecondGreenBoundaryFluxDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenBoundaryFlux φ = ∫ x in Set.Icc a b, ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (hsecondGreenFaceTrace : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (b i) x) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (a i) x) i)) = ∫ y, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y ∂boundaryMeasure) (hsecondGreenTestTraceZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, secondGreenTestTrace φ y = 0) (htestLaplacian : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → testLaplacianAction φ = laplacian φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ := by have hsecondGreenTraceProductZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y = 0 := generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero boundaryMeasure secondGreenTestTrace secondGreenNormalFluxTrace Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior hsecondGreenTestTraceZero exact generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFlux a b weightedField weightedFieldDeriv exceptionSet boundaryMeasure secondGreenTestTrace secondGreenNormalFluxTrace Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux sigmaCoeff hdiffusionAction hdiffusionLaplacianTerm hfirstGreenResidual hfirstGreenDivergence hfirstGreenZeroBoundary hsecondGreenResidual hsecondGreenDivergence hle hsecondGreenWeightedFieldContinuous hsecondGreenExceptionCountable hsecondGreenDeriv hsecondGreenDivIntegrable hsecondGreenBoundaryFluxDivergence hsecondGreenFaceTrace hsecondGreenTraceProductZero htestLaplacian /-- Second-Green diffusion-source handoff with the zero trace narrowed to a trace-identification theorem plus the standard admissible-test zero trace. This lower-scout continuation does not prove the analytic trace theorem. It only replaces the direct `hsecondGreenTestTraceZero` input by two smaller source-facing facts: the second-Green test trace agrees a.e. with the boundary trace used for admissible weak tests, and that boundary trace is zero a.e. under the no-boundary/compact-support condition. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTraceEqTestTraceZero {n : Nat} {Boundary Test : Type*} [MeasurableSpace Boundary] (a b : Fin (n + 1) → Real) (weightedField : Test → (Fin (n + 1) → Real) → Fin (n + 1) → Real) (weightedFieldDeriv : Test → (Fin (n + 1) → Real) → (Fin (n + 1) → Real) →L[Real] Fin (n + 1) → Real) (exceptionSet : Test → Set (Fin (n + 1) → Real)) (boundaryMeasure : MeasureTheory.Measure Boundary) (secondGreenTestTrace testTrace : Test → Boundary → Real) (secondGreenNormalFluxTrace : Boundary → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux : Test → Real) (sigmaCoeff : Real) (hdiffusionAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = weakDiffusionAction φ) (hdiffusionLaplacianTerm : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • densityLaplacianAction φ) (hfirstGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ - negativeGradientPairAction φ = firstGreenTotal φ) (hfirstGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenTotal φ = firstGreenBoundaryFlux φ) (hfirstGreenZeroBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenBoundaryFlux φ = 0) (hsecondGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → negativeGradientPairAction φ - testLaplacianAction φ = secondGreenTotal φ) (hsecondGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenTotal φ = secondGreenBoundaryFlux φ) (hle : a ≤ b) (hsecondGreenWeightedFieldContinuous : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ContinuousOn (weightedField φ) (Set.Icc a b)) (hsecondGreenExceptionCountable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (exceptionSet φ).Countable) (hsecondGreenDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ exceptionSet φ, HasFDerivAt (weightedField φ) (weightedFieldDeriv φ x) x) (hsecondGreenDivIntegrable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.IntegrableOn (fun x => ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (Set.Icc a b) MeasureTheory.volume) (hsecondGreenBoundaryFluxDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenBoundaryFlux φ = ∫ x in Set.Icc a b, ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (hsecondGreenFaceTrace : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (b i) x) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (a i) x) i)) = ∫ y, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y ∂boundaryMeasure) (hsecondGreenTestTraceEq : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, secondGreenTestTrace φ y = testTrace φ y) (htestTraceZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, testTrace φ y = 0) (htestLaplacian : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → testLaplacianAction φ = laplacian φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ := by have hsecondGreenTestTraceZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, secondGreenTestTrace φ y = 0 := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary exact ((hsecondGreenTestTraceEq φ hφ hcommon hkernel hdrift hdensity htests hboundary).and (htestTraceZero φ hφ hcommon hkernel hdrift hdensity htests hboundary)).mono (by intro y hy rw [hy.1, hy.2]) exact generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTestTraceZero a b weightedField weightedFieldDeriv exceptionSet boundaryMeasure secondGreenTestTrace secondGreenNormalFluxTrace Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux sigmaCoeff hdiffusionAction hdiffusionLaplacianTerm hfirstGreenResidual hfirstGreenDivergence hfirstGreenZeroBoundary hsecondGreenResidual hsecondGreenDivergence hle hsecondGreenWeightedFieldContinuous hsecondGreenExceptionCountable hsecondGreenDeriv hsecondGreenDivIntegrable hsecondGreenBoundaryFluxDivergence hsecondGreenFaceTrace hsecondGreenTestTraceZero htestLaplacian /-- Second-Green diffusion-source handoff with trace identification narrowed from an a.e. boundary statement to pointwise equality of the selected traces. This lower_2 continuation keeps the analytic source task as the pointwise trace-definition theorem saying that the second-Green boundary trace is the same boundary trace used for admissible weak tests. It derives the a.e. identification required by the lower_1 consumer and leaves the zero boundary trace theorem, box-divergence package, second-Green residual/divergence facts, and test-Laplacian normalization explicit. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqTestTraceZero {n : Nat} {Boundary Test : Type*} [MeasurableSpace Boundary] (a b : Fin (n + 1) → Real) (weightedField : Test → (Fin (n + 1) → Real) → Fin (n + 1) → Real) (weightedFieldDeriv : Test → (Fin (n + 1) → Real) → (Fin (n + 1) → Real) →L[Real] Fin (n + 1) → Real) (exceptionSet : Test → Set (Fin (n + 1) → Real)) (boundaryMeasure : MeasureTheory.Measure Boundary) (secondGreenTestTrace testTrace : Test → Boundary → Real) (secondGreenNormalFluxTrace : Boundary → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux : Test → Real) (sigmaCoeff : Real) (hdiffusionAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = weakDiffusionAction φ) (hdiffusionLaplacianTerm : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • densityLaplacianAction φ) (hfirstGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ - negativeGradientPairAction φ = firstGreenTotal φ) (hfirstGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenTotal φ = firstGreenBoundaryFlux φ) (hfirstGreenZeroBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenBoundaryFlux φ = 0) (hsecondGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → negativeGradientPairAction φ - testLaplacianAction φ = secondGreenTotal φ) (hsecondGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenTotal φ = secondGreenBoundaryFlux φ) (hle : a ≤ b) (hsecondGreenWeightedFieldContinuous : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ContinuousOn (weightedField φ) (Set.Icc a b)) (hsecondGreenExceptionCountable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (exceptionSet φ).Countable) (hsecondGreenDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ exceptionSet φ, HasFDerivAt (weightedField φ) (weightedFieldDeriv φ x) x) (hsecondGreenDivIntegrable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.IntegrableOn (fun x => ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (Set.Icc a b) MeasureTheory.volume) (hsecondGreenBoundaryFluxDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenBoundaryFlux φ = ∫ x in Set.Icc a b, ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (hsecondGreenFaceTrace : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (b i) x) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (a i) x) i)) = ∫ y, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y ∂boundaryMeasure) (hsecondGreenTestTracePointwiseEq : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ y, secondGreenTestTrace φ y = testTrace φ y) (htestTraceZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, testTrace φ y = 0) (htestLaplacian : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → testLaplacianAction φ = laplacian φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ := by have hsecondGreenTestTraceEq : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, secondGreenTestTrace φ y = testTrace φ y := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary exact Filter.Eventually.of_forall (hsecondGreenTestTracePointwiseEq φ hφ hcommon hkernel hdrift hdensity htests hboundary) exact generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTraceEqTestTraceZero a b weightedField weightedFieldDeriv exceptionSet boundaryMeasure secondGreenTestTrace testTrace secondGreenNormalFluxTrace Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux sigmaCoeff hdiffusionAction hdiffusionLaplacianTerm hfirstGreenResidual hfirstGreenDivergence hfirstGreenZeroBoundary hsecondGreenResidual hsecondGreenDivergence hle hsecondGreenWeightedFieldContinuous hsecondGreenExceptionCountable hsecondGreenDeriv hsecondGreenDivIntegrable hsecondGreenBoundaryFluxDivergence hsecondGreenFaceTrace hsecondGreenTestTraceEq htestTraceZero htestLaplacian /-- Second-Green diffusion-source handoff with admissible-test zero trace narrowed from an a.e. boundary statement to pointwise zero trace. This cycle-133 continuation keeps the selected second-Green trace identification as the pointwise source-definition equality from cycle 132. It removes the direct a.e. `htestTraceZero` premise by exposing the smaller source theorem that the admissible-test boundary trace itself vanishes pointwise on the boundary. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero {n : Nat} {Boundary Test : Type*} [MeasurableSpace Boundary] (a b : Fin (n + 1) → Real) (weightedField : Test → (Fin (n + 1) → Real) → Fin (n + 1) → Real) (weightedFieldDeriv : Test → (Fin (n + 1) → Real) → (Fin (n + 1) → Real) →L[Real] Fin (n + 1) → Real) (exceptionSet : Test → Set (Fin (n + 1) → Real)) (boundaryMeasure : MeasureTheory.Measure Boundary) (secondGreenTestTrace testTrace : Test → Boundary → Real) (secondGreenNormalFluxTrace : Boundary → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux : Test → Real) (sigmaCoeff : Real) (hdiffusionAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = weakDiffusionAction φ) (hdiffusionLaplacianTerm : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • densityLaplacianAction φ) (hfirstGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ - negativeGradientPairAction φ = firstGreenTotal φ) (hfirstGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenTotal φ = firstGreenBoundaryFlux φ) (hfirstGreenZeroBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenBoundaryFlux φ = 0) (hsecondGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → negativeGradientPairAction φ - testLaplacianAction φ = secondGreenTotal φ) (hsecondGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenTotal φ = secondGreenBoundaryFlux φ) (hle : a ≤ b) (hsecondGreenWeightedFieldContinuous : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ContinuousOn (weightedField φ) (Set.Icc a b)) (hsecondGreenExceptionCountable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (exceptionSet φ).Countable) (hsecondGreenDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ exceptionSet φ, HasFDerivAt (weightedField φ) (weightedFieldDeriv φ x) x) (hsecondGreenDivIntegrable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.IntegrableOn (fun x => ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (Set.Icc a b) MeasureTheory.volume) (hsecondGreenBoundaryFluxDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenBoundaryFlux φ = ∫ x in Set.Icc a b, ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (hsecondGreenFaceTrace : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (b i) x) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (a i) x) i)) = ∫ y, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y ∂boundaryMeasure) (hsecondGreenTestTracePointwiseEq : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ y, secondGreenTestTrace φ y = testTrace φ y) (htestTracePointwiseZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ y, testTrace φ y = 0) (htestLaplacian : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → testLaplacianAction φ = laplacian φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ := by have htestTraceZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, testTrace φ y = 0 := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary exact Filter.Eventually.of_forall (htestTracePointwiseZero φ hφ hcommon hkernel hdrift hdensity htests hboundary) exact generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqTestTraceZero a b weightedField weightedFieldDeriv exceptionSet boundaryMeasure secondGreenTestTrace testTrace secondGreenNormalFluxTrace Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux sigmaCoeff hdiffusionAction hdiffusionLaplacianTerm hfirstGreenResidual hfirstGreenDivergence hfirstGreenZeroBoundary hsecondGreenResidual hsecondGreenDivergence hle hsecondGreenWeightedFieldContinuous hsecondGreenExceptionCountable hsecondGreenDeriv hsecondGreenDivIntegrable hsecondGreenBoundaryFluxDivergence hsecondGreenFaceTrace hsecondGreenTestTracePointwiseEq htestTraceZero htestLaplacian /-- Second-Green diffusion-source handoff with test-Laplacian normalization narrowed to a test-local premise. This lower-scout continuation removes the broad `htestLaplacian` premise from the cycle-133 pointwise-trace consumer. The replacement is the source-facing normalization of the chosen admissible test calculus: once `φ` is admissible and the test class is regular, `testLaplacianAction φ` is the abstract `laplacian φ`. Law, kernel, density, and boundary hypotheses are no longer arguments to this normalization leaf. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfTestLaplacianNormalization {n : Nat} {Boundary Test : Type*} [MeasurableSpace Boundary] (a b : Fin (n + 1) → Real) (weightedField : Test → (Fin (n + 1) → Real) → Fin (n + 1) → Real) (weightedFieldDeriv : Test → (Fin (n + 1) → Real) → (Fin (n + 1) → Real) →L[Real] Fin (n + 1) → Real) (exceptionSet : Test → Set (Fin (n + 1) → Real)) (boundaryMeasure : MeasureTheory.Measure Boundary) (secondGreenTestTrace testTrace : Test → Boundary → Real) (secondGreenNormalFluxTrace : Boundary → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux : Test → Real) (sigmaCoeff : Real) (hdiffusionAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = weakDiffusionAction φ) (hdiffusionLaplacianTerm : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • densityLaplacianAction φ) (hfirstGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ - negativeGradientPairAction φ = firstGreenTotal φ) (hfirstGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenTotal φ = firstGreenBoundaryFlux φ) (hfirstGreenZeroBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenBoundaryFlux φ = 0) (hsecondGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → negativeGradientPairAction φ - testLaplacianAction φ = secondGreenTotal φ) (hsecondGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenTotal φ = secondGreenBoundaryFlux φ) (hle : a ≤ b) (hsecondGreenWeightedFieldContinuous : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ContinuousOn (weightedField φ) (Set.Icc a b)) (hsecondGreenExceptionCountable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (exceptionSet φ).Countable) (hsecondGreenDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ exceptionSet φ, HasFDerivAt (weightedField φ) (weightedFieldDeriv φ x) x) (hsecondGreenDivIntegrable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.IntegrableOn (fun x => ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (Set.Icc a b) MeasureTheory.volume) (hsecondGreenBoundaryFluxDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenBoundaryFlux φ = ∫ x in Set.Icc a b, ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (hsecondGreenFaceTrace : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (b i) x) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (a i) x) i)) = ∫ y, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y ∂boundaryMeasure) (hsecondGreenTestTracePointwiseEq : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ y, secondGreenTestTrace φ y = testTrace φ y) (htestTracePointwiseZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ y, testTrace φ y = 0) (htestLaplacianLocal : ∀ φ, Admissible φ → testRegular → testLaplacianAction φ = laplacian φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ := by have htestLaplacian : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → testLaplacianAction φ = laplacian φ := by intro φ hφ _hcommon _hkernel _hdrift _hdensity htests _hboundary exact htestLaplacianLocal φ hφ htests exact generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero a b weightedField weightedFieldDeriv exceptionSet boundaryMeasure secondGreenTestTrace testTrace secondGreenNormalFluxTrace Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux sigmaCoeff hdiffusionAction hdiffusionLaplacianTerm hfirstGreenResidual hfirstGreenDivergence hfirstGreenZeroBoundary hsecondGreenResidual hsecondGreenDivergence hle hsecondGreenWeightedFieldContinuous hsecondGreenExceptionCountable hsecondGreenDeriv hsecondGreenDivIntegrable hsecondGreenBoundaryFluxDivergence hsecondGreenFaceTrace hsecondGreenTestTracePointwiseEq htestTracePointwiseZero htestLaplacian /-- Test-local Laplacian normalization from an operator-level source identity. This lower_2 theorem narrows the remaining analytic `htestLaplacianLocal` leaf from the cycle-133 second-Green route. Instead of asking for a per-test theorem with an explicit admissibility argument, it asks only that the selected regular test calculus identify the test-Laplacian operator with the abstract Laplacian operator. The admissibility argument is then inert, matching the source rewrite in `appendix.tex:1379-1427`. -/ theorem generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization {Test : Type*} (Admissible : Test → Prop) (testRegular : Prop) (testLaplacianAction laplacian : Test → Real) (htestLaplacianOperator : testRegular → testLaplacianAction = laplacian) : ∀ φ, Admissible φ → testRegular → testLaplacianAction φ = laplacian φ := by intro φ _hφ htests exact congrFun (htestLaplacianOperator htests) φ /-- Operator-level test-Laplacian normalization from shared source definitions. This cycle-134 helper is the next source-facing boundary below `generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization`. It does not ask directly for `testLaplacianAction = laplacian`; instead both abstract actions must be identified with the same source Laplacian action on the selected weak-test representation from `appendix.tex:1379-1427`. -/ theorem generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback {Test SourceTest : Type*} (selectedTest : Test → SourceTest) (sourceLaplacianAction : SourceTest → Real) (testRegular : Prop) (testLaplacianAction laplacian : Test → Real) (htestLaplacianActionDef : testRegular → testLaplacianAction = (fun φ => sourceLaplacianAction (selectedTest φ))) (hweakFpLaplacianDef : testRegular → laplacian = (fun φ => sourceLaplacianAction (selectedTest φ))) : testRegular → testLaplacianAction = laplacian := by intro htests calc testLaplacianAction = (fun φ => sourceLaplacianAction (selectedTest φ)) := htestLaplacianActionDef htests _ = laplacian := (hweakFpLaplacianDef htests).symm /-- Pointwise test-Laplacian normalization from source-pullback definitions. This lower_1 scout bridge adapts the cycle-134 operator-level source-pullback normalization to the pointwise `htestLaplacianPointwise` leaf exposed by the cycle-137 Green split. The direct pointwise theorem is no longer primitive: it follows once the test-calculus action and the weak-FP abstract Laplacian action are both identified with the same selected source-Laplacian pullback from `appendix.tex:1379-1427`. -/ theorem generalMovingTargetDiscreteTestLaplacianPointwiseOfSourcePullback {Test SourceTest : Type*} (selectedTest : Test → SourceTest) (sourceLaplacianAction : SourceTest → Real) (testRegular : Prop) (testLaplacianAction laplacian : Test → Real) (htestLaplacianActionDef : testRegular → testLaplacianAction = (fun φ => sourceLaplacianAction (selectedTest φ))) (hweakFpLaplacianDef : testRegular → laplacian = (fun φ => sourceLaplacianAction (selectedTest φ))) : testRegular → ∀ φ, testLaplacianAction φ = laplacian φ := by intro htests φ exact congrFun (generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback selectedTest sourceLaplacianAction testRegular testLaplacianAction laplacian htestLaplacianActionDef hweakFpLaplacianDef htests) φ /-- Mathlib source formula for the selected weak-test Laplacian. This cycle-134 scout theorem narrows the remaining source-definition leaves `htestLaplacianActionDef` and `hweakFpLaplacianDef`: once the selected source test is represented as a smooth real-valued function on a finite-dimensional real inner-product space, the source Laplacian can be instantiated by Mathlib's standard-basis second-derivative formula. It is a local Mathlib bridge for `appendix.tex:1379-1427`, not a proof of the EM weak-FP or Green identities. -/ theorem generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv {E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (sourceTest : E → Real) : Laplacian.laplacian sourceTest = fun x => ∑ i, iteratedFDeriv Real 2 sourceTest x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis sourceTest /-- Weak-FP Laplacian definition from the standard-basis source formula. This lower_2 helper narrows the `hweakFpLaplacianDef` leaf left by the cycle-134 source-pullback packet. Once the selected weak test is represented as a real-valued test on the source Euclidean state space, it is enough to identify the weak-FP abstract Laplacian action with the standard-basis second-derivative formula; the Mathlib Laplacian source action follows from the compiled lower_1 bridge. This is still only the test-calculus definition boundary for `appendix.tex:1379-1427`, not a proof of the second-Green or diffusion leaves. -/ theorem generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (laplacian : Test → Real) (hweakFpStdBasisDef : testRegular → laplacian = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ)) := by intro htests calc laplacian = (fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) := hweakFpStdBasisDef htests _ = (fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ))) := by funext φ exact congrArg sourceLaplacianFunctional (generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv (sourceTest := selectedTest φ)).symm /-- Density-Laplacian source formula from a pointwise standard-basis field. This cycle-136 lower_1 scout theorem narrows the remaining `hdensityLaplacianStdBasisDef` source boundary. It separates the weak-action definition of the density-Laplacian source term from the pointwise calculus identity saying that the selected source field is the standard-basis second-derivative formula from `appendix.tex:1379-1427`. -/ theorem generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (sourceDensityLaplacian : Test → E → Real) (testRegular : Prop) (densityLaplacianAction : Test → Real) (hdensityLaplacianActionDef : testRegular → densityLaplacianAction = fun φ => sourceLaplacianFunctional (sourceDensityLaplacian φ)) (hsourceDensityLaplacianStdBasis : testRegular → sourceDensityLaplacian = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → densityLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := by intro htests calc densityLaplacianAction = (fun φ => sourceLaplacianFunctional (sourceDensityLaplacian φ)) := hdensityLaplacianActionDef htests _ = (fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) := by funext φ exact congrArg sourceLaplacianFunctional (congrFun (hsourceDensityLaplacianStdBasis htests) φ) /-- Pointwise density-Laplacian source formula from the Mathlib Laplacian. This lower_2 helper narrows the remaining `hsourceDensityLaplacianStdBasis` source boundary from lower_1. It is enough to identify the named source density-Laplacian field with Mathlib's Laplacian of the selected weak test; the already-compiled standard-basis theorem then supplies the second-derivative formula used in `appendix.tex:1379-1427`. -/ theorem generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (sourceDensityLaplacian : Test → E → Real) (testRegular : Prop) (hsourceDensityLaplacianEqLaplacian : testRegular → sourceDensityLaplacian = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → sourceDensityLaplacian = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := by intro htests calc sourceDensityLaplacian = (fun φ => Laplacian.laplacian (selectedTest φ)) := hsourceDensityLaplacianEqLaplacian htests _ = (fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := by funext φ exact generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv (sourceTest := selectedTest φ) /-- Weak-FP standard-basis source formula from the density-Laplacian action. This cycle-136 helper targets the remaining weak-FP side of the standard-basis source pair. It derives the `hweakFpStdBasisDef` shape consumed by the cycle-135 second-Green theorem from two smaller source-facing facts: the weak-FP abstract Laplacian action is the selected density-Laplacian action, and that density-Laplacian action has the standard-basis second-derivative source formula from `appendix.tex:1379-1427`. -/ theorem generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (laplacian densityLaplacianAction : Test → Real) (hweakFpDensityLaplacianAction : testRegular → laplacian = densityLaplacianAction) (hdensityLaplacianStdBasisDef : testRegular → densityLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := by intro htests exact (hweakFpDensityLaplacianAction htests).trans (hdensityLaplacianStdBasisDef htests) /-- Weak-FP density-Laplacian action from pointwise weak Laplacian IBP. This cycle-137 helper narrows the remaining `hweakFpDensityLaplacianAction` boundary from cycle 136. The source-facing analytic leaf is now the pointwise weak Laplacian integration-by-parts identity for each selected weak test; this theorem only turns that pointwise statement into the function equality consumed by the standard-basis source formula. -/ theorem generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP {Test : Type*} (testRegular : Prop) (laplacian densityLaplacianAction : Test → Real) (hpointwiseWeakLaplacianIbP : testRegular → ∀ φ, densityLaplacianAction φ = laplacian φ) : testRegular → laplacian = densityLaplacianAction := by intro htests funext φ exact (hpointwiseWeakLaplacianIbP htests φ).symm /-- Weak-FP standard-basis source formula from pointwise weak Laplacian IBP. This cycle-137 downstream bridge removes the direct `hweakFpDensityLaplacianAction` premise from the cycle-136 standard-basis consumer. The remaining source-cited inputs are the pointwise weak Laplacian IBP equality and the density-Laplacian standard-basis source formula. -/ theorem generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (laplacian densityLaplacianAction : Test → Real) (hpointwiseWeakLaplacianIbP : testRegular → ∀ φ, densityLaplacianAction φ = laplacian φ) (hdensityLaplacianStdBasisDef : testRegular → densityLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := by have hweakFpDensityLaplacianAction : testRegular → laplacian = densityLaplacianAction := generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP testRegular laplacian densityLaplacianAction hpointwiseWeakLaplacianIbP exact generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula selectedTest sourceLaplacianFunctional testRegular laplacian densityLaplacianAction hweakFpDensityLaplacianAction hdensityLaplacianStdBasisDef /-- Pointwise weak Laplacian IBP from the Green identity chain. This lower_1 proof-scout helper narrows the pointwise weak Laplacian integration-by-parts leaf exposed in cycle 137. The source-facing theorem to prove next is no longer a monolithic pointwise equality; it is the three pointwise Green/test-calculus identities used in `appendix.tex:1379-1427`. -/ theorem generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity {Test : Type*} (testRegular : Prop) (densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian : Test → Real) (hfirstGreenPointwise : testRegular → ∀ φ, densityLaplacianAction φ = negativeGradientPairAction φ) (hsecondGreenPointwise : testRegular → ∀ φ, negativeGradientPairAction φ = testLaplacianAction φ) (htestLaplacianPointwise : testRegular → ∀ φ, testLaplacianAction φ = laplacian φ) : testRegular → ∀ φ, densityLaplacianAction φ = laplacian φ := by intro htests φ calc densityLaplacianAction φ = negativeGradientPairAction φ := hfirstGreenPointwise htests φ _ = testLaplacianAction φ := hsecondGreenPointwise htests φ _ = laplacian φ := htestLaplacianPointwise htests φ /-- Weak-FP standard-basis formula from pointwise Green identities. This downstream bridge feeds the pointwise Green/test-calculus split directly into the cycle-137 standard-basis consumer. The remaining analytic leaves are the first-Green identity, second-Green identity, and test-Laplacian normalization for each selected weak test; the density source-definition facts remain separate. -/ theorem generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (laplacian densityLaplacianAction negativeGradientPairAction testLaplacianAction : Test → Real) (hfirstGreenPointwise : testRegular → ∀ φ, densityLaplacianAction φ = negativeGradientPairAction φ) (hsecondGreenPointwise : testRegular → ∀ φ, negativeGradientPairAction φ = testLaplacianAction φ) (htestLaplacianPointwise : testRegular → ∀ φ, testLaplacianAction φ = laplacian φ) (hdensityLaplacianStdBasisDef : testRegular → densityLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := by have hpointwiseWeakLaplacianIbP : testRegular → ∀ φ, densityLaplacianAction φ = laplacian φ := generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity testRegular densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian hfirstGreenPointwise hsecondGreenPointwise htestLaplacianPointwise exact generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP selectedTest sourceLaplacianFunctional testRegular laplacian densityLaplacianAction hpointwiseWeakLaplacianIbP hdensityLaplacianStdBasisDef /-- Pointwise second-Green identity from box-divergence and zero test trace. This lower_2 helper narrows the direct pointwise `hsecondGreenPointwise` leaf exposed by the cycle-137 Green-identity scout. It reconstructs the pointwise second Green identity from the same source-facing pieces already isolated for the EM backend: the residual identity, the boundary-flux divergence formula, Mathlib's box divergence theorem hypotheses, the signed-face trace identification, pointwise equality with the admissible test trace, and pointwise zero boundary trace. It does not prove the first-Green identity or test-Laplacian normalization. -/ theorem generalMovingTargetDiscreteSecondGreenPointwiseOfBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero {n : Nat} {Boundary Test : Type*} [MeasurableSpace Boundary] (a b : Fin (n + 1) → Real) (weightedField : Test → (Fin (n + 1) → Real) → Fin (n + 1) → Real) (weightedFieldDeriv : Test → (Fin (n + 1) → Real) → (Fin (n + 1) → Real) →L[Real] Fin (n + 1) → Real) (exceptionSet : Test → Set (Fin (n + 1) → Real)) (boundaryMeasure : MeasureTheory.Measure Boundary) (secondGreenTestTrace testTrace : Test → Boundary → Real) (secondGreenNormalFluxTrace : Boundary → Real) (testRegular : Prop) (negativeGradientPairAction testLaplacianAction secondGreenTotal secondGreenBoundaryFlux : Test → Real) (hsecondGreenResidual : testRegular → ∀ φ, negativeGradientPairAction φ - testLaplacianAction φ = secondGreenTotal φ) (hsecondGreenDivergence : testRegular → ∀ φ, secondGreenTotal φ = secondGreenBoundaryFlux φ) (hle : a ≤ b) (hsecondGreenWeightedFieldContinuous : testRegular → ∀ φ, ContinuousOn (weightedField φ) (Set.Icc a b)) (hsecondGreenExceptionCountable : testRegular → ∀ φ, (exceptionSet φ).Countable) (hsecondGreenDeriv : testRegular → ∀ φ, ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ exceptionSet φ, HasFDerivAt (weightedField φ) (weightedFieldDeriv φ x) x) (hsecondGreenDivIntegrable : testRegular → ∀ φ, MeasureTheory.IntegrableOn (fun x => ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (Set.Icc a b) MeasureTheory.volume) (hsecondGreenBoundaryFluxDivergence : testRegular → ∀ φ, secondGreenBoundaryFlux φ = ∫ x in Set.Icc a b, ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (hsecondGreenFaceTrace : testRegular → ∀ φ, (∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (b i) x) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (a i) x) i)) = ∫ y, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y ∂boundaryMeasure) (hsecondGreenTestTracePointwiseEq : testRegular → ∀ φ y, secondGreenTestTrace φ y = testTrace φ y) (htestTracePointwiseZero : testRegular → ∀ φ y, testTrace φ y = 0) : testRegular → ∀ φ, negativeGradientPairAction φ = testLaplacianAction φ := by intro htests φ have hbox : (∫ x in Set.Icc a b, ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) = ∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (b i) x) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (a i) x) i) := MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable a b hle (weightedField φ) (weightedFieldDeriv φ) (exceptionSet φ) (hsecondGreenExceptionCountable htests φ) (hsecondGreenWeightedFieldContinuous htests φ) (hsecondGreenDeriv htests φ) (hsecondGreenDivIntegrable htests φ) have hsecondGreenBoundaryFluxIntegral : secondGreenBoundaryFlux φ = ∫ y, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y ∂boundaryMeasure := by calc secondGreenBoundaryFlux φ = ∫ x in Set.Icc a b, ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i := hsecondGreenBoundaryFluxDivergence htests φ _ = ∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (b i) x) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (a i) x) i) := hbox _ = ∫ y, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y ∂boundaryMeasure := hsecondGreenFaceTrace htests φ have hsecondGreenTraceProductZero : ∀ᵐ y ∂boundaryMeasure, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y = 0 := Filter.Eventually.of_forall fun y => by rw [ hsecondGreenTestTracePointwiseEq htests φ y, htestTracePointwiseZero htests φ y ] simp have hsecondGreenZeroBoundary : secondGreenBoundaryFlux φ = 0 := by calc secondGreenBoundaryFlux φ = ∫ y, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y ∂boundaryMeasure := hsecondGreenBoundaryFluxIntegral _ = ∫ _y, (0 : Real) ∂boundaryMeasure := MeasureTheory.integral_congr_ae hsecondGreenTraceProductZero _ = 0 := by simp have hresidualZero : negativeGradientPairAction φ - testLaplacianAction φ = 0 := by calc negativeGradientPairAction φ - testLaplacianAction φ = secondGreenTotal φ := hsecondGreenResidual htests φ _ = secondGreenBoundaryFlux φ := hsecondGreenDivergence htests φ _ = 0 := hsecondGreenZeroBoundary linarith /-- Weak-FP standard-basis formula with the second-Green pointwise leaf factored through box-divergence and pointwise zero trace. This downstream bridge instantiates `generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity` after deriving the second-Green pointwise identity from the source-facing box-divergence/test-trace boundary. The first-Green pointwise identity, test-Laplacian pointwise normalization, and density-Laplacian source formula remain explicit. -/ theorem generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux {n : Nat} {Boundary Test E : Type*} [MeasurableSpace Boundary] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (a b : Fin (n + 1) → Real) (weightedField : Test → (Fin (n + 1) → Real) → Fin (n + 1) → Real) (weightedFieldDeriv : Test → (Fin (n + 1) → Real) → (Fin (n + 1) → Real) →L[Real] Fin (n + 1) → Real) (exceptionSet : Test → Set (Fin (n + 1) → Real)) (boundaryMeasure : MeasureTheory.Measure Boundary) (secondGreenTestTrace testTrace : Test → Boundary → Real) (secondGreenNormalFluxTrace : Boundary → Real) (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (laplacian densityLaplacianAction negativeGradientPairAction testLaplacianAction secondGreenTotal secondGreenBoundaryFlux : Test → Real) (hfirstGreenPointwise : testRegular → ∀ φ, densityLaplacianAction φ = negativeGradientPairAction φ) (hsecondGreenResidual : testRegular → ∀ φ, negativeGradientPairAction φ - testLaplacianAction φ = secondGreenTotal φ) (hsecondGreenDivergence : testRegular → ∀ φ, secondGreenTotal φ = secondGreenBoundaryFlux φ) (hle : a ≤ b) (hsecondGreenWeightedFieldContinuous : testRegular → ∀ φ, ContinuousOn (weightedField φ) (Set.Icc a b)) (hsecondGreenExceptionCountable : testRegular → ∀ φ, (exceptionSet φ).Countable) (hsecondGreenDeriv : testRegular → ∀ φ, ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ exceptionSet φ, HasFDerivAt (weightedField φ) (weightedFieldDeriv φ x) x) (hsecondGreenDivIntegrable : testRegular → ∀ φ, MeasureTheory.IntegrableOn (fun x => ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (Set.Icc a b) MeasureTheory.volume) (hsecondGreenBoundaryFluxDivergence : testRegular → ∀ φ, secondGreenBoundaryFlux φ = ∫ x in Set.Icc a b, ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (hsecondGreenFaceTrace : testRegular → ∀ φ, (∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (b i) x) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (a i) x) i)) = ∫ y, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y ∂boundaryMeasure) (hsecondGreenTestTracePointwiseEq : testRegular → ∀ φ y, secondGreenTestTrace φ y = testTrace φ y) (htestTracePointwiseZero : testRegular → ∀ φ y, testTrace φ y = 0) (htestLaplacianPointwise : testRegular → ∀ φ, testLaplacianAction φ = laplacian φ) (hdensityLaplacianStdBasisDef : testRegular → densityLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := by have hsecondGreenPointwise : testRegular → ∀ φ, negativeGradientPairAction φ = testLaplacianAction φ := generalMovingTargetDiscreteSecondGreenPointwiseOfBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero a b weightedField weightedFieldDeriv exceptionSet boundaryMeasure secondGreenTestTrace testTrace secondGreenNormalFluxTrace testRegular negativeGradientPairAction testLaplacianAction secondGreenTotal secondGreenBoundaryFlux hsecondGreenResidual hsecondGreenDivergence hle hsecondGreenWeightedFieldContinuous hsecondGreenExceptionCountable hsecondGreenDeriv hsecondGreenDivIntegrable hsecondGreenBoundaryFluxDivergence hsecondGreenFaceTrace hsecondGreenTestTracePointwiseEq htestTracePointwiseZero exact generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity selectedTest sourceLaplacianFunctional testRegular laplacian densityLaplacianAction negativeGradientPairAction testLaplacianAction hfirstGreenPointwise hsecondGreenPointwise htestLaplacianPointwise hdensityLaplacianStdBasisDef /-- Pointwise first-Green identity from boundary-flux cancellation. This cycle-138 helper narrows the direct `hfirstGreenPointwise` leaf exposed by the cycle-137 Green-identity scout. The pointwise first Green equality is no longer primitive: it follows from the density-Laplacian residual identity, the first-Green divergence/boundary-flux identity, and the zero boundary-flux fact for the selected weak tests from `appendix.tex:1392-1427`. -/ theorem generalMovingTargetDiscreteFirstGreenPointwiseOfBoundaryFluxZero {Test : Type*} (testRegular : Prop) (densityLaplacianAction negativeGradientPairAction firstGreenTotal firstGreenBoundaryFlux : Test → Real) (hfirstGreenResidual : testRegular → ∀ φ, densityLaplacianAction φ - negativeGradientPairAction φ = firstGreenTotal φ) (hfirstGreenDivergence : testRegular → ∀ φ, firstGreenTotal φ = firstGreenBoundaryFlux φ) (hfirstGreenZeroBoundary : testRegular → ∀ φ, firstGreenBoundaryFlux φ = 0) : testRegular → ∀ φ, densityLaplacianAction φ = negativeGradientPairAction φ := by intro htests φ have hresidualZero : densityLaplacianAction φ - negativeGradientPairAction φ = 0 := by calc densityLaplacianAction φ - negativeGradientPairAction φ = firstGreenTotal φ := hfirstGreenResidual htests φ _ = firstGreenBoundaryFlux φ := hfirstGreenDivergence htests φ _ = 0 := hfirstGreenZeroBoundary htests φ linarith /-- Weak-FP standard-basis formula with both Green pointwise leaves narrowed. This downstream bridge removes the direct `hfirstGreenPointwise` premise from `generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux`. It derives first-Green pointwise from the residual/divergence/zero-boundary facts, while reusing the cycle-137 second-Green box-boundary narrowing and leaving the test-Laplacian pointwise normalization plus density-source facts explicit. -/ theorem generalMovingTargetDiscreteWeakFpStdBasisDefOfFirstGreenBoundaryFluxAndSecondGreenBoxBoundaryFlux {n : Nat} {Boundary Test E : Type*} [MeasurableSpace Boundary] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (a b : Fin (n + 1) → Real) (weightedField : Test → (Fin (n + 1) → Real) → Fin (n + 1) → Real) (weightedFieldDeriv : Test → (Fin (n + 1) → Real) → (Fin (n + 1) → Real) →L[Real] Fin (n + 1) → Real) (exceptionSet : Test → Set (Fin (n + 1) → Real)) (boundaryMeasure : MeasureTheory.Measure Boundary) (secondGreenTestTrace testTrace : Test → Boundary → Real) (secondGreenNormalFluxTrace : Boundary → Real) (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (laplacian densityLaplacianAction negativeGradientPairAction testLaplacianAction firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux : Test → Real) (hfirstGreenResidual : testRegular → ∀ φ, densityLaplacianAction φ - negativeGradientPairAction φ = firstGreenTotal φ) (hfirstGreenDivergence : testRegular → ∀ φ, firstGreenTotal φ = firstGreenBoundaryFlux φ) (hfirstGreenZeroBoundary : testRegular → ∀ φ, firstGreenBoundaryFlux φ = 0) (hsecondGreenResidual : testRegular → ∀ φ, negativeGradientPairAction φ - testLaplacianAction φ = secondGreenTotal φ) (hsecondGreenDivergence : testRegular → ∀ φ, secondGreenTotal φ = secondGreenBoundaryFlux φ) (hle : a ≤ b) (hsecondGreenWeightedFieldContinuous : testRegular → ∀ φ, ContinuousOn (weightedField φ) (Set.Icc a b)) (hsecondGreenExceptionCountable : testRegular → ∀ φ, (exceptionSet φ).Countable) (hsecondGreenDeriv : testRegular → ∀ φ, ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ exceptionSet φ, HasFDerivAt (weightedField φ) (weightedFieldDeriv φ x) x) (hsecondGreenDivIntegrable : testRegular → ∀ φ, MeasureTheory.IntegrableOn (fun x => ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (Set.Icc a b) MeasureTheory.volume) (hsecondGreenBoundaryFluxDivergence : testRegular → ∀ φ, secondGreenBoundaryFlux φ = ∫ x in Set.Icc a b, ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (hsecondGreenFaceTrace : testRegular → ∀ φ, (∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (b i) x) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (a i) x) i)) = ∫ y, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y ∂boundaryMeasure) (hsecondGreenTestTracePointwiseEq : testRegular → ∀ φ y, secondGreenTestTrace φ y = testTrace φ y) (htestTracePointwiseZero : testRegular → ∀ φ y, testTrace φ y = 0) (htestLaplacianPointwise : testRegular → ∀ φ, testLaplacianAction φ = laplacian φ) (hdensityLaplacianStdBasisDef : testRegular → densityLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := by have hfirstGreenPointwise : testRegular → ∀ φ, densityLaplacianAction φ = negativeGradientPairAction φ := generalMovingTargetDiscreteFirstGreenPointwiseOfBoundaryFluxZero testRegular densityLaplacianAction negativeGradientPairAction firstGreenTotal firstGreenBoundaryFlux hfirstGreenResidual hfirstGreenDivergence hfirstGreenZeroBoundary exact generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux a b weightedField weightedFieldDeriv exceptionSet boundaryMeasure secondGreenTestTrace testTrace secondGreenNormalFluxTrace selectedTest sourceLaplacianFunctional testRegular laplacian densityLaplacianAction negativeGradientPairAction testLaplacianAction secondGreenTotal secondGreenBoundaryFlux hfirstGreenPointwise hsecondGreenResidual hsecondGreenDivergence hle hsecondGreenWeightedFieldContinuous hsecondGreenExceptionCountable hsecondGreenDeriv hsecondGreenDivIntegrable hsecondGreenBoundaryFluxDivergence hsecondGreenFaceTrace hsecondGreenTestTracePointwiseEq htestTracePointwiseZero htestLaplacianPointwise hdensityLaplacianStdBasisDef /-- Test-calculus Laplacian definition from the standard-basis source formula. This cycle-135 helper is the test-action sibling of `generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula`. It narrows the remaining `htestLaplacianActionDef` leaf left by the cycle-134 source-pullback packet: the local test-calculus action only has to be identified with the standard-basis second-derivative source formula, after which Mathlib's Laplacian source action follows from the same `appendix.tex:1379-1427` bridge. It does not prove the weak-FP, Green, trace, or diffusion leaves. -/ theorem generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (testLaplacianAction : Test → Real) (htestLaplacianStdBasisDef : testRegular → testLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) : testRegular → testLaplacianAction = fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ)) := by intro htests calc testLaplacianAction = (fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) := htestLaplacianStdBasisDef htests _ = (fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ))) := by funext φ exact congrArg sourceLaplacianFunctional (generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv (sourceTest := selectedTest φ)).symm /-- Pointwise test-Laplacian normalization from the test standard-basis formula. This lower_2 bridge continues the cycle-138 pointwise source-pullback scout without using the weak-FP standard-basis conclusion as an input. It discharges the test-calculus side of the pointwise normalization from `htestLaplacianStdBasisDef`, while keeping the weak-FP source-Laplacian definition explicit as a separate non-circular source boundary. -/ theorem generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (testLaplacianAction laplacian : Test → Real) (htestLaplacianStdBasisDef : testRegular → testLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) (hweakFpLaplacianDef : testRegular → laplacian = fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ))) : testRegular → ∀ φ, testLaplacianAction φ = laplacian φ := by have htestLaplacianActionDef : testRegular → testLaplacianAction = fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ)) := generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula selectedTest sourceLaplacianFunctional testRegular testLaplacianAction htestLaplacianStdBasisDef exact generalMovingTargetDiscreteTestLaplacianPointwiseOfSourcePullback (fun φ => Laplacian.laplacian (selectedTest φ)) sourceLaplacianFunctional testRegular testLaplacianAction laplacian htestLaplacianActionDef hweakFpLaplacianDef /-- Weak-FP source-action definition from the EM state-law integral. This lower_1 scout bridge narrows `hweakFpSourceActionDef` to the concrete state-law integral interface behind `appendix.tex:1379-1387`. Once `\hat\rho_s` is represented as the map law of the EM interpolation state, the source functional is the integral against that law, and the abstract weak-FP Laplacian action is supplied as the sample-space pullback integral, the function-level source-action equality follows by Mathlib's `integral_map`. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (laplacian : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (weakFpLaplacianSourceField φ) hatRhoS) (hlaplacianStateIntegral : testRegular → ∀ φ, laplacian φ = ∫ ω, weakFpLaplacianSourceField φ (hatXAtS ω) ∂P) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by intro htests funext φ have hfieldMap : MeasureTheory.AEStronglyMeasurable (weakFpLaplacianSourceField φ) (MeasureTheory.Measure.map hatXAtS P) := by simpa [hhatRhoS] using hweakFpFieldMeas htests φ calc laplacian φ = ∫ ω, weakFpLaplacianSourceField φ (hatXAtS ω) ∂P := hlaplacianStateIntegral htests φ _ = ∫ x, weakFpLaplacianSourceField φ x ∂hatRhoS := by rw [hhatRhoS] exact (MeasureTheory.integral_map hhatX hfieldMap).symm _ = sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by rw [hsourceLaplacianFunctional] /-- Weak-FP source-action definition from the selected source Laplacian field. This lower_2 continuation narrows the lower_1 state-integral inputs. Once the named weak-FP source field is identified with Mathlib's Laplacian of the selected weak-test representative, it is enough to supply measurability and the sample-space pullback integral formula for that concrete Laplacian field. The map-law transport is still the compiled lower_1 `integral_map` bridge, and the Green, trace, standard-basis, and diffusion leaves remain explicit. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (laplacian : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianSourceStateIntegral : testRegular → ∀ φ, laplacian φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hweakFpFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (weakFpLaplacianSourceField φ) hatRhoS := by intro htests φ simpa [congrFun (hweakFpSourceFieldEqLaplacian htests) φ] using hsourceLaplacianFieldMeas htests φ have hlaplacianStateIntegral : testRegular → ∀ φ, laplacian φ = ∫ ω, weakFpLaplacianSourceField φ (hatXAtS ω) ∂P := by intro htests φ calc laplacian φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P := hlaplacianSourceStateIntegral htests φ _ = ∫ ω, weakFpLaplacianSourceField φ (hatXAtS ω) ∂P := by rw [← congrFun (hweakFpSourceFieldEqLaplacian htests) φ] exact generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral P hatRhoS hatXAtS weakFpLaplacianSourceField sourceLaplacianFunctional testRegular laplacian hhatRhoS hsourceLaplacianFunctional hhatX hweakFpFieldMeas hlaplacianStateIntegral /-- Source-Laplacian field measurability from ordinary measurability. Cycle 199 narrows the direct weak-Fokker--Planck side condition `hsourceLaplacianFieldMeas` to a source-facing test-class regularity premise that does not mention the EM interpolation law. The remaining analytic source work is to prove measurability of the selected-test Laplacian itself. -/ theorem generalMovingTargetDiscreteSourceLaplacianFieldMeasOfSelectedTestLaplacianMeasurable {Test E : Type*} [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (hatRhoS : MeasureTheory.Measure E) (selectedTest : Test → E → Real) (testRegular : Prop) (hSelectedTestLaplacianMeasurable : testRegular → ∀ φ, Measurable (Laplacian.laplacian (selectedTest φ))) : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS := by intro htests φ exact (hSelectedTestLaplacianMeasurable htests φ).aestronglyMeasurable /-- Selected-test Laplacian measurability from continuity. Lower_3's cycle 199 API bridge keeps the remaining source-facing regularity honest: if the original test class supplies continuity of the selected-test Laplacian, Mathlib immediately yields the measurability premise consumed by `generalMovingTargetDiscreteSourceLaplacianFieldMeasOfSelectedTestLaplacianMeasurable`. -/ theorem generalMovingTargetDiscreteSelectedTestLaplacianMeasurableOfContinuous {Test E : Type*} [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] [BorelSpace E] (selectedTest : Test → E → Real) (testRegular : Prop) (hSelectedTestLaplacianContinuous : testRegular → ∀ φ, Continuous (Laplacian.laplacian (selectedTest φ))) : testRegular → ∀ φ, Measurable (Laplacian.laplacian (selectedTest φ)) := by intro htests φ exact (hSelectedTestLaplacianContinuous htests φ).measurable /-- Source-Laplacian state integral from the frozen EM generator component. This cycle-140 bridge narrows the remaining `hlaplacianSourceStateIntegral` input exposed by the cycle-139 lower_2 theorem. It separates the source proof into the EM frozen-generator action identity and the sample-space state integral for that generator component over `appendix.tex:984-995` and `appendix.tex:1379-1387`. -/ theorem generalMovingTargetDiscreteLaplacianSourceStateIntegralOfEmGeneratorStateIntegral {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorStateIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P) : testRegular → ∀ φ, laplacian φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P := by intro htests φ calc laplacian φ = emGeneratorLaplacianAction φ := hlaplacianEqEmGenerator htests φ _ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P := hemGeneratorStateIntegral htests φ /-- Frozen EM generator Laplacian state integral from its law-integral form. This lower_1 scout bridge narrows the `hemGeneratorStateIntegral` premise to a law-space source fact. Once the paper-selected marginal is represented by `hatRhoS = Measure.map hatXAtS P`, Mathlib's `integral_map` turns the law-integral generator Laplacian action into the sample-space integral along the frozen EM interpolation state. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hhatX : AEMeasurable hatXAtS P) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hemGeneratorLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P := by intro htests φ have hfieldMap : MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) (MeasureTheory.Measure.map hatXAtS P) := by simpa [hhatRhoS] using hsourceLaplacianFieldMeas htests φ calc emGeneratorLaplacianAction φ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := hemGeneratorLawIntegral htests φ _ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P := by rw [hhatRhoS] exact MeasureTheory.integral_map hhatX hfieldMap /-- Frozen EM generator Laplacian state integral from its source-functional form. This cycle-153 bridge narrows the direct `hemGeneratorLaplacianStateIntegral` leaf selected by the EM conditional-law/state-event illness area. The remaining source-facing input is the definition of the frozen generator Laplacian action as the source Laplacian functional applied to Mathlib's selected-test Laplacian; the existing map-law transport theorem then recovers the sample-space integral along `hatXAtS`. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfSourceFunctional {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hemGeneratorSourceActionDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ))) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P := by have hemGeneratorLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := by intro htests φ calc emGeneratorLaplacianAction φ = sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ)) := congrFun (hemGeneratorSourceActionDef htests) φ _ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := by rw [hsourceLaplacianFunctional] exact generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral P hatRhoS hatXAtS selectedTest testRegular emGeneratorLaplacianAction hhatRhoS hhatX hsourceLaplacianFieldMeas hemGeneratorLawIntegral /-- Weak-FP source-action definition from the frozen EM generator state integral. This consumer feeds the cycle-140 source-integral bridge into `generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral`. Only `hlaplacianSourceStateIntegral` is replaced; the source-field equality, source-Laplacian measurability, source-functional definition, Green/trace, standard-basis, and diffusion leaves remain explicit. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStateIntegral {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorStateIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hlaplacianSourceStateIntegral : testRegular → ∀ φ, laplacian φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P := generalMovingTargetDiscreteLaplacianSourceStateIntegralOfEmGeneratorStateIntegral P hatXAtS selectedTest testRegular laplacian emGeneratorLaplacianAction hlaplacianEqEmGenerator hemGeneratorStateIntegral exact generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional testRegular laplacian hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianSourceStateIntegral /-- Weak-FP source-action definition from the frozen EM generator law integral. This lower_1 consumer removes the sample-space `hemGeneratorStateIntegral` premise from the cycle-140 bridge. The remaining source-cited analytic fact is the law-space generator Laplacian identity against `hatRhoS`; the sample-space version is recovered by `generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral`. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorLawIntegral {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorStateIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P := generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral P hatRhoS hatXAtS selectedTest testRegular emGeneratorLaplacianAction hhatRhoS hhatX hsourceLaplacianFieldMeas hemGeneratorLawIntegral exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStateIntegral P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorStateIntegral /-- Weak-FP source-action definition from the frozen EM generator source functional. This lower_2 continuation narrows lower_1's remaining law-space `hemGeneratorLawIntegral` fact. If the frozen EM generator's Laplacian component is defined by applying the source Laplacian functional to Mathlib's selected-test Laplacian, and that source functional is integration against `\hat\rho_s`, the law-integral premise follows by unfolding those source definitions. The map-law transport, source-field equality, source-field measurability, and the generator/action equality remain the existing explicit inputs. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorSourceActionDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ))) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := by intro htests φ calc emGeneratorLaplacianAction φ = sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ)) := congrFun (hemGeneratorSourceActionDef htests) φ _ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := by rw [hsourceLaplacianFunctional] exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorLawIntegral P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorLawIntegral /-- Frozen EM generator source action from the standard-basis source formula. This cycle-141 illness-area bridge narrows the remaining `hemGeneratorSourceActionDef` leaf from the cycle-140 source-functional route. For the frozen EM interpolation in `appendix.tex:984-995`, it is enough to show that the generator's Laplacian component is the source functional applied to the standard-basis second derivatives of the selected weak test. The existing Mathlib Laplacian bridge then recovers the `Laplacian.laplacian` source-action shape used by the law-integral consumer. This does not prove the analytic generator calculus fact itself. -/ theorem generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hemGeneratorStdBasisDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ)) := by intro htests calc emGeneratorLaplacianAction = (fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) := hemGeneratorStdBasisDef htests _ = (fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ))) := by funext φ exact congrArg sourceLaplacianFunctional (generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv (sourceTest := selectedTest φ)).symm /-- Frozen EM generator Laplacian state integral from the standard-basis source formula. This lower_2 continuation narrows the cycle-153 `hemGeneratorSourceActionDef` input to the source-cited standard-basis formula for the frozen EM generator Laplacian component. The law-to-state transport and source-functional law-integral construction remain the already explicit inputs. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hemGeneratorStdBasisDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P := by have hemGeneratorSourceActionDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ)) := generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula selectedTest sourceLaplacianFunctional testRegular emGeneratorLaplacianAction hemGeneratorStdBasisDef exact generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfSourceFunctional P hatRhoS hatXAtS selectedTest sourceLaplacianFunctional testRegular emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hsourceLaplacianFieldMeas hemGeneratorSourceActionDef /-- Weak-FP source-action route from the EM generator standard-basis formula. This consumer feeds the cycle-141 split into `generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional`. It replaces the direct `hemGeneratorSourceActionDef` premise by the smaller source-facing standard-basis formula for the frozen EM generator Laplacian component, while keeping `hlaplacianEqEmGenerator`, `hweakFpSourceFieldEqLaplacian`, `hsourceLaplacianFieldMeas`, and `hsourceLaplacianFunctional` explicit. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStdBasisSourceFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorStdBasisDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorSourceActionDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ)) := generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula selectedTest sourceLaplacianFunctional testRegular emGeneratorLaplacianAction hemGeneratorStdBasisDef exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorSourceActionDef /-- Frozen EM generator standard-basis formula from a named trace field. This lower_1 scout bridge narrows the remaining `hemGeneratorStdBasisDef` source theorem. The analytic Brownian-generator work is now the smaller pair: the frozen EM Laplacian action is the source functional applied to a named trace field, and that trace field is pointwise the standard-basis second derivative of the selected weak test. The bridge itself is only function extensionality and source-functional congruence. -/ theorem generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hemGeneratorTraceActionDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (emGeneratorTraceField φ)) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := by intro htests calc emGeneratorLaplacianAction = (fun φ => sourceLaplacianFunctional (emGeneratorTraceField φ)) := hemGeneratorTraceActionDef htests _ = (fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) := by funext φ exact congrArg sourceLaplacianFunctional (congrFun (htraceFieldStdBasis htests) φ) /-- Frozen EM generator Laplacian state integral from the trace-field split. This cycle-154 bridge discharges the `hemGeneratorStdBasisDef` premise left by the cycle-153 state-integral packet. The remaining source-facing facts are the trace-action definition for the frozen EM Laplacian component and the trace-field standard-basis formula; the already compiled cycle-153 source-functional and law-to-state transport bridges then recover the sample-space selected-test Laplacian integral. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceFieldSourceFunctional {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hemGeneratorTraceActionDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (emGeneratorTraceField φ)) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P := by have hemGeneratorStdBasisDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField selectedTest sourceLaplacianFunctional emGeneratorTraceField testRegular emGeneratorLaplacianAction hemGeneratorTraceActionDef htraceFieldStdBasis exact generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional P hatRhoS hatXAtS selectedTest sourceLaplacianFunctional testRegular emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hsourceLaplacianFieldMeas hemGeneratorStdBasisDef /-- Weak-FP source-action route from the EM generator trace-field split. This feeds the lower_1 trace-field narrowing into the existing cycle-141 standard-basis consumer. It replaces `hemGeneratorStdBasisDef` by the smaller source-cited trace-action definition plus the pointwise trace-field formula. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceFieldSourceFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorTraceActionDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (emGeneratorTraceField φ)) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorStdBasisDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField selectedTest sourceLaplacianFunctional emGeneratorTraceField testRegular emGeneratorLaplacianAction hemGeneratorTraceActionDef htraceFieldStdBasis exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStdBasisSourceFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorStdBasisDef /-- Weak-FP source-action route from a law-space EM generator trace integral. This lower_2 continuation narrows the trace-action source boundary exposed by the lower_1 trace-field split. Instead of assuming directly that the frozen EM generator Laplacian action is `sourceLaplacianFunctional` applied to the named trace field, it is enough to prove the source law-integral formula for that trace field against `hatRhoS`; the existing source-functional definition then recovers the trace-action premise consumed by the trace-field route. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorTraceLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, emGeneratorTraceField φ x ∂hatRhoS) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorTraceActionDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (emGeneratorTraceField φ) := by intro htests funext φ calc emGeneratorLaplacianAction φ = ∫ x, emGeneratorTraceField φ x ∂hatRhoS := hemGeneratorTraceLawIntegral htests φ _ = sourceLaplacianFunctional (emGeneratorTraceField φ) := by rw [hsourceLaplacianFunctional] exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceFieldSourceFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorTraceActionDef htraceFieldStdBasis /-- Frozen EM generator trace law integral from the EM state integral. This cycle-142 middle bridge narrows the remaining `hemGeneratorTraceLawIntegral` boundary to the sample-space trace integral along the frozen EM interpolation state. The only local proof step is the source-law transport `hatRhoS = Measure.map hatXAtS P` by `MeasureTheory.integral_map`; the analytic trace integral itself remains a source-cited EM fact. -/ theorem generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (emGeneratorTraceField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hhatX : AEMeasurable hatXAtS P) (htraceFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (emGeneratorTraceField φ) hatRhoS) (hemGeneratorTraceStateIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, emGeneratorTraceField φ (hatXAtS ω) ∂P) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, emGeneratorTraceField φ x ∂hatRhoS := by intro htests φ have hfieldMap : MeasureTheory.AEStronglyMeasurable (emGeneratorTraceField φ) (MeasureTheory.Measure.map hatXAtS P) := by simpa [hhatRhoS] using htraceFieldMeas htests φ calc emGeneratorLaplacianAction φ = ∫ ω, emGeneratorTraceField φ (hatXAtS ω) ∂P := hemGeneratorTraceStateIntegral htests φ _ = ∫ x, emGeneratorTraceField φ x ∂hatRhoS := by rw [hhatRhoS] exact (MeasureTheory.integral_map hhatX hfieldMap).symm /-- Weak-FP source-action route from the EM trace state integral. This cycle-142 consumer feeds the state-integral trace narrowing into the cycle-141 trace-law route. It replaces the law-space `hemGeneratorTraceLawIntegral` input by the smaller sample-space EM trace integral plus trace-field measurability under the EM law. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceStateIntegralSourceFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (htraceFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (emGeneratorTraceField φ) hatRhoS) (hemGeneratorTraceStateIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, emGeneratorTraceField φ (hatXAtS ω) ∂P) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorTraceLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, emGeneratorTraceField φ x ∂hatRhoS := generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral P hatRhoS hatXAtS emGeneratorTraceField testRegular emGeneratorLaplacianAction hhatRhoS hhatX htraceFieldMeas hemGeneratorTraceStateIntegral exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorTraceLawIntegral htraceFieldStdBasis /-- Trace-field measurability from the selected-test Laplacian field. This lower_1 scout bridge narrows the trace-field measurability side condition left by the state-integral EM generator route. Once the named trace field is pointwise the standard-basis trace of the selected test, Mathlib's Laplacian standard-basis formula identifies it with `Laplacian.laplacian`; the existing source-Laplacian measurability hypothesis is therefore enough. -/ theorem generalMovingTargetDiscreteEmGeneratorTraceFieldMeasOfSourceLaplacianFieldMeas {Test E : Type*} [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (hatRhoS : MeasureTheory.Measure E) (selectedTest : Test → E → Real) (emGeneratorTraceField : Test → E → Real) (testRegular : Prop) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (emGeneratorTraceField φ) hatRhoS := by intro htests φ have htraceEqLaplacian : emGeneratorTraceField φ = Laplacian.laplacian (selectedTest φ) := by calc emGeneratorTraceField φ = (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := congrFun (htraceFieldStdBasis htests) φ _ = Laplacian.laplacian (selectedTest φ) := (generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv (sourceTest := selectedTest φ)).symm simpa [htraceEqLaplacian] using hsourceLaplacianFieldMeas htests φ /-- Trace state integral from the selected-test Laplacian state integral. This lower_1 scout bridge narrows the remaining `hemGeneratorTraceStateIntegral` source theorem. It does not prove the EM generator state integral itself; it only replaces the named trace-field integrand by Mathlib's selected-test Laplacian using the already-local standard-basis formula. -/ theorem generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegral {Ω Test E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (emGeneratorTraceField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hemGeneratorLaplacianStateIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, emGeneratorTraceField φ (hatXAtS ω) ∂P := by intro htests φ have htraceEqLaplacian : emGeneratorTraceField φ = Laplacian.laplacian (selectedTest φ) := by calc emGeneratorTraceField φ = (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := congrFun (htraceFieldStdBasis htests) φ _ = Laplacian.laplacian (selectedTest φ) := (generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv (sourceTest := selectedTest φ)).symm calc emGeneratorLaplacianAction φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P := hemGeneratorLaplacianStateIntegral htests φ _ = ∫ ω, emGeneratorTraceField φ (hatXAtS ω) ∂P := by rw [htraceEqLaplacian] /-- Trace state integral from a direct trace-field/Laplacian identity. This cycle-155 dynamic-leaf bridge narrows the remaining `hemGeneratorTraceStateIntegral` boundary exposed by the cycle-154 state integral route. It does not introduce a downstream consumer: it only rewrites the sample-space trace integrand using the already explicit `htraceFieldEqLaplacian` source identity. -/ theorem generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegralLaplacianField {Ω Test E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (emGeneratorTraceField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hemGeneratorLaplacianStateIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P) (htraceFieldEqLaplacian : testRegular → emGeneratorTraceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, emGeneratorTraceField φ (hatXAtS ω) ∂P := by intro htests φ have htraceEqLaplacian : emGeneratorTraceField φ = Laplacian.laplacian (selectedTest φ) := congrFun (htraceFieldEqLaplacian htests) φ calc emGeneratorLaplacianAction φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P := hemGeneratorLaplacianStateIntegral htests φ _ = ∫ ω, emGeneratorTraceField φ (hatXAtS ω) ∂P := by rw [htraceEqLaplacian] /-- Weak-FP source-action route from the EM Laplacian state integral. This lower_1 consumer removes the trace-specific state-integral and measurability premises exposed by the cycle-142 middle packet. The remaining EM analytic input is the selected-test Laplacian state integral; the trace field is connected to it by the compiled standard-basis source formula. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorLaplacianStateIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have htraceFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (emGeneratorTraceField φ) hatRhoS := generalMovingTargetDiscreteEmGeneratorTraceFieldMeasOfSourceLaplacianFieldMeas hatRhoS selectedTest emGeneratorTraceField testRegular hsourceLaplacianFieldMeas htraceFieldStdBasis have hemGeneratorTraceStateIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, emGeneratorTraceField φ (hatXAtS ω) ∂P := generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegral P hatXAtS selectedTest emGeneratorTraceField testRegular emGeneratorLaplacianAction hemGeneratorLaplacianStateIntegral htraceFieldStdBasis exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceStateIntegralSourceFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator htraceFieldMeas hemGeneratorTraceStateIntegral htraceFieldStdBasis /-- Weak-FP source-action route from the EM Laplacian law integral. This lower_2 continuation narrows the cycle-142 `hemGeneratorLaplacianStateIntegral` leaf exposed by the trace-state route. The remaining source-cited EM generator fact is now the law-space Laplacian integral against `hatRhoS`; the sample-space integral along `hatXAtS` is recovered by the existing `MeasureTheory.integral_map` transport. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorLaplacianLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorLaplacianStateIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P := generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral P hatRhoS hatXAtS selectedTest testRegular emGeneratorLaplacianAction hhatRhoS hhatX hsourceLaplacianFieldMeas hemGeneratorLaplacianLawIntegral exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorLaplacianStateIntegral htraceFieldStdBasis /-- Frozen EM generator Laplacian law integral from a state-event formula. This cycle-143 bridge narrows the remaining `hemGeneratorLaplacianLawIntegral` boundary. Instead of assuming the law-space selected-test Laplacian integral directly, it asks for a named frozen-generator event field whose total action is the generator Laplacian action and whose set integrals agree with the selected-test Laplacian on every measurable state event. Taking the event to be the whole state space recovers the law-integral premise used by the cycle-142 trace-Laplacian consumer. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula {Test E : Type*} [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (hatRhoS : MeasureTheory.Measure E) (selectedTest : Test → E → Real) (emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hemGeneratorLaplacianTotalEventIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS) (hemGeneratorLaplacianStateEventEqLaplacian : testRegular → ∀ φ (t : Set E), MeasurableSet t → (∫ x in t, emGeneratorLaplacianEventField φ x ∂hatRhoS) = ∫ x in t, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := by intro htests φ calc emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := hemGeneratorLaplacianTotalEventIntegral htests φ _ = ∫ x in Set.univ, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := hemGeneratorLaplacianStateEventEqLaplacian htests φ Set.univ MeasurableSet.univ _ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := by simp /-- State-event Laplacian equality from a pointwise event-field definition. This lower-1 cycle-143 bridge narrows the remaining measurable-state-event boundary. If the named frozen-generator event field is pointwise the selected-test Laplacian field, then its integral over every measurable state event agrees with the selected-test Laplacian integral. The total-event generator action is deliberately kept separate. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise {Test E : Type*} [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (hatRhoS : MeasureTheory.Measure E) (selectedTest : Test → E → Real) (emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (hemGeneratorLaplacianEventFieldEqLaplacian : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x) : testRegular → ∀ φ (t : Set E), MeasurableSet t → (∫ x in t, emGeneratorLaplacianEventField φ x ∂hatRhoS) = ∫ x in t, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := by intro htests φ t _ht simp [hemGeneratorLaplacianEventFieldEqLaplacian htests φ] /-- Total-event generator formula from the source action definition. This lower-2 cycle-143 helper narrows the remaining `hemGeneratorLaplacianTotalEventIntegral` premise. The source-facing boundary is now the function-level definition of the frozen EM generator Laplacian action as integration of the named event field over the full state event; the old per-test total-event formula follows by function extensionality. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef {Test E : Type*} [MeasurableSpace E] (hatRhoS : MeasureTheory.Measure E) (emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hemGeneratorLaplacianActionDef : testRegular → emGeneratorLaplacianAction = fun φ => ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := by intro htests φ exact congrFun (hemGeneratorLaplacianActionDef htests) φ /-- Total-event generator formula from the source-functional Laplacian action. This lower-2 cycle-148 helper targets the genuine source leaf left by the state-event route. The total-event action formula follows from the paper's source-functional definition of the frozen EM Laplacian action and the pointwise identification of the named event field with Mathlib's selected-test Laplacian. The state-event equality and trace-field Laplacian identity remain separate source boundaries. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfSourceFunctional {Test E : Type*} [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (hatRhoS : MeasureTheory.Measure E) (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hemGeneratorSourceActionDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ))) (hemGeneratorLaplacianEventFieldEqLaplacian : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := by intro htests φ calc emGeneratorLaplacianAction φ = sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ)) := congrFun (hemGeneratorSourceActionDef htests) φ _ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := by rw [hsourceLaplacianFunctional] _ = ∫ x in Set.univ, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := by simp _ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := by rw [← hemGeneratorLaplacianEventFieldEqLaplacian htests φ] /-- Total-event generator formula from the standard-basis source action. This cycle-149 dynamic-leaf bridge narrows the direct `hemGeneratorSourceActionDef` input left by the cycle-148 source-functional route. The remaining source-facing EM calculus fact is the standard-basis source formula for the frozen generator Laplacian component; the already-local Mathlib Laplacian bridge recovers the source-functional action definition consumed by `generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfSourceFunctional`. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceFunctional {Test E : Type*} [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (hatRhoS : MeasureTheory.Measure E) (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hemGeneratorStdBasisDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) (hemGeneratorLaplacianEventFieldEqLaplacian : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := by have hemGeneratorSourceActionDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ)) := generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula selectedTest sourceLaplacianFunctional testRegular emGeneratorLaplacianAction hemGeneratorStdBasisDef exact generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfSourceFunctional hatRhoS selectedTest sourceLaplacianFunctional emGeneratorLaplacianEventField testRegular emGeneratorLaplacianAction hsourceLaplacianFunctional hemGeneratorSourceActionDef hemGeneratorLaplacianEventFieldEqLaplacian /-- Frozen EM generator Laplacian event-field identity from the source standard-basis formula. This cycle-144 bridge narrows the remaining pointwise event-field identity. Instead of requiring the named frozen-generator Laplacian event field to be identified directly with Mathlib's `Laplacian.laplacian`, it asks for the source-facing standard-basis second-derivative definition exposed by the EM generator paragraph. The existing Mathlib Laplacian standard-basis theorem then recovers the pointwise identity consumed by the cycle-143 state-event route. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (hemGeneratorLaplacianEventFieldStdBasisDef : testRegular → emGeneratorLaplacianEventField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x := by intro htests φ calc emGeneratorLaplacianEventField φ = (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := congrFun (hemGeneratorLaplacianEventFieldStdBasisDef htests) φ _ = Laplacian.laplacian (selectedTest φ) := (generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv (sourceTest := selectedTest φ)).symm /-- Total-event generator formula from standard-basis source and event fields. This cycle-149 lower_1 bridge removes the direct `hemGeneratorLaplacianEventFieldEqLaplacian` premise left by the cycle-149 standard-basis source-functional route. It keeps the source-functional law integral explicit, but asks for the paper-facing standard-basis definition of the named frozen EM Laplacian event field instead of the Mathlib Laplacian identity itself. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceAndEventFormula {Test E : Type*} [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (hatRhoS : MeasureTheory.Measure E) (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hemGeneratorStdBasisDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) (hemGeneratorLaplacianEventFieldStdBasisDef : testRegular → emGeneratorLaplacianEventField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := by have hemGeneratorLaplacianEventFieldEqLaplacian : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x := generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula selectedTest emGeneratorLaplacianEventField testRegular hemGeneratorLaplacianEventFieldStdBasisDef exact generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceFunctional hatRhoS selectedTest sourceLaplacianFunctional emGeneratorLaplacianEventField testRegular emGeneratorLaplacianAction hsourceLaplacianFunctional hemGeneratorStdBasisDef hemGeneratorLaplacianEventFieldEqLaplacian /-- Weak-FP source-action route from an EM generator state-event formula. This cycle-143 consumer feeds the state-event narrowing into the cycle-142 trace-Laplacian law-integral route. The old `hemGeneratorLaplacianLawIntegral` premise is reconstructed from the total state event and the measurable-state-event Laplacian formula, while `hlaplacianEqEmGenerator`, `htraceFieldStdBasis`, source-field equality, source-Laplacian measurability, and the source-functional definition remain explicit. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateEventFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorLaplacianTotalEventIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS) (hemGeneratorLaplacianStateEventEqLaplacian : testRegular → ∀ φ (t : Set E), MeasurableSet t → (∫ x in t, emGeneratorLaplacianEventField φ x ∂hatRhoS) = ∫ x in t, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorLaplacianLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula hatRhoS selectedTest emGeneratorLaplacianEventField testRegular emGeneratorLaplacianAction hemGeneratorLaplacianTotalEventIntegral hemGeneratorLaplacianStateEventEqLaplacian exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorLaplacianLawIntegral htraceFieldStdBasis /-- Weak-FP source-action route from a pointwise EM event-field formula. This lower-1 cycle-143 consumer removes the all-state-events integral equality as a primitive premise. It reconstructs that equality from the pointwise definition of the named frozen-generator Laplacian event field, then reuses the cycle-143 state-event consumer. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorLaplacianTotalEventIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS) (hemGeneratorLaplacianEventFieldEqLaplacian : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorLaplacianStateEventEqLaplacian : testRegular → ∀ φ (t : Set E), MeasurableSet t → (∫ x in t, emGeneratorLaplacianEventField φ x ∂hatRhoS) = ∫ x in t, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise hatRhoS selectedTest emGeneratorLaplacianEventField testRegular hemGeneratorLaplacianEventFieldEqLaplacian exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateEventFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField emGeneratorLaplacianEventField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorLaplacianTotalEventIntegral hemGeneratorLaplacianStateEventEqLaplacian htraceFieldStdBasis /-- Weak-FP source-action route from a pointwise EM event-field definition and the source action definition. This lower-2 cycle-143 consumer removes the total-event formula as a primitive premise. It derives that total-event formula from the source definition of `emGeneratorLaplacianAction`, then reuses the lower-1 pointwise event-field consumer. The pointwise event-field identity and sibling EM/weak-FP leaves remain explicit. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorLaplacianActionDef : testRegular → emGeneratorLaplacianAction = fun φ => ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS) (hemGeneratorLaplacianEventFieldEqLaplacian : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorLaplacianTotalEventIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef hatRhoS emGeneratorLaplacianEventField testRegular emGeneratorLaplacianAction hemGeneratorLaplacianActionDef exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField emGeneratorLaplacianEventField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorLaplacianTotalEventIntegral hemGeneratorLaplacianEventFieldEqLaplacian htraceFieldStdBasis /-- Weak-FP source-action route from a standard-basis EM event-field definition and the source action definition. This cycle-144 consumer removes the pointwise `hemGeneratorLaplacianEventFieldEqLaplacian` premise left by the cycle-143 action-definition route. The remaining source-facing event-field boundary is the standard-basis second-derivative definition for the named frozen EM Laplacian event field; `hemGeneratorLaplacianActionDef` and sibling EM/weak-FP source leaves remain explicit. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorLaplacianActionDef : testRegular → emGeneratorLaplacianAction = fun φ => ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS) (hemGeneratorLaplacianEventFieldStdBasisDef : testRegular → emGeneratorLaplacianEventField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorLaplacianEventFieldEqLaplacian : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x := generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula selectedTest emGeneratorLaplacianEventField testRegular hemGeneratorLaplacianEventFieldStdBasisDef exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField emGeneratorLaplacianEventField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorLaplacianActionDef hemGeneratorLaplacianEventFieldEqLaplacian htraceFieldStdBasis /-- Frozen EM generator Laplacian action definition from its standard-basis event-action formula. This lower-2 cycle-144 helper narrows the remaining `hemGeneratorLaplacianActionDef` premise. If the paper source gives the frozen EM Laplacian action as the state integral of the same standard-basis second-derivative expression used to define the named event field, then the older action definition with `emGeneratorLaplacianEventField` follows by rewriting the integrand. The standard-basis action formula and event-field definition remain the explicit source-facing facts. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionFormula {Test E : Type*} [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (hatRhoS : MeasureTheory.Measure E) (selectedTest : Test → E → Real) (emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hemGeneratorLaplacianStdBasisActionDef : testRegular → emGeneratorLaplacianAction = fun φ => ∫ x in Set.univ, (∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) ∂hatRhoS) (hemGeneratorLaplacianEventFieldStdBasisDef : testRegular → emGeneratorLaplacianEventField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → emGeneratorLaplacianAction = fun φ => ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := by intro htests calc emGeneratorLaplacianAction = (fun φ => ∫ x in Set.univ, (∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) ∂hatRhoS) := hemGeneratorLaplacianStdBasisActionDef htests _ = (fun φ => ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS) := by funext φ rw [congrFun (hemGeneratorLaplacianEventFieldStdBasisDef htests) φ] /-- Weak-FP source-action route from standard-basis formulas for both the frozen EM event field and action. This lower-2 cycle-144 consumer removes `hemGeneratorLaplacianActionDef` as a primitive premise under the standard-basis event-field route. The remaining source-facing EM facts are now the standard-basis action integral and the standard-basis event-field definition for the named frozen EM Laplacian component. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorLaplacianStdBasisActionDef : testRegular → emGeneratorLaplacianAction = fun φ => ∫ x in Set.univ, (∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) ∂hatRhoS) (hemGeneratorLaplacianEventFieldStdBasisDef : testRegular → emGeneratorLaplacianEventField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorLaplacianActionDef : testRegular → emGeneratorLaplacianAction = fun φ => ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionFormula hatRhoS selectedTest emGeneratorLaplacianEventField testRegular emGeneratorLaplacianAction hemGeneratorLaplacianStdBasisActionDef hemGeneratorLaplacianEventFieldStdBasisDef exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField emGeneratorLaplacianEventField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorLaplacianActionDef hemGeneratorLaplacianEventFieldStdBasisDef htraceFieldStdBasis /-- Frozen EM generator standard-basis action definition from the law-space Laplacian integral. This cycle-145 bridge narrows the remaining `hemGeneratorLaplacianStdBasisActionDef` boundary from the cycle-144 lower_2 packet. Instead of asking directly for the standard-basis state integral, it asks for the already exposed law-space selected-test Laplacian integral against `hatRhoS`; Mathlib's standard-basis Laplacian formula rewrites the integrand to the source display used by the frozen EM generator paragraph. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula {Test E : Type*} [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (hatRhoS : MeasureTheory.Measure E) (selectedTest : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hemGeneratorLaplacianLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS) : testRegular → emGeneratorLaplacianAction = fun φ => ∫ x in Set.univ, (∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) ∂hatRhoS := by intro htests funext φ calc emGeneratorLaplacianAction φ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := hemGeneratorLaplacianLawIntegral htests φ _ = ∫ x in Set.univ, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := by simp _ = ∫ x in Set.univ, (∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) ∂hatRhoS := by rw [generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv (sourceTest := selectedTest φ)] /-- Weak-FP source-action route from the EM law integral and standard-basis event-field formula. This cycle-145 consumer removes `hemGeneratorLaplacianStdBasisActionDef` as a primitive premise under the cycle-144 standard-basis action route. The remaining source-facing EM inputs are the law-space Laplacian action integral and the named event-field standard-basis definition; `hlaplacianEqEmGenerator`, `htraceFieldStdBasis`, and sibling weak-FP/source leaves remain explicit. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventLawIntegralFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorLaplacianLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS) (hemGeneratorLaplacianEventFieldStdBasisDef : testRegular → emGeneratorLaplacianEventField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorLaplacianStdBasisActionDef : testRegular → emGeneratorLaplacianAction = fun φ => ∫ x in Set.univ, (∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) ∂hatRhoS := generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula hatRhoS selectedTest testRegular emGeneratorLaplacianAction hemGeneratorLaplacianLawIntegral exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField emGeneratorLaplacianEventField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorLaplacianStdBasisActionDef hemGeneratorLaplacianEventFieldStdBasisDef htraceFieldStdBasis /-- Frozen EM generator Laplacian event-field standard-basis definition from the named trace field. This lower_2 cycle-145 bridge narrows the remaining `hemGeneratorLaplacianEventFieldStdBasisDef` premise. If the paper identifies the named Laplacian event field with the already tracked frozen-generator trace field, then the existing trace-field standard-basis source formula supplies the event-field standard-basis definition. The law-space generator Laplacian action integral remains a separate EM source boundary. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (hemGeneratorLaplacianEventFieldEqTraceField : testRegular → emGeneratorLaplacianEventField = emGeneratorTraceField) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → emGeneratorLaplacianEventField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := by intro htests exact (hemGeneratorLaplacianEventFieldEqTraceField htests).trans (htraceFieldStdBasis htests) /-- Trace-field standard-basis formula from the Mathlib Laplacian field. This cycle-146 lower_1 scout helper narrows the remaining `htraceFieldStdBasis` boundary. It is enough to identify the named frozen EM trace field with Mathlib's selected-test Laplacian; the existing local standard-basis theorem supplies the source second-derivative display. -/ theorem generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (emGeneratorTraceField : Test → E → Real) (testRegular : Prop) (htraceFieldEqLaplacian : testRegular → emGeneratorTraceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := by intro htests calc emGeneratorTraceField = (fun φ => Laplacian.laplacian (selectedTest φ)) := htraceFieldEqLaplacian htests _ = (fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := by funext φ exact generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv (sourceTest := selectedTest φ) /-- Trace-field equality from the pointwise trace-field identity. This cycle-156 illness-area refiner narrows the direct `htraceFieldEqLaplacian` boundary. The field-level equality for the named frozen EM trace field is no longer primitive: it follows from the smaller source-facing pointwise identity for each selected weak test. The analytic proof of that pointwise identity remains the exact source-cited boundary from `appendix.tex:1379-1387`. -/ theorem generalMovingTargetDiscreteEmGeneratorTraceFieldEqLaplacianOfPointwise {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (emGeneratorTraceField : Test → E → Real) (testRegular : Prop) (htraceFieldPointwiseEqLaplacian : testRegular → ∀ φ, emGeneratorTraceField φ = Laplacian.laplacian (selectedTest φ)) : testRegular → emGeneratorTraceField = fun φ => Laplacian.laplacian (selectedTest φ) := by intro htests funext φ exact htraceFieldPointwiseEqLaplacian htests φ /-- Pointwise trace-field standard-basis display from the event-field display. This lower_2 illness-area refiner narrows the remaining `htraceFieldPointwiseStdBasis` leaf. It is enough to prove that the named frozen EM Laplacian event field is the same field as the trace field, and that the event field has the source standard-basis Hessian-trace display. Both facts are still source-cited EM generator boundaries; this theorem only transfers the display to the trace-field name. -/ theorem generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseStdBasisOfEventFieldStdBasis {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (hemGeneratorLaplacianEventFieldEqTraceField : testRegular → emGeneratorLaplacianEventField = emGeneratorTraceField) (hemGeneratorLaplacianEventFieldStdBasisDef : testRegular → emGeneratorLaplacianEventField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → ∀ φ, emGeneratorTraceField φ = fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := by intro htests φ calc emGeneratorTraceField φ = emGeneratorLaplacianEventField φ := congrFun (hemGeneratorLaplacianEventFieldEqTraceField htests).symm φ _ = (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := congrFun (hemGeneratorLaplacianEventFieldStdBasisDef htests) φ /-- Pointwise trace-field Laplacian identity from the standard-basis display. This lower_1 scout bridge narrows the remaining cycle-156 pointwise leaf. It is enough to prove the paper's explicit Hessian-trace formula for each selected weak test; Mathlib's finite-dimensional Laplacian formula then identifies that standard-basis trace with `Laplacian.laplacian`. -/ theorem generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseEqLaplacianOfStdBasis {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (emGeneratorTraceField : Test → E → Real) (testRegular : Prop) (htraceFieldPointwiseStdBasis : testRegular → ∀ φ, emGeneratorTraceField φ = fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → ∀ φ, emGeneratorTraceField φ = Laplacian.laplacian (selectedTest φ) := by intro htests φ calc emGeneratorTraceField φ = (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := htraceFieldPointwiseStdBasis htests φ _ = Laplacian.laplacian (selectedTest φ) := by exact (generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv (sourceTest := selectedTest φ)).symm /-- Event-field standard-basis definition from a pointwise event-field display. This cycle-157 illness-area refiner narrows the direct `hemGeneratorLaplacianEventFieldStdBasisDef` boundary. The field-level definition of the named frozen EM Laplacian event field is no longer primitive: it follows from the smaller source-facing pointwise Hessian-trace display for each selected weak test. The analytic proof of that pointwise display remains the exact source-cited boundary from `appendix.tex:984-995` and `appendix.tex:1368-1387`. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfPointwise {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (hEmGeneratorLaplacianEventFieldPointwiseStdBasisDef : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → emGeneratorLaplacianEventField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := by intro htests funext φ exact hEmGeneratorLaplacianEventFieldPointwiseStdBasisDef htests φ /-- Pointwise event-field standard-basis display from a pointwise Laplacian identity. This lower_1 scout split keeps the remaining analytic source fact at the paper's `Delta` notation. Once the named frozen EM Laplacian event field is identified pointwise with Mathlib's `Laplacian.laplacian` of the selected weak test, the already compiled finite-dimensional Laplacian bridge gives the standard-basis Hessian-trace display needed by the cycle-157 field-level theorem. The pointwise Laplacian identity must still be proved directly from the frozen EM/Fokker--Planck display, not by cycling through trace/source-field consumers. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisDefOfPointwiseLaplacian {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (hemGeneratorLaplacianEventFieldEqLaplacian : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = Laplacian.laplacian (selectedTest φ)) : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := by intro htests φ calc emGeneratorLaplacianEventField φ = Laplacian.laplacian (selectedTest φ) := hemGeneratorLaplacianEventFieldEqLaplacian htests φ _ = fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := by exact generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv (sourceTest := selectedTest φ) /-- Event-field Laplacian identity from a statewise pointwise source display. This lower_2 cycle-157 bridge narrows the remaining `hemGeneratorLaplacianEventFieldEqLaplacian` boundary without using the older weak-FP source-field or trace-field consumer routes. It is enough to prove the paper-facing scalar identity at each selected weak test and state point; function extensionality then recovers the per-test field equality consumed by the lower_1 standard-basis bridge. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfPointwiseScalar {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian : testRegular → ∀ φ x, emGeneratorLaplacianEventField φ x = Laplacian.laplacian (selectedTest φ) x) : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = Laplacian.laplacian (selectedTest φ) := by intro htests φ funext x exact hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian htests φ x /-- Source-facing frozen Brownian generator Laplacian event field. For the EM interpolation in `appendix.tex:984-995`, the Brownian diffusion generator contributes the selected-test Laplacian field appearing in the Fokker--Planck line `appendix.tex:1379-1387`. The scalar diffusion coefficient is handled by the surrounding generator action, so this event field records only the statewise `Delta` factor. -/ noncomputable def emFrozenBrownianLaplacianEventField {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) : Test → E → Real := fun φ x => Laplacian.laplacian (selectedTest φ) x /-- Pointwise event-field Delta identity from the Brownian event-field definition. This lower_2 cycle-158 theorem implements the lower_1 scout route. It narrows `hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian` to the smaller source-facing definition `hEmGeneratorLaplacianEventFieldBrownianDef` for the named frozen Brownian-generator Laplacian event field. It does not use the older weak-FP source-field, trace-field, or downstream total-event routes. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseEqLaplacianOfBrownianDef {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (hEmGeneratorLaplacianEventFieldBrownianDef : testRegular → emGeneratorLaplacianEventField = emFrozenBrownianLaplacianEventField selectedTest) : testRegular → ∀ φ x, emGeneratorLaplacianEventField φ x = Laplacian.laplacian (selectedTest φ) x := by intro htests φ x calc emGeneratorLaplacianEventField φ x = emFrozenBrownianLaplacianEventField selectedTest φ x := congrFun (congrFun (hEmGeneratorLaplacianEventFieldBrownianDef htests) φ) x _ = Laplacian.laplacian (selectedTest φ) x := rfl /-- Brownian event-field definition from a pointwise source display. This cycle-159 middle bridge narrows the remaining `hEmGeneratorLaplacianEventFieldBrownianDef` boundary to the pointwise Brownian-generator event-field display supplied by the frozen EM interpolation and Fokker--Planck diffusion line. The source work is now the statewise definition itself; function extensionality recovers the named field equality. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefOfPointwise {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (hEmGeneratorLaplacianEventFieldBrownianPointwiseDef : testRegular → ∀ φ x, emGeneratorLaplacianEventField φ x = emFrozenBrownianLaplacianEventField selectedTest φ x) : testRegular → emGeneratorLaplacianEventField = emFrozenBrownianLaplacianEventField selectedTest := by intro htests funext φ x exact hEmGeneratorLaplacianEventFieldBrownianPointwiseDef htests φ x /-- Pointwise Brownian event-field definition from the coordinate trace display. This lower_2 cycle-159 bridge narrows the remaining pointwise Brownian-generator event-field boundary to the paper's coordinate Hessian-trace formula for the scalar Brownian diffusion generator. The only local proof content is the existing Mathlib finite-dimensional Laplacian normalization; the source-cited coordinate trace display remains the exact analytic boundary from the frozen EM interpolation and Fokker--Planck diffusion line. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseDefOfStdBasis {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef : testRegular → ∀ φ x, emGeneratorLaplacianEventField φ x = ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → ∀ φ x, emGeneratorLaplacianEventField φ x = emFrozenBrownianLaplacianEventField selectedTest φ x := by intro htests φ x calc emGeneratorLaplacianEventField φ x = ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef htests φ x _ = emFrozenBrownianLaplacianEventField selectedTest φ x := by simpa [emFrozenBrownianLaplacianEventField] using (congrFun (generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv (sourceTest := selectedTest φ)) x).symm /-- Source-facing frozen scalar Brownian Ito generator event field. For the Brownian increment in `appendix.tex:984-995`, the scalar diffusion coefficient is handled by the surrounding weak-FP action. This named event field records only the coordinate Hessian trace supplied by the scalar Brownian/Ito generator before the local Laplacian-normalization bridge rewrites it to `SALD.emFrozenBrownianLaplacianEventField`. -/ noncomputable def emFrozenScalarBrownianItoGeneratorEventField {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) : Test → E → Real := fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] /-- Brownian pointwise standard-basis display from the named scalar Ito generator. This cycle-160 middle bridge narrows the remaining `hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef` boundary to one named source-cited stochastic-generator interface for the frozen scalar Brownian increment in `eq:general_moving_target_SALD_frozen_interp`. The lower source task is now to identify the abstract EM Laplacian event field with `SALD.emFrozenScalarBrownianItoGeneratorEventField selectedTest`; this theorem only unfolds that named interface and does not route through weak-FP source-field, trace-field, or downstream total-event consumers. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDefOfFrozenScalarBrownianItoGenerator {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (hFrozenScalarBrownianItoGeneratorEventFieldDef : testRegular → emGeneratorLaplacianEventField = emFrozenScalarBrownianItoGeneratorEventField selectedTest) : testRegular → ∀ φ x, emGeneratorLaplacianEventField φ x = ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := by intro htests φ x calc emGeneratorLaplacianEventField φ x = emFrozenScalarBrownianItoGeneratorEventField selectedTest φ x := congrFun (congrFun (hFrozenScalarBrownianItoGeneratorEventFieldDef htests) φ) x _ = ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := rfl /-- Function-level frozen scalar Ito generator definition from its pointwise form. This lower-scout bridge keeps the active boundary on the Brownian/Ito generator source theorem. It narrows the remaining function equality `hFrozenScalarBrownianItoGeneratorEventFieldDef` to the pointwise equality for each selected weak test and state; the hard stochastic step remains proving that pointwise equality from the frozen Brownian increment in the paper. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoGeneratorDefOfPointwise {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (hFrozenScalarBrownianItoGeneratorEventFieldPointwiseDef : testRegular → ∀ φ x, emGeneratorLaplacianEventField φ x = emFrozenScalarBrownianItoGeneratorEventField selectedTest φ x) : testRegular → emGeneratorLaplacianEventField = emFrozenScalarBrownianItoGeneratorEventField selectedTest := by intro htests funext φ x exact hFrozenScalarBrownianItoGeneratorEventFieldPointwiseDef htests φ x /-- Pointwise frozen scalar Ito generator display from coordinate-generator data. This lower_2 cycle-160 bridge narrows the remaining pointwise Brownian/Ito generator boundary to the paper's two smaller stochastic-generator facts: the abstract EM Laplacian event field is the sum of its frozen Brownian coordinate generator contributions, and each coordinate contribution is the corresponding diagonal second derivative. The theorem only assembles those source facts into `SALD.emFrozenScalarBrownianItoGeneratorEventField`; it does not route through weak-FP source fields, trace fields, or downstream total-event consumers. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (emGeneratorLaplacianEventField : Test → E → Real) (brownianCoordinateGenerator : Test → E → Fin (Module.finrank Real E) → Real) (testRegular : Prop) (hFrozenScalarBrownianItoEventFieldCoordinateSum : testRegular → ∀ φ x, emGeneratorLaplacianEventField φ x = ∑ i, brownianCoordinateGenerator φ x i) (hFrozenScalarBrownianItoCoordinateGeneratorDef : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → ∀ φ x, emGeneratorLaplacianEventField φ x = emFrozenScalarBrownianItoGeneratorEventField selectedTest φ x := by intro htests φ x calc emGeneratorLaplacianEventField φ x = ∑ i, brownianCoordinateGenerator φ x i := hFrozenScalarBrownianItoEventFieldCoordinateSum htests φ x _ = ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := by exact Finset.sum_congr rfl (fun i _ => hFrozenScalarBrownianItoCoordinateGeneratorDef htests φ x i) _ = emFrozenScalarBrownianItoGeneratorEventField selectedTest φ x := rfl /-- One-dimensional scalar Brownian Ito/Taylor second-order term. For a fixed coordinate direction `e`, this is the diagonal second derivative term produced by the scalar Brownian second moment in the frozen interpolation. The diffusion coefficient remains in the surrounding weak-FP action. -/ noncomputable def emFrozenScalarBrownianItoOneDimTaylorGenerator {E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] (sourceTest : E → Real) (x e : E) : Real := iteratedFDeriv Real 2 sourceTest x ![e, e] /-- Centered real Gaussian second moment for the scalar Brownian coordinate. This is the Mathlib-backed moment fact needed below the one-dimensional Brownian/Ito Taylor boundary for `eq:general_moving_target_SALD_frozen_interp`. The remaining source work is the Taylor/generator passage that applies this moment to the selected weak test; this lemma only discharges the centered Gaussian second moment itself. -/ theorem gaussianRealZeroSecondMoment (v : NNReal) : (∫ z : Real, z ^ 2 ∂(ProbabilityTheory.gaussianReal (0 : Real) v)) = (v : Real) := by have hvar := ProbabilityTheory.variance_id_gaussianReal (μ := (0 : Real)) (v := v) have hvarIntegral := ProbabilityTheory.variance_eq_integral (X := (id : Real → Real)) (μ := ProbabilityTheory.gaussianReal (0 : Real) v) measurable_id'.aemeasurable rw [hvarIntegral] at hvar simpa [ProbabilityTheory.integral_id_gaussianReal] using hvar /-- Centered scalar Gaussian Taylor moment contribution. This lower_2 bridge combines the zero first moment of `ProbabilityTheory.gaussianReal 0 v` with the compiled centered second moment. It is the local moment-algebra step below the one-dimensional Brownian/Ito Taylor boundary; the remaining source theorem is still the dominated Taylor-remainder and generator-limit passage for the frozen Brownian increment. -/ theorem gaussianRealZeroOneDimTaylorMomentContribution (v : NNReal) (linearCoeff quadraticCoeff : Real) : linearCoeff * (∫ z : Real, z ∂(ProbabilityTheory.gaussianReal (0 : Real) v)) + quadraticCoeff * (∫ z : Real, z ^ 2 ∂(ProbabilityTheory.gaussianReal (0 : Real) v)) = quadraticCoeff * (v : Real) := by simp [ProbabilityTheory.integral_id_gaussianReal, gaussianRealZeroSecondMoment] /-- Gaussian integrability of the linear and quadratic scalar Taylor summands. This removes the polynomial-summand integrability bookkeeping from the Taylor-integral source boundary. The normalized-remainder integrability and the source Taylor integral definition remain separate source-facing fields. -/ theorem gaussianRealLinearQuadraticTaylorSummandsIntegrable (v : NNReal) (linearCoeff quadraticCoeff : Real) : MeasureTheory.Integrable (fun z : Real => linearCoeff * z) (ProbabilityTheory.gaussianReal (0 : Real) v) ∧ MeasureTheory.Integrable (fun z : Real => quadraticCoeff * z ^ 2) (ProbabilityTheory.gaussianReal (0 : Real) v) := by have hExpPos : MeasureTheory.Integrable (fun z : Real => Real.exp ((1 : Real) * z)) (ProbabilityTheory.gaussianReal (0 : Real) v) := by simpa using ProbabilityTheory.integrable_exp_mul_gaussianReal (μ := (0 : Real)) (v := v) (t := (1 : Real)) have hExpNeg : MeasureTheory.Integrable (fun z : Real => Real.exp (-(1 : Real) * z)) (ProbabilityTheory.gaussianReal (0 : Real) v) := by simpa using ProbabilityTheory.integrable_exp_mul_gaussianReal (μ := (0 : Real)) (v := v) (t := (-(1 : Real))) have hPowOne : MeasureTheory.Integrable (fun z : Real => z ^ 1) (ProbabilityTheory.gaussianReal (0 : Real) v) := by exact ProbabilityTheory.integrable_pow_of_integrable_exp_mul (X := fun z : Real => z) (μ := ProbabilityTheory.gaussianReal (0 : Real) v) (t := (1 : Real)) (by norm_num) hExpPos hExpNeg 1 have hPowTwo : MeasureTheory.Integrable (fun z : Real => z ^ 2) (ProbabilityTheory.gaussianReal (0 : Real) v) := by exact ProbabilityTheory.integrable_pow_of_integrable_exp_mul (X := fun z : Real => z) (μ := ProbabilityTheory.gaussianReal (0 : Real) v) (t := (1 : Real)) (by norm_num) hExpPos hExpNeg 2 refine ⟨?_, ?_⟩ · simpa [pow_one] using hPowOne.const_mul linearCoeff · exact hPowTwo.const_mul quadraticCoeff /-- Taylor integral split for the frozen scalar Brownian coordinate. This bridge narrows the source-facing `hFrozenScalarBrownianItoTaylorMomentDecomposition` input to a direct Taylor integral definition of the coordinate generator, explicit integrability of the three summands, and the definition of the normalized remainder contribution. The stochastic Taylor/source step that supplies those fields remains separate. -/ theorem selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefs {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (brownianCoordinateGenerator : Test → E → Fin (Module.finrank Real E) → Real) (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (linearCoeff quadraticCoeff remainderGeneratorLimit : Test → E → Fin (Module.finrank Real E) → Real) (normalizedRemainder : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hBrownianCoordinateGeneratorTaylorIntegralDef : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ z : Real, (linearCoeff φ x i * z + quadraticCoeff φ x i * z ^ 2 + normalizedRemainder φ x i z) ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) (hLinearInt : testRegular → ∀ φ x i, MeasureTheory.Integrable (fun z : Real => linearCoeff φ x i * z) (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) (hQuadraticInt : testRegular → ∀ φ x i, MeasureTheory.Integrable (fun z : Real => quadraticCoeff φ x i * z ^ 2) (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) (hRemainderInt : testRegular → ∀ φ x i, MeasureTheory.Integrable (fun z : Real => normalizedRemainder φ x i z) (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) (hRemainderGeneratorLimitDef : testRegular → ∀ φ x i, remainderGeneratorLimit φ x i = ∫ z : Real, normalizedRemainder φ x i z ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = linearCoeff φ x i * (∫ z : Real, z ∂(ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) + quadraticCoeff φ x i * (∫ z : Real, z ^ 2 ∂(ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) + remainderGeneratorLimit φ x i := by intro htests φ x i let μ := ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i) have hlin : MeasureTheory.Integrable (fun z : Real => linearCoeff φ x i * z) μ := by simpa [μ] using hLinearInt htests φ x i have hquad : MeasureTheory.Integrable (fun z : Real => quadraticCoeff φ x i * z ^ 2) μ := by simpa [μ] using hQuadraticInt htests φ x i have hrem : MeasureTheory.Integrable (fun z : Real => normalizedRemainder φ x i z) μ := by simpa [μ] using hRemainderInt htests φ x i calc brownianCoordinateGenerator φ x i = ∫ z : Real, (linearCoeff φ x i * z + quadraticCoeff φ x i * z ^ 2 + normalizedRemainder φ x i z) ∂μ := by simpa [μ] using hBrownianCoordinateGeneratorTaylorIntegralDef htests φ x i _ = (∫ z : Real, linearCoeff φ x i * z ∂μ) + (∫ z : Real, quadraticCoeff φ x i * z ^ 2 ∂μ) + (∫ z : Real, normalizedRemainder φ x i z ∂μ) := by have hsplit₁ := MeasureTheory.integral_add (hlin.add hquad) hrem have hsplit₂ := MeasureTheory.integral_add hlin hquad simp only [Pi.add_apply] at hsplit₁ rw [hsplit₂] at hsplit₁ simpa [add_assoc] using hsplit₁ _ = linearCoeff φ x i * (∫ z : Real, z ∂(ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) + quadraticCoeff φ x i * (∫ z : Real, z ^ 2 ∂(ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) + remainderGeneratorLimit φ x i := by rw [MeasureTheory.integral_const_mul, MeasureTheory.integral_const_mul, hRemainderGeneratorLimitDef htests φ x i] /-- Taylor integral split with Gaussian polynomial summand integrability. This bridge removes the two polynomial summand integrability hypotheses from `selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefs`. The remaining source-facing fields are the coordinate-generator Taylor integral definition, normalized-remainder integrability, and the remainder-integral definition. -/ theorem selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndGaussianPolynomialIntegrability {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (brownianCoordinateGenerator : Test → E → Fin (Module.finrank Real E) → Real) (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (linearCoeff quadraticCoeff remainderGeneratorLimit : Test → E → Fin (Module.finrank Real E) → Real) (normalizedRemainder : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hBrownianCoordinateGeneratorTaylorIntegralDef : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ z : Real, (linearCoeff φ x i * z + quadraticCoeff φ x i * z ^ 2 + normalizedRemainder φ x i z) ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) (hRemainderInt : testRegular → ∀ φ x i, MeasureTheory.Integrable (fun z : Real => normalizedRemainder φ x i z) (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) (hRemainderGeneratorLimitDef : testRegular → ∀ φ x i, remainderGeneratorLimit φ x i = ∫ z : Real, normalizedRemainder φ x i z ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = linearCoeff φ x i * (∫ z : Real, z ∂(ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) + quadraticCoeff φ x i * (∫ z : Real, z ^ 2 ∂(ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) + remainderGeneratorLimit φ x i := by refine selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefs brownianCoordinateGenerator variance linearCoeff quadraticCoeff remainderGeneratorLimit normalizedRemainder testRegular hBrownianCoordinateGeneratorTaylorIntegralDef ?_ ?_ hRemainderInt hRemainderGeneratorLimitDef · intro htests φ x i exact (gaussianRealLinearQuadraticTaylorSummandsIntegrable (variance φ x i) (linearCoeff φ x i) (quadraticCoeff φ x i)).1 · intro htests φ x i exact (gaussianRealLinearQuadraticTaylorSummandsIntegrable (variance φ x i) (linearCoeff φ x i) (quadraticCoeff φ x i)).2 /-- Taylor integral split with dominated remainder integrability. This lower_2 bridge removes the remaining `hRemainderInt` bookkeeping field from the Taylor moment split. The stochastic source equality and the remainder-generator definition stay explicit; only the normalized-remainder integrability is derived from a.e. measurability, a.e. domination, and an integrable dominating bound. -/ theorem selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndDominatedRemainder {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (brownianCoordinateGenerator : Test → E → Fin (Module.finrank Real E) → Real) (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (linearCoeff quadraticCoeff remainderGeneratorLimit : Test → E → Fin (Module.finrank Real E) → Real) (normalizedRemainder remainderBound : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hBrownianCoordinateGeneratorTaylorIntegralDef : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ z : Real, (linearCoeff φ x i * z + quadraticCoeff φ x i * z ^ 2 + normalizedRemainder φ x i z) ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) (hRemainderMeas : testRegular → ∀ φ x i, MeasureTheory.AEStronglyMeasurable (fun z : Real => normalizedRemainder φ x i z) (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) (hRemainderBound : testRegular → ∀ φ x i, ∀ᵐ z ∂ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i), ‖normalizedRemainder φ x i z‖ ≤ remainderBound φ x i z) (hRemainderBoundInt : testRegular → ∀ φ x i, MeasureTheory.Integrable (fun z : Real => remainderBound φ x i z) (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) (hRemainderGeneratorLimitDef : testRegular → ∀ φ x i, remainderGeneratorLimit φ x i = ∫ z : Real, normalizedRemainder φ x i z ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = linearCoeff φ x i * (∫ z : Real, z ∂(ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) + quadraticCoeff φ x i * (∫ z : Real, z ^ 2 ∂(ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) + remainderGeneratorLimit φ x i := by refine selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndGaussianPolynomialIntegrability brownianCoordinateGenerator variance linearCoeff quadraticCoeff remainderGeneratorLimit normalizedRemainder testRegular hBrownianCoordinateGeneratorTaylorIntegralDef ?_ hRemainderGeneratorLimitDef intro htests φ x i exact MeasureTheory.Integrable.mono' (hRemainderBoundInt htests φ x i) (hRemainderMeas htests φ x i) (hRemainderBound htests φ x i) /-- Source-integrand/a.e. equality bridge for the Brownian coordinate integral. This middle packet narrows the remaining coordinate-generator Taylor integral definition to two source-facing fields: a source integral definition for the paper's frozen Brownian scalar integrand, and an a.e. equality identifying that source integrand with the local Taylor sum. It is only the `MeasureTheory.integral_congr_ae` transport step; the scalar Taylor identity and stochastic source integral remain obligations. -/ theorem selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralAndAE {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (brownianCoordinateGenerator : Test → E → Fin (Module.finrank Real E) → Real) (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (linearCoeff quadraticCoeff : Test → E → Fin (Module.finrank Real E) → Real) (normalizedRemainder sourceTaylorIntegrand : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hBrownianCoordinateGeneratorSourceIntegralDef : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ z : Real, sourceTaylorIntegrand φ x i z ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) (hBrownianCoordinateGeneratorTaylorIntegrandAE : testRegular → ∀ φ x i, sourceTaylorIntegrand φ x i =ᵐ[ ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)] fun z : Real => linearCoeff φ x i * z + quadraticCoeff φ x i * z ^ 2 + normalizedRemainder φ x i z) : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ z : Real, (linearCoeff φ x i * z + quadraticCoeff φ x i * z ^ 2 + normalizedRemainder φ x i z) ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := by intro htests φ x i calc brownianCoordinateGenerator φ x i = ∫ z : Real, sourceTaylorIntegrand φ x i z ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := hBrownianCoordinateGeneratorSourceIntegralDef htests φ x i _ = ∫ z : Real, (linearCoeff φ x i * z + quadraticCoeff φ x i * z ^ 2 + normalizedRemainder φ x i z) ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := MeasureTheory.integral_congr_ae (hBrownianCoordinateGeneratorTaylorIntegrandAE htests φ x i) /-- Pointwise source Taylor identity supplies the Brownian-coordinate a.e. integrand equality. This lower packet is the narrow pointwise-to-a.e. adapter for the source Taylor integrand. The analytic content remains the source-facing pointwise identity `hSourceTaylorIntegrandPointwise`; this theorem only turns it into the a.e. equality required by the cycle-183 integral-congruence bridge. -/ theorem selectedWeakTestBrownianCoordinateGeneratorTaylorIntegrandAEOfPointwise {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (linearCoeff quadraticCoeff : Test → E → Fin (Module.finrank Real E) → Real) (normalizedRemainder sourceTaylorIntegrand : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hSourceTaylorIntegrandPointwise : testRegular → ∀ φ x i z, sourceTaylorIntegrand φ x i z = linearCoeff φ x i * z + quadraticCoeff φ x i * z ^ 2 + normalizedRemainder φ x i z) : testRegular → ∀ φ x i, sourceTaylorIntegrand φ x i =ᵐ[ ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)] fun z : Real => linearCoeff φ x i * z + quadraticCoeff φ x i * z ^ 2 + normalizedRemainder φ x i z := by intro htests φ x i exact Filter.Eventually.of_forall (hSourceTaylorIntegrandPointwise htests φ x i) /-- Raw source Taylor integrand from selected-line increment naming. This lower_2 bridge narrows the source-facing field `hSourceTaylorIntegrandRawDef`. The remaining paper content is split into two smaller source-cited fields: the source integrand is the selected weak-test increment for the frozen scalar Brownian coordinate, and that selected increment is the normalized coordinate-line expression. The weak-FP diffusion factor `sigma_eta^2 / 2` remains outside this raw scalar integrand. -/ theorem selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementAndLineDef {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (sourceTaylorIntegrand sourceSelectedLineIncrement : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hSourceTaylorIntegrandSelectedIncrementDef : testRegular → ∀ φ x i z, sourceTaylorIntegrand φ x i z = sourceSelectedLineIncrement φ x i z) (hSelectedIncrementCoordinateLineDef : testRegular → ∀ φ x i z, sourceSelectedLineIncrement φ x i z = selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x) : testRegular → ∀ φ x i z, sourceTaylorIntegrand φ x i z = selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x := by intro htests φ x i z calc sourceTaylorIntegrand φ x i z = sourceSelectedLineIncrement φ x i z := hSourceTaylorIntegrandSelectedIncrementDef htests φ x i z _ = selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x := hSelectedIncrementCoordinateLineDef htests φ x i z /-- Selected increment coordinate-line identity from endpoint naming. This cycle-190 bridge narrows the source-facing `hSelectedIncrementCoordinateLineDef` field. The remaining paper content is split into the selected-endpoint definition of the weak-test increment and the coordinate-line identity for that normalized frozen Brownian endpoint. The proof is only the local rewrite from the endpoint name to `x + z • stdOrthonormalBasis Real E i`. -/ theorem selectedWeakTestSelectedIncrementCoordinateLineDefOfEndpointAndLineDef {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (sourceSelectedLineIncrement : Test → E → Fin (Module.finrank Real E) → Real → Real) (sourceSelectedEndpoint : Test → E → Fin (Module.finrank Real E) → Real → E) (testRegular : Prop) (hSelectedIncrementEndpointDef : testRegular → ∀ φ x i z, sourceSelectedLineIncrement φ x i z = selectedTest φ (sourceSelectedEndpoint φ x i z) - selectedTest φ x) (hSelectedEndpointCoordinateLineDef : testRegular → ∀ φ x i z, sourceSelectedEndpoint φ x i z = x + z • (stdOrthonormalBasis Real E i)) : testRegular → ∀ φ x i z, sourceSelectedLineIncrement φ x i z = selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x := by intro htests φ x i z calc sourceSelectedLineIncrement φ x i z = selectedTest φ (sourceSelectedEndpoint φ x i z) - selectedTest φ x := hSelectedIncrementEndpointDef htests φ x i z _ = selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x := by rw [hSelectedEndpointCoordinateLineDef htests φ x i z] /-- Raw source Taylor integrand from selected-increment endpoint fields. This cycle-190 bridge removes the older supplied field `hSelectedIncrementCoordinateLineDef` from the raw source-integrand route by deriving it from the selected endpoint definition and the normalized coordinate-line endpoint identity. -/ theorem selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementEndpointAndLineDef {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (sourceTaylorIntegrand sourceSelectedLineIncrement : Test → E → Fin (Module.finrank Real E) → Real → Real) (sourceSelectedEndpoint : Test → E → Fin (Module.finrank Real E) → Real → E) (testRegular : Prop) (hSourceTaylorIntegrandSelectedIncrementDef : testRegular → ∀ φ x i z, sourceTaylorIntegrand φ x i z = sourceSelectedLineIncrement φ x i z) (hSelectedIncrementEndpointDef : testRegular → ∀ φ x i z, sourceSelectedLineIncrement φ x i z = selectedTest φ (sourceSelectedEndpoint φ x i z) - selectedTest φ x) (hSelectedEndpointCoordinateLineDef : testRegular → ∀ φ x i z, sourceSelectedEndpoint φ x i z = x + z • (stdOrthonormalBasis Real E i)) : testRegular → ∀ φ x i z, sourceTaylorIntegrand φ x i z = selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x := by have hSelectedIncrementCoordinateLineDef : testRegular → ∀ φ x i z, sourceSelectedLineIncrement φ x i z = selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x := selectedWeakTestSelectedIncrementCoordinateLineDefOfEndpointAndLineDef selectedTest sourceSelectedLineIncrement sourceSelectedEndpoint testRegular hSelectedIncrementEndpointDef hSelectedEndpointCoordinateLineDef exact selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementAndLineDef selectedTest sourceTaylorIntegrand sourceSelectedLineIncrement testRegular hSourceTaylorIntegrandSelectedIncrementDef hSelectedIncrementCoordinateLineDef /-- Source Taylor integrand definition from the raw selected-line increment. This cycle-188 bridge narrows the source-facing field `hSourceTaylorIntegrandDef`. The source correspondence below it now has two smaller pieces: the paper integrand is the selected weak-test increment along the normalized scalar Brownian coordinate line, and that selected-line increment splits into the source linear Taylor term, source quadratic Taylor term, and normalized remainder. The theorem only composes these two source definitions; it does not prove the Taylor expansion or any Hessian regularity. -/ theorem selectedWeakTestSourceTaylorIntegrandDefOfRawAndLineTaylorSplit {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (sourceLinearTerm sourceQuadraticTerm normalizedRemainder sourceTaylorIntegrand : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hSourceTaylorIntegrandRawDef : testRegular → ∀ φ x i z, sourceTaylorIntegrand φ x i z = selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x) (hSelectedLineTaylorSplitDef : testRegular → ∀ φ x i z, selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x = sourceLinearTerm φ x i z + sourceQuadraticTerm φ x i z + normalizedRemainder φ x i z) : testRegular → ∀ φ x i z, sourceTaylorIntegrand φ x i z = sourceLinearTerm φ x i z + sourceQuadraticTerm φ x i z + normalizedRemainder φ x i z := by intro htests φ x i z calc sourceTaylorIntegrand φ x i z = selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x := hSourceTaylorIntegrandRawDef htests φ x i z _ = sourceLinearTerm φ x i z + sourceQuadraticTerm φ x i z + normalizedRemainder φ x i z := hSelectedLineTaylorSplitDef htests φ x i z /-- Selected-line Taylor split from raw Taylor terms and source-term naming. This lower_2 bridge narrows the source-facing `hSelectedLineTaylorSplitDef` leaf. The remaining analytic Taylor content is the raw scalar expansion of `q ↦ selectedTest φ (x + q • e_i)`; this theorem only replaces its first and second Taylor terms by the separately tracked source-term names. -/ theorem selectedWeakTestSelectedLineTaylorSplitDefOfRawTaylorAndTermDefs {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (sourceLinearTerm sourceQuadraticTerm normalizedRemainder : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hSelectedLineTaylorRawSplitDef : testRegular → ∀ φ x i z, selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x = deriv (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0 * z + ((2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0) * z ^ 2 + normalizedRemainder φ x i z) (hSourceLinearTermTaylorDef : testRegular → ∀ φ x i z, sourceLinearTerm φ x i z = deriv (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0 * z) (hSourceQuadraticTermTaylorDef : testRegular → ∀ φ x i z, sourceQuadraticTerm φ x i z = ((2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0) * z ^ 2) : testRegular → ∀ φ x i z, selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x = sourceLinearTerm φ x i z + sourceQuadraticTerm φ x i z + normalizedRemainder φ x i z := by intro htests φ x i z calc selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x = deriv (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0 * z + ((2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0) * z ^ 2 + normalizedRemainder φ x i z := hSelectedLineTaylorRawSplitDef htests φ x i z _ = sourceLinearTerm φ x i z + sourceQuadraticTerm φ x i z + normalizedRemainder φ x i z := by rw [← hSourceLinearTermTaylorDef htests φ x i z, ← hSourceQuadraticTermTaylorDef htests φ x i z] /-- Source-term split for the Brownian-coordinate pointwise Taylor integrand. This cycle-186 bridge narrows the source-facing identity `hSourceTaylorIntegrandPointwise`. It keeps the actual scalar Taylor correspondence explicit as three smaller source fields: the paper's integrand is the sum of its source linear term, source quadratic term, and normalized remainder; the linear and quadratic source terms are then identified with the local coefficient conventions. This theorem is only that definitional/algebraic assembly and does not prove the scalar Taylor source fields themselves. -/ theorem selectedWeakTestSourceTaylorIntegrandPointwiseOfLineTermDefs {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (linearCoeff quadraticCoeff : Test → E → Fin (Module.finrank Real E) → Real) (sourceLinearTerm sourceQuadraticTerm normalizedRemainder sourceTaylorIntegrand : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hSourceTaylorIntegrandDef : testRegular → ∀ φ x i z, sourceTaylorIntegrand φ x i z = sourceLinearTerm φ x i z + sourceQuadraticTerm φ x i z + normalizedRemainder φ x i z) (hSourceLinearTermDef : testRegular → ∀ φ x i z, sourceLinearTerm φ x i z = linearCoeff φ x i * z) (hSourceQuadraticTermDef : testRegular → ∀ φ x i z, sourceQuadraticTerm φ x i z = quadraticCoeff φ x i * z ^ 2) : testRegular → ∀ φ x i z, sourceTaylorIntegrand φ x i z = linearCoeff φ x i * z + quadraticCoeff φ x i * z ^ 2 + normalizedRemainder φ x i z := by intro htests φ x i z calc sourceTaylorIntegrand φ x i z = sourceLinearTerm φ x i z + sourceQuadraticTerm φ x i z + normalizedRemainder φ x i z := hSourceTaylorIntegrandDef htests φ x i z _ = linearCoeff φ x i * z + quadraticCoeff φ x i * z ^ 2 + normalizedRemainder φ x i z := by rw [hSourceLinearTermDef htests φ x i z, hSourceQuadraticTermDef htests φ x i z] /-- Source linear term from the scalar line Taylor term and coefficient convention. This lower_2 bridge narrows the source-facing field `hSourceLinearTermDef`. The analytic paper content remains in two smaller source-cited fields: the first-order Taylor term for the selected scalar line and the convention identifying `linearCoeff` with that first derivative. The proof below is only the two-rewrite algebraic assembly. -/ theorem selectedWeakTestSourceLinearTermDefOfScalarLineFirstCoeffDef {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (linearCoeff : Test → E → Fin (Module.finrank Real E) → Real) (sourceLinearTerm : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hSourceLinearTermTaylorDef : testRegular → ∀ φ x i z, sourceLinearTerm φ x i z = deriv (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0 * z) (hScalarLineFirstCoeffDef : testRegular → ∀ φ x i, linearCoeff φ x i = deriv (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0) : testRegular → ∀ φ x i z, sourceLinearTerm φ x i z = linearCoeff φ x i * z := by intro htests φ x i z calc sourceLinearTerm φ x i z = deriv (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0 * z := hSourceLinearTermTaylorDef htests φ x i z _ = linearCoeff φ x i * z := by rw [← hScalarLineFirstCoeffDef htests φ x i] /-- Source quadratic term from the scalar Taylor quadratic term and coefficient convention. This cycle-187 bridge is the quadratic analogue of `selectedWeakTestSourceLinearTermDefOfScalarLineFirstCoeffDef`. It narrows the source-facing field `hSourceQuadraticTermDef` to two smaller fields: the paper's scalar order-two Taylor term along the selected Brownian coordinate line, and the existing local convention identifying `quadraticCoeff` with that same scalar Taylor coefficient. The proof is only the two-rewrite algebraic assembly; it does not prove the scalar Taylor coefficient source correspondence or any Hessian regularity. -/ theorem selectedWeakTestSourceQuadraticTermDefOfScalarLineTaylorCoeffDef {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (quadraticCoeff : Test → E → Fin (Module.finrank Real E) → Real) (sourceQuadraticTerm : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hSourceQuadraticTermTaylorDef : testRegular → ∀ φ x i z, sourceQuadraticTerm φ x i z = ((2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0) * z ^ 2) (hScalarLineTaylorCoeffDef : testRegular → ∀ φ x i, quadraticCoeff φ x i = (2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0) : testRegular → ∀ φ x i z, sourceQuadraticTerm φ x i z = quadraticCoeff φ x i * z ^ 2 := by intro htests φ x i z calc sourceQuadraticTerm φ x i z = ((2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0) * z ^ 2 := hSourceQuadraticTermTaylorDef htests φ x i z _ = quadraticCoeff φ x i * z ^ 2 := by rw [← hScalarLineTaylorCoeffDef htests φ x i] /-- Brownian coordinate Taylor integral from the raw selected-line Taylor data. This cycle-189 bridge composes the already compiled source-integral, a.e., source-term, and raw selected-line Taylor bridges. It narrows the active `hBrownianCoordinateGeneratorTaylorIntegralDef` leaf to the source integral definition plus the raw selected-line Taylor fields; it does not prove the raw Taylor expansion, scalar coefficient conventions, or any Hessian regularity. -/ theorem selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralRawTaylorAndTermDefs {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (brownianCoordinateGenerator : Test → E → Fin (Module.finrank Real E) → Real) (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (linearCoeff quadraticCoeff : Test → E → Fin (Module.finrank Real E) → Real) (sourceLinearTerm sourceQuadraticTerm normalizedRemainder sourceTaylorIntegrand : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hBrownianCoordinateGeneratorSourceIntegralDef : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ z : Real, sourceTaylorIntegrand φ x i z ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) (hSourceTaylorIntegrandRawDef : testRegular → ∀ φ x i z, sourceTaylorIntegrand φ x i z = selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x) (hSelectedLineTaylorRawSplitDef : testRegular → ∀ φ x i z, selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x = deriv (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0 * z + ((2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0) * z ^ 2 + normalizedRemainder φ x i z) (hSourceLinearTermTaylorDef : testRegular → ∀ φ x i z, sourceLinearTerm φ x i z = deriv (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0 * z) (hScalarLineFirstCoeffDef : testRegular → ∀ φ x i, linearCoeff φ x i = deriv (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0) (hSourceQuadraticTermTaylorDef : testRegular → ∀ φ x i z, sourceQuadraticTerm φ x i z = ((2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0) * z ^ 2) (hScalarLineTaylorCoeffDef : testRegular → ∀ φ x i, quadraticCoeff φ x i = (2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0) : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ z : Real, (linearCoeff φ x i * z + quadraticCoeff φ x i * z ^ 2 + normalizedRemainder φ x i z) ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := by have hSelectedLineTaylorSplitDef : testRegular → ∀ φ x i z, selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x = sourceLinearTerm φ x i z + sourceQuadraticTerm φ x i z + normalizedRemainder φ x i z := selectedWeakTestSelectedLineTaylorSplitDefOfRawTaylorAndTermDefs selectedTest sourceLinearTerm sourceQuadraticTerm normalizedRemainder testRegular hSelectedLineTaylorRawSplitDef hSourceLinearTermTaylorDef hSourceQuadraticTermTaylorDef have hSourceTaylorIntegrandDef : testRegular → ∀ φ x i z, sourceTaylorIntegrand φ x i z = sourceLinearTerm φ x i z + sourceQuadraticTerm φ x i z + normalizedRemainder φ x i z := selectedWeakTestSourceTaylorIntegrandDefOfRawAndLineTaylorSplit selectedTest sourceLinearTerm sourceQuadraticTerm normalizedRemainder sourceTaylorIntegrand testRegular hSourceTaylorIntegrandRawDef hSelectedLineTaylorSplitDef have hSourceLinearTermDef : testRegular → ∀ φ x i z, sourceLinearTerm φ x i z = linearCoeff φ x i * z := selectedWeakTestSourceLinearTermDefOfScalarLineFirstCoeffDef selectedTest linearCoeff sourceLinearTerm testRegular hSourceLinearTermTaylorDef hScalarLineFirstCoeffDef have hSourceQuadraticTermDef : testRegular → ∀ φ x i z, sourceQuadraticTerm φ x i z = quadraticCoeff φ x i * z ^ 2 := selectedWeakTestSourceQuadraticTermDefOfScalarLineTaylorCoeffDef selectedTest quadraticCoeff sourceQuadraticTerm testRegular hSourceQuadraticTermTaylorDef hScalarLineTaylorCoeffDef have hSourceTaylorIntegrandPointwise : testRegular → ∀ φ x i z, sourceTaylorIntegrand φ x i z = linearCoeff φ x i * z + quadraticCoeff φ x i * z ^ 2 + normalizedRemainder φ x i z := selectedWeakTestSourceTaylorIntegrandPointwiseOfLineTermDefs linearCoeff quadraticCoeff sourceLinearTerm sourceQuadraticTerm normalizedRemainder sourceTaylorIntegrand testRegular hSourceTaylorIntegrandDef hSourceLinearTermDef hSourceQuadraticTermDef have hBrownianCoordinateGeneratorTaylorIntegrandAE : testRegular → ∀ φ x i, sourceTaylorIntegrand φ x i =ᵐ[ ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)] fun z : Real => linearCoeff φ x i * z + quadraticCoeff φ x i * z ^ 2 + normalizedRemainder φ x i z := selectedWeakTestBrownianCoordinateGeneratorTaylorIntegrandAEOfPointwise variance linearCoeff quadraticCoeff normalizedRemainder sourceTaylorIntegrand testRegular hSourceTaylorIntegrandPointwise exact selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralAndAE brownianCoordinateGenerator variance linearCoeff quadraticCoeff normalizedRemainder sourceTaylorIntegrand testRegular hBrownianCoordinateGeneratorSourceIntegralDef hBrownianCoordinateGeneratorTaylorIntegrandAE /-- Quadratic-coefficient definition from the diagonal second Taylor coefficient. This cycle-175 lower_2 bridge only unfolds the local name `emFrozenScalarBrownianItoOneDimTaylorGenerator`. The source-facing analytic identity remains `hSecondTaylorCoeffDef`: the normalized scalar Brownian quadratic coefficient is the diagonal second Frechet derivative in the standard Brownian coordinate. The paper's `sigma_eta^2 / 2` factor stays outside this event-field coefficient. -/ theorem selectedWeakTestQuadraticCoeffDefOfSecondTaylorCoeffDef {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (quadraticCoeff : Test → E → Fin (Module.finrank Real E) → Real) (testRegular : Prop) (hSecondTaylorCoeffDef : testRegular → ∀ φ x i, quadraticCoeff φ x i = iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → ∀ φ x i, quadraticCoeff φ x i = emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest φ) x ((stdOrthonormalBasis Real E) i) := by intro htests φ x i simpa [emFrozenScalarBrownianItoOneDimTaylorGenerator] using hSecondTaylorCoeffDef htests φ x i /-- Real-valued variance-one field from the normalized Brownian variance definition. This bridge discharges the downstream `hVarianceOne` shape once the source correspondence has defined the normalized scalar Brownian coordinate variance as the constant `1 : NNReal`. The stochastic source fact remains the standard Gaussian coordinate normalization from the EM increment; this theorem only handles the `NNReal`-to-`Real` coercion used by the local Brownian/Ito algebra. -/ theorem selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (testRegular : Prop) (hNormalizedVarianceDef : testRegular → ∀ φ x i, variance φ x i = (1 : NNReal)) : testRegular → ∀ φ x i, (variance φ x i : Real) = 1 := by intro htests φ x i rw [hNormalizedVarianceDef htests φ x i] norm_num /-- Normalized Brownian coordinate law supplies the `NNReal` variance definition. This cycle-178 middle bridge narrows the source-facing `hNormalizedVarianceDef` field. Once the source correspondence identifies the normalized scalar Brownian coordinate law with `gaussianReal 0 1`, and records that the local `variance` field is the `NNReal` package of the real variance of that law, Mathlib's Gaussian variance theorem gives the unit `NNReal` variance used by the Brownian/Ito normalization. This theorem does not move the paper's `sigma_eta^2 / 2` coefficient into the event field. -/ theorem selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (normalizedCoordinateLaw : Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real) (testRegular : Prop) (hVarianceDef : testRegular → ∀ φ x i, (variance φ x i : Real) = ProbabilityTheory.variance (id : Real → Real) (normalizedCoordinateLaw φ x i)) (hNormalizedCoordinateLaw : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal)) : testRegular → ∀ φ x i, variance φ x i = (1 : NNReal) := by intro htests φ x i exact NNReal.coe_injective (by calc (variance φ x i : Real) = ProbabilityTheory.variance (id : Real → Real) (normalizedCoordinateLaw φ x i) := hVarianceDef htests φ x i _ = ProbabilityTheory.variance (id : Real → Real) (ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal)) := by rw [hNormalizedCoordinateLaw htests φ x i] _ = (1 : Real) := by simp) /-- Normalized scalar coordinate law from a vector standard Gaussian law. This cycle-178 lower_2 bridge narrows the scalar coordinate-law source field: once the paper correspondence supplies the normalized vector increment law as `stdGaussian E` and defines the scalar coordinate law as the map by a standard orthonormal coordinate, Mathlib's multivariate Gaussian API gives `gaussianReal 0 1`. The variance packaging field remains separate. -/ theorem selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] [MeasurableSpace E] [BorelSpace E] (normalizedVectorLaw : Test → E → MeasureTheory.Measure E) (normalizedCoordinateLaw : Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real) (testRegular : Prop) (hNormalizedVectorLaw : testRegular → ∀ φ x, normalizedVectorLaw φ x = ProbabilityTheory.stdGaussian E) (hCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = (normalizedVectorLaw φ x).map (fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y)) : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal) := by intro htests φ x i calc normalizedCoordinateLaw φ x i = (normalizedVectorLaw φ x).map (fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y) := hCoordinateLawDef htests φ x i _ = (ProbabilityTheory.stdGaussian E).map (fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y) := by rw [hNormalizedVectorLaw htests φ x] _ = ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal) := by let L : StrongDual Real E := InnerProductSpace.toDual Real E ((stdOrthonormalBasis Real E) i) change (ProbabilityTheory.stdGaussian E).map L = ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal) rw [ProbabilityTheory.IsGaussian.map_eq_gaussianReal L] rw [ProbabilityTheory.integral_strongDual_stdGaussian L] rw [ProbabilityTheory.variance_dual_stdGaussian L] simp [L] /-- Source coordinate-generator integral from the normalized Brownian law. This cycle-184 bridge narrows `hBrownianCoordinateGeneratorSourceIntegralDef`. The source work below it is the stochastic definition of the scalar Brownian coordinate generator as an integral under the normalized scalar-coordinate law, plus the already local Gaussian-coordinate/variance packaging fields from the frozen EM Brownian increment. It does not prove the scalar Taylor identity or the normalized-remainder domination package. -/ theorem selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfStdGaussianVectorLaw {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] [MeasurableSpace E] [BorelSpace E] (brownianCoordinateGenerator : Test → E → Fin (Module.finrank Real E) → Real) (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (normalizedVectorLaw : Test → E → MeasureTheory.Measure E) (normalizedCoordinateLaw : Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real) (sourceTaylorIntegrand : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hBrownianCoordinateGeneratorNormalizedLawDef : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ z : Real, sourceTaylorIntegrand φ x i z ∂ (normalizedCoordinateLaw φ x i)) (hNormalizedVectorLaw : testRegular → ∀ φ x, normalizedVectorLaw φ x = ProbabilityTheory.stdGaussian E) (hCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = (normalizedVectorLaw φ x).map (fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y)) (hVarianceDef : testRegular → ∀ φ x i, (variance φ x i : Real) = ProbabilityTheory.variance (id : Real → Real) (normalizedCoordinateLaw φ x i)) : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ z : Real, sourceTaylorIntegrand φ x i z ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := by intro htests φ x i have hNormalizedCoordinateLaw : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal) := selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw normalizedVectorLaw normalizedCoordinateLaw testRegular hNormalizedVectorLaw hCoordinateLawDef have hVarianceOne : testRegular → ∀ φ x i, variance φ x i = (1 : NNReal) := selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw variance normalizedCoordinateLaw testRegular hVarianceDef hNormalizedCoordinateLaw calc brownianCoordinateGenerator φ x i = ∫ z : Real, sourceTaylorIntegrand φ x i z ∂ (normalizedCoordinateLaw φ x i) := hBrownianCoordinateGeneratorNormalizedLawDef htests φ x i _ = ∫ z : Real, sourceTaylorIntegrand φ x i z ∂ (ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal)) := by rw [hNormalizedCoordinateLaw htests φ x i] _ = ∫ z : Real, sourceTaylorIntegrand φ x i z ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := by rw [hVarianceOne htests φ x i] /-- Remainder generator integral from the normalized Brownian coordinate law. This cycle-185 lower_2 bridge narrows `hRemainderGeneratorLimitDef`: once the source correspondence defines the normalized remainder contribution as an integral under the normalized scalar-coordinate law, the already compiled standard-Gaussian coordinate-law and variance-packaging bridges rewrite that integral under `gaussianReal 0 (variance phi x i)`. -/ theorem selectedWeakTestRemainderGeneratorLimitDefOfStdGaussianVectorLaw {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] [MeasurableSpace E] [BorelSpace E] (remainderGeneratorLimit : Test → E → Fin (Module.finrank Real E) → Real) (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (normalizedVectorLaw : Test → E → MeasureTheory.Measure E) (normalizedCoordinateLaw : Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real) (normalizedRemainder : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hRemainderGeneratorNormalizedLawDef : testRegular → ∀ φ x i, remainderGeneratorLimit φ x i = ∫ z : Real, normalizedRemainder φ x i z ∂ (normalizedCoordinateLaw φ x i)) (hNormalizedVectorLaw : testRegular → ∀ φ x, normalizedVectorLaw φ x = ProbabilityTheory.stdGaussian E) (hCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = (normalizedVectorLaw φ x).map (fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y)) (hVarianceDef : testRegular → ∀ φ x i, (variance φ x i : Real) = ProbabilityTheory.variance (id : Real → Real) (normalizedCoordinateLaw φ x i)) : testRegular → ∀ φ x i, remainderGeneratorLimit φ x i = ∫ z : Real, normalizedRemainder φ x i z ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := by intro htests φ x i have hNormalizedCoordinateLaw : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal) := selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw normalizedVectorLaw normalizedCoordinateLaw testRegular hNormalizedVectorLaw hCoordinateLawDef have hVarianceOne : testRegular → ∀ φ x i, variance φ x i = (1 : NNReal) := selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw variance normalizedCoordinateLaw testRegular hVarianceDef hNormalizedCoordinateLaw calc remainderGeneratorLimit φ x i = ∫ z : Real, normalizedRemainder φ x i z ∂ (normalizedCoordinateLaw φ x i) := hRemainderGeneratorNormalizedLawDef htests φ x i _ = ∫ z : Real, normalizedRemainder φ x i z ∂ (ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal)) := by rw [hNormalizedCoordinateLaw htests φ x i] _ = ∫ z : Real, normalizedRemainder φ x i z ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := by rw [hVarianceOne htests φ x i] /-- Remainder measurability transported from the normalized Brownian coordinate law. This cycle-194 lower_2 bridge discharges the downstream `hRemainderMeas` shape once the source correspondence has supplied measurability under the normalized scalar-coordinate law. The law transport uses only the already compiled standard-Gaussian coordinate-law bridge and the variance packaging bridge; domination and integrability remain separate leaves. -/ theorem selectedWeakTestRemainderMeasOfStdGaussianVectorLaw {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] [MeasurableSpace E] [BorelSpace E] (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (normalizedVectorLaw : Test → E → MeasureTheory.Measure E) (normalizedCoordinateLaw : Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real) (normalizedRemainder : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hNormalizedRemainderMeas : testRegular → ∀ φ x i, MeasureTheory.AEStronglyMeasurable (normalizedRemainder φ x i) (normalizedCoordinateLaw φ x i)) (hNormalizedVectorLaw : testRegular → ∀ φ x, normalizedVectorLaw φ x = ProbabilityTheory.stdGaussian E) (hCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = (normalizedVectorLaw φ x).map (fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y)) (hVarianceDef : testRegular → ∀ φ x i, (variance φ x i : Real) = ProbabilityTheory.variance (id : Real → Real) (normalizedCoordinateLaw φ x i)) : testRegular → ∀ φ x i, MeasureTheory.AEStronglyMeasurable (fun z : Real => normalizedRemainder φ x i z) (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := by intro htests φ x i have hNormalizedCoordinateLaw : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal) := selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw normalizedVectorLaw normalizedCoordinateLaw testRegular hNormalizedVectorLaw hCoordinateLawDef have hVarianceOne : testRegular → ∀ φ x i, variance φ x i = (1 : NNReal) := selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw variance normalizedCoordinateLaw testRegular hVarianceDef hNormalizedCoordinateLaw simpa [hNormalizedCoordinateLaw htests φ x i, hVarianceOne htests φ x i] using hNormalizedRemainderMeas htests φ x i /-- Remainder domination transported from the normalized Brownian coordinate law. This cycle-194 bridge narrows the downstream `hRemainderBound` leaf to the same source-side normalized scalar-coordinate law used for the remainder measurability bridge. It only rewrites the ambient measure using the compiled standard-Gaussian coordinate-law bridge and the variance packaging bridge; integrability of the dominating bound remains a separate leaf. -/ theorem selectedWeakTestRemainderBoundOfStdGaussianVectorLaw {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] [MeasurableSpace E] [BorelSpace E] (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (normalizedVectorLaw : Test → E → MeasureTheory.Measure E) (normalizedCoordinateLaw : Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real) (normalizedRemainder remainderBound : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hNormalizedRemainderBound : testRegular → ∀ φ x i, ∀ᵐ z ∂normalizedCoordinateLaw φ x i, ‖normalizedRemainder φ x i z‖ ≤ remainderBound φ x i z) (hNormalizedVectorLaw : testRegular → ∀ φ x, normalizedVectorLaw φ x = ProbabilityTheory.stdGaussian E) (hCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = (normalizedVectorLaw φ x).map (fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y)) (hVarianceDef : testRegular → ∀ φ x i, (variance φ x i : Real) = ProbabilityTheory.variance (id : Real → Real) (normalizedCoordinateLaw φ x i)) : testRegular → ∀ φ x i, ∀ᵐ z ∂ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i), ‖normalizedRemainder φ x i z‖ ≤ remainderBound φ x i z := by intro htests φ x i have hNormalizedCoordinateLaw : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal) := selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw normalizedVectorLaw normalizedCoordinateLaw testRegular hNormalizedVectorLaw hCoordinateLawDef have hVarianceOne : testRegular → ∀ φ x i, variance φ x i = (1 : NNReal) := selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw variance normalizedCoordinateLaw testRegular hVarianceDef hNormalizedCoordinateLaw simpa [hNormalizedCoordinateLaw htests φ x i, hVarianceOne htests φ x i] using hNormalizedRemainderBound htests φ x i /-- Remainder-bound integrability transported from the normalized Brownian coordinate law. This cycle-195 bridge narrows the downstream `hRemainderBoundInt` leaf to the same source-side normalized scalar-coordinate law used for the cycle-194 measurability and domination transports. It only rewrites the ambient measure using the compiled standard-Gaussian coordinate-law bridge and variance packaging bridge; it does not prove the concrete Taylor domination bound. -/ theorem selectedWeakTestRemainderBoundIntegrableOfStdGaussianVectorLaw {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] [MeasurableSpace E] [BorelSpace E] (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (normalizedVectorLaw : Test → E → MeasureTheory.Measure E) (normalizedCoordinateLaw : Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real) (remainderBound : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hNormalizedRemainderBoundInt : testRegular → ∀ φ x i, MeasureTheory.Integrable (fun z : Real => remainderBound φ x i z) (normalizedCoordinateLaw φ x i)) (hNormalizedVectorLaw : testRegular → ∀ φ x, normalizedVectorLaw φ x = ProbabilityTheory.stdGaussian E) (hCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = (normalizedVectorLaw φ x).map (fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y)) (hVarianceDef : testRegular → ∀ φ x i, (variance φ x i : Real) = ProbabilityTheory.variance (id : Real → Real) (normalizedCoordinateLaw φ x i)) : testRegular → ∀ φ x i, MeasureTheory.Integrable (fun z : Real => remainderBound φ x i z) (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := by intro htests φ x i have hNormalizedCoordinateLaw : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal) := selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw normalizedVectorLaw normalizedCoordinateLaw testRegular hNormalizedVectorLaw hCoordinateLawDef have hVarianceOne : testRegular → ∀ φ x i, variance φ x i = (1 : NNReal) := selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw variance normalizedCoordinateLaw testRegular hVarianceDef hNormalizedCoordinateLaw simpa [hNormalizedCoordinateLaw htests φ x i, hVarianceOne htests φ x i] using hNormalizedRemainderBoundInt htests φ x i /-- Normalized-law coordinate generator from a scalar pushforward law. This cycle-184 lower_2 bridge narrows the source-facing `hBrownianCoordinateGeneratorNormalizedLawDef`: once the frozen-interpolation source correspondence supplies the scalar Brownian coordinate as a measurable random variable, defines the normalized scalar law as its pushforward, and defines the coordinate generator as the sample-space expectation of the source Taylor integrand, the law-space integral follows by `MeasureTheory.integral_map`. It does not identify that law with a Gaussian and does not prove the scalar Taylor or normalized-remainder leaves. -/ theorem selectedWeakTestBrownianCoordinateGeneratorNormalizedLawDefOfScalarPushforward {Ω Test E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (scalarBrownianCoordinate : Test → E → Fin (Module.finrank Real E) → Ω → Real) (brownianCoordinateGenerator : Test → E → Fin (Module.finrank Real E) → Real) (normalizedCoordinateLaw : Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real) (sourceTaylorIntegrand : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hScalarMeas : testRegular → ∀ φ x i, AEMeasurable (scalarBrownianCoordinate φ x i) P) (hNormalizedCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = MeasureTheory.Measure.map (scalarBrownianCoordinate φ x i) P) (hSourceTaylorIntegrandMeas : testRegular → ∀ φ x i, MeasureTheory.AEStronglyMeasurable (sourceTaylorIntegrand φ x i) (normalizedCoordinateLaw φ x i)) (hGeneratorPullbackDef : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ ω : Ω, sourceTaylorIntegrand φ x i (scalarBrownianCoordinate φ x i ω) ∂P) : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ z : Real, sourceTaylorIntegrand φ x i z ∂ (normalizedCoordinateLaw φ x i) := by intro htests φ x i have hfieldMap : MeasureTheory.AEStronglyMeasurable (sourceTaylorIntegrand φ x i) (MeasureTheory.Measure.map (scalarBrownianCoordinate φ x i) P) := by simpa [hNormalizedCoordinateLawDef htests φ x i] using hSourceTaylorIntegrandMeas htests φ x i calc brownianCoordinateGenerator φ x i = ∫ ω : Ω, sourceTaylorIntegrand φ x i (scalarBrownianCoordinate φ x i ω) ∂P := hGeneratorPullbackDef htests φ x i _ = ∫ z : Real, sourceTaylorIntegrand φ x i z ∂ (normalizedCoordinateLaw φ x i) := by rw [hNormalizedCoordinateLawDef htests φ x i] exact (MeasureTheory.integral_map (hScalarMeas htests φ x i) hfieldMap).symm /-- Source coordinate-generator integral from scalar pushforward and Gaussian coordinate law. This cycle-191 bridge discharges the supplied `hBrownianCoordinateGeneratorNormalizedLawDef` field from the source-facing `hBrownianCoordinateGeneratorSourceIntegralDef` backend. It first derives the normalized-law coordinate-generator integral from the scalar Brownian coordinate pushforward, then reuses the standard-Gaussian vector coordinate-law bridge and variance packaging. -/ theorem selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfScalarPushforwardAndStdGaussianVectorLaw {Ω Test E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] [MeasurableSpace E] [BorelSpace E] (P : MeasureTheory.Measure Ω) (scalarBrownianCoordinate : Test → E → Fin (Module.finrank Real E) → Ω → Real) (brownianCoordinateGenerator : Test → E → Fin (Module.finrank Real E) → Real) (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (normalizedVectorLaw : Test → E → MeasureTheory.Measure E) (normalizedCoordinateLaw : Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real) (sourceTaylorIntegrand : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hScalarMeas : testRegular → ∀ φ x i, AEMeasurable (scalarBrownianCoordinate φ x i) P) (hNormalizedCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = MeasureTheory.Measure.map (scalarBrownianCoordinate φ x i) P) (hSourceTaylorIntegrandMeas : testRegular → ∀ φ x i, MeasureTheory.AEStronglyMeasurable (sourceTaylorIntegrand φ x i) (normalizedCoordinateLaw φ x i)) (hGeneratorPullbackDef : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ ω : Ω, sourceTaylorIntegrand φ x i (scalarBrownianCoordinate φ x i ω) ∂P) (hNormalizedVectorLaw : testRegular → ∀ φ x, normalizedVectorLaw φ x = ProbabilityTheory.stdGaussian E) (hCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = (normalizedVectorLaw φ x).map (fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y)) (hVarianceDef : testRegular → ∀ φ x i, (variance φ x i : Real) = ProbabilityTheory.variance (id : Real → Real) (normalizedCoordinateLaw φ x i)) : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ z : Real, sourceTaylorIntegrand φ x i z ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := by exact selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfStdGaussianVectorLaw brownianCoordinateGenerator variance normalizedVectorLaw normalizedCoordinateLaw sourceTaylorIntegrand testRegular (selectedWeakTestBrownianCoordinateGeneratorNormalizedLawDefOfScalarPushforward P scalarBrownianCoordinate brownianCoordinateGenerator normalizedCoordinateLaw sourceTaylorIntegrand testRegular hScalarMeas hNormalizedCoordinateLawDef hSourceTaylorIntegrandMeas hGeneratorPullbackDef) hNormalizedVectorLaw hCoordinateLawDef hVarianceDef /-- Brownian coordinate Taylor integral from scalar pushforward and raw Taylor fields. This cycle-192 bridge discharges the supplied `hBrownianCoordinateGeneratorSourceIntegralDef` field from the `hBrownianCoordinateGeneratorTaylorIntegralDef` backend. It first derives the source-integral generator identity from scalar pushforward plus the standard-Gaussian vector-law bridge, then reuses the raw selected-line Taylor term bridge from cycle 189. It does not prove the scalar pushforward fields, raw Taylor split, coefficient conventions, remainder domination, or selected weak-test Hessian regularity. -/ theorem selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfScalarPushforwardRawTaylorAndTermDefs {Ω Test E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] [MeasurableSpace E] [BorelSpace E] (P : MeasureTheory.Measure Ω) (selectedTest : Test → E → Real) (scalarBrownianCoordinate : Test → E → Fin (Module.finrank Real E) → Ω → Real) (brownianCoordinateGenerator : Test → E → Fin (Module.finrank Real E) → Real) (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (normalizedVectorLaw : Test → E → MeasureTheory.Measure E) (normalizedCoordinateLaw : Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real) (linearCoeff quadraticCoeff : Test → E → Fin (Module.finrank Real E) → Real) (sourceLinearTerm sourceQuadraticTerm normalizedRemainder sourceTaylorIntegrand : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hScalarMeas : testRegular → ∀ φ x i, AEMeasurable (scalarBrownianCoordinate φ x i) P) (hNormalizedCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = MeasureTheory.Measure.map (scalarBrownianCoordinate φ x i) P) (hSourceTaylorIntegrandMeas : testRegular → ∀ φ x i, MeasureTheory.AEStronglyMeasurable (sourceTaylorIntegrand φ x i) (normalizedCoordinateLaw φ x i)) (hGeneratorPullbackDef : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ ω : Ω, sourceTaylorIntegrand φ x i (scalarBrownianCoordinate φ x i ω) ∂P) (hNormalizedVectorLaw : testRegular → ∀ φ x, normalizedVectorLaw φ x = ProbabilityTheory.stdGaussian E) (hCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = (normalizedVectorLaw φ x).map (fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y)) (hVarianceDef : testRegular → ∀ φ x i, (variance φ x i : Real) = ProbabilityTheory.variance (id : Real → Real) (normalizedCoordinateLaw φ x i)) (hSourceTaylorIntegrandRawDef : testRegular → ∀ φ x i z, sourceTaylorIntegrand φ x i z = selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x) (hSelectedLineTaylorRawSplitDef : testRegular → ∀ φ x i z, selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x = deriv (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0 * z + ((2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0) * z ^ 2 + normalizedRemainder φ x i z) (hSourceLinearTermTaylorDef : testRegular → ∀ φ x i z, sourceLinearTerm φ x i z = deriv (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0 * z) (hScalarLineFirstCoeffDef : testRegular → ∀ φ x i, linearCoeff φ x i = deriv (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0) (hSourceQuadraticTermTaylorDef : testRegular → ∀ φ x i z, sourceQuadraticTerm φ x i z = ((2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0) * z ^ 2) (hScalarLineTaylorCoeffDef : testRegular → ∀ φ x i, quadraticCoeff φ x i = (2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0) : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ z : Real, (linearCoeff φ x i * z + quadraticCoeff φ x i * z ^ 2 + normalizedRemainder φ x i z) ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := by have hSourceIntegral : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ z : Real, sourceTaylorIntegrand φ x i z ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfScalarPushforwardAndStdGaussianVectorLaw P scalarBrownianCoordinate brownianCoordinateGenerator variance normalizedVectorLaw normalizedCoordinateLaw sourceTaylorIntegrand testRegular hScalarMeas hNormalizedCoordinateLawDef hSourceTaylorIntegrandMeas hGeneratorPullbackDef hNormalizedVectorLaw hCoordinateLawDef hVarianceDef exact selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralRawTaylorAndTermDefs selectedTest brownianCoordinateGenerator variance linearCoeff quadraticCoeff sourceLinearTerm sourceQuadraticTerm normalizedRemainder sourceTaylorIntegrand testRegular hSourceIntegral hSourceTaylorIntegrandRawDef hSelectedLineTaylorRawSplitDef hSourceLinearTermTaylorDef hScalarLineFirstCoeffDef hSourceQuadraticTermTaylorDef hScalarLineTaylorCoeffDef /-- Normalized-law remainder generator from a scalar pushforward law. This cycle-185 lower_2 bridge narrows the source-facing `hRemainderGeneratorNormalizedLawDef`: once the frozen-interpolation source correspondence supplies the scalar Brownian coordinate as a measurable random variable, defines the normalized scalar law as its pushforward, and defines the remainder contribution as the sample-space expectation of the normalized Taylor remainder, the law-space integral follows by `MeasureTheory.integral_map`. It does not identify that law with a Gaussian and does not prove the domination/DCT remainder package. -/ theorem selectedWeakTestRemainderGeneratorNormalizedLawDefOfScalarPushforward {Ω Test E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (scalarBrownianCoordinate : Test → E → Fin (Module.finrank Real E) → Ω → Real) (remainderGeneratorLimit : Test → E → Fin (Module.finrank Real E) → Real) (normalizedCoordinateLaw : Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real) (normalizedRemainder : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hScalarMeas : testRegular → ∀ φ x i, AEMeasurable (scalarBrownianCoordinate φ x i) P) (hNormalizedCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = MeasureTheory.Measure.map (scalarBrownianCoordinate φ x i) P) (hNormalizedRemainderMeas : testRegular → ∀ φ x i, MeasureTheory.AEStronglyMeasurable (normalizedRemainder φ x i) (normalizedCoordinateLaw φ x i)) (hRemainderPullbackDef : testRegular → ∀ φ x i, remainderGeneratorLimit φ x i = ∫ ω : Ω, normalizedRemainder φ x i (scalarBrownianCoordinate φ x i ω) ∂P) : testRegular → ∀ φ x i, remainderGeneratorLimit φ x i = ∫ z : Real, normalizedRemainder φ x i z ∂ (normalizedCoordinateLaw φ x i) := by intro htests φ x i have hfieldMap : MeasureTheory.AEStronglyMeasurable (normalizedRemainder φ x i) (MeasureTheory.Measure.map (scalarBrownianCoordinate φ x i) P) := by simpa [hNormalizedCoordinateLawDef htests φ x i] using hNormalizedRemainderMeas htests φ x i calc remainderGeneratorLimit φ x i = ∫ ω : Ω, normalizedRemainder φ x i (scalarBrownianCoordinate φ x i ω) ∂P := hRemainderPullbackDef htests φ x i _ = ∫ z : Real, normalizedRemainder φ x i z ∂ (normalizedCoordinateLaw φ x i) := by rw [hNormalizedCoordinateLawDef htests φ x i] exact (MeasureTheory.integral_map (hScalarMeas htests φ x i) hfieldMap).symm /-- Remainder generator integral from scalar pushforward and Gaussian coordinate law. This cycle-190 lower_2 bridge discharges the supplied `hRemainderGeneratorNormalizedLawDef` field from the source-facing `hRemainderGeneratorLimitDef` backend. It first derives the normalized-law remainder integral from the scalar Brownian coordinate pushforward, then reuses the standard-Gaussian vector coordinate-law bridge and variance packaging. -/ theorem selectedWeakTestRemainderGeneratorLimitDefOfScalarPushforwardAndStdGaussianVectorLaw {Ω Test E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] [MeasurableSpace E] [BorelSpace E] (P : MeasureTheory.Measure Ω) (scalarBrownianCoordinate : Test → E → Fin (Module.finrank Real E) → Ω → Real) (remainderGeneratorLimit : Test → E → Fin (Module.finrank Real E) → Real) (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (normalizedVectorLaw : Test → E → MeasureTheory.Measure E) (normalizedCoordinateLaw : Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real) (normalizedRemainder : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hScalarMeas : testRegular → ∀ φ x i, AEMeasurable (scalarBrownianCoordinate φ x i) P) (hNormalizedCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = MeasureTheory.Measure.map (scalarBrownianCoordinate φ x i) P) (hNormalizedRemainderMeas : testRegular → ∀ φ x i, MeasureTheory.AEStronglyMeasurable (normalizedRemainder φ x i) (normalizedCoordinateLaw φ x i)) (hRemainderPullbackDef : testRegular → ∀ φ x i, remainderGeneratorLimit φ x i = ∫ ω : Ω, normalizedRemainder φ x i (scalarBrownianCoordinate φ x i ω) ∂P) (hNormalizedVectorLaw : testRegular → ∀ φ x, normalizedVectorLaw φ x = ProbabilityTheory.stdGaussian E) (hCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = (normalizedVectorLaw φ x).map (fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y)) (hVarianceDef : testRegular → ∀ φ x i, (variance φ x i : Real) = ProbabilityTheory.variance (id : Real → Real) (normalizedCoordinateLaw φ x i)) : testRegular → ∀ φ x i, remainderGeneratorLimit φ x i = ∫ z : Real, normalizedRemainder φ x i z ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := by exact selectedWeakTestRemainderGeneratorLimitDefOfStdGaussianVectorLaw remainderGeneratorLimit variance normalizedVectorLaw normalizedCoordinateLaw normalizedRemainder testRegular (selectedWeakTestRemainderGeneratorNormalizedLawDefOfScalarPushforward P scalarBrownianCoordinate remainderGeneratorLimit normalizedCoordinateLaw normalizedRemainder testRegular hScalarMeas hNormalizedCoordinateLawDef hNormalizedRemainderMeas hRemainderPullbackDef) hNormalizedVectorLaw hCoordinateLawDef hVarianceDef /-- Taylor moment split from an explicit Taylor integral and scalar-pushforward remainder law. This cycle-193 lower_2 bridge removes only the primitive `hRemainderGeneratorLimitDef` supplied hypothesis from the dominated-remainder Taylor moment consumer. The Taylor integral identity for the Brownian coordinate generator remains explicit; the remainder integral definition is derived from the scalar Brownian coordinate pushforward, normalized-remainder pullback, and standard-Gaussian coordinate-law fields. -/ theorem selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsDominatedRemainderAndRemainderLimitScalarPushforward {Ω Test E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] [MeasurableSpace E] [BorelSpace E] (P : MeasureTheory.Measure Ω) (scalarBrownianCoordinate : Test → E → Fin (Module.finrank Real E) → Ω → Real) (brownianCoordinateGenerator : Test → E → Fin (Module.finrank Real E) → Real) (remainderGeneratorLimit : Test → E → Fin (Module.finrank Real E) → Real) (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (normalizedVectorLaw : Test → E → MeasureTheory.Measure E) (normalizedCoordinateLaw : Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real) (linearCoeff quadraticCoeff : Test → E → Fin (Module.finrank Real E) → Real) (normalizedRemainder remainderBound : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hBrownianCoordinateGeneratorTaylorIntegralDef : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ z : Real, (linearCoeff φ x i * z + quadraticCoeff φ x i * z ^ 2 + normalizedRemainder φ x i z) ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) (hScalarMeas : testRegular → ∀ φ x i, AEMeasurable (scalarBrownianCoordinate φ x i) P) (hNormalizedCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = MeasureTheory.Measure.map (scalarBrownianCoordinate φ x i) P) (hNormalizedRemainderMeas : testRegular → ∀ φ x i, MeasureTheory.AEStronglyMeasurable (normalizedRemainder φ x i) (normalizedCoordinateLaw φ x i)) (hRemainderPullbackDef : testRegular → ∀ φ x i, remainderGeneratorLimit φ x i = ∫ ω : Ω, normalizedRemainder φ x i (scalarBrownianCoordinate φ x i ω) ∂P) (hNormalizedVectorLaw : testRegular → ∀ φ x, normalizedVectorLaw φ x = ProbabilityTheory.stdGaussian E) (hCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = (normalizedVectorLaw φ x).map (fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y)) (hVarianceDef : testRegular → ∀ φ x i, (variance φ x i : Real) = ProbabilityTheory.variance (id : Real → Real) (normalizedCoordinateLaw φ x i)) (hRemainderMeas : testRegular → ∀ φ x i, MeasureTheory.AEStronglyMeasurable (fun z : Real => normalizedRemainder φ x i z) (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) (hRemainderBound : testRegular → ∀ φ x i, ∀ᵐ z ∂ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i), ‖normalizedRemainder φ x i z‖ ≤ remainderBound φ x i z) (hRemainderBoundInt : testRegular → ∀ φ x i, MeasureTheory.Integrable (fun z : Real => remainderBound φ x i z) (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = linearCoeff φ x i * (∫ z : Real, z ∂(ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) + quadraticCoeff φ x i * (∫ z : Real, z ^ 2 ∂(ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) + remainderGeneratorLimit φ x i := by have hRemainderDef : testRegular → ∀ φ x i, remainderGeneratorLimit φ x i = ∫ z : Real, normalizedRemainder φ x i z ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := selectedWeakTestRemainderGeneratorLimitDefOfScalarPushforwardAndStdGaussianVectorLaw P scalarBrownianCoordinate remainderGeneratorLimit variance normalizedVectorLaw normalizedCoordinateLaw normalizedRemainder testRegular hScalarMeas hNormalizedCoordinateLawDef hNormalizedRemainderMeas hRemainderPullbackDef hNormalizedVectorLaw hCoordinateLawDef hVarianceDef exact selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndDominatedRemainder brownianCoordinateGenerator variance linearCoeff quadraticCoeff remainderGeneratorLimit normalizedRemainder remainderBound testRegular hBrownianCoordinateGeneratorTaylorIntegralDef hRemainderMeas hRemainderBound hRemainderBoundInt hRemainderDef /-- Taylor moment decomposition from scalar pushforward, raw Taylor fields, and dominated remainder. This cycle-193 bridge removes the primitive `hBrownianCoordinateGeneratorTaylorIntegralDef` and `hRemainderGeneratorLimitDef` supplied hypotheses from the Taylor moment split. It composes the already compiled scalar-pushforward Taylor-integral bridge, the scalar-pushforward remainder-limit bridge, and the dominated remainder integrability bridge. The source-facing scalar law fields, raw selected-line Taylor fields, and domination fields remain explicit. -/ theorem selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfScalarPushforwardRawTaylorAndDominatedRemainder {Ω Test E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] [MeasurableSpace E] [BorelSpace E] (P : MeasureTheory.Measure Ω) (selectedTest : Test → E → Real) (scalarBrownianCoordinate : Test → E → Fin (Module.finrank Real E) → Ω → Real) (brownianCoordinateGenerator : Test → E → Fin (Module.finrank Real E) → Real) (remainderGeneratorLimit : Test → E → Fin (Module.finrank Real E) → Real) (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (normalizedVectorLaw : Test → E → MeasureTheory.Measure E) (normalizedCoordinateLaw : Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real) (linearCoeff quadraticCoeff : Test → E → Fin (Module.finrank Real E) → Real) (sourceLinearTerm sourceQuadraticTerm normalizedRemainder sourceTaylorIntegrand : Test → E → Fin (Module.finrank Real E) → Real → Real) (remainderBound : Test → E → Fin (Module.finrank Real E) → Real → Real) (testRegular : Prop) (hScalarMeas : testRegular → ∀ φ x i, AEMeasurable (scalarBrownianCoordinate φ x i) P) (hNormalizedCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = MeasureTheory.Measure.map (scalarBrownianCoordinate φ x i) P) (hSourceTaylorIntegrandMeas : testRegular → ∀ φ x i, MeasureTheory.AEStronglyMeasurable (sourceTaylorIntegrand φ x i) (normalizedCoordinateLaw φ x i)) (hGeneratorPullbackDef : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ ω : Ω, sourceTaylorIntegrand φ x i (scalarBrownianCoordinate φ x i ω) ∂P) (hNormalizedRemainderMeas : testRegular → ∀ φ x i, MeasureTheory.AEStronglyMeasurable (normalizedRemainder φ x i) (normalizedCoordinateLaw φ x i)) (hRemainderPullbackDef : testRegular → ∀ φ x i, remainderGeneratorLimit φ x i = ∫ ω : Ω, normalizedRemainder φ x i (scalarBrownianCoordinate φ x i ω) ∂P) (hNormalizedVectorLaw : testRegular → ∀ φ x, normalizedVectorLaw φ x = ProbabilityTheory.stdGaussian E) (hCoordinateLawDef : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = (normalizedVectorLaw φ x).map (fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y)) (hVarianceDef : testRegular → ∀ φ x i, (variance φ x i : Real) = ProbabilityTheory.variance (id : Real → Real) (normalizedCoordinateLaw φ x i)) (hSourceTaylorIntegrandRawDef : testRegular → ∀ φ x i z, sourceTaylorIntegrand φ x i z = selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x) (hSelectedLineTaylorRawSplitDef : testRegular → ∀ φ x i z, selectedTest φ (x + z • (stdOrthonormalBasis Real E i)) - selectedTest φ x = deriv (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0 * z + ((2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0) * z ^ 2 + normalizedRemainder φ x i z) (hSourceLinearTermTaylorDef : testRegular → ∀ φ x i z, sourceLinearTerm φ x i z = deriv (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0 * z) (hScalarLineFirstCoeffDef : testRegular → ∀ φ x i, linearCoeff φ x i = deriv (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0) (hSourceQuadraticTermTaylorDef : testRegular → ∀ φ x i z, sourceQuadraticTerm φ x i z = ((2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0) * z ^ 2) (hScalarLineTaylorCoeffDef : testRegular → ∀ φ x i, quadraticCoeff φ x i = (2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0) (hRemainderMeas : testRegular → ∀ φ x i, MeasureTheory.AEStronglyMeasurable (fun z : Real => normalizedRemainder φ x i z) (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) (hRemainderBound : testRegular → ∀ φ x i, ∀ᵐ z ∂ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i), ‖normalizedRemainder φ x i z‖ ≤ remainderBound φ x i z) (hRemainderBoundInt : testRegular → ∀ φ x i, MeasureTheory.Integrable (fun z : Real => remainderBound φ x i z) (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = linearCoeff φ x i * (∫ z : Real, z ∂(ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) + quadraticCoeff φ x i * (∫ z : Real, z ^ 2 ∂(ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) + remainderGeneratorLimit φ x i := by have hTaylorIntegral : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = ∫ z : Real, (linearCoeff φ x i * z + quadraticCoeff φ x i * z ^ 2 + normalizedRemainder φ x i z) ∂ (ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfScalarPushforwardRawTaylorAndTermDefs P selectedTest scalarBrownianCoordinate brownianCoordinateGenerator variance normalizedVectorLaw normalizedCoordinateLaw linearCoeff quadraticCoeff sourceLinearTerm sourceQuadraticTerm normalizedRemainder sourceTaylorIntegrand testRegular hScalarMeas hNormalizedCoordinateLawDef hSourceTaylorIntegrandMeas hGeneratorPullbackDef hNormalizedVectorLaw hCoordinateLawDef hVarianceDef hSourceTaylorIntegrandRawDef hSelectedLineTaylorRawSplitDef hSourceLinearTermTaylorDef hScalarLineFirstCoeffDef hSourceQuadraticTermTaylorDef hScalarLineTaylorCoeffDef exact selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsDominatedRemainderAndRemainderLimitScalarPushforward P scalarBrownianCoordinate brownianCoordinateGenerator remainderGeneratorLimit variance normalizedVectorLaw normalizedCoordinateLaw linearCoeff quadraticCoeff normalizedRemainder remainderBound testRegular hTaylorIntegral hScalarMeas hNormalizedCoordinateLawDef hNormalizedRemainderMeas hRemainderPullbackDef hNormalizedVectorLaw hCoordinateLawDef hVarianceDef hRemainderMeas hRemainderBound hRemainderBoundInt /-- Quadratic-variation normalization from source coefficient and variance fields. This cycle-174 lower_2 bridge is only the algebraic assembly below `hFrozenScalarBrownianItoQuadraticVariationNormalization`: once the source correspondence supplies the normalized quadratic coefficient and the unit variance for the scalar Brownian coordinate, the downstream normalization is a rewrite. The source-facing fields themselves remain explicit. -/ theorem selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (quadraticCoeff : Test → E → Fin (Module.finrank Real E) → Real) (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (testRegular : Prop) (hQuadraticCoeffDef : testRegular → ∀ φ x i, quadraticCoeff φ x i = emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest φ) x ((stdOrthonormalBasis Real E) i)) (hVarianceOne : testRegular → ∀ φ x i, (variance φ x i : Real) = 1) : testRegular → ∀ φ x i, quadraticCoeff φ x i * (variance φ x i : Real) = emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest φ) x ((stdOrthonormalBasis Real E) i) := by intro htests φ x i rw [hQuadraticCoeffDef htests φ x i, hVarianceOne htests φ x i, mul_one] /-- Quadratic-variation normalization from second Taylor and normalized variance fields. This cycle-176 lower_2 bridge composes the cycle-175 coefficient bridge with the normalized Brownian variance bridge. It removes the older intermediate `hQuadraticCoeffDef`/`hVarianceOne` pair from this local normalization step: the remaining source-facing fields are the diagonal second Taylor coefficient identity and the `NNReal` unit-variance definition for the normalized scalar Brownian coordinate. -/ theorem selectedWeakTestQuadraticVariationNormalizationOfSecondTaylorCoeffAndNormalizedVarianceDef {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (quadraticCoeff : Test → E → Fin (Module.finrank Real E) → Real) (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (testRegular : Prop) (hSecondTaylorCoeffDef : testRegular → ∀ φ x i, quadraticCoeff φ x i = iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) (hNormalizedVarianceDef : testRegular → ∀ φ x i, variance φ x i = (1 : NNReal)) : testRegular → ∀ φ x i, quadraticCoeff φ x i * (variance φ x i : Real) = emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest φ) x ((stdOrthonormalBasis Real E) i) := by have hQuadraticCoeffDef : testRegular → ∀ φ x i, quadraticCoeff φ x i = emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest φ) x ((stdOrthonormalBasis Real E) i) := selectedWeakTestQuadraticCoeffDefOfSecondTaylorCoeffDef selectedTest quadraticCoeff testRegular hSecondTaylorCoeffDef have hVarianceOne : testRegular → ∀ φ x i, (variance φ x i : Real) = 1 := selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef variance testRegular hNormalizedVarianceDef exact selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne selectedTest quadraticCoeff variance testRegular hQuadraticCoeffDef hVarianceOne /-- Dominated-convergence handoff for the normalized scalar Taylor remainder. This lower_2 cycle-162 theorem is the integral-limit block below `hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes`. It isolates the Mathlib dominated-convergence step from the later source Taylor proof that will instantiate `normalizedRemainder` with the selected one-dimensional Brownian coordinate path. -/ theorem gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT (v : NNReal) (normalizedRemainder : Real → Real → Real) (bound : Real → Real) (hMeas : ∀ᶠ h in 𝓝[Set.Ioi (0 : Real)] (0 : Real), MeasureTheory.AEStronglyMeasurable (normalizedRemainder h) (ProbabilityTheory.gaussianReal (0 : Real) v)) (hBound : ∀ᶠ h in 𝓝[Set.Ioi (0 : Real)] (0 : Real), ∀ᵐ z ∂ProbabilityTheory.gaussianReal (0 : Real) v, ‖normalizedRemainder h z‖ ≤ bound z) (hBoundInt : MeasureTheory.Integrable bound (ProbabilityTheory.gaussianReal (0 : Real) v)) (hPoint : ∀ᵐ z ∂ProbabilityTheory.gaussianReal (0 : Real) v, Filter.Tendsto (fun h => normalizedRemainder h z) (𝓝[Set.Ioi (0 : Real)] (0 : Real)) (𝓝 0)) : Filter.Tendsto (fun h => ∫ z : Real, normalizedRemainder h z ∂ProbabilityTheory.gaussianReal (0 : Real) v) (𝓝[Set.Ioi (0 : Real)] (0 : Real)) (𝓝 0) := by let μ := ProbabilityTheory.gaussianReal (0 : Real) v have hDct : Filter.Tendsto (fun h => ∫ z : Real, normalizedRemainder h z ∂μ) (𝓝[Set.Ioi (0 : Real)] (0 : Real)) (𝓝 (∫ z : Real, (0 : Real) ∂μ)) := by exact MeasureTheory.tendsto_integral_filter_of_dominated_convergence (μ := μ) (l := 𝓝[Set.Ioi (0 : Real)] (0 : Real)) (F := normalizedRemainder) (f := fun _ : Real => (0 : Real)) bound (by simpa [μ] using hMeas) (by simpa [μ] using hBound) (by simpa [μ] using hBoundInt) (by simpa [μ] using hPoint) simpa [μ] using hDct /-- Source-shaped scalar Taylor pointwise limit for the normalized remainder. Cycle 163 narrows the remaining `hPoint` input of `gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT`. For each fixed scalar Brownian coordinate `z`, Mathlib's one-dimensional `Real.taylor_tendsto` proves that the second-order remainder of the selected line `r ↦ sourceTest (x + r • e)` tends to zero after the source scaling `r = h * z`. This theorem only supplies the pointwise Taylor limit; the paper-specific Taylor moment decomposition, quadratic-variation normalization, and domination hypotheses remain explicit. -/ theorem gaussianRealSelectedTestLineSecondOrderTaylorRemainderPointwiseAE {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (v : NNReal) (sourceTest : E → Real) (x e : E) (hLine : ContDiffOn Real 2 (fun r : Real => sourceTest (x + r • e)) Set.univ) : ∀ᵐ z ∂ProbabilityTheory.gaussianReal (0 : Real) v, Filter.Tendsto (fun h : Real => ((sourceTest (x + (h * z) • e) - taylorWithinEval (fun r : Real => sourceTest (x + r • e)) 2 Set.univ 0 (h * z)) / (h * z) ^ 2) * z ^ 2) (𝓝[Set.Ioi (0 : Real)] (0 : Real)) (𝓝 0) := by refine Filter.Eventually.of_forall ?_ intro z let f : Real → Real := fun r => sourceTest (x + r • e) have hTaylor : Filter.Tendsto (fun r : Real => (f r - taylorWithinEval f 2 Set.univ 0 r) / r ^ 2) (𝓝 (0 : Real)) (𝓝 0) := by have h := Real.taylor_tendsto (f := f) (x₀ := (0 : Real)) (n := 2) (s := Set.univ) convex_univ (Set.mem_univ _) hLine simpa [nhdsWithin_univ, sub_zero] using h have hscale : Filter.Tendsto (fun h : Real => h * z) (𝓝[Set.Ioi (0 : Real)] (0 : Real)) (𝓝 (0 : Real)) := by have hid : Filter.Tendsto (fun h : Real => h) (𝓝[Set.Ioi (0 : Real)] (0 : Real)) (𝓝 (0 : Real)) := Filter.Tendsto.mono_left Filter.tendsto_id nhdsWithin_le_nhds simpa using hid.mul tendsto_const_nhds simpa [f] using (hTaylor.comp hscale).mul tendsto_const_nhds /-- DCT integral limit from identifying the paper remainder with the source line. This lower_1 scout bridge keeps the remaining analytic content at the source boundary. Once the paper's `normalizedRemainder` is eventually a.e. equal to the scaled selected-test line Taylor remainder, the compiled pointwise Taylor theorem supplies the `hPoint` input to the dominated-convergence theorem. Measurability, domination, and integrability remain explicit hypotheses. -/ theorem gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfSelectedTestLineEq {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (v : NNReal) (sourceTest : E → Real) (x e : E) (normalizedRemainder : Real → Real → Real) (bound : Real → Real) (hLine : ContDiffOn Real 2 (fun r : Real => sourceTest (x + r • e)) Set.univ) (hSourceEq : ∀ᵐ z ∂ProbabilityTheory.gaussianReal (0 : Real) v, (fun h : Real => normalizedRemainder h z) =ᶠ[ 𝓝[Set.Ioi (0 : Real)] (0 : Real)] fun h : Real => ((sourceTest (x + (h * z) • e) - taylorWithinEval (fun r : Real => sourceTest (x + r • e)) 2 Set.univ 0 (h * z)) / (h * z) ^ 2) * z ^ 2) (hMeas : ∀ᶠ h in 𝓝[Set.Ioi (0 : Real)] (0 : Real), MeasureTheory.AEStronglyMeasurable (normalizedRemainder h) (ProbabilityTheory.gaussianReal (0 : Real) v)) (hBound : ∀ᶠ h in 𝓝[Set.Ioi (0 : Real)] (0 : Real), ∀ᵐ z ∂ProbabilityTheory.gaussianReal (0 : Real) v, ‖normalizedRemainder h z‖ ≤ bound z) (hBoundInt : MeasureTheory.Integrable bound (ProbabilityTheory.gaussianReal (0 : Real) v)) : Filter.Tendsto (fun h => ∫ z : Real, normalizedRemainder h z ∂ProbabilityTheory.gaussianReal (0 : Real) v) (𝓝[Set.Ioi (0 : Real)] (0 : Real)) (𝓝 0) := by refine gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT v normalizedRemainder bound hMeas hBound hBoundInt ?_ have hPointSource := gaussianRealSelectedTestLineSecondOrderTaylorRemainderPointwiseAE v sourceTest x e hLine filter_upwards [hSourceEq, hPointSource] with z hzEq hzPoint exact hzPoint.congr' hzEq.symm /-- Source-shaped selected-test scalar normalized Taylor remainder. This is the concrete normalized remainder used by the paper's scalar Brownian/Ito Taylor line after writing the Brownian coordinate increment as `r = h * z`. It is intentionally local to the cycle-163 source-identification boundary and does not fold in the separate moment decomposition, quadratic variation normalization, or domination hypotheses. -/ noncomputable def gaussianRealSelectedTestLineSecondOrderNormalizedRemainder {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (x e : E) (h z : Real) : Real := ((sourceTest (x + (h * z) • e) - taylorWithinEval (fun r : Real => sourceTest (x + r • e)) 2 Set.univ 0 (h * z)) / (h * z) ^ 2) * z ^ 2 /-- The source-shaped selected-test remainder supplies the `hSourceEq` input. Cycle 163 lower_2 removes the source-equality placeholder for the concrete selected scalar line: once `normalizedRemainder` is the source-shaped remainder above, the eventual Gaussian-a.e. equality required by `gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfSelectedTestLineEq` is definitional. The remaining DCT inputs are only measurability, domination, and integrability for this concrete expression. -/ theorem gaussianRealSelectedTestLineSecondOrderNormalizedRemainderSourceEq {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (v : NNReal) (sourceTest : E → Real) (x e : E) : ∀ᵐ z ∂ProbabilityTheory.gaussianReal (0 : Real) v, (fun h : Real => gaussianRealSelectedTestLineSecondOrderNormalizedRemainder sourceTest x e h z) =ᶠ[ 𝓝[Set.Ioi (0 : Real)] (0 : Real)] fun h : Real => ((sourceTest (x + (h * z) • e) - taylorWithinEval (fun r : Real => sourceTest (x + r • e)) 2 Set.univ 0 (h * z)) / (h * z) ^ 2) * z ^ 2 := by refine Filter.Eventually.of_forall ?_ intro z refine Filter.Eventually.of_forall ?_ intro h rfl /-- DCT limit for the concrete selected-test normalized remainder. This lower_2 bridge discharges the `hSourceEq` placeholder from the lower_1 source-identification theorem for the actual selected scalar Taylor remainder. It narrows the remaining normalized-remainder boundary to the DCT measurability/domination/integrability package for this concrete source expression. -/ theorem gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (v : NNReal) (sourceTest : E → Real) (x e : E) (bound : Real → Real) (hLine : ContDiffOn Real 2 (fun r : Real => sourceTest (x + r • e)) Set.univ) (hMeas : ∀ᶠ h in 𝓝[Set.Ioi (0 : Real)] (0 : Real), MeasureTheory.AEStronglyMeasurable (gaussianRealSelectedTestLineSecondOrderNormalizedRemainder sourceTest x e h) (ProbabilityTheory.gaussianReal (0 : Real) v)) (hBound : ∀ᶠ h in 𝓝[Set.Ioi (0 : Real)] (0 : Real), ∀ᵐ z ∂ProbabilityTheory.gaussianReal (0 : Real) v, ‖gaussianRealSelectedTestLineSecondOrderNormalizedRemainder sourceTest x e h z‖ ≤ bound z) (hBoundInt : MeasureTheory.Integrable bound (ProbabilityTheory.gaussianReal (0 : Real) v)) : Filter.Tendsto (fun h => ∫ z : Real, gaussianRealSelectedTestLineSecondOrderNormalizedRemainder sourceTest x e h z ∂ProbabilityTheory.gaussianReal (0 : Real) v) (𝓝[Set.Ioi (0 : Real)] (0 : Real)) (𝓝 0) := by exact gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfSelectedTestLineEq v sourceTest x e (fun h z => gaussianRealSelectedTestLineSecondOrderNormalizedRemainder sourceTest x e h z) bound hLine (gaussianRealSelectedTestLineSecondOrderNormalizedRemainderSourceEq v sourceTest x e) hMeas hBound hBoundInt /-- Measurability of the concrete selected-test normalized Taylor remainder. Cycle 164 discharges the `hMeas` input for the source-shaped scalar remainder below `gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero`. The proof uses only the selected one-dimensional `ContDiffOn` line regularity, continuity of `taylorWithinEval` in the evaluation variable, and Mathlib closure of `AEStronglyMeasurable` under arithmetic operations. -/ theorem gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (v : NNReal) (sourceTest : E → Real) (x e : E) (hLine : ContDiffOn Real 2 (fun r : Real => sourceTest (x + r • e)) Set.univ) : ∀ᶠ h in 𝓝[Set.Ioi (0 : Real)] (0 : Real), MeasureTheory.AEStronglyMeasurable (gaussianRealSelectedTestLineSecondOrderNormalizedRemainder sourceTest x e h) (ProbabilityTheory.gaussianReal (0 : Real) v) := by refine Filter.Eventually.of_forall ?_ intro h let μ := ProbabilityTheory.gaussianReal (0 : Real) v change MeasureTheory.AEStronglyMeasurable (fun z : Real => ((sourceTest (x + (h * z) • e) - taylorWithinEval (fun r : Real => sourceTest (x + r • e)) 2 Set.univ 0 (h * z)) / (h * z) ^ 2) * z ^ 2) μ let f : Real → Real := fun r => sourceTest (x + r • e) have hfContOn : ContinuousOn f Set.univ := by simpa [f] using hLine.continuousOn have hfCont : Continuous f := by rw [continuous_iff_continuousAt] intro r change Filter.Tendsto f (𝓝 r) (𝓝 (f r)) exact (continuousWithinAt_univ f r).mp (hfContOn r (Set.mem_univ r)) have hz : MeasureTheory.AEStronglyMeasurable (fun z : Real => h * z) μ := by exact (continuous_const.mul continuous_id).aestronglyMeasurable have hSource : MeasureTheory.AEStronglyMeasurable (fun z : Real => sourceTest (x + (h * z) • e)) μ := by simpa [μ, f] using hfCont.comp_aestronglyMeasurable hz have hTaylorCont : Continuous (fun r : Real => taylorWithinEval f 2 Set.univ 0 r) := by rw [continuous_iff_continuousAt] intro r exact (hasDerivAt_taylorWithinEval_succ (f := f) (n := 1) (s := Set.univ) (x₀ := (0 : Real)) (x := r)).continuousAt have hTaylor : MeasureTheory.AEStronglyMeasurable (fun z : Real => taylorWithinEval f 2 Set.univ 0 (h * z)) μ := by exact hTaylorCont.comp_aestronglyMeasurable hz have hDenom : MeasureTheory.AEStronglyMeasurable (fun z : Real => (h * z) ^ 2) μ := by exact hz.pow 2 have hZsq : MeasureTheory.AEStronglyMeasurable (fun z : Real => z ^ 2) μ := by exact (continuous_id.aestronglyMeasurable (μ := μ)).pow 2 change MeasureTheory.AEStronglyMeasurable ((((fun z : Real => sourceTest (x + (h * z) • e)) - fun z => taylorWithinEval f 2 Set.univ 0 (h * z)) / fun z => (h * z) ^ 2) * fun z => z ^ 2) μ exact ((hSource.sub hTaylor).div₀ hDenom).mul hZsq /-- Split the selected scalar second-order Taylor quotient bound. Cycle 165 lower_2 narrows the deterministic `hTaylorQuotientBound` input for the concrete normalized-remainder domination theorem. It is enough to supply the first-order quadratic remainder estimate for the selected line together with a bound on the second Taylor coefficient; the remaining source-facing analytic work is proving that first-order estimate from the paper's selected test regularity. -/ theorem gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (x e : E) (C1 C2 C : Real) (hFirst : ∀ r : Real, ‖(sourceTest (x + r • e) - taylorWithinEval (fun r : Real => sourceTest (x + r • e)) 1 Set.univ 0 r) / r ^ 2‖ ≤ C1) (hSecondCoeff : ‖taylorCoeffWithin (fun r : Real => sourceTest (x + r • e)) 2 Set.univ 0‖ ≤ C2) (hCnonneg : 0 ≤ C) (hC : C1 + C2 ≤ C) : ∀ r : Real, ‖(sourceTest (x + r • e) - taylorWithinEval (fun r : Real => sourceTest (x + r • e)) 2 Set.univ 0 r) / r ^ 2‖ ≤ C := by intro r by_cases hr : r = 0 · subst r simpa using hCnonneg · let f : Real → Real := fun r : Real => sourceTest (x + r • e) change ‖(f r - taylorWithinEval f 2 Set.univ 0 r) / r ^ 2‖ ≤ C have hr2 : r ^ 2 ≠ 0 := pow_ne_zero 2 hr have hT2 : taylorWithinEval f 2 Set.univ 0 r = taylorWithinEval f 1 Set.univ 0 r + r ^ 2 * taylorCoeffWithin f 2 Set.univ 0 := by rw [taylorWithinEval_succ] simp [taylorCoeffWithin] ring have hquot : (f r - taylorWithinEval f 2 Set.univ 0 r) / r ^ 2 = (f r - taylorWithinEval f 1 Set.univ 0 r) / r ^ 2 - taylorCoeffWithin f 2 Set.univ 0 := by rw [hT2] field_simp [hr2] ring have htriangle : ‖(f r - taylorWithinEval f 2 Set.univ 0 r) / r ^ 2‖ ≤ ‖(f r - taylorWithinEval f 1 Set.univ 0 r) / r ^ 2‖ + ‖taylorCoeffWithin f 2 Set.univ 0‖ := by rw [hquot] exact norm_sub_le _ _ have hparts : ‖(f r - taylorWithinEval f 1 Set.univ 0 r) / r ^ 2‖ + ‖taylorCoeffWithin f 2 Set.univ 0‖ ≤ C1 + C2 := by exact add_le_add (by simpa [f] using hFirst r) (by simpa [f] using hSecondCoeff) exact htriangle.trans (hparts.trans hC) /-- Quadratic domination of the concrete selected-test normalized remainder. This cycle-165 bridge narrows the remaining DCT `hBound` input to the deterministic scalar Taylor quotient estimate for the selected one-dimensional line. If the quotient `(sourceTest (x + r • e) - taylorWithinEval ... r) / r^2` is uniformly bounded by `C`, then the source-shaped normalized remainder is eventually, hence a.e., dominated by the quadratic Gaussian bound `fun z => C * z^2`. -/ theorem gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (v : NNReal) (sourceTest : E → Real) (x e : E) (C : Real) (hTaylorQuotientBound : ∀ r : Real, ‖(sourceTest (x + r • e) - taylorWithinEval (fun r : Real => sourceTest (x + r • e)) 2 Set.univ 0 r) / r ^ 2‖ ≤ C) : ∀ᶠ h in 𝓝[Set.Ioi (0 : Real)] (0 : Real), ∀ᵐ z ∂ProbabilityTheory.gaussianReal (0 : Real) v, ‖gaussianRealSelectedTestLineSecondOrderNormalizedRemainder sourceTest x e h z‖ ≤ C * z ^ 2 := by refine Filter.Eventually.of_forall ?_ intro h refine Filter.Eventually.of_forall ?_ intro z have hquot := hTaylorQuotientBound (h * z) have hmul : ‖(sourceTest (x + (h * z) • e) - taylorWithinEval (fun r : Real => sourceTest (x + r • e)) 2 Set.univ 0 (h * z)) / (h * z) ^ 2‖ * ‖z ^ 2‖ ≤ C * ‖z ^ 2‖ := mul_le_mul_of_nonneg_right hquot (norm_nonneg (z ^ 2)) calc ‖gaussianRealSelectedTestLineSecondOrderNormalizedRemainder sourceTest x e h z‖ = ‖(sourceTest (x + (h * z) • e) - taylorWithinEval (fun r : Real => sourceTest (x + r • e)) 2 Set.univ 0 (h * z)) / (h * z) ^ 2‖ * ‖z ^ 2‖ := by simp [gaussianRealSelectedTestLineSecondOrderNormalizedRemainder, norm_mul] _ ≤ C * ‖z ^ 2‖ := hmul _ = C * z ^ 2 := by simp [Real.norm_eq_abs] /-- First-order selected-test quotient bound from a quadratic remainder bound. Cycle 166 narrows the `hFirst` input of `gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff`. The source-facing Taylor estimate is the non-quotient quadratic bound on the degree-one remainder. This lemma performs only the division by `r ^ 2` and the `r = 0` bookkeeping; the analytic proof of that quadratic remainder estimate from the paper's selected-test second-derivative bound remains separate. -/ theorem gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (x e : E) (C1 : Real) (hC1 : 0 ≤ C1) (hRemainder : ∀ r : Real, ‖sourceTest (x + r • e) - taylorWithinEval (fun r : Real => sourceTest (x + r • e)) 1 Set.univ 0 r‖ ≤ C1 * r ^ 2) : ∀ r : Real, ‖(sourceTest (x + r • e) - taylorWithinEval (fun r : Real => sourceTest (x + r • e)) 1 Set.univ 0 r) / r ^ 2‖ ≤ C1 := by intro r by_cases hr : r = 0 · subst r simpa using hC1 · let f : Real → Real := fun r : Real => sourceTest (x + r • e) change ‖(f r - taylorWithinEval f 1 Set.univ 0 r) / r ^ 2‖ ≤ C1 have hr2ne : r ^ 2 ≠ 0 := pow_ne_zero 2 hr have hr2nonneg : 0 ≤ r ^ 2 := sq_nonneg r have hnormdenom : ‖r ^ 2‖ = r ^ 2 := by simp [Real.norm_eq_abs] have hdiv : ‖(f r - taylorWithinEval f 1 Set.univ 0 r) / r ^ 2‖ = ‖f r - taylorWithinEval f 1 Set.univ 0 r‖ / r ^ 2 := by rw [norm_div, hnormdenom] have hrem : ‖f r - taylorWithinEval f 1 Set.univ 0 r‖ ≤ C1 * r ^ 2 := by simpa [f] using hRemainder r have hquot : ‖f r - taylorWithinEval f 1 Set.univ 0 r‖ / r ^ 2 ≤ (C1 * r ^ 2) / r ^ 2 := div_le_div_of_nonneg_right hrem hr2nonneg have hcancel : (C1 * r ^ 2) / r ^ 2 = C1 := by field_simp [hr2ne] rw [hdiv] exact hquot.trans_eq hcancel /-- Nonnegative-side first-order selected-line Taylor remainder from Mathlib Taylor. This is the source-facing positive half of the remaining cycle-166 `hFirst` boundary. Mathlib's `taylor_mean_remainder_bound` is interval-based, so the theorem keeps the required `Icc 0 r` regularity, second-derivative bound, and the compatibility rewrite from the interval Taylor polynomial to the `Set.univ` Taylor polynomial explicit. The negative/reflection half and the source proof of the compatibility hypothesis remain separate. -/ theorem gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (x e : E) (C1 r : Real) (hr : 0 ≤ r) (hCont : ContDiffOn Real 2 (fun q : Real => sourceTest (x + q • e)) (Set.Icc 0 r)) (hSecond : ∀ y ∈ Set.Icc (0 : Real) r, ‖iteratedDerivWithin 2 (fun q : Real => sourceTest (x + q • e)) (Set.Icc 0 r) y‖ ≤ C1) (hTaylorCompat : taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 (Set.Icc 0 r) 0 r = taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 Set.univ 0 r) : ‖sourceTest (x + r • e) - taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 Set.univ 0 r‖ ≤ C1 * r ^ 2 := by have hx : r ∈ Set.Icc (0 : Real) r := ⟨hr, le_rfl⟩ have hTaylor := taylor_mean_remainder_bound (f := fun q : Real => sourceTest (x + q • e)) (a := (0 : Real)) (b := r) (C := C1) (x := r) (n := 1) hr hCont hx hSecond rw [← hTaylorCompat] simpa [pow_two] using hTaylor /-- Signed first-order selected-line Taylor remainder from interval Taylor data. This lower_2 bridge combines the compiled nonnegative interval Taylor lemma with the reflected line `q ↦ sourceTest (x + q • (-e))` for negative `r`. The remaining source-facing work is now the signed interval regularity, second-derivative domination, and the Taylor-polynomial compatibility hypotheses; the all-`r` non-quotient quadratic remainder is no longer a primitive input. -/ theorem gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (x e : E) (C1 : Real) (hNonnegCont : ∀ r : Real, 0 ≤ r → ContDiffOn Real 2 (fun q : Real => sourceTest (x + q • e)) (Set.Icc 0 r)) (hNonnegSecond : ∀ r : Real, 0 ≤ r → ∀ y ∈ Set.Icc (0 : Real) r, ‖iteratedDerivWithin 2 (fun q : Real => sourceTest (x + q • e)) (Set.Icc 0 r) y‖ ≤ C1) (hNonnegTaylorCompat : ∀ r : Real, 0 ≤ r → taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 (Set.Icc 0 r) 0 r = taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 Set.univ 0 r) (hNegCont : ∀ u : Real, 0 ≤ u → ContDiffOn Real 2 (fun q : Real => sourceTest (x + q • (-e))) (Set.Icc 0 u)) (hNegSecond : ∀ u : Real, 0 ≤ u → ∀ y ∈ Set.Icc (0 : Real) u, ‖iteratedDerivWithin 2 (fun q : Real => sourceTest (x + q • (-e))) (Set.Icc 0 u) y‖ ≤ C1) (hNegTaylorCompat : ∀ u : Real, 0 ≤ u → taylorWithinEval (fun q : Real => sourceTest (x + q • (-e))) 1 (Set.Icc 0 u) 0 u = taylorWithinEval (fun q : Real => sourceTest (x + q • (-e))) 1 Set.univ 0 u) (hNegTaylorReflect : ∀ r : Real, r < 0 → taylorWithinEval (fun q : Real => sourceTest (x + q • (-e))) 1 Set.univ 0 (-r) = taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 Set.univ 0 r) : ∀ r : Real, ‖sourceTest (x + r • e) - taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 Set.univ 0 r‖ ≤ C1 * r ^ 2 := by intro r by_cases hr : 0 ≤ r · exact gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor (sourceTest := sourceTest) (x := x) (e := e) (C1 := C1) (r := r) hr (hNonnegCont r hr) (hNonnegSecond r hr) (hNonnegTaylorCompat r hr) · have hrlt : r < 0 := lt_of_not_ge hr have hu : 0 ≤ -r := neg_nonneg.mpr (le_of_lt hrlt) have hbound : ‖sourceTest (x + (-r) • (-e)) - taylorWithinEval (fun q : Real => sourceTest (x + q • (-e))) 1 Set.univ 0 (-r)‖ ≤ C1 * (-r) ^ 2 := gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor (sourceTest := sourceTest) (x := x) (e := -e) (C1 := C1) (r := -r) hu (hNegCont (-r) hu) (hNegSecond (-r) hu) (hNegTaylorCompat (-r) hu) have hline : x + (-r) • (-e) = x + r • e := by simp calc ‖sourceTest (x + r • e) - taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 Set.univ 0 r‖ = ‖sourceTest (x + (-r) • (-e)) - taylorWithinEval (fun q : Real => sourceTest (x + q • (-e))) 1 Set.univ 0 (-r)‖ := by rw [← hNegTaylorReflect r hrlt, hline] _ ≤ C1 * (-r) ^ 2 := hbound _ = C1 * r ^ 2 := by ring /-- Reflection compatibility for the first-order selected-line Taylor polynomial on `Set.univ`. The negative-side interval Taylor argument uses the line `q ↦ sourceTest (x + q • (-e))` at `-r`. For the first-order Taylor polynomial over `Set.univ`, this is definitionally the original selected line after the domain reflection `q ↦ -q`; Mathlib's `deriv_comp_neg` supplies the linear-coefficient sign change. -/ theorem gaussianRealSelectedTestLineFirstOrderTaylorReflectSetUniv {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (x e : E) (r : Real) : taylorWithinEval (fun q : Real => sourceTest (x + q • (-e))) 1 Set.univ 0 (-r) = taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 Set.univ 0 r := by let f : Real → Real := fun q : Real => sourceTest (x + q • e) have hfun : (fun q : Real => sourceTest (x + q • (-e))) = fun q : Real => f (-q) := by funext q simp [f] have hfun' : (fun q : Real => sourceTest (x + -(q • e))) = fun q : Real => f (-q) := by funext q simp [f] have hderiv : deriv (fun q : Real => sourceTest (x + -(q • e))) 0 = -deriv (fun q : Real => sourceTest (x + q • e)) 0 := by rw [hfun'] change deriv (fun q : Real => f (-q)) 0 = -deriv f 0 rw [deriv_comp_neg] simp rw [hfun] rw [taylorWithinEval_succ, taylorWithinEval_succ] simp [f] rw [hderiv] ring /-- First-order interval Taylor compatibility from differentiability at the base point. For the selected scalar line, the order-one Taylor polynomial on `Icc 0 r` agrees at the endpoint `r` with the `Set.univ` Taylor polynomial once the line is differentiable at the expansion point `0`. The case `r = 0` is purely algebraic; for `0 < r`, Mathlib's `uniqueDiffOn_Icc` identifies the one-sided derivative inside the interval with the ordinary derivative. -/ theorem gaussianRealSelectedTestLineFirstOrderTaylorCompatOfDifferentiableAt {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (x e : E) (r : Real) (hr : 0 ≤ r) (hDiff : DifferentiableAt Real (fun q : Real => sourceTest (x + q • e)) 0) : taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 (Set.Icc 0 r) 0 r = taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 Set.univ 0 r := by let f : Real → Real := fun q : Real => sourceTest (x + q • e) by_cases hr0 : r = 0 · subst r rw [taylorWithinEval_succ, taylorWithinEval_succ] simp · have hpos : 0 < r := lt_of_le_of_ne hr (Ne.symm hr0) have hmem : (0 : Real) ∈ Set.Icc (0 : Real) r := ⟨le_rfl, hr⟩ have hderiv : iteratedDerivWithin 1 f (Set.Icc (0 : Real) r) 0 = iteratedDerivWithin 1 f Set.univ 0 := by have hDiffF : DifferentiableAt Real f 0 := by simpa [f] using hDiff rw [iteratedDerivWithin_one, iteratedDerivWithin_one] rw [DifferentiableAt.derivWithin hDiffF ((uniqueDiffOn_Icc hpos).uniqueDiffWithinAt hmem)] simp [derivWithin_univ] rw [show (fun q : Real => sourceTest (x + q • e)) = f by rfl] rw [taylorWithinEval_succ, taylorWithinEval_succ] rw [hderiv] simp /-- Signed first-order selected-line Taylor remainder without a reflected Taylor premise. Cycle 167 removes the explicit `hNegTaylorReflect` input from the cycle-166 signed interval theorem. The only remaining negative-side Taylor data are the reflected interval regularity, reflected second-derivative domination, and the reflected interval-to-`Set.univ` compatibility; the `Set.univ` reflection compatibility follows from `gaussianRealSelectedTestLineFirstOrderTaylorReflectSetUniv`. -/ theorem gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (x e : E) (C1 : Real) (hNonnegCont : ∀ r : Real, 0 ≤ r → ContDiffOn Real 2 (fun q : Real => sourceTest (x + q • e)) (Set.Icc 0 r)) (hNonnegSecond : ∀ r : Real, 0 ≤ r → ∀ y ∈ Set.Icc (0 : Real) r, ‖iteratedDerivWithin 2 (fun q : Real => sourceTest (x + q • e)) (Set.Icc 0 r) y‖ ≤ C1) (hNonnegTaylorCompat : ∀ r : Real, 0 ≤ r → taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 (Set.Icc 0 r) 0 r = taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 Set.univ 0 r) (hNegCont : ∀ u : Real, 0 ≤ u → ContDiffOn Real 2 (fun q : Real => sourceTest (x + q • (-e))) (Set.Icc 0 u)) (hNegSecond : ∀ u : Real, 0 ≤ u → ∀ y ∈ Set.Icc (0 : Real) u, ‖iteratedDerivWithin 2 (fun q : Real => sourceTest (x + q • (-e))) (Set.Icc 0 u) y‖ ≤ C1) (hNegTaylorCompat : ∀ u : Real, 0 ≤ u → taylorWithinEval (fun q : Real => sourceTest (x + q • (-e))) 1 (Set.Icc 0 u) 0 u = taylorWithinEval (fun q : Real => sourceTest (x + q • (-e))) 1 Set.univ 0 u) : ∀ r : Real, ‖sourceTest (x + r • e) - taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 Set.univ 0 r‖ ≤ C1 * r ^ 2 := gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor (sourceTest := sourceTest) (x := x) (e := e) (C1 := C1) hNonnegCont hNonnegSecond hNonnegTaylorCompat hNegCont hNegSecond hNegTaylorCompat (fun r _hr => gaussianRealSelectedTestLineFirstOrderTaylorReflectSetUniv (sourceTest := sourceTest) (x := x) (e := e) r) /-- Signed first-order selected-line Taylor remainder from interval data and base differentiability. This narrows the cycle-167 signed interval boundary by discharging the interval-to-`Set.univ` Taylor-compatibility hypotheses from differentiability of the selected and reflected scalar lines at the expansion point. The remaining source-facing analytic data are the signed interval `ContDiffOn` facts and signed interval second-derivative domination. -/ theorem gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (x e : E) (C1 : Real) (hNonnegCont : ∀ r : Real, 0 ≤ r → ContDiffOn Real 2 (fun q : Real => sourceTest (x + q • e)) (Set.Icc 0 r)) (hNonnegSecond : ∀ r : Real, 0 ≤ r → ∀ y ∈ Set.Icc (0 : Real) r, ‖iteratedDerivWithin 2 (fun q : Real => sourceTest (x + q • e)) (Set.Icc 0 r) y‖ ≤ C1) (hNegCont : ∀ u : Real, 0 ≤ u → ContDiffOn Real 2 (fun q : Real => sourceTest (x + q • (-e))) (Set.Icc 0 u)) (hNegSecond : ∀ u : Real, 0 ≤ u → ∀ y ∈ Set.Icc (0 : Real) u, ‖iteratedDerivWithin 2 (fun q : Real => sourceTest (x + q • (-e))) (Set.Icc 0 u) y‖ ≤ C1) (hNonnegBaseDiff : DifferentiableAt Real (fun q : Real => sourceTest (x + q • e)) 0) (hNegBaseDiff : DifferentiableAt Real (fun q : Real => sourceTest (x + q • (-e))) 0) : ∀ r : Real, ‖sourceTest (x + r • e) - taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 Set.univ 0 r‖ ≤ C1 * r ^ 2 := gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect (sourceTest := sourceTest) (x := x) (e := e) (C1 := C1) hNonnegCont hNonnegSecond (fun r hr => gaussianRealSelectedTestLineFirstOrderTaylorCompatOfDifferentiableAt (sourceTest := sourceTest) (x := x) (e := e) r hr hNonnegBaseDiff) hNegCont hNegSecond (fun u hu => gaussianRealSelectedTestLineFirstOrderTaylorCompatOfDifferentiableAt (sourceTest := sourceTest) (x := x) (e := -e) u hu hNegBaseDiff) /-- Global selected-test regularity supplies global regularity of each scalar line. Cycle 168 narrows the `hLine` input exposed by the cycle-167 global-line Taylor bridge. If the selected source test is globally `C^2` on the ambient state space, then its affine scalar slice `q ↦ sourceTest (x + q • e)` is globally `C^2` on `Real`. The signed second-derivative domination hypotheses remain separate bounded-Hessian leaves. -/ theorem gaussianRealSelectedTestLineContDiffOnOfSourceContDiffOn {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (x e : E) (hSource : ContDiffOn Real 2 sourceTest Set.univ) : ContDiffOn Real 2 (fun q : Real => sourceTest (x + q • e)) Set.univ := by have hSourceGlobal : ContDiff Real 2 sourceTest := by simpa using contDiffOn_univ.1 hSource have hLineMap : ContDiff Real 2 (fun q : Real => x + q • e) := by exact contDiff_const.add (contDiff_id.smul_const e) exact contDiffOn_univ.2 (hSourceGlobal.comp hLineMap) /-- Signed first-order selected-line Taylor remainder from global line regularity. This lower_2 bridge removes the signed interval `ContDiffOn` inputs and the two base differentiability inputs from the lower_1 `BaseDiff` theorem. A single global `ContDiffOn Real 2` assumption for the selected scalar line supplies the nonnegative interval regularity, the reflected negative-line regularity by composing with `q ↦ -q`, and differentiability at the expansion point. The bounded-Hessian content is still kept in the two explicit interval second-derivative domination hypotheses. -/ theorem gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (x e : E) (C1 : Real) (hLine : ContDiffOn Real 2 (fun q : Real => sourceTest (x + q • e)) Set.univ) (hNonnegSecond : ∀ r : Real, 0 ≤ r → ∀ y ∈ Set.Icc (0 : Real) r, ‖iteratedDerivWithin 2 (fun q : Real => sourceTest (x + q • e)) (Set.Icc 0 r) y‖ ≤ C1) (hNegSecond : ∀ u : Real, 0 ≤ u → ∀ y ∈ Set.Icc (0 : Real) u, ‖iteratedDerivWithin 2 (fun q : Real => sourceTest (x + q • (-e))) (Set.Icc 0 u) y‖ ≤ C1) : ∀ r : Real, ‖sourceTest (x + r • e) - taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 Set.univ 0 r‖ ≤ C1 * r ^ 2 := by let f : Real → Real := fun q : Real => sourceTest (x + q • e) have hLineGlobal : ContDiff Real 2 f := by simpa [f] using contDiffOn_univ.1 hLine have hNegGlobal : ContDiffOn Real 2 (fun q : Real => sourceTest (x + q • (-e))) Set.univ := by have hReflect : ContDiff Real 2 (fun q : Real => f (-q)) := by change ContDiff Real 2 (f ∘ fun q : Real => -q) exact hLineGlobal.comp (contDiff_neg.comp contDiff_id) simpa [f] using contDiffOn_univ.2 hReflect have hNonnegBaseDiff : DifferentiableAt Real (fun q : Real => sourceTest (x + q • e)) 0 := by exact (hLine.contDiffAt (by simp)).differentiableAt (by norm_num) have hNegBaseDiff : DifferentiableAt Real (fun q : Real => sourceTest (x + q • (-e))) 0 := by exact (hNegGlobal.contDiffAt (by simp)).differentiableAt (by norm_num) exact gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff (sourceTest := sourceTest) (x := x) (e := e) (C1 := C1) (fun r _hr => hLine.mono (Set.subset_univ _)) hNonnegSecond (fun u _hu => hNegGlobal.mono (Set.subset_univ _)) hNegSecond hNonnegBaseDiff hNegBaseDiff /-- Signed first-order selected-line Taylor remainder from ambient source-test regularity. This cycle-168 bridge supplies the cycle-167 `hLine` input from the paper-facing global selected-test `C^2` hypothesis. It deliberately keeps the nonnegative and reflected interval second-derivative domination assumptions explicit, so the remaining bounded-Hessian route is not hidden in a broad wrapper. -/ theorem gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (x e : E) (C1 : Real) (hSource : ContDiffOn Real 2 sourceTest Set.univ) (hNonnegSecond : ∀ r : Real, 0 ≤ r → ∀ y ∈ Set.Icc (0 : Real) r, ‖iteratedDerivWithin 2 (fun q : Real => sourceTest (x + q • e)) (Set.Icc 0 r) y‖ ≤ C1) (hNegSecond : ∀ u : Real, 0 ≤ u → ∀ y ∈ Set.Icc (0 : Real) u, ‖iteratedDerivWithin 2 (fun q : Real => sourceTest (x + q • (-e))) (Set.Icc 0 u) y‖ ≤ C1) : ∀ r : Real, ‖sourceTest (x + r • e) - taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 Set.univ 0 r‖ ≤ C1 * r ^ 2 := gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff (sourceTest := sourceTest) (x := x) (e := e) (C1 := C1) (gaussianRealSelectedTestLineContDiffOnOfSourceContDiffOn (sourceTest := sourceTest) (x := x) (e := e) hSource) hNonnegSecond hNegSecond /-- Signed first-order selected-line Taylor remainder from global line second-derivative bounds. This lower_2 bridge narrows the signed interval second-derivative domination inputs left by the source-`ContDiffOn` bridge. Instead of requiring `iteratedDerivWithin 2` bounds separately on every signed interval, it derives those interval hypotheses from global selected and reflected scalar-line `iteratedDeriv 2` bounds. The remaining bounded-Hessian source work is the ambient chain-rule step producing the two global line bounds. -/ theorem gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (x e : E) (C1 : Real) (hSource : ContDiffOn Real 2 sourceTest Set.univ) (hC1 : 0 ≤ C1) (hLineSecond : ∀ y : Real, ‖iteratedDeriv 2 (fun q : Real => sourceTest (x + q • e)) y‖ ≤ C1) (hNegLineSecond : ∀ y : Real, ‖iteratedDeriv 2 (fun q : Real => sourceTest (x + q • (-e))) y‖ ≤ C1) : ∀ r : Real, ‖sourceTest (x + r • e) - taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 Set.univ 0 r‖ ≤ C1 * r ^ 2 := by let f : Real → Real := fun q : Real => sourceTest (x + q • e) let g : Real → Real := fun q : Real => sourceTest (x + q • (-e)) have hLine : ContDiffOn Real 2 f Set.univ := by simpa [f] using gaussianRealSelectedTestLineContDiffOnOfSourceContDiffOn (sourceTest := sourceTest) (x := x) (e := e) hSource have hNegLine : ContDiffOn Real 2 g Set.univ := by simpa [g] using gaussianRealSelectedTestLineContDiffOnOfSourceContDiffOn (sourceTest := sourceTest) (x := x) (e := -e) hSource have hLineGlobal : ContDiff Real 2 f := by simpa using contDiffOn_univ.1 hLine have hNegLineGlobal : ContDiff Real 2 g := by simpa using contDiffOn_univ.1 hNegLine have hNonnegSecond : ∀ r : Real, 0 ≤ r → ∀ y ∈ Set.Icc (0 : Real) r, ‖iteratedDerivWithin 2 (fun q : Real => sourceTest (x + q • e)) (Set.Icc 0 r) y‖ ≤ C1 := by intro r hr y hy by_cases hr0 : r = 0 · subst r have hy0 : y = 0 := by exact le_antisymm hy.2 hy.1 subst y have hnot : ¬AccPt (0 : Real) (Filter.principal (Set.Icc (0 : Real) 0)) := by rw [Set.Icc_self] rw [accPt_iff_nhds] intro h rcases h Set.univ (by simp) with ⟨y, hy, hyne⟩ exact hyne hy.2 have hzero : iteratedDerivWithin 2 f (Set.Icc (0 : Real) 0) 0 = 0 := by rw [iteratedDerivWithin_succ] rw [derivWithin_zero_of_not_accPt hnot] calc ‖iteratedDerivWithin 2 (fun q : Real => sourceTest (x + q • e)) (Set.Icc 0 (0 : Real)) (0 : Real)‖ = 0 := by simpa [f] using congrArg norm hzero _ ≤ C1 := hC1 · have hpos : 0 < r := lt_of_le_of_ne hr (Ne.symm hr0) have hEq : iteratedDerivWithin 2 f (Set.Icc (0 : Real) r) y = iteratedDeriv 2 f y := iteratedDerivWithin_eq_iteratedDeriv (uniqueDiffOn_Icc hpos) hLineGlobal.contDiffAt hy calc ‖iteratedDerivWithin 2 (fun q : Real => sourceTest (x + q • e)) (Set.Icc 0 r) y‖ = ‖iteratedDeriv 2 f y‖ := by simpa [f] using congrArg norm hEq _ ≤ C1 := by simpa [f] using hLineSecond y have hNegSecond : ∀ u : Real, 0 ≤ u → ∀ y ∈ Set.Icc (0 : Real) u, ‖iteratedDerivWithin 2 (fun q : Real => sourceTest (x + q • (-e))) (Set.Icc 0 u) y‖ ≤ C1 := by intro u hu y hy by_cases hu0 : u = 0 · subst u have hy0 : y = 0 := by exact le_antisymm hy.2 hy.1 subst y have hnot : ¬AccPt (0 : Real) (Filter.principal (Set.Icc (0 : Real) 0)) := by rw [Set.Icc_self] rw [accPt_iff_nhds] intro h rcases h Set.univ (by simp) with ⟨y, hy, hyne⟩ exact hyne hy.2 have hzero : iteratedDerivWithin 2 g (Set.Icc (0 : Real) 0) 0 = 0 := by rw [iteratedDerivWithin_succ] rw [derivWithin_zero_of_not_accPt hnot] calc ‖iteratedDerivWithin 2 (fun q : Real => sourceTest (x + q • (-e))) (Set.Icc 0 (0 : Real)) (0 : Real)‖ = 0 := by simpa [g] using congrArg norm hzero _ ≤ C1 := hC1 · have hpos : 0 < u := lt_of_le_of_ne hu (Ne.symm hu0) have hEq : iteratedDerivWithin 2 g (Set.Icc (0 : Real) u) y = iteratedDeriv 2 g y := iteratedDerivWithin_eq_iteratedDeriv (uniqueDiffOn_Icc hpos) hNegLineGlobal.contDiffAt hy calc ‖iteratedDerivWithin 2 (fun q : Real => sourceTest (x + q • (-e))) (Set.Icc 0 u) y‖ = ‖iteratedDeriv 2 g y‖ := by simpa [g] using congrArg norm hEq _ ≤ C1 := by simpa [g] using hNegLineSecond y exact gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn (sourceTest := sourceTest) (x := x) (e := e) (C1 := C1) hSource hNonnegSecond hNegSecond /-- Scalar-line second derivative as an ambient directional Hessian. This cycle-169 chain-rule bridge is the local Mathlib step below the selected-test bounded-Hessian source hypothesis. It rewrites the second ordinary derivative of the affine scalar slice `q ↦ sourceTest (x + q • e)` as the ambient second Frechet derivative of `sourceTest` applied twice to the same direction. The proof uses `iteratedDeriv_vcomp_two`; the affine line has first derivative `e` and second derivative `0`. -/ theorem gaussianRealSelectedTestLineSecondDerivEqDirectionalFDeriv {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (x e : E) (hSource : ContDiffOn Real 2 sourceTest Set.univ) : ∀ y : Real, iteratedDeriv 2 (fun q : Real => sourceTest (x + q • e)) y = iteratedFDeriv Real 2 sourceTest (x + y • e) (fun _ : Fin 2 => e) := by intro y let line : Real → E := fun q : Real => x + q • e have hSourceAt : ContDiffAt Real 2 sourceTest (line y) := by exact hSource.contDiffAt (by simp) have hLineAt : ContDiffAt Real 2 line y := by have hLineGlobal : ContDiff Real 2 line := by exact contDiff_const.add (contDiff_id.smul_const e) exact hLineGlobal.contDiffAt have hderivLine : deriv line y = e := by simp [line, deriv_smul_const] have hsecondLine : iteratedDeriv 2 line y = 0 := by rw [show line = (fun q : Real => x + q • e) by rfl] rw [iteratedDeriv_const_add (show 0 < 2 by norm_num) x] rw [iteratedDeriv_smul_const (f := fun q : Real => q) (x := y) (n := 2) (v := e)] · simp [iteratedDeriv_fun_id] · exact contDiffAt_id have hchain := iteratedDeriv_vcomp_two (g := sourceTest) (f := line) (x := y) hSourceAt hLineAt simpa [line, Function.comp_def, hderivLine, hsecondLine] using hchain /-- Scalar-line second derivative from Mathlib's Taylor coefficient convention. This cycle-179 bridge narrows the source-facing scalar Brownian coefficient boundary one step further. If the paper correspondence supplies the scalar line coefficient as twice the order-two `taylorCoeffWithin` term, then Mathlib's definition of `taylorCoeffWithin` over `Set.univ` identifies it with the ordinary second derivative at the expansion point. -/ theorem selectedWeakTestScalarLineSecondCoeffDefOfTaylorCoeffWithin {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (quadraticCoeff : Test → E → Fin (Module.finrank Real E) → Real) (testRegular : Prop) (hScalarLineTaylorCoeffDef : testRegular → ∀ φ x i, quadraticCoeff φ x i = (2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0) : testRegular → ∀ φ x i, quadraticCoeff φ x i = iteratedDeriv 2 (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0 := by intro htests φ x i rw [hScalarLineTaylorCoeffDef htests φ x i] simp [taylorCoeffWithin] /-- Ambient diagonal second Taylor coefficient from a scalar-line coefficient. This cycle-177 bridge is the local Mathlib part below the source-facing `hSecondTaylorCoeffDef` field. Once the source correspondence identifies the normalized scalar Brownian quadratic coefficient with the ordinary second derivative of the selected one-dimensional line at the expansion point, the existing affine-line chain rule converts it to the ambient diagonal second Frechet derivative in the standard Brownian coordinate. -/ theorem selectedWeakTestSecondTaylorCoeffDefOfScalarLineSecondCoeffDef {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (quadraticCoeff : Test → E → Fin (Module.finrank Real E) → Real) (testRegular : Prop) (hSource : testRegular → ∀ φ, ContDiffOn Real 2 (selectedTest φ) Set.univ) (hScalarLineSecondCoeffDef : testRegular → ∀ φ x i, quadraticCoeff φ x i = iteratedDeriv 2 (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0) : testRegular → ∀ φ x i, quadraticCoeff φ x i = iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := by intro htests φ x i rw [hScalarLineSecondCoeffDef htests φ x i] have hline := gaussianRealSelectedTestLineSecondDerivEqDirectionalFDeriv (sourceTest := selectedTest φ) (x := x) (e := stdOrthonormalBasis Real E i) (hSource htests φ) 0 have hvec : (fun _ : Fin 2 => (stdOrthonormalBasis Real E) i) = ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := by funext j fin_cases j <;> rfl simpa [hvec] using hline /-- Quadratic-variation normalization from scalar-line second coefficient data. This cycle-177 lower_2 bridge composes the scalar-line coefficient bridge with the cycle-176 normalization bridge. It removes the older primitive `hSecondTaylorCoeffDef` field from this downstream normalization step: the remaining source-facing coefficient boundary is the ordinary second derivative of the selected Brownian coordinate line, together with the separate normalized variance definition. -/ theorem selectedWeakTestQuadraticVariationNormalizationOfScalarLineSecondCoeffAndNormalizedVarianceDef {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (quadraticCoeff : Test → E → Fin (Module.finrank Real E) → Real) (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (testRegular : Prop) (hSource : testRegular → ∀ φ, ContDiffOn Real 2 (selectedTest φ) Set.univ) (hScalarLineSecondCoeffDef : testRegular → ∀ φ x i, quadraticCoeff φ x i = iteratedDeriv 2 (fun q : Real => selectedTest φ (x + q • (stdOrthonormalBasis Real E i))) 0) (hNormalizedVarianceDef : testRegular → ∀ φ x i, variance φ x i = (1 : NNReal)) : testRegular → ∀ φ x i, quadraticCoeff φ x i * (variance φ x i : Real) = emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest φ) x ((stdOrthonormalBasis Real E) i) := by have hSecondTaylorCoeffDef : testRegular → ∀ φ x i, quadraticCoeff φ x i = iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := selectedWeakTestSecondTaylorCoeffDefOfScalarLineSecondCoeffDef selectedTest quadraticCoeff testRegular hSource hScalarLineSecondCoeffDef exact selectedWeakTestQuadraticVariationNormalizationOfSecondTaylorCoeffAndNormalizedVarianceDef selectedTest quadraticCoeff variance testRegular hSecondTaylorCoeffDef hNormalizedVarianceDef /-- Selected/reflected global line-second bounds from one ambient directional bound. The reflected line has direction `-e`; because the ambient second Frechet derivative is bilinear, applying it to `(-e, -e)` agrees with applying it to `(e, e)`. Thus a single global diagonal directional-Hessian bound supplies both `hLineSecond` and `hNegLineSecond` for the cycle-168 Taylor bridge. -/ theorem gaussianRealSelectedTestLineSecondBoundsOfDirectionalSecondBound {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (x e : E) (C1 : Real) (hSource : ContDiffOn Real 2 sourceTest Set.univ) (hDirectionalSecond : ∀ z : E, ‖iteratedFDeriv Real 2 sourceTest z (fun _ : Fin 2 => e)‖ ≤ C1) : (∀ y : Real, ‖iteratedDeriv 2 (fun q : Real => sourceTest (x + q • e)) y‖ ≤ C1) ∧ (∀ y : Real, ‖iteratedDeriv 2 (fun q : Real => sourceTest (x + q • (-e))) y‖ ≤ C1) := by constructor · intro y rw [gaussianRealSelectedTestLineSecondDerivEqDirectionalFDeriv (sourceTest := sourceTest) (x := x) (e := e) hSource y] exact hDirectionalSecond (x + y • e) · intro y rw [gaussianRealSelectedTestLineSecondDerivEqDirectionalFDeriv (sourceTest := sourceTest) (x := x) (e := -e) hSource y] have hsign : iteratedFDeriv Real 2 sourceTest (x + y • (-e)) (fun _ : Fin 2 => -e) = iteratedFDeriv Real 2 sourceTest (x + y • (-e)) (fun _ : Fin 2 => e) := by have h := (iteratedFDeriv Real 2 sourceTest (x + y • (-e))).map_smul_univ (fun _ : Fin 2 => (-1 : Real)) (fun _ : Fin 2 => e) simpa using h rw [hsign] exact hDirectionalSecond (x + y • (-e)) /-- Ambient second-derivative operator norm bound supplies the selected diagonal bound. This is the local bounded-Hessian leaf below the cycle-169 directional-Hessian interface: a uniform operator-norm bound on the ambient second Frechet derivative controls its evaluation on the repeated unit direction `e`. -/ theorem gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (e : E) (C1 : Real) (hSecondFDerivOpNorm : ∀ z : E, ‖iteratedFDeriv Real 2 sourceTest z‖ ≤ C1) (heUnit : ‖e‖ ≤ 1) : ∀ z : E, ‖iteratedFDeriv Real 2 sourceTest z (fun _ : Fin 2 => e)‖ ≤ C1 := by intro z have h := ContinuousMultilinearMap.le_mul_prod_of_opNorm_le_of_le (f := iteratedFDeriv Real 2 sourceTest z) (m := fun _ : Fin 2 => e) (C := C1) (b := fun _ : Fin 2 => (1 : Real)) (hSecondFDerivOpNorm z) (fun _ => heUnit) simpa using h /-- Hessian-as-derivative-of-gradient bound supplies the iterated-Frechet bound. This is a source-facing reformulation of the selected-test bounded-Hessian leaf: if the paper supplies the uniform operator-norm bound on `fderiv Real (fderiv Real sourceTest)`, Mathlib's iterated Frechet derivative API converts it to the `hSecondFDerivOpNorm` shape consumed downstream. -/ theorem gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (C1 : Real) (hHessianOpNorm : ∀ z : E, ‖fderiv Real (fderiv Real sourceTest) z‖ ≤ C1) : ∀ z : E, ‖iteratedFDeriv Real 2 sourceTest z‖ ≤ C1 := by intro z have hnorm : ‖iteratedFDeriv Real 2 sourceTest z‖ = ‖fderiv Real (fderiv Real sourceTest) z‖ := by rw [← one_add_one_eq_two] rw [← norm_iteratedFDeriv_fderiv (𝕜 := Real) (f := sourceTest) (x := z) (n := 1)] rw [norm_iteratedFDeriv_one] rw [hnorm] exact hHessianOpNorm z /-- A source-backed Hessian field supplies the selected-test Hessian operator bound. This is the narrow cycle-173 source-contract bridge: once the faithful source correspondence provides a Hessian representative for `sourceTest` together with its uniform operator-norm bound, the downstream `hHessianOpNorm` assumption is obtained by Mathlib's uniqueness of the Frechet derivative. -/ theorem selectedWeakTestHessianOpNormOfSourceHessianField {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (sourceHessian : E → E →L[Real] (E →L[Real] Real)) (C1 : Real) (hSourceHasHessian : ∀ z : E, HasFDerivAt (fderiv Real sourceTest) (sourceHessian z) z) (hSourceHessianBound : ∀ z : E, ‖sourceHessian z‖ ≤ C1) : ∀ z : E, ‖fderiv Real (fderiv Real sourceTest) z‖ ≤ C1 := by intro z rw [(hSourceHasHessian z).fderiv] exact hSourceHessianBound z /-- Mathlib standard orthonormal basis directions are unit directions. This discharges the Brownian coordinate side condition `heUnit` for the standard-basis scalar Ito branch. -/ theorem gaussianRealStdOrthonormalBasisUnit {E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (i : Fin (Module.finrank Real E)) : ‖(stdOrthonormalBasis Real E) i‖ ≤ (1 : Real) := by rw [OrthonormalBasis.norm_eq_one] /-- Operator-norm Hessian bound specialized to Brownian standard-basis directions. Cycle 170 removes the separate unit-direction hypothesis for `e = (stdOrthonormalBasis Real E) i`; the remaining analytic boundary is the selected-test second Frechet derivative operator-norm bound. -/ theorem gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis {E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (sourceTest : E → Real) (C1 : Real) (i : Fin (Module.finrank Real E)) (hSecondFDerivOpNorm : ∀ z : E, ‖iteratedFDeriv Real 2 sourceTest z‖ ≤ C1) : ∀ z : E, ‖iteratedFDeriv Real 2 sourceTest z (fun _ : Fin 2 => (stdOrthonormalBasis Real E) i)‖ ≤ C1 := by exact gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm (sourceTest := sourceTest) (e := (stdOrthonormalBasis Real E) i) (C1 := C1) hSecondFDerivOpNorm (gaussianRealStdOrthonormalBasisUnit i) /-- First-order selected-line Taylor remainder from an ambient directional Hessian bound. Cycle 169 narrows the remaining `hLineSecond`/`hNegLineSecond` boundary left by `gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds`. The new source-facing analytic leaf is the single ambient diagonal bound `∀ z, ‖iteratedFDeriv Real 2 sourceTest z (fun _ => e)‖ ≤ C1`; the older nonnegative constant input follows from this norm bound at `x`. -/ theorem gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound {E : Type*} [NormedAddCommGroup E] [NormedSpace Real E] (sourceTest : E → Real) (x e : E) (C1 : Real) (hSource : ContDiffOn Real 2 sourceTest Set.univ) (hDirectionalSecond : ∀ z : E, ‖iteratedFDeriv Real 2 sourceTest z (fun _ : Fin 2 => e)‖ ≤ C1) : ∀ r : Real, ‖sourceTest (x + r • e) - taylorWithinEval (fun q : Real => sourceTest (x + q • e)) 1 Set.univ 0 r‖ ≤ C1 * r ^ 2 := by have hC1 : 0 ≤ C1 := by exact le_trans (norm_nonneg _) (hDirectionalSecond x) have hBounds := gaussianRealSelectedTestLineSecondBoundsOfDirectionalSecondBound (sourceTest := sourceTest) (x := x) (e := e) (C1 := C1) hSource hDirectionalSecond exact gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds (sourceTest := sourceTest) (x := x) (e := e) (C1 := C1) hSource hC1 hBounds.1 hBounds.2 /-- Standard-basis selected-line Taylor remainder from source Hessian fields. Cycle 175 narrows the selected-line Taylor-domination leaf for the Brownian coordinate direction. Once the source correspondence supplies a Hessian representative for the selected weak test and its uniform operator-norm bound, the already compiled Hessian/operator-norm/directional chain supplies the degree-one quadratic Taylor remainder for `e = (stdOrthonormalBasis Real E) i`. The source-facing Hessian fields remain explicit; this theorem does not derive them from `testRegular`. -/ theorem gaussianRealSelectedTestStdOrthonormalFirstOrderQuadraticRemainderBoundOfSourceHessianField {E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (sourceTest : E → Real) (sourceHessian : E → E →L[Real] (E →L[Real] Real)) (x : E) (i : Fin (Module.finrank Real E)) (C1 : Real) (hSource : ContDiffOn Real 2 sourceTest Set.univ) (hSourceHasHessian : ∀ z : E, HasFDerivAt (fderiv Real sourceTest) (sourceHessian z) z) (hSourceHessianBound : ∀ z : E, ‖sourceHessian z‖ ≤ C1) : ∀ r : Real, ‖sourceTest (x + r • ((stdOrthonormalBasis Real E) i)) - taylorWithinEval (fun q : Real => sourceTest (x + q • ((stdOrthonormalBasis Real E) i))) 1 Set.univ 0 r‖ ≤ C1 * r ^ 2 := by have hHessianOpNorm : ∀ z : E, ‖fderiv Real (fderiv Real sourceTest) z‖ ≤ C1 := selectedWeakTestHessianOpNormOfSourceHessianField sourceTest sourceHessian C1 hSourceHasHessian hSourceHessianBound have hSecondFDerivOpNorm : ∀ z : E, ‖iteratedFDeriv Real 2 sourceTest z‖ ≤ C1 := gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm sourceTest C1 hHessianOpNorm have hDirectionalSecond : ∀ z : E, ‖iteratedFDeriv Real 2 sourceTest z (fun _ : Fin 2 => (stdOrthonormalBasis Real E) i)‖ ≤ C1 := gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis sourceTest C1 i hSecondFDerivOpNorm exact gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound (sourceTest := sourceTest) (x := x) (e := (stdOrthonormalBasis Real E) i) (C1 := C1) hSource hDirectionalSecond /-- Gaussian integrability of the quadratic domination bound. For the concrete selected-test normalized remainder, the source Taylor domination leaf is expected to use a quadratic Gaussian bound `fun z => C * z ^ 2`. This theorem discharges only the DCT `hBoundInt` input for that chosen bound; the pointwise domination `hBound` remains a separate source Taylor estimate. -/ theorem gaussianRealSelectedTestLineSecondOrderQuadraticBoundIntegrable (v : NNReal) (C : Real) : MeasureTheory.Integrable (fun z : Real => C * z ^ 2) (ProbabilityTheory.gaussianReal (0 : Real) v) := by have hExpPos : MeasureTheory.Integrable (fun z : Real => Real.exp ((1 : Real) * z)) (ProbabilityTheory.gaussianReal (0 : Real) v) := by simpa using ProbabilityTheory.integrable_exp_mul_gaussianReal (μ := (0 : Real)) (v := v) (t := (1 : Real)) have hExpNeg : MeasureTheory.Integrable (fun z : Real => Real.exp (-(1 : Real) * z)) (ProbabilityTheory.gaussianReal (0 : Real) v) := by simpa using ProbabilityTheory.integrable_exp_mul_gaussianReal (μ := (0 : Real)) (v := v) (t := (-(1 : Real))) have hPow : MeasureTheory.Integrable (fun z : Real => z ^ 2) (ProbabilityTheory.gaussianReal (0 : Real) v) := by exact ProbabilityTheory.integrable_pow_of_integrable_exp_mul (X := fun z : Real => z) (μ := ProbabilityTheory.gaussianReal (0 : Real) v) (t := (1 : Real)) (by norm_num) hExpPos hExpNeg 2 exact hPow.const_mul C /-- Normalized remainder-bound integrability from a concrete quadratic bound. This cycle-196 bridge narrows the normalized-law integrability leaf itself: once the source correspondence identifies the scalar dominating remainder bound as `C * z ^ 2` under the normalized scalar-coordinate law, the remaining analytic input is exactly the already compiled Gaussian quadratic integrability theorem. The concrete Taylor quotient identity `hNormalizedRemainderBoundDef` remains the source-cited boundary supplied by the paper transcript. -/ theorem selectedWeakTestNormalizedRemainderBoundIntOfQuadraticBound {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (normalizedCoordinateLaw : Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real) (remainderBound : Test → E → Fin (Module.finrank Real E) → Real → Real) (remainderBoundC : Test → E → Fin (Module.finrank Real E) → Real) (testRegular : Prop) (hNormalizedCoordinateLaw : testRegular → ∀ φ x i, normalizedCoordinateLaw φ x i = ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal)) (hNormalizedRemainderBoundDef : testRegular → ∀ φ x i z, remainderBound φ x i z = remainderBoundC φ x i * z ^ 2) : testRegular → ∀ φ x i, MeasureTheory.Integrable (fun z : Real => remainderBound φ x i z) (normalizedCoordinateLaw φ x i) := by intro htests φ x i simpa [hNormalizedCoordinateLaw htests φ x i, hNormalizedRemainderBoundDef htests φ x i] using gaussianRealSelectedTestLineSecondOrderQuadraticBoundIntegrable (1 : NNReal) (remainderBoundC φ x i) /-- One-dimensional scalar Brownian Ito generator from a Taylor moment split. This cycle-162 bridge narrows the remaining one-dimensional Brownian/Ito generator boundary to three source-facing scalar inputs: a Taylor moment decomposition for the coordinate generator, the Brownian quadratic-variation normalization after the paper's `sigma_eta^2/2` coefficient has been factored outside the event field, and vanishing of the normalized Taylor remainder. The proof uses the compiled centered Gaussian first/second moment collapse; it does not discharge the analytic dominated-remainder theorem itself. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (brownianCoordinateGenerator : Test → E → Fin (Module.finrank Real E) → Real) (variance : Test → E → Fin (Module.finrank Real E) → NNReal) (linearCoeff quadraticCoeff remainderGeneratorLimit : Test → E → Fin (Module.finrank Real E) → Real) (testRegular : Prop) (hFrozenScalarBrownianItoTaylorMomentDecomposition : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = linearCoeff φ x i * (∫ z : Real, z ∂(ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) + quadraticCoeff φ x i * (∫ z : Real, z ^ 2 ∂(ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) + remainderGeneratorLimit φ x i) (hFrozenScalarBrownianItoQuadraticVariationNormalization : testRegular → ∀ φ x i, quadraticCoeff φ x i * (variance φ x i : Real) = emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest φ) x ((stdOrthonormalBasis Real E) i)) (hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes : testRegular → ∀ φ x i, remainderGeneratorLimit φ x i = 0) : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest φ) x ((stdOrthonormalBasis Real E) i) := by intro htests φ x i calc brownianCoordinateGenerator φ x i = linearCoeff φ x i * (∫ z : Real, z ∂(ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) + quadraticCoeff φ x i * (∫ z : Real, z ^ 2 ∂(ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i))) + remainderGeneratorLimit φ x i := hFrozenScalarBrownianItoTaylorMomentDecomposition htests φ x i _ = quadraticCoeff φ x i * (variance φ x i : Real) + remainderGeneratorLimit φ x i := by rw [gaussianRealZeroOneDimTaylorMomentContribution] _ = emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest φ) x ((stdOrthonormalBasis Real E) i) + 0 := by rw [hFrozenScalarBrownianItoQuadraticVariationNormalization htests φ x i, hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes htests φ x i] _ = emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest φ) x ((stdOrthonormalBasis Real E) i) := by rw [add_zero] /-- Per-coordinate frozen scalar Ito generator from a one-dimensional Taylor term. This cycle-161 bridge narrows the supplied boundary `hFrozenScalarBrownianItoCoordinateGeneratorDef` to the smaller source-cited one-dimensional Brownian/Ito Taylor statement `hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor`. It only unfolds the named one-dimensional term along the paper's standard Brownian coordinate and does not route through weak-FP source fields, trace fields, or downstream total-event consumers. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (brownianCoordinateGenerator : Test → E → Fin (Module.finrank Real E) → Real) (testRegular : Prop) (hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest φ) x ((stdOrthonormalBasis Real E) i)) : testRegular → ∀ φ x i, brownianCoordinateGenerator φ x i = iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := by intro htests φ x i calc brownianCoordinateGenerator φ x i = emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest φ) x ((stdOrthonormalBasis Real E) i) := hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor htests φ x i _ = iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := rfl /-- Total-event generator formula from trace-field source/event identities. This cycle-149 lower_2 bridge narrows the two standard-basis inputs left by the lower_1 total-event route. The total-event formula no longer needs `hemGeneratorStdBasisDef` or `hemGeneratorLaplacianEventFieldStdBasisDef` as primitive assumptions: both are recovered from the source-facing trace-action definition, the equality between the Laplacian event field and the trace field, and the trace-field Laplacian identity. The source-functional law integral remains explicit. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceFieldSourceAndEventFormula {Test E : Type*} [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (hatRhoS : MeasureTheory.Measure E) (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hemGeneratorTraceActionDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (emGeneratorTraceField φ)) (hemGeneratorLaplacianEventFieldEqTraceField : testRegular → emGeneratorLaplacianEventField = emGeneratorTraceField) (htraceFieldEqLaplacian : testRegular → emGeneratorTraceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := by have htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField selectedTest emGeneratorTraceField testRegular htraceFieldEqLaplacian have hemGeneratorStdBasisDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField selectedTest sourceLaplacianFunctional emGeneratorTraceField testRegular emGeneratorLaplacianAction hemGeneratorTraceActionDef htraceFieldStdBasis have hemGeneratorLaplacianEventFieldStdBasisDef : testRegular → emGeneratorLaplacianEventField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField selectedTest emGeneratorTraceField emGeneratorLaplacianEventField testRegular hemGeneratorLaplacianEventFieldEqTraceField htraceFieldStdBasis exact generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceAndEventFormula hatRhoS selectedTest sourceLaplacianFunctional emGeneratorLaplacianEventField testRegular emGeneratorLaplacianAction hsourceLaplacianFunctional hemGeneratorStdBasisDef hemGeneratorLaplacianEventFieldStdBasisDef /-- Trace-action definition from the law-space trace integral. This cycle-150 helper narrows the remaining `hemGeneratorTraceActionDef` premise in the total-event trace-field route. The source-cited law-space trace integral against `hatRhoS`, together with the already explicit definition of `sourceLaplacianFunctional`, recovers the trace-action source formula without changing the event-field or trace-field Laplacian leaves. -/ theorem generalMovingTargetDiscreteEmGeneratorTraceActionDefOfTraceLawIntegral {Test E : Type*} [MeasurableSpace E] (hatRhoS : MeasureTheory.Measure E) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hemGeneratorTraceLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, emGeneratorTraceField φ x ∂hatRhoS) : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (emGeneratorTraceField φ) := by intro htests funext φ calc emGeneratorLaplacianAction φ = ∫ x, emGeneratorTraceField φ x ∂hatRhoS := hemGeneratorTraceLawIntegral htests φ _ = sourceLaplacianFunctional (emGeneratorTraceField φ) := by rw [hsourceLaplacianFunctional] /-- Frozen EM generator Laplacian state integral from trace law and trace Laplacian source fields. This lower_1 scout bridge feeds the existing trace-action and trace-field Laplacian narrowings into the cycle-154 state-integral route. The state integral no longer needs the trace-action source definition or the trace-field standard-basis formula as primitive assumptions: those follow from the law-space trace integral, the source-functional law definition, and the named trace-field/Laplacian identity. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceLawIntegralLaplacianField {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hemGeneratorTraceLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, emGeneratorTraceField φ x ∂hatRhoS) (htraceFieldEqLaplacian : testRegular → emGeneratorTraceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P := by have hemGeneratorTraceActionDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (emGeneratorTraceField φ) := generalMovingTargetDiscreteEmGeneratorTraceActionDefOfTraceLawIntegral hatRhoS sourceLaplacianFunctional emGeneratorTraceField testRegular emGeneratorLaplacianAction hsourceLaplacianFunctional hemGeneratorTraceLawIntegral have htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField selectedTest emGeneratorTraceField testRegular htraceFieldEqLaplacian exact generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceFieldSourceFunctional P hatRhoS hatXAtS selectedTest sourceLaplacianFunctional emGeneratorTraceField testRegular emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hsourceLaplacianFieldMeas hemGeneratorTraceActionDef htraceFieldStdBasis /-- Frozen EM generator Laplacian state integral from the trace state integral. This lower_2 continuation narrows the `hemGeneratorTraceLawIntegral` input left by the lower_1 state-integral bridge. The law-space trace integral follows from the sample-space trace integral along `hatXAtS`, the map-law identity for `hatRhoS`, and trace-field measurability, then the lower_1 trace-law/Laplacian bridge recovers the selected-test Laplacian state integral. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceStateIntegralLaplacianField {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (htraceFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (emGeneratorTraceField φ) hatRhoS) (hemGeneratorTraceStateIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, emGeneratorTraceField φ (hatXAtS ω) ∂P) (htraceFieldEqLaplacian : testRegular → emGeneratorTraceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P := by have hemGeneratorTraceLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, emGeneratorTraceField φ x ∂hatRhoS := generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral P hatRhoS hatXAtS emGeneratorTraceField testRegular emGeneratorLaplacianAction hhatRhoS hhatX htraceFieldMeas hemGeneratorTraceStateIntegral exact generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceLawIntegralLaplacianField P hatRhoS hatXAtS selectedTest sourceLaplacianFunctional emGeneratorTraceField testRegular emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hsourceLaplacianFieldMeas hemGeneratorTraceLawIntegral htraceFieldEqLaplacian /-- Total-event generator formula from a trace law-integral source leaf. This cycle-150 bridge feeds the trace-action law-integral narrowing into the cycle-149 lower_2 total-event theorem. It removes only the direct `hemGeneratorTraceActionDef` premise from the current total-event route; the event-field/trace-field equality, the trace-field Laplacian identity, and the source-functional law integral remain explicit. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLawIntegralSourceAndEventFormula {Test E : Type*} [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (hatRhoS : MeasureTheory.Measure E) (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hemGeneratorTraceLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, emGeneratorTraceField φ x ∂hatRhoS) (hemGeneratorLaplacianEventFieldEqTraceField : testRegular → emGeneratorLaplacianEventField = emGeneratorTraceField) (htraceFieldEqLaplacian : testRegular → emGeneratorTraceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := by have hemGeneratorTraceActionDef : testRegular → emGeneratorLaplacianAction = fun φ => sourceLaplacianFunctional (emGeneratorTraceField φ) := generalMovingTargetDiscreteEmGeneratorTraceActionDefOfTraceLawIntegral hatRhoS sourceLaplacianFunctional emGeneratorTraceField testRegular emGeneratorLaplacianAction hsourceLaplacianFunctional hemGeneratorTraceLawIntegral exact generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceFieldSourceAndEventFormula hatRhoS selectedTest sourceLaplacianFunctional emGeneratorTraceField emGeneratorLaplacianEventField testRegular emGeneratorLaplacianAction hsourceLaplacianFunctional hemGeneratorTraceActionDef hemGeneratorLaplacianEventFieldEqTraceField htraceFieldEqLaplacian /-- Total-event generator formula from a trace state-integral source leaf. This lower_1 cycle-150 continuation narrows the remaining `hemGeneratorTraceLawIntegral` premise exposed by the trace-law total-event route. The law-space trace integral is recovered from the sample-space trace integral along `hatXAtS` by the existing map-integral transport helper; the event-field/trace-field equality and trace-field Laplacian identity remain explicit. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceStateIntegralSourceAndEventFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (htraceFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (emGeneratorTraceField φ) hatRhoS) (hemGeneratorTraceStateIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, emGeneratorTraceField φ (hatXAtS ω) ∂P) (hemGeneratorLaplacianEventFieldEqTraceField : testRegular → emGeneratorLaplacianEventField = emGeneratorTraceField) (htraceFieldEqLaplacian : testRegular → emGeneratorTraceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := by have hemGeneratorTraceLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, emGeneratorTraceField φ x ∂hatRhoS := generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral P hatRhoS hatXAtS emGeneratorTraceField testRegular emGeneratorLaplacianAction hhatRhoS hhatX htraceFieldMeas hemGeneratorTraceStateIntegral exact generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLawIntegralSourceAndEventFormula hatRhoS selectedTest sourceLaplacianFunctional emGeneratorTraceField emGeneratorLaplacianEventField testRegular emGeneratorLaplacianAction hsourceLaplacianFunctional hemGeneratorTraceLawIntegral hemGeneratorLaplacianEventFieldEqTraceField htraceFieldEqLaplacian /-- Total-event generator formula from the selected-test Laplacian state integral. This lower_2 cycle-150 continuation narrows the trace-state total-event route one step further. The trace-field measurability and trace-state integral premises are reconstructed from the source-Laplacian measurability and the sample-space selected-test Laplacian integral, using the already explicit trace-field/Laplacian identity. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndEventFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hemGeneratorLaplacianStateIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P) (hemGeneratorLaplacianEventFieldEqTraceField : testRegular → emGeneratorLaplacianEventField = emGeneratorTraceField) (htraceFieldEqLaplacian : testRegular → emGeneratorTraceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := by have htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField selectedTest emGeneratorTraceField testRegular htraceFieldEqLaplacian have htraceFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (emGeneratorTraceField φ) hatRhoS := generalMovingTargetDiscreteEmGeneratorTraceFieldMeasOfSourceLaplacianFieldMeas hatRhoS selectedTest emGeneratorTraceField testRegular hsourceLaplacianFieldMeas htraceFieldStdBasis have hemGeneratorTraceStateIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, emGeneratorTraceField φ (hatXAtS ω) ∂P := generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegral P hatXAtS selectedTest emGeneratorTraceField testRegular emGeneratorLaplacianAction hemGeneratorLaplacianStateIntegral htraceFieldStdBasis exact generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceStateIntegralSourceAndEventFormula P hatRhoS hatXAtS selectedTest sourceLaplacianFunctional emGeneratorTraceField emGeneratorLaplacianEventField testRegular emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX htraceFieldMeas hemGeneratorTraceStateIntegral hemGeneratorLaplacianEventFieldEqTraceField htraceFieldEqLaplacian /-- Weak-FP source-action route from the EM law integral and trace-field event identification. This cycle-145 lower_2 consumer removes the direct `hemGeneratorLaplacianEventFieldStdBasisDef` premise from the law-integral route. The remaining source-facing event-field boundary is the equality between the named Laplacian event field and the already tracked trace field, plus `htraceFieldStdBasis`; `hemGeneratorLaplacianLawIntegral` remains explicit. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventLawIntegralFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorLaplacianLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS) (hemGeneratorLaplacianEventFieldEqTraceField : testRegular → emGeneratorLaplacianEventField = emGeneratorTraceField) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorLaplacianEventFieldStdBasisDef : testRegular → emGeneratorLaplacianEventField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField selectedTest emGeneratorTraceField emGeneratorLaplacianEventField testRegular hemGeneratorLaplacianEventFieldEqTraceField htraceFieldStdBasis exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventLawIntegralFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField emGeneratorLaplacianEventField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorLaplacianLawIntegral hemGeneratorLaplacianEventFieldStdBasisDef htraceFieldStdBasis /-- Weak-FP source-action route from a trace-event total-state formula. This cycle-146 consumer removes the direct `hemGeneratorLaplacianLawIntegral` premise from the cycle-145 trace-event route. The source-facing EM boundary is now the total state-event formula for the named frozen-generator Laplacian event field. The state-event equality with the selected-test Laplacian is reconstructed from the trace-event identification and the existing trace-field standard-basis formula. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorLaplacianTotalEventIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS) (hemGeneratorLaplacianEventFieldEqTraceField : testRegular → emGeneratorLaplacianEventField = emGeneratorTraceField) (htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorLaplacianEventFieldStdBasisDef : testRegular → emGeneratorLaplacianEventField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField selectedTest emGeneratorTraceField emGeneratorLaplacianEventField testRegular hemGeneratorLaplacianEventFieldEqTraceField htraceFieldStdBasis have hemGeneratorLaplacianEventFieldEqLaplacian : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x := generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula selectedTest emGeneratorLaplacianEventField testRegular hemGeneratorLaplacianEventFieldStdBasisDef have hemGeneratorLaplacianStateEventEqLaplacian : testRegular → ∀ φ (t : Set E), MeasurableSet t → (∫ x in t, emGeneratorLaplacianEventField φ x ∂hatRhoS) = ∫ x in t, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise hatRhoS selectedTest emGeneratorLaplacianEventField testRegular hemGeneratorLaplacianEventFieldEqLaplacian have hemGeneratorLaplacianLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula hatRhoS selectedTest emGeneratorLaplacianEventField testRegular emGeneratorLaplacianAction hemGeneratorLaplacianTotalEventIntegral hemGeneratorLaplacianStateEventEqLaplacian exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventLawIntegralFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField emGeneratorLaplacianEventField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorLaplacianLawIntegral hemGeneratorLaplacianEventFieldEqTraceField htraceFieldStdBasis /-- Weak-FP source-action route from the trace-event total-state formula and a Mathlib-Laplacian trace-field definition. This cycle-146 lower_1 continuation removes `htraceFieldStdBasis` as a primitive premise from the current trace-event total-event route. The smaller source boundary is the identification of the named trace field with Mathlib's selected-test Laplacian; the local standard-basis bridge supplies the trace formula consumed by the existing cycle-146 theorem. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventTraceLaplacianFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorLaplacianTotalEventIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS) (hemGeneratorLaplacianEventFieldEqTraceField : testRegular → emGeneratorLaplacianEventField = emGeneratorTraceField) (htraceFieldEqLaplacian : testRegular → emGeneratorTraceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have htraceFieldStdBasis : testRegular → emGeneratorTraceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField selectedTest emGeneratorTraceField testRegular htraceFieldEqLaplacian exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField emGeneratorLaplacianEventField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorLaplacianTotalEventIntegral hemGeneratorLaplacianEventFieldEqTraceField htraceFieldStdBasis /-- Trace-event equality from pointwise Laplacian identities. This cycle-146 lower_2 helper narrows the source equality between the named frozen EM Laplacian event field and the trace field. It is enough to identify both fields with Mathlib's selected-test Laplacian; no integral or weak-FP source facts are discharged here. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (hemGeneratorLaplacianEventFieldEqLaplacian : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x) (htraceFieldEqLaplacian : testRegular → emGeneratorTraceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → emGeneratorLaplacianEventField = emGeneratorTraceField := by intro htests funext φ calc emGeneratorLaplacianEventField φ = (fun x => Laplacian.laplacian (selectedTest φ) x) := hemGeneratorLaplacianEventFieldEqLaplacian htests φ _ = emGeneratorTraceField φ := (congrFun (htraceFieldEqLaplacian htests) φ).symm /-- Event-field source equality from common standard-basis source fields. This cycle-152 direct-leaf helper narrows `hemGeneratorLaplacianEventFieldEqSourceField` itself. Instead of identifying the frozen EM Laplacian event field directly with the weak-FP source field, it keeps two source-facing definitions explicit: both named fields are the same standard-basis second-derivative field for the selected weak test. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqSourceFieldOfStdBasisFields {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (weakFpLaplacianSourceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (hemGeneratorLaplacianEventFieldStdBasisDef : testRegular → emGeneratorLaplacianEventField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) (hweakFpSourceFieldStdBasisDef : testRegular → weakFpLaplacianSourceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) : testRegular → emGeneratorLaplacianEventField = weakFpLaplacianSourceField := by intro htests calc emGeneratorLaplacianEventField = (fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := hemGeneratorLaplacianEventFieldStdBasisDef htests _ = weakFpLaplacianSourceField := (hweakFpSourceFieldStdBasisDef htests).symm /-- Weak-FP source-field standard-basis formula from the Mathlib Laplacian field. This cycle-152 lower_1 scout helper narrows one of the two remaining standard-basis leaves exposed by `generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqSourceFieldOfStdBasisFields`. It does not identify the frozen EM event field; it only says that once the paper's weak-FP source field has been identified with Mathlib's selected-test Laplacian, the local standard-basis Laplacian theorem supplies the source second-derivative display. -/ theorem generalMovingTargetDiscreteWeakFpSourceFieldStdBasisDefOfLaplacianField {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (testRegular : Prop) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → weakFpLaplacianSourceField = fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] := by intro htests calc weakFpLaplacianSourceField = (fun φ => Laplacian.laplacian (selectedTest φ)) := hweakFpSourceFieldEqLaplacian htests _ = (fun φ x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) := by funext φ exact generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv (sourceTest := selectedTest φ) /-- Weak-FP source-field equality from the pointwise source identity. This cycle-152 lower_2 helper narrows the remaining `hweakFpSourceFieldEqLaplacian` leaf from the weak-FP source display. The field-level equality is no longer primitive: it follows from the pointwise source-facing identity for each selected weak test. The analytic proof of that pointwise identity remains the exact source-cited boundary from `appendix.tex:1379-1387`. -/ theorem generalMovingTargetDiscreteWeakFpSourceFieldEqLaplacianOfPointwise {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (testRegular : Prop) (hweakFpSourceFieldPointwiseEqLaplacian : testRegular → ∀ φ, weakFpLaplacianSourceField φ = Laplacian.laplacian (selectedTest φ)) : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ) := by intro htests funext φ exact hweakFpSourceFieldPointwiseEqLaplacian htests φ /-- Event-field Laplacian identity from the named weak-FP source field. This lower_2 cycle-151 direct-leaf theorem narrows `hemGeneratorLaplacianEventFieldEqLaplacian` itself. It is enough to identify the frozen EM Laplacian event field with the already named weak-FP source Laplacian field, while keeping that source field's selected-test Laplacian definition as a separate source boundary. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfWeakFpSourceField {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (weakFpLaplacianSourceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (hemGeneratorLaplacianEventFieldEqSourceField : testRegular → emGeneratorLaplacianEventField = weakFpLaplacianSourceField) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x := by intro htests φ calc emGeneratorLaplacianEventField φ = weakFpLaplacianSourceField φ := congrFun (hemGeneratorLaplacianEventFieldEqSourceField htests) φ _ = (fun φ => Laplacian.laplacian (selectedTest φ)) φ := congrFun (hweakFpSourceFieldEqLaplacian htests) φ _ = (fun x => Laplacian.laplacian (selectedTest φ) x) := rfl /-- Total-event generator formula from pointwise event-field and trace-field Laplacian identities. This cycle-151 worker packet removes `hemGeneratorLaplacianEventFieldEqTraceField` from the current cycle-150 trace-Laplacian state total-event route. The smaller source boundary is the pointwise identity for the named frozen EM Laplacian event field together with the trace-field/Laplacian identity; the selected-test Laplacian state integral, source-Laplacian measurability, and source-functional definition remain explicit. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndPointwiseEventFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hemGeneratorLaplacianStateIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ ω, Laplacian.laplacian (selectedTest φ) (hatXAtS ω) ∂P) (hemGeneratorLaplacianEventFieldEqLaplacian : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x) (htraceFieldEqLaplacian : testRegular → emGeneratorTraceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := by have hemGeneratorLaplacianEventFieldEqTraceField : testRegular → emGeneratorLaplacianEventField = emGeneratorTraceField := generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields selectedTest emGeneratorTraceField emGeneratorLaplacianEventField testRegular hemGeneratorLaplacianEventFieldEqLaplacian htraceFieldEqLaplacian exact generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndEventFormula P hatRhoS hatXAtS selectedTest sourceLaplacianFunctional emGeneratorTraceField emGeneratorLaplacianEventField testRegular emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hsourceLaplacianFieldMeas hemGeneratorLaplacianStateIntegral hemGeneratorLaplacianEventFieldEqTraceField htraceFieldEqLaplacian /-- Weak-FP source-action route from pointwise event-field and trace-field Laplacian identities. This cycle-146 lower_2 continuation removes `hemGeneratorLaplacianEventFieldEqTraceField` as a primitive premise from the latest trace-event total-event route. The smaller source boundary is the pointwise identity for the named EM Laplacian event field together with the trace-field/Laplacian identity introduced by lower_1. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorLaplacianTotalEventIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS) (hemGeneratorLaplacianEventFieldEqLaplacian : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x) (htraceFieldEqLaplacian : testRegular → emGeneratorTraceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorLaplacianEventFieldEqTraceField : testRegular → emGeneratorLaplacianEventField = emGeneratorTraceField := generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields selectedTest emGeneratorTraceField emGeneratorLaplacianEventField testRegular hemGeneratorLaplacianEventFieldEqLaplacian htraceFieldEqLaplacian exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventTraceLaplacianFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField emGeneratorLaplacianEventField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorLaplacianTotalEventIntegral hemGeneratorLaplacianEventFieldEqTraceField htraceFieldEqLaplacian /-- Weak-FP source-action route from pointwise event/trace Laplacian identities and the source action definition. This cycle-147 continuation removes `hemGeneratorLaplacianTotalEventIntegral` as a primitive premise from the current pointwise trace-event route. The smaller source boundary is the function-level action definition for the named frozen EM Laplacian event field; the event-field and trace-field Laplacian identities remain explicit. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefTraceLaplacianFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorLaplacianActionDef : testRegular → emGeneratorLaplacianAction = fun φ => ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS) (hemGeneratorLaplacianEventFieldEqLaplacian : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x) (htraceFieldEqLaplacian : testRegular → emGeneratorTraceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorLaplacianTotalEventIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef hatRhoS emGeneratorLaplacianEventField testRegular emGeneratorLaplacianAction hemGeneratorLaplacianActionDef exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField emGeneratorLaplacianEventField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorLaplacianTotalEventIntegral hemGeneratorLaplacianEventFieldEqLaplacian htraceFieldEqLaplacian /-- Frozen EM generator action definition from the source standard-basis action and the pointwise event-field Laplacian identity. This lower-1 cycle-147 proof-scout helper narrows the remaining `hemGeneratorLaplacianActionDef` premise on the current trace-Laplacian route. The source-facing action formula is the standard-basis second-derivative integral from the EM generator paragraph; the already explicit pointwise event-field/Laplacian identity rewrites that integral back to the named event field. No conditional-law or SLT result is used. -/ theorem generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionPointwiseEventFormula {Test E : Type*} [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (hatRhoS : MeasureTheory.Measure E) (selectedTest : Test → E → Real) (emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (emGeneratorLaplacianAction : Test → Real) (hemGeneratorLaplacianStdBasisActionDef : testRegular → emGeneratorLaplacianAction = fun φ => ∫ x in Set.univ, (∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) ∂hatRhoS) (hemGeneratorLaplacianEventFieldEqLaplacian : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x) : testRegular → emGeneratorLaplacianAction = fun φ => ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := by intro htests funext φ calc emGeneratorLaplacianAction φ = ∫ x in Set.univ, (∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) ∂hatRhoS := congrFun (hemGeneratorLaplacianStdBasisActionDef htests) φ _ = ∫ x in Set.univ, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := by rw [← generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv (sourceTest := selectedTest φ)] _ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := by rw [← hemGeneratorLaplacianEventFieldEqLaplacian htests φ] /-- Weak-FP source-action route from the source standard-basis action, pointwise event-field Laplacian identity, and trace-field Laplacian identity. This lower-1 cycle-147 scout continuation removes `hemGeneratorLaplacianActionDef` as a primitive premise from the current pointwise event/trace Laplacian consumer. The remaining EM action boundary is the source standard-basis action integral; the event-field and trace-field Laplacian identities remain explicit. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStdBasisActionTraceLaplacianFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorLaplacianStdBasisActionDef : testRegular → emGeneratorLaplacianAction = fun φ => ∫ x in Set.univ, (∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) ∂hatRhoS) (hemGeneratorLaplacianEventFieldEqLaplacian : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x) (htraceFieldEqLaplacian : testRegular → emGeneratorTraceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorLaplacianActionDef : testRegular → emGeneratorLaplacianAction = fun φ => ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS := generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionPointwiseEventFormula hatRhoS selectedTest emGeneratorLaplacianEventField testRegular emGeneratorLaplacianAction hemGeneratorLaplacianStdBasisActionDef hemGeneratorLaplacianEventFieldEqLaplacian exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefTraceLaplacianFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField emGeneratorLaplacianEventField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorLaplacianActionDef hemGeneratorLaplacianEventFieldEqLaplacian htraceFieldEqLaplacian /-- Weak-FP source-action route from the EM law integral, pointwise event-field Laplacian identity, and trace-field Laplacian identity. This lower-2 cycle-147 continuation removes the `hemGeneratorLaplacianStdBasisActionDef` premise left by the lower-1 pointwise-event route. The source-facing law-space selected-test Laplacian integral is enough, through the existing standard-basis Laplacian bridge, to recover the standard-basis action premise consumed by the current theorem. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventLawIntegralTraceLaplacianFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorLaplacianLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS) (hemGeneratorLaplacianEventFieldEqLaplacian : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x) (htraceFieldEqLaplacian : testRegular → emGeneratorTraceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorLaplacianStdBasisActionDef : testRegular → emGeneratorLaplacianAction = fun φ => ∫ x in Set.univ, (∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]) ∂hatRhoS := generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula hatRhoS selectedTest testRegular emGeneratorLaplacianAction hemGeneratorLaplacianLawIntegral exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStdBasisActionTraceLaplacianFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField emGeneratorLaplacianEventField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorLaplacianStdBasisActionDef hemGeneratorLaplacianEventFieldEqLaplacian htraceFieldEqLaplacian /-- Weak-FP source-action route from the state-event source formula on the current pointwise event/trace-Laplacian branch. This cycle-148 middle continuation removes `hemGeneratorLaplacianLawIntegral` as a primitive premise from the current pointwise law-integral consumer. The smaller source-facing boundary is the EM conditional-law/state-event formula: the named frozen-generator Laplacian event field has the total generator action over `Set.univ`, and its integrals over measurable state events agree with the selected-test Laplacian integrals. The pointwise event-field and trace-field Laplacian identities remain explicit. -/ theorem generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula {Ω Test E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (P : MeasureTheory.Measure Ω) (hatRhoS : MeasureTheory.Measure E) (hatXAtS : Ω → E) (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (emGeneratorTraceField emGeneratorLaplacianEventField : Test → E → Real) (testRegular : Prop) (laplacian emGeneratorLaplacianAction : Test → Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hsourceLaplacianFunctional : sourceLaplacianFunctional = fun ψ : E → Real => ∫ x, ψ x ∂hatRhoS) (hhatX : AEMeasurable hatXAtS P) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) (hsourceLaplacianFieldMeas : testRegular → ∀ φ, MeasureTheory.AEStronglyMeasurable (Laplacian.laplacian (selectedTest φ)) hatRhoS) (hlaplacianEqEmGenerator : testRegular → ∀ φ, laplacian φ = emGeneratorLaplacianAction φ) (hemGeneratorLaplacianTotalEventIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x in Set.univ, emGeneratorLaplacianEventField φ x ∂hatRhoS) (hemGeneratorLaplacianStateEventEqLaplacian : testRegular → ∀ φ (t : Set E), MeasurableSet t → (∫ x in t, emGeneratorLaplacianEventField φ x ∂hatRhoS) = ∫ x in t, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS) (hemGeneratorLaplacianEventFieldEqLaplacian : testRegular → ∀ φ, emGeneratorLaplacianEventField φ = fun x => Laplacian.laplacian (selectedTest φ) x) (htraceFieldEqLaplacian : testRegular → emGeneratorTraceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ) := by have hemGeneratorLaplacianLawIntegral : testRegular → ∀ φ, emGeneratorLaplacianAction φ = ∫ x, Laplacian.laplacian (selectedTest φ) x ∂hatRhoS := generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula hatRhoS selectedTest emGeneratorLaplacianEventField testRegular emGeneratorLaplacianAction hemGeneratorLaplacianTotalEventIntegral hemGeneratorLaplacianStateEventEqLaplacian exact generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventLawIntegralTraceLaplacianFormula P hatRhoS hatXAtS selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional emGeneratorTraceField emGeneratorLaplacianEventField testRegular laplacian emGeneratorLaplacianAction hhatRhoS hsourceLaplacianFunctional hhatX hweakFpSourceFieldEqLaplacian hsourceLaplacianFieldMeas hlaplacianEqEmGenerator hemGeneratorLaplacianLawIntegral hemGeneratorLaplacianEventFieldEqLaplacian htraceFieldEqLaplacian /-- Weak-FP source-Laplacian definition from a named source field. This cycle-139 helper targets the non-circular weak-FP side left by `generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula`. Instead of deriving the weak-FP source pullback from the standard-basis conclusion being proved by the Green route, it asks for two source-facing facts from `appendix.tex:1379-1427`: the weak-FP abstract Laplacian action is the source functional applied to a named source field, and that field is Mathlib's Laplacian of the selected weak-test representative. -/ theorem generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (laplacian : Test → Real) (hweakFpSourceActionDef : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ)) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → laplacian = fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ)) := by intro htests calc laplacian = (fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ)) := hweakFpSourceActionDef htests _ = (fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ))) := by funext φ exact congrArg sourceLaplacianFunctional (congrFun (hweakFpSourceFieldEqLaplacian htests) φ) /-- Pointwise test-Laplacian normalization from a non-circular weak-FP source field. This cycle-139 downstream bridge feeds the weak-FP source-field split into the cycle-138 pointwise test-Laplacian route. The test-calculus side still comes from `htestLaplacianStdBasisDef`, while the weak-FP side comes from the named source-action and source-field equalities above. It does not reintroduce `htestLaplacianActionDef`, `htestLaplacianOperator`, or the circular `hweakFpStdBasisDef` premise. -/ theorem generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (weakFpLaplacianSourceField : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (testLaplacianAction laplacian : Test → Real) (htestLaplacianStdBasisDef : testRegular → testLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) (hweakFpSourceActionDef : testRegular → laplacian = fun φ => sourceLaplacianFunctional (weakFpLaplacianSourceField φ)) (hweakFpSourceFieldEqLaplacian : testRegular → weakFpLaplacianSourceField = fun φ => Laplacian.laplacian (selectedTest φ)) : testRegular → ∀ φ, testLaplacianAction φ = laplacian φ := by have hweakFpLaplacianDef : testRegular → laplacian = fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ)) := generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField selectedTest weakFpLaplacianSourceField sourceLaplacianFunctional testRegular laplacian hweakFpSourceActionDef hweakFpSourceFieldEqLaplacian exact generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula selectedTest sourceLaplacianFunctional testRegular testLaplacianAction laplacian htestLaplacianStdBasisDef hweakFpLaplacianDef /-- Operator-level normalization from the two standard-basis source formulas. This cycle-135 scout helper composes the weak-FP and test-calculus standard-basis leaves. It removes the older source-pullback hypotheses from the operator normalization boundary, while keeping the real source-facing work as the two explicit standard-basis formulas from `appendix.tex:1392-1427`. -/ theorem generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula {Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (testRegular : Prop) (testLaplacianAction laplacian : Test → Real) (htestLaplacianStdBasisDef : testRegular → testLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) (hweakFpStdBasisDef : testRegular → laplacian = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) : testRegular → testLaplacianAction = laplacian := by have htestLaplacianActionDef : testRegular → testLaplacianAction = fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ)) := generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula selectedTest sourceLaplacianFunctional testRegular testLaplacianAction htestLaplacianStdBasisDef have hweakFpLaplacianDef : testRegular → laplacian = fun φ => sourceLaplacianFunctional (Laplacian.laplacian (selectedTest φ)) := generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula selectedTest sourceLaplacianFunctional testRegular laplacian hweakFpStdBasisDef exact generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback (fun φ => Laplacian.laplacian (selectedTest φ)) sourceLaplacianFunctional testRegular testLaplacianAction laplacian htestLaplacianActionDef hweakFpLaplacianDef /-- Second-Green diffusion-source handoff from standard-basis source formulas. This lower_2 continuation pushes the cycle-135 standard-basis source formulas into the downstream second-Green consumer. The older source-pullback premises `htestLaplacianActionDef` and `hweakFpLaplacianDef` are derived locally from the smaller source-facing leaves `htestLaplacianStdBasisDef` and `hweakFpStdBasisDef`; all Green, trace, box-divergence, and diffusion leaves remain explicit. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula {n : Nat} {Boundary Test E : Type*} [MeasurableSpace Boundary] [NormedAddCommGroup E] [InnerProductSpace Real E] [FiniteDimensional Real E] (a b : Fin (n + 1) → Real) (weightedField : Test → (Fin (n + 1) → Real) → Fin (n + 1) → Real) (weightedFieldDeriv : Test → (Fin (n + 1) → Real) → (Fin (n + 1) → Real) →L[Real] Fin (n + 1) → Real) (exceptionSet : Test → Set (Fin (n + 1) → Real)) (boundaryMeasure : MeasureTheory.Measure Boundary) (secondGreenTestTrace testTrace : Test → Boundary → Real) (secondGreenNormalFluxTrace : Boundary → Real) (selectedTest : Test → E → Real) (sourceLaplacianFunctional : (E → Real) → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux : Test → Real) (sigmaCoeff : Real) (hdiffusionAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = weakDiffusionAction φ) (hdiffusionLaplacianTerm : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • densityLaplacianAction φ) (hfirstGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ - negativeGradientPairAction φ = firstGreenTotal φ) (hfirstGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenTotal φ = firstGreenBoundaryFlux φ) (hfirstGreenZeroBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenBoundaryFlux φ = 0) (hsecondGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → negativeGradientPairAction φ - testLaplacianAction φ = secondGreenTotal φ) (hsecondGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenTotal φ = secondGreenBoundaryFlux φ) (hle : a ≤ b) (hsecondGreenWeightedFieldContinuous : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ContinuousOn (weightedField φ) (Set.Icc a b)) (hsecondGreenExceptionCountable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (exceptionSet φ).Countable) (hsecondGreenDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ exceptionSet φ, HasFDerivAt (weightedField φ) (weightedFieldDeriv φ x) x) (hsecondGreenDivIntegrable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.IntegrableOn (fun x => ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (Set.Icc a b) MeasureTheory.volume) (hsecondGreenBoundaryFluxDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenBoundaryFlux φ = ∫ x in Set.Icc a b, ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (hsecondGreenFaceTrace : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (b i) x) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (a i) x) i)) = ∫ y, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y ∂boundaryMeasure) (hsecondGreenTestTracePointwiseEq : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ y, secondGreenTestTrace φ y = testTrace φ y) (htestTracePointwiseZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ y, testTrace φ y = 0) (htestLaplacianStdBasisDef : testRegular → testLaplacianAction = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) (hweakFpStdBasisDef : testRegular → laplacian = fun φ => sourceLaplacianFunctional (fun x => ∑ i, iteratedFDeriv Real 2 (selectedTest φ) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i])) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ := by have htestLaplacianOperator : testRegular → testLaplacianAction = laplacian := generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula selectedTest sourceLaplacianFunctional testRegular testLaplacianAction laplacian htestLaplacianStdBasisDef hweakFpStdBasisDef have htestLaplacianLocal : ∀ φ, Admissible φ → testRegular → testLaplacianAction φ = laplacian φ := generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization Admissible testRegular testLaplacianAction laplacian htestLaplacianOperator exact generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfTestLaplacianNormalization a b weightedField weightedFieldDeriv exceptionSet boundaryMeasure secondGreenTestTrace testTrace secondGreenNormalFluxTrace Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux sigmaCoeff hdiffusionAction hdiffusionLaplacianTerm hfirstGreenResidual hfirstGreenDivergence hfirstGreenZeroBoundary hsecondGreenResidual hsecondGreenDivergence hle hsecondGreenWeightedFieldContinuous hsecondGreenExceptionCountable hsecondGreenDeriv hsecondGreenDivIntegrable hsecondGreenBoundaryFluxDivergence hsecondGreenFaceTrace hsecondGreenTestTracePointwiseEq htestTracePointwiseZero htestLaplacianLocal /-- Second-Green diffusion-source handoff with operator normalization narrowed to shared source-Laplacian pullback definitions. This dynamic-leaf worker packet replaces the direct `htestLaplacianOperator : testRegular → testLaplacianAction = laplacian` boundary by two definition-identification facts: the local test-Laplacian action and the weak-FP abstract Laplacian action are both the pullback of the same source Laplacian action along the selected weak-test representation. The analytic content remains the source calculus in `appendix.tex:1379-1427`; the proof below only composes those source-facing facts with the cycle-133 second-Green chain. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfSourceLaplacianPullback {n : Nat} {Boundary Test SourceTest : Type*} [MeasurableSpace Boundary] (a b : Fin (n + 1) → Real) (weightedField : Test → (Fin (n + 1) → Real) → Fin (n + 1) → Real) (weightedFieldDeriv : Test → (Fin (n + 1) → Real) → (Fin (n + 1) → Real) →L[Real] Fin (n + 1) → Real) (exceptionSet : Test → Set (Fin (n + 1) → Real)) (boundaryMeasure : MeasureTheory.Measure Boundary) (secondGreenTestTrace testTrace : Test → Boundary → Real) (secondGreenNormalFluxTrace : Boundary → Real) (selectedTest : Test → SourceTest) (sourceLaplacianAction : SourceTest → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux : Test → Real) (sigmaCoeff : Real) (hdiffusionAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = weakDiffusionAction φ) (hdiffusionLaplacianTerm : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakDiffusionAction φ = sigmaCoeff • densityLaplacianAction φ) (hfirstGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → densityLaplacianAction φ - negativeGradientPairAction φ = firstGreenTotal φ) (hfirstGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenTotal φ = firstGreenBoundaryFlux φ) (hfirstGreenZeroBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → firstGreenBoundaryFlux φ = 0) (hsecondGreenResidual : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → negativeGradientPairAction φ - testLaplacianAction φ = secondGreenTotal φ) (hsecondGreenDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenTotal φ = secondGreenBoundaryFlux φ) (hle : a ≤ b) (hsecondGreenWeightedFieldContinuous : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ContinuousOn (weightedField φ) (Set.Icc a b)) (hsecondGreenExceptionCountable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (exceptionSet φ).Countable) (hsecondGreenDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ exceptionSet φ, HasFDerivAt (weightedField φ) (weightedFieldDeriv φ x) x) (hsecondGreenDivIntegrable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.IntegrableOn (fun x => ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (Set.Icc a b) MeasureTheory.volume) (hsecondGreenBoundaryFluxDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → secondGreenBoundaryFlux φ = ∫ x in Set.Icc a b, ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (hsecondGreenFaceTrace : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (b i) x) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), weightedField φ (i.insertNth (a i) x) i)) = ∫ y, secondGreenTestTrace φ y * secondGreenNormalFluxTrace y ∂boundaryMeasure) (hsecondGreenTestTracePointwiseEq : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ y, secondGreenTestTrace φ y = testTrace φ y) (htestTracePointwiseZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ y, testTrace φ y = 0) (htestLaplacianActionDef : testRegular → testLaplacianAction = (fun φ => sourceLaplacianAction (selectedTest φ))) (hweakFpLaplacianDef : testRegular → laplacian = (fun φ => sourceLaplacianAction (selectedTest φ))) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ := by have htestLaplacianOperator : testRegular → testLaplacianAction = laplacian := generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback selectedTest sourceLaplacianAction testRegular testLaplacianAction laplacian htestLaplacianActionDef hweakFpLaplacianDef have htestLaplacianLocal : ∀ φ, Admissible φ → testRegular → testLaplacianAction φ = laplacian φ := generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization Admissible testRegular testLaplacianAction laplacian htestLaplacianOperator exact generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfTestLaplacianNormalization a b weightedField weightedFieldDeriv exceptionSet boundaryMeasure secondGreenTestTrace testTrace secondGreenNormalFluxTrace Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior diffusionAction weakDiffusionAction densityLaplacianAction negativeGradientPairAction testLaplacianAction laplacian firstGreenTotal firstGreenBoundaryFlux secondGreenTotal secondGreenBoundaryFlux sigmaCoeff hdiffusionAction hdiffusionLaplacianTerm hfirstGreenResidual hfirstGreenDivergence hfirstGreenZeroBoundary hsecondGreenResidual hsecondGreenDivergence hle hsecondGreenWeightedFieldContinuous hsecondGreenExceptionCountable hsecondGreenDeriv hsecondGreenDivIntegrable hsecondGreenBoundaryFluxDivergence hsecondGreenFaceTrace hsecondGreenTestTracePointwiseEq htestTracePointwiseZero htestLaplacianLocal /-- Continuity of the concrete product flux `hatRhoS * barB` on a box. This is the first concrete sub-boundary below the cycle-107 box theorem: once the density representative for `hatRhoS` and the conditional drift `barB` are continuous on the source box, the weighted Euclidean flux `x ↦ hatRhoDensity x • barB x` is continuous there. -/ theorem generalMovingTargetDiscreteHatRhoBarBWeightedFieldContinuousOnBox {n : Nat} (a b : Fin (n + 1) → Real) (hatRhoDensity : (Fin (n + 1) → Real) → Real) (barB : (Fin (n + 1) → Real) → Fin (n + 1) → Real) (hrhoContinuous : ContinuousOn hatRhoDensity (Set.Icc a b)) (hbarBContinuous : ContinuousOn barB (Set.Icc a b)) : ContinuousOn (fun x => hatRhoDensity x • barB x) (Set.Icc a b) := by exact hrhoContinuous.smul hbarBContinuous /-- Pointwise Frechet derivative of the concrete product flux `hatRhoS * barB`. This is the local Mathlib product-rule component below the cycle-108 box handoff. It converts separate derivatives of the density representative and the conditional drift into the derivative of `x ↦ hatRhoDensity x • barB x`, without proving the paper-specific integrability or boundary-trace identifications. -/ theorem generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAt {n : Nat} (hatRhoDensity : (Fin (n + 1) → Real) → Real) (barB : (Fin (n + 1) → Real) → Fin (n + 1) → Real) (hatRhoDeriv : (Fin (n + 1) → Real) →L[Real] Real) (barBDeriv : (Fin (n + 1) → Real) →L[Real] Fin (n + 1) → Real) (x : Fin (n + 1) → Real) (hrhoDeriv : HasFDerivAt hatRhoDensity hatRhoDeriv x) (hbarBDeriv : HasFDerivAt barB barBDeriv x) : HasFDerivAt (fun y => hatRhoDensity y • barB y) (hatRhoDensity x • barBDeriv + hatRhoDeriv.smulRight (barB x)) x := by change HasFDerivAt (hatRhoDensity • barB) (hatRhoDensity x • barBDeriv + hatRhoDeriv.smulRight (barB x)) x exact hrhoDeriv.smul hbarBDeriv /-- Off-countable derivative of the concrete product flux from separate density and drift derivative exception sets. The remaining cycle-108 box-trace boundary asks for Frechet differentiability of `x ↦ hatRhoDensity x • barB x` away from a countable interior exception set. This theorem reduces that input to separate density and drift derivative facts away from their own exception sets; the combined exception set is their union. -/ theorem generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAtOffUnion {n : Nat} (a b : Fin (n + 1) → Real) (hatRhoDensity : (Fin (n + 1) → Real) → Real) (barB : (Fin (n + 1) → Real) → Fin (n + 1) → Real) (hatRhoDeriv : (Fin (n + 1) → Real) → (Fin (n + 1) → Real) →L[Real] Real) (barBDeriv : (Fin (n + 1) → Real) → (Fin (n + 1) → Real) →L[Real] Fin (n + 1) → Real) (rhoException barBException : Set (Fin (n + 1) → Real)) (hrhoDeriv : ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ rhoException, HasFDerivAt hatRhoDensity (hatRhoDeriv x) x) (hbarBDeriv : ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ barBException, HasFDerivAt barB (barBDeriv x) x) : ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ (rhoException ∪ barBException), HasFDerivAt (fun y => hatRhoDensity y • barB y) (hatRhoDensity x • barBDeriv x + (hatRhoDeriv x).smulRight (barB x)) x := by intro x hx exact generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAt hatRhoDensity barB (hatRhoDeriv x) (barBDeriv x) x (hrhoDeriv x ⟨hx.1, fun hmem => hx.2 (Or.inl hmem)⟩) (hbarBDeriv x ⟨hx.1, fun hmem => hx.2 (Or.inr hmem)⟩) /-- Boundary-flux integral for the concrete product flux `hatRhoS * barB`. Compared with `generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox`, this specializes `weightedField` to the Euclidean product `x ↦ hatRhoDensity x • barB x`. It discharges the generic continuity premise from separate continuity of the density and `barB`, while leaving the remaining source-cited product-flux derivative, divergence-integrability, boundaryFlux/interior-divergence, and signed-face-to-trace identifications explicit. -/ theorem generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox {n : Nat} {Test Boundary : Type*} [MeasurableSpace Boundary] (a b : Fin (n + 1) → Real) (hatRhoDensity : (Fin (n + 1) → Real) → Real) (barB : (Fin (n + 1) → Real) → Fin (n + 1) → Real) (weightedFieldDeriv : Test → (Fin (n + 1) → Real) → (Fin (n + 1) → Real) →L[Real] Fin (n + 1) → Real) (exceptionSet : Test → Set (Fin (n + 1) → Real)) (boundaryMeasure : MeasureTheory.Measure Boundary) (boundaryFlux : Test → Real) (testTrace : Test → Boundary → Real) (normalFluxTrace : Boundary → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (hle : a ≤ b) (hrhoContinuous : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ContinuousOn hatRhoDensity (Set.Icc a b)) (hbarBContinuous : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ContinuousOn barB (Set.Icc a b)) (hexceptionCountable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (exceptionSet φ).Countable) (hderiv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ exceptionSet φ, HasFDerivAt (fun x => hatRhoDensity x • barB x) (weightedFieldDeriv φ x) x) (hdivIntegrable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.IntegrableOn (fun x => ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (Set.Icc a b) MeasureTheory.volume) (hboundaryFluxDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → boundaryFlux φ = ∫ x in Set.Icc a b, ∑ i, weightedFieldDeriv φ x (Pi.single i (1 : Real)) i) (hfaceTrace : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), (hatRhoDensity (i.insertNth (b i) x) • barB (i.insertNth (b i) x)) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), (hatRhoDensity (i.insertNth (a i) x) • barB (i.insertNth (a i) x)) i)) = ∫ y, testTrace φ y * normalFluxTrace y ∂boundaryMeasure) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → boundaryFlux φ = ∫ y, testTrace φ y * normalFluxTrace y ∂boundaryMeasure := by have hcontinuous : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ContinuousOn ((fun _ x => hatRhoDensity x • barB x) φ) (Set.Icc a b) := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary exact generalMovingTargetDiscreteHatRhoBarBWeightedFieldContinuousOnBox a b hatRhoDensity barB (hrhoContinuous φ hφ hcommon hkernel hdrift hdensity htests hboundary) (hbarBContinuous φ hφ hcommon hkernel hdrift hdensity htests hboundary) exact generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox a b (fun _ x => hatRhoDensity x • barB x) weightedFieldDeriv exceptionSet boundaryMeasure boundaryFlux testTrace normalFluxTrace Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior hle hcontinuous hexceptionCountable hderiv hdivIntegrable hboundaryFluxDivergence hfaceTrace /-- Boundary-flux integral for `hatRhoS * barB` with the product derivative instantiated from separate density and drift derivatives. This narrows the cycle-108 remaining box-trace boundary by discharging the generic off-countable Frechet differentiability input for the concrete product flux. Divergence integrability, the `boundaryFlux`/interior-divergence identity, and the signed-face-to-trace identification remain explicit. -/ theorem generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBoxProductDeriv {n : Nat} {Test Boundary : Type*} [MeasurableSpace Boundary] (a b : Fin (n + 1) → Real) (hatRhoDensity : (Fin (n + 1) → Real) → Real) (barB : (Fin (n + 1) → Real) → Fin (n + 1) → Real) (hatRhoDeriv : (Fin (n + 1) → Real) → (Fin (n + 1) → Real) →L[Real] Real) (barBDeriv : (Fin (n + 1) → Real) → (Fin (n + 1) → Real) →L[Real] Fin (n + 1) → Real) (rhoException barBException : Test → Set (Fin (n + 1) → Real)) (boundaryMeasure : MeasureTheory.Measure Boundary) (boundaryFlux : Test → Real) (testTrace : Test → Boundary → Real) (normalFluxTrace : Boundary → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (hle : a ≤ b) (hrhoContinuous : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ContinuousOn hatRhoDensity (Set.Icc a b)) (hbarBContinuous : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ContinuousOn barB (Set.Icc a b)) (hrhoExceptionCountable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (rhoException φ).Countable) (hbarBExceptionCountable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (barBException φ).Countable) (hrhoDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ rhoException φ, HasFDerivAt hatRhoDensity (hatRhoDeriv x) x) (hbarBDeriv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ barBException φ, HasFDerivAt barB (barBDeriv x) x) (hdivIntegrable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.IntegrableOn (fun x => ∑ i, (hatRhoDensity x • barBDeriv x + (hatRhoDeriv x).smulRight (barB x)) (Pi.single i (1 : Real)) i) (Set.Icc a b) MeasureTheory.volume) (hboundaryFluxDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → boundaryFlux φ = ∫ x in Set.Icc a b, ∑ i, (hatRhoDensity x • barBDeriv x + (hatRhoDeriv x).smulRight (barB x)) (Pi.single i (1 : Real)) i) (hfaceTrace : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (∑ i : Fin (n + 1), ((∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), (hatRhoDensity (i.insertNth (b i) x) • barB (i.insertNth (b i) x)) i) - ∫ (x : Fin n → Real) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), (hatRhoDensity (i.insertNth (a i) x) • barB (i.insertNth (a i) x)) i)) = ∫ y, testTrace φ y * normalFluxTrace y ∂boundaryMeasure) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → boundaryFlux φ = ∫ y, testTrace φ y * normalFluxTrace y ∂boundaryMeasure := by have hexceptionCountable : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → (rhoException φ ∪ barBException φ).Countable := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary exact (hrhoExceptionCountable φ hφ hcommon hkernel hdrift hdensity htests hboundary).union (hbarBExceptionCountable φ hφ hcommon hkernel hdrift hdensity htests hboundary) have hderiv : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ (rhoException φ ∪ barBException φ), HasFDerivAt (fun x => hatRhoDensity x • barB x) (hatRhoDensity x • barBDeriv x + (hatRhoDeriv x).smulRight (barB x)) x := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary x hx exact generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAtOffUnion a b hatRhoDensity barB hatRhoDeriv barBDeriv (rhoException φ) (barBException φ) (hrhoDeriv φ hφ hcommon hkernel hdrift hdensity htests hboundary) (hbarBDeriv φ hφ hcommon hkernel hdrift hdensity htests hboundary) x hx exact generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox a b hatRhoDensity barB (fun _ x => hatRhoDensity x • barBDeriv x + (hatRhoDeriv x).smulRight (barB x)) (fun φ => rhoException φ ∪ barBException φ) boundaryMeasure boundaryFlux testTrace normalFluxTrace Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior hle hrhoContinuous hbarBContinuous hexceptionCountable hderiv hdivIntegrable hboundaryFluxDivergence hfaceTrace /-- Drift source action with the zero-boundary-flux input narrowed to a boundary trace-product condition. This version keeps the cycle-101 product-rule and divergence-theorem inputs, but replaces the raw `hzeroBoundary : boundaryFlux phi = 0` premise by the smaller source-cited boundary statement that the boundary-flux integral of the test trace times the normal trace of `hatRhoS * barB` vanishes. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundary {State Vec Test Boundary : Type*} [MeasurableSpace State] [MeasurableSpace Boundary] [NormedAddCommGroup Vec] [InnerProductSpace Real Vec] (hatRhoS : MeasureTheory.Measure State) (barB : State → Vec) (testGrad : Test → State → Vec) (boundaryMeasure : MeasureTheory.Measure Boundary) (testTrace : Test → Boundary → Real) (normalFluxTrace : Boundary → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction driftDiv : Test → Real) (pairBound : Test → Real) (divTotal boundaryFlux : Test → Real) (hdriftBarBAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = (fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂hatRhoS) barB φ) (hbarBIntegrable : MeasureTheory.Integrable barB hatRhoS) (hpairMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun x => inner Real (testGrad φ x) (barB x)) hatRhoS) (hgradNormBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂hatRhoS, ‖testGrad φ x‖ ≤ pairBound φ) (hproductRule : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ + (∫ x, inner Real (testGrad φ x) (barB x) ∂hatRhoS) = divTotal φ) (hdivergenceTheorem : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → divTotal φ = boundaryFlux φ) (hboundaryFluxIntegral : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → boundaryFlux φ = ∫ y, testTrace φ y * normalFluxTrace y ∂boundaryMeasure) (htraceProductZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, testTrace φ y * normalFluxTrace y = 0) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ) := by have hzeroBoundary : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → boundaryFlux φ = 0 := generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero boundaryMeasure boundaryFlux testTrace normalFluxTrace Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior hboundaryFluxIntegral htraceProductZero exact generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientProductRuleBoundary hatRhoS barB testGrad Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction driftDiv pairBound divTotal boundaryFlux hdriftBarBAction hbarBIntegrable hpairMeas hgradNormBound hproductRule hdivergenceTheorem hzeroBoundary /-- Drift source action with the trace-product input narrowed to zero admissible-test trace on the boundary. This lower packet discharges the supplied `htraceProductZero` premise of `generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundary` from the smaller source-facing condition that admissible tests have zero boundary trace a.e. The boundary-flux integral representation, product rule, divergence theorem, and weak-test gradient bound remain explicit. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero {State Vec Test Boundary : Type*} [MeasurableSpace State] [MeasurableSpace Boundary] [NormedAddCommGroup Vec] [InnerProductSpace Real Vec] (hatRhoS : MeasureTheory.Measure State) (barB : State → Vec) (testGrad : Test → State → Vec) (boundaryMeasure : MeasureTheory.Measure Boundary) (testTrace : Test → Boundary → Real) (normalFluxTrace : Boundary → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction driftDiv : Test → Real) (pairBound : Test → Real) (divTotal boundaryFlux : Test → Real) (hdriftBarBAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = (fun f ψ => ∫ x, inner Real (testGrad ψ x) (f x) ∂hatRhoS) barB φ) (hbarBIntegrable : MeasureTheory.Integrable barB hatRhoS) (hpairMeas : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.AEStronglyMeasurable (fun x => inner Real (testGrad φ x) (barB x)) hatRhoS) (hgradNormBound : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ x ∂hatRhoS, ‖testGrad φ x‖ ≤ pairBound φ) (hproductRule : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftDiv φ + (∫ x, inner Real (testGrad φ x) (barB x) ∂hatRhoS) = divTotal φ) (hdivergenceTheorem : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → divTotal φ = boundaryFlux φ) (hboundaryFluxIntegral : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → boundaryFlux φ = ∫ y, testTrace φ y * normalFluxTrace y ∂boundaryMeasure) (htestTraceZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, testTrace φ y = 0) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ) := by have htraceProductZero : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → ∀ᵐ y ∂boundaryMeasure, testTrace φ y * normalFluxTrace y = 0 := generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero boundaryMeasure testTrace normalFluxTrace Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior htestTraceZero exact generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundary hatRhoS barB testGrad boundaryMeasure testTrace normalFluxTrace Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction driftDiv pairBound divTotal boundaryFlux hdriftBarBAction hbarBIntegrable hpairMeas hgradNormBound hproductRule hdivergenceTheorem hboundaryFluxIntegral htraceProductZero /-- Drift weak-action pairing from conditional-drift component pairings. This cycle-95 lower theorem narrows the first half of `ASTIS.SALD.cycle94.remaining_barB_divergence_boundary`. Once `barB` is the paper component field `dotTk • condC + sigmaCoeff • condScore`, and once the weak gradient pairing is linear and congruent under pointwise equality, it is enough to prove the drift generator action against the two conditional component fields. The remaining source-cited work is the conditional-expectation theorem identifying those component pairings, and the separate no-boundary divergence theorem for `hatRhoS * barB`. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings {State Vec Test F : Type*} [AddCommGroup Vec] [Module Real Vec] [AddCommGroup F] [Module Real F] (barB condC condScore : State → Vec) (weakGradPairing : (State → Vec) → Test → F) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction : Test → F) (dotTk sigmaCoeff : Real) (hbar : ∀ x, barB x = dotTk • condC x + sigmaCoeff • condScore x) (hweakCongr : ∀ {f g : State → Vec}, (∀ x, f x = g x) → ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing f φ = weakGradPairing g φ) (hweakAdd : ∀ (f g : State → Vec) (φ : Test), Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing (fun x => f x + g x) φ = weakGradPairing f φ + weakGradPairing g φ) (hweakSmul : ∀ (a : Real) (f : State → Vec) (φ : Test), Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing (fun x => a • f x) φ = a • weakGradPairing f φ) (hdriftComponents : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = dotTk • weakGradPairing condC φ + sigmaCoeff • weakGradPairing condScore φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = weakGradPairing barB φ := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary have hcombo : weakGradPairing (fun x => dotTk • condC x + sigmaCoeff • condScore x) φ = dotTk • weakGradPairing condC φ + sigmaCoeff • weakGradPairing condScore φ := by rw [ hweakAdd (fun x => dotTk • condC x) (fun x => sigmaCoeff • condScore x) φ hφ hcommon hkernel hdrift hdensity htests hboundary, hweakSmul dotTk condC φ hφ hcommon hkernel hdrift hdensity htests hboundary, hweakSmul sigmaCoeff condScore φ hφ hcommon hkernel hdrift hdensity htests hboundary ] calc driftAction φ = dotTk • weakGradPairing condC φ + sigmaCoeff • weakGradPairing condScore φ := by exact hdriftComponents φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = weakGradPairing (fun x => dotTk • condC x + sigmaCoeff • condScore x) φ := by rw [hcombo] _ = weakGradPairing barB φ := by exact hweakCongr (fun x => (hbar x).symm) φ hφ hcommon hkernel hdrift hdensity htests hboundary /-- Drift source action with the `barB` weak action reduced to component conditional pairings. This composes the cycle-95 component-pairing theorem with the cycle-94 `barB` drift-source handoff. It removes the direct supplied `hdriftBarBAction` premise: lower/future work can now target the two component conditional-expectation generator pairings plus the still-separate divergence and no-boundary theorem. -/ theorem generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBComponentPairings {State Vec Test F : Type*} [AddCommGroup Vec] [Module Real Vec] [AddCommGroup F] [Module Real F] (barB condC condScore : State → Vec) (weakGradPairing : (State → Vec) → Test → F) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction driftDiv : Test → F) (dotTk sigmaCoeff : Real) (hbar : ∀ x, barB x = dotTk • condC x + sigmaCoeff • condScore x) (hweakCongr : ∀ {f g : State → Vec}, (∀ x, f x = g x) → ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing f φ = weakGradPairing g φ) (hweakAdd : ∀ (f g : State → Vec) (φ : Test), Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing (fun x => f x + g x) φ = weakGradPairing f φ + weakGradPairing g φ) (hweakSmul : ∀ (a : Real) (f : State → Vec) (φ : Test), Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing (fun x => a • f x) φ = a • weakGradPairing f φ) (hdriftComponents : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = dotTk • weakGradPairing condC φ + sigmaCoeff • weakGradPairing condScore φ) (hbarBWeakDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing barB φ = -(driftDiv φ)) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ) := by have hdriftBarBAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = weakGradPairing barB φ := generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings barB condC condScore weakGradPairing Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction dotTk sigmaCoeff hbar hweakCongr hweakAdd hweakSmul hdriftComponents exact generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction barB weakGradPairing Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction driftDiv hdriftBarBAction hbarBWeakDivergence /-- One-component `condDistrib` generator-pairing handoff. This cycle-96 lower theorem targets the conditional-expectation half of `appendix.tex:1368-1377`. If a named component field such as `condC_{k,s}` or `condScore_{k,s}` is chosen as a `hatRhoS`-a.e. version of the canonical conditional integral against `condDistrib X_k^eta hatX_s P`, and if the weak test-gradient pairing is congruent under that `hatRhoS`-a.e. equality, then it is enough to prove the generator action against the canonical integral field. The remaining analytic boundary is that canonical `condDistrib`/`condexp` generator identity; this lemma does not prove the regular conditional law, weak FP theorem, or divergence/no-boundary identity. -/ theorem generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion {Ω State Vec Test F : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace ℝ Vec] [StandardBorelSpace State] [Nonempty State] [AddCommGroup F] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {componentIntegrand : State × State → Vec} {componentField : State → Vec} (componentAction : Test → F) (weakGradPairing : (State → Vec) → Test → F) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (hfieldAe : (fun x => ∫ y, componentIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) =ᵐ[hatRhoS] componentField) (hweakAeCongr : ∀ {f g : State → Vec}, f =ᵐ[hatRhoS] g → ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing f φ = weakGradPairing g φ) (hcanonical : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → componentAction φ = weakGradPairing (fun x => ∫ y, componentIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → componentAction φ = weakGradPairing componentField φ := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary calc componentAction φ = weakGradPairing (fun x => ∫ y, componentIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) φ := by exact hcanonical φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = weakGradPairing componentField φ := by exact hweakAeCongr hfieldAe φ hφ hcommon hkernel hdrift hdensity htests hboundary /-- Canonical component pairing from the named-law `condDistrib` integral. This cycle-97 lower theorem is the compiled use of `AutoSamplingTheory.condDistribIntegralNamedLawIntegral` requested after the middle disintegration backfill. It removes the raw cycle-96 `hcanonical` premise by reducing it to the paper-definition equalities for the sample component action and weak test-gradient pairing, plus integrability of the paired test component. It still does not prove those definition equalities, the component-field version equality, the divergence/no-boundary theorem, or the weak Fokker--Planck equation. -/ theorem generalMovingTargetDiscreteCondDistribComponentWeakPairingOfIntegralAction {Ω State Vec Test F : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [NormedAddCommGroup Vec] [NormedSpace Real Vec] [StandardBorelSpace State] [Nonempty State] [NormedAddCommGroup F] [NormedSpace Real F] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {hatRhoS : MeasureTheory.Measure State} {hatXAtS Xk : Ω → State} {componentIntegrand : State × State → Vec} {componentField : State → Vec} {pairedTestIntegrand : Test → State × State → F} (componentAction : Test → F) (weakGradPairing : (State → Vec) → Test → F) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map hatXAtS P) (hhatXAtS : AEMeasurable hatXAtS P) (hXk : AEMeasurable Xk P) (hfieldAe : (fun x => ∫ y, componentIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) =ᵐ[hatRhoS] componentField) (hpairedInt : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → MeasureTheory.Integrable (pairedTestIntegrand φ) (MeasureTheory.Measure.map (fun ω => (hatXAtS ω, Xk ω)) P)) (hcomponentActionDef : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → componentAction φ = ∫ ω, pairedTestIntegrand φ (hatXAtS ω, Xk ω) ∂P) (hweakCanonicalDef : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing (fun x => ∫ y, componentIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) φ = ∫ x, ∫ y, pairedTestIntegrand φ (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x ∂hatRhoS) (hweakAeCongr : ∀ {f g : State → Vec}, f =ᵐ[hatRhoS] g → ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing f φ = weakGradPairing g φ) : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → componentAction φ = weakGradPairing componentField φ := by have hcanonical : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → componentAction φ = weakGradPairing (fun x => ∫ y, componentIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) φ := by intro φ hφ hcommon hkernel hdrift hdensity htests hboundary have hdisintegration : (∫ x, ∫ y, pairedTestIntegrand φ (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x ∂hatRhoS) = ∫ ω, pairedTestIntegrand φ (hatXAtS ω, Xk ω) ∂P := by exact AutoSamplingTheory.condDistribIntegralNamedLawIntegral (μ := P) (hatRho := hatRhoS) (X := hatXAtS) (Y := Xk) (f := pairedTestIntegrand φ) hhatRhoS hhatXAtS hXk (hpairedInt φ hφ hcommon hkernel hdrift hdensity htests hboundary) calc componentAction φ = ∫ ω, pairedTestIntegrand φ (hatXAtS ω, Xk ω) ∂P := by exact hcomponentActionDef φ hφ hcommon hkernel hdrift hdensity htests hboundary _ = (∫ x, ∫ y, pairedTestIntegrand φ (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x ∂hatRhoS) := by exact hdisintegration.symm _ = weakGradPairing (fun x => ∫ y, componentIntegrand (x, y) ∂ProbabilityTheory.condDistrib Xk hatXAtS P x) φ := by exact (hweakCanonicalDef φ hφ hcommon hkernel hdrift hdensity htests hboundary).symm exact generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion (P := P) (hatRhoS := hatRhoS) (hatXAtS := hatXAtS) (Xk := Xk) (componentIntegrand := componentIntegrand) (componentField := componentField) componentAction weakGradPairing Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior hfieldAe hweakAeCongr hcanonical /-- Mapped-law weak derivative with the drift source action factored through the conditional drift `barB`. This cycle-94 refinement composes the cycle-92 direct law-derivative route with `generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction`. It no longer takes the primitive `hdriftSource` hypothesis. The exact remaining source-cited drift boundary is: identify the frozen EM drift generator action with the weak pairing against `barB`, then prove the no-boundary integration-by-parts/divergence identity for that pairing. -/ theorem generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff {Ω State Vec Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) (hatX : Real → Ω → State) (hatRhoS : MeasureTheory.Measure State) (barB : State → Vec) (testEval : Test → State → Real) (weakGradPairing : (State → Vec) → Test → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (driftAction diffusionAction driftDiv laplacian : Test → Real) (s0 sigmaEta sigmaCoeff : Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map (hatX s0) P) (hhatX : ∀ s, AEMeasurable (hatX s) P) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (MeasureTheory.Measure.map (hatX s) P)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hsampleGenerator : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → HasDerivAt (fun s => ∫ ω, testEval φ (hatX s ω) ∂P) (driftAction φ + diffusionAction φ) s0) (hdriftBarBAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = weakGradPairing barB φ) (hbarBWeakDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing barB φ = -(driftDiv φ)) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : ∀ φ, Admissible φ → HasDerivAt (fun s => ∫ x, testEval φ x ∂MeasureTheory.Measure.map (hatX s) P) (-(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) s0 := by have _ := hhatRhoS have hdriftSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ) := generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction barB weakGradPairing Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction driftDiv hdriftBarBAction hbarBWeakDivergence exact generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff P hatX testEval Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction diffusionAction driftDiv laplacian s0 sigmaEta sigmaCoeff hhatX htestMeas hcommon hkernel hdrift hdensity htests hboundary hcoeff hsampleGenerator hdriftSource hdiffusionSource /-- Source-sign handoff with the drift source action factored through the conditional drift `barB`. This cycle-94 lower refinement applies the same `barB` weak-action boundary to the normalized weak Fokker--Planck source-sign route. It removes the primitive `hdriftSource` premise from the source-sign consumer itself: the remaining drift obligations are exactly the conditional-expectation generator pairing against `barB` and the no-boundary divergence theorem for `hat rho_s * barB`. -/ theorem generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff {Ω State Vec Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) (hatX : Real → Ω → State) (hatRhoS : MeasureTheory.Measure State) (barB : State → Vec) (testEval : Test → State → Real) (weakGradPairing : (State → Vec) → Test → Real) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (partialS driftAction diffusionAction driftDiv laplacian : Test → Real) (s0 sigmaEta sigmaCoeff : Real) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map (hatX s0) P) (hhatX : ∀ s, AEMeasurable (hatX s) P) (htestMeas : ∀ φ s, Admissible φ → MeasureTheory.AEStronglyMeasurable (testEval φ) (MeasureTheory.Measure.map (hatX s) P)) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hsampleGenerator : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → HasDerivAt (fun s => ∫ ω, testEval φ (hatX s ω) ∂P) (driftAction φ + diffusionAction φ) s0) (hlawDerivative : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → HasDerivAt (fun s => ∫ x, testEval φ x ∂MeasureTheory.Measure.map (hatX s) P) (partialS φ) s0) (hdriftBarBAction : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = weakGradPairing barB φ) (hbarBWeakDivergence : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → weakGradPairing barB φ = -(driftDiv φ)) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := by have _ := hhatRhoS have hdriftSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ) := generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction barB weakGradPairing Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior driftAction driftDiv hdriftBarBAction hbarBWeakDivergence exact generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff P hatX testEval Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior partialS driftAction diffusionAction driftDiv laplacian s0 sigmaEta sigmaCoeff hhatX htestMeas hcommon hkernel hdrift hdensity htests hboundary hcoeff hsampleGenerator hlawDerivative hdriftSource hdiffusionSource /-- Cycle-82 middle bridge from endpoint/conditional weak-FP readiness to the generator-piece source-sign handoff. Cycle 81 packaged the endpoint laws, named `hatRhoS` marginal, kernel orientation, and `barB` regularity into an abstract `WeakFpPrereq`. The weak Fokker--Planck theorem itself is still analytic input: this wrapper only exposes how that readiness predicate supplies the common-space, conditional-kernel, and drift-regularity hypotheses consumed by the existing generator-piece source-sign handoff, while density/time regularity, admissible tests, and boundary behavior remain explicit hypotheses. -/ theorem generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfReadinessAndGeneratorPiecesHandoff {State Vec Kernel Test F : Type*} [MeasurableSpace State] [AddCommGroup F] [Module Real F] (WeakFpPrereq : MeasureTheory.Measure State → Kernel → (State → Vec) → Prop) (hatRhoS : MeasureTheory.Measure State) (kernel : Kernel) (barB : State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (partialS generatorAction driftAction diffusionAction driftDiv laplacian : Test → F) (sigmaEta sigmaCoeff : Real) (hready : WeakFpPrereq hatRhoS kernel barB) (hcommonOfReady : WeakFpPrereq hatRhoS kernel barB → commonSpace) (hkernelOfReady : WeakFpPrereq hatRhoS kernel barB → conditionalKernel) (hdriftOfReady : WeakFpPrereq hatRhoS kernel barB → driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hgenerator : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → partialS φ = generatorAction φ) (hgeneratorSplit : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → generatorAction φ = driftAction φ + diffusionAction φ) (hdriftSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ)) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior partialS generatorAction driftAction diffusionAction driftDiv laplacian sigmaEta sigmaCoeff (hcommonOfReady hready) (hkernelOfReady hready) (hdriftOfReady hready) hdensity htests hboundary hcoeff hgenerator hgeneratorSplit hdriftSource hdiffusionSource /-- Cycle-82 lower bridge from endpoint/conditional readiness data all the way to the weak-test source signs. This composes the cycle-81 endpoint/conditional `WeakFpPrereq` readiness wrapper with the cycle-82 readiness-to-generator-piece source-sign wrapper. It still does not prove the analytic weak Fokker-Planck theorem: density/time regularity, admissible tests, boundary behavior, the generator/time-derivative identity, and the drift/diffusion source actions remain supplied hypotheses. -/ theorem generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff {Ω State Vec Kernel Test F : Type*} [MeasurableSpace Ω] [MeasurableSpace State] [AddCommGroup State] [Module Real State] [AddCommGroup Vec] [Module Real Vec] [AddCommGroup F] [Module Real F] (P : MeasureTheory.Measure Ω) (hatX : Real → Ω → State) (Xk XNext frozenDrift W noise : Ω → State) (rhoK rhoNext hatRhoS : MeasureTheory.Measure State) (kernel : Kernel) (KernelCompatibleSwapped : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (KernelCompatibleOriginal : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (KernelIntegralField : Kernel → (State → Vec) → Prop) (FieldMeasurable FieldIntegrable : (State → Vec) → Prop) (WeakFpPrereq : MeasureTheory.Measure State → Kernel → (State → Vec) → Prop) (barB condC condScore : State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (partialS generatorAction driftAction diffusionAction driftDiv laplacian : Test → F) (s sK sNext eta sigmaEta dotTk barCoeff fpCoeff : Real) (hXk : Measurable Xk) (hhatAtS : Measurable (hatX s)) (hhatLeft : hatX sK =ᵐ[P] fun ω => Xk ω + (sK - sK) • frozenDrift ω + sigmaEta • (W ω - W ω)) (hhatRight : hatX sNext =ᵐ[P] fun ω => Xk ω + (sNext - sK) • frozenDrift ω + sigmaEta • noise ω) (hstep : sNext - sK = eta) (hupdate : XNext =ᵐ[P] fun ω => Xk ω + eta • frozenDrift ω + sigmaEta • noise ω) (hrhoK : rhoK = MeasureTheory.Measure.map Xk P) (hrhoNext : rhoNext = MeasureTheory.Measure.map XNext P) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map (hatX s) P) (hcompatSwapped : KernelCompatibleSwapped (MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P) hatRhoS kernel) (hcompatBridge : ∀ {joint jointSwapped : MeasureTheory.Measure (State × State)} {μ : MeasureTheory.Measure State}, MeasureTheory.Measure.map Prod.swap joint = jointSwapped → KernelCompatibleSwapped jointSwapped μ kernel → KernelCompatibleOriginal joint μ kernel) (hcondC : KernelIntegralField kernel condC) (hcondScore : KernelIntegralField kernel condScore) (hmeasOfKernel : ∀ {f : State → Vec}, KernelIntegralField kernel f → FieldMeasurable f) (hintOfKernel : ∀ {f : State → Vec}, KernelIntegralField kernel f → FieldIntegrable f) (hbar : ∀ x, barB x = dotTk • condC x + barCoeff • condScore x) (hmeasAdd : ∀ {f g : State → Vec}, FieldMeasurable f → FieldMeasurable g → FieldMeasurable (fun x => f x + g x)) (hmeasSmul : ∀ (a : Real) {f : State → Vec}, FieldMeasurable f → FieldMeasurable (fun x => a • f x)) (hintAdd : ∀ {f g : State → Vec}, FieldIntegrable f → FieldIntegrable g → FieldIntegrable (fun x => f x + g x)) (hintSmul : ∀ (a : Real) {f : State → Vec}, FieldIntegrable f → FieldIntegrable (fun x => a • f x)) (hmeasCongr : ∀ {f g : State → Vec}, (∀ x, f x = g x) → FieldMeasurable f → FieldMeasurable g) (hintCongr : ∀ {f g : State → Vec}, (∀ x, f x = g x) → FieldIntegrable f → FieldIntegrable g) (hweakReady : MeasureTheory.Measure.map (hatX sK) P = rhoK → MeasureTheory.Measure.map (hatX sNext) P = rhoNext → MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) = hatRhoS → MeasureTheory.Measure.map Prod.fst (MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P) = hatRhoS → MeasureTheory.Measure.map Prod.swap (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) = MeasureTheory.Measure.map (fun ω => (hatX s ω, Xk ω)) P → KernelCompatibleOriginal (MeasureTheory.Measure.map (fun ω => (Xk ω, hatX s ω)) P) hatRhoS kernel → FieldMeasurable barB → FieldIntegrable barB → WeakFpPrereq hatRhoS kernel barB) (hcommonOfReady : WeakFpPrereq hatRhoS kernel barB → commonSpace) (hkernelOfReady : WeakFpPrereq hatRhoS kernel barB → conditionalKernel) (hdriftOfReady : WeakFpPrereq hatRhoS kernel barB → driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : fpCoeff = sigmaEta ^ 2 / 2) (hgenerator : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → partialS φ = generatorAction φ) (hgeneratorSplit : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → generatorAction φ = driftAction φ + diffusionAction φ) (hdriftSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ)) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = fpCoeff • laplacian φ) : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := by have hreadyPack := generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff P hatX Xk XNext frozenDrift W noise rhoK rhoNext hatRhoS kernel KernelCompatibleSwapped KernelCompatibleOriginal KernelIntegralField FieldMeasurable FieldIntegrable WeakFpPrereq barB condC condScore s sK sNext eta sigmaEta dotTk barCoeff hXk hhatAtS hhatLeft hhatRight hstep hupdate hrhoK hrhoNext hhatRhoS hcompatSwapped hcompatBridge hcondC hcondScore hmeasOfKernel hintOfKernel hbar hmeasAdd hmeasSmul hintAdd hintSmul hmeasCongr hintCongr hweakReady exact generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfReadinessAndGeneratorPiecesHandoff WeakFpPrereq hatRhoS kernel barB Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior partialS generatorAction driftAction diffusionAction driftDiv laplacian sigmaEta fpCoeff hreadyPack.2.2.2.2.2.2.2.2 hcommonOfReady hkernelOfReady hdriftOfReady hdensity htests hboundary hcoeff hgenerator hgeneratorSplit hdriftSource hdiffusionSource /-- KL-derivative handoff after weak conditional Fokker--Planck substitution. This is the cycle-73 proof-producing wrapper for the first handoff in `appendix.tex:1358-1387`. Once the analytic backend has supplied the differentiated KL identity and the weak conditional Fokker--Planck identity for the admissible log-density-ratio test, this theorem substitutes the weak FP right-hand side while preserving the paper's negative drift-divergence sign and positive `sigma_eta^2/2` Laplacian coefficient. It does not prove density, admissibility of the log-ratio test, integration by parts, or the Fisher identity. -/ theorem generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar {Test : Type*} (Admissible : Test → Prop) (logRatioTest : Test) (dK targetTimeTerm : Real) (partialS driftDiv laplacian : Test → Real) (sigmaEta sigmaCoeff : Real) (hlog : Admissible logRatioTest) (hkl : dK = partialS logRatioTest - targetTimeTerm) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hweak : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + sigmaCoeff * laplacian φ) : dK = (-(driftDiv logRatioTest) + (sigmaEta ^ 2 / 2) * laplacian logRatioTest) - targetTimeTerm := by rw [hkl, hweak logRatioTest hlog, hcoeff] /-- KL-derivative handoff from normalized weak-FP source signs. This lower cycle-78 wrapper isolates the final substitution after the weak conditional Fokker--Planck backend has already produced the paper-normalized source signs for admissible tests. It consumes only the differentiated KL formula and the normalized weak-FP identity at the admissible log-ratio test; all density, admissibility, weak-FP, boundary, and integration-by-parts facts remain explicit upstream obligations. -/ theorem generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns {Test : Type*} (Admissible : Test → Prop) (logRatioTest : Test) (dK targetTimeTerm : Real) (partialS driftDiv laplacian : Test → Real) (sigmaEta : Real) (hlog : Admissible logRatioTest) (hkl : dK = partialS logRatioTest - targetTimeTerm) (hsourceSigns : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) : dK = (-(driftDiv logRatioTest) + (sigmaEta ^ 2 / 2) * laplacian logRatioTest) - targetTimeTerm := by have hlogWeak : partialS logRatioTest = -(driftDiv logRatioTest) + (sigmaEta ^ 2 / 2) * laplacian logRatioTest := by simpa [smul_eq_mul] using hsourceSigns logRatioTest hlog rw [hkl, hlogWeak] /-- KL-derivative handoff together with the log-ratio weak-FP action. This lower cycle-83 companion keeps the two source-cited steps adjacent: the weak conditional Fokker--Planck source signs evaluated at the admissible log-density-ratio test, and the substitution of that action into `eq:general_KL_derivative_0_discrete`. It is still only local equality bookkeeping under supplied analytic hypotheses. -/ theorem generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction {Test : Type*} (Admissible : Test → Prop) (logRatioTest : Test) (dK targetTimeTerm : Real) (partialS driftDiv laplacian : Test → Real) (sigmaEta : Real) (hlog : Admissible logRatioTest) (hkl : dK = partialS logRatioTest - targetTimeTerm) (hsourceSigns : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) : partialS logRatioTest = -(driftDiv logRatioTest) + (sigmaEta ^ 2 / 2) * laplacian logRatioTest ∧ dK = (-(driftDiv logRatioTest) + (sigmaEta ^ 2 / 2) * laplacian logRatioTest) - targetTimeTerm := by have hlogWeak : partialS logRatioTest = -(driftDiv logRatioTest) + (sigmaEta ^ 2 / 2) * laplacian logRatioTest := by simpa [smul_eq_mul] using hsourceSigns logRatioTest hlog refine ⟨hlogWeak, ?_⟩ rw [hkl, hlogWeak] /-- Scalar mass-conservation drop for the discrete general KL derivative. In `appendix.tex:1358-1366`, differentiating `KL(hat rho_s || tilde pi_s)` first gives the log-ratio action plus the scalar mass derivative `int partial_s hat rho_s dx`, and the paper drops that mass term because it is zero. This lemma records only that Real-algebra handoff; KL differentiability, density/absolute-continuity, and mass conservation remain analytic hypotheses. -/ theorem generalMovingTargetDiscreteKlDerivativeMassConservationDropScalar (dK logAction massTerm targetTimeTerm : Real) (hklRaw : dK = logAction + massTerm - targetTimeTerm) (hmass : massTerm = 0) : dK = logAction - targetTimeTerm := by rw [hklRaw, hmass] ring /-- KL weak-FP handoff from a raw differentiated KL display. This cycle-87 companion removes the older supplied post-mass-drop `hkl` display from the weak-FP-to-`dK` equality bookkeeping. It starts instead from the raw differentiated KL display with the explicit mass term from `appendix.tex:1358-1366`, drops that term using the supplied mass-conservation identity, and then applies the existing source-sign handoff at the admissible log-ratio test. It does not prove the raw KL differentiation theorem, mass conservation, or log-ratio admissibility. -/ theorem generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction {Test : Type*} (Admissible : Test → Prop) (logRatioTest : Test) (dK massTerm targetTimeTerm : Real) (partialS driftDiv laplacian : Test → Real) (sigmaEta : Real) (hlog : Admissible logRatioTest) (hklRaw : dK = partialS logRatioTest + massTerm - targetTimeTerm) (hmass : massTerm = 0) (hsourceSigns : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) : partialS logRatioTest = -(driftDiv logRatioTest) + (sigmaEta ^ 2 / 2) * laplacian logRatioTest ∧ dK = (-(driftDiv logRatioTest) + (sigmaEta ^ 2 / 2) * laplacian logRatioTest) - targetTimeTerm := by have hkl : dK = partialS logRatioTest - targetTimeTerm := generalMovingTargetDiscreteKlDerivativeMassConservationDropScalar dK (partialS logRatioTest) massTerm targetTimeTerm hklRaw hmass exact generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction Admissible logRatioTest dK targetTimeTerm partialS driftDiv laplacian sigmaEta hlog hkl hsourceSigns /-- General moving-target mapped-law constant-test mass conservation. For `appendix.tex:1358-1366`, the source drops `int partial_s hat rho_s dx` after naming `hat rho_s = Law(hat X_s)`. This local theorem proves the constant-test part of that sentence: once the raw KL backend identifies `massTerm` as the derivative of the mapped-law pairing against the constant test `1`, the derivative is zero. It does not prove the raw KL differentiation theorem or the identification of `massTerm` with that pairing. -/ theorem generalMovingTargetDiscreteKlMassTermZeroOfLawConstantTestDerivative {Ω State : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] (hatX : Real → Ω → State) (s0 massTerm : Real) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hmassDeriv : HasDerivAt (fun s => ∫ _, (1 : Real) ∂MeasureTheory.Measure.map (hatX s) P) massTerm s0) : massTerm = 0 := by have hzeroDeriv : HasDerivAt (fun s => ∫ _, (1 : Real) ∂MeasureTheory.Measure.map (hatX s) P) 0 s0 := by refine lawMapIntegralHasDerivAtOfSample (P := P) (X := hatX) (φ := fun _ : State => (1 : Real)) hhatX ?_ ?_ · intro s exact MeasureTheory.aestronglyMeasurable_const · simpa using hasDerivAt_const s0 (∫ _, (1 : Real) ∂P) exact hmassDeriv.unique hzeroDeriv /-- General moving-target raw KL handoff with mapped-law mass conservation. This cycle-93 refinement removes the primitive `hmass : massTerm = 0` input from `generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction` when the mass term is the derivative of the constant weak-test pairing for `hat rho_s = Law(hat X_s)`. The remaining analytic inputs are still the raw KL differentiation display, the log-ratio admissible test, the target-time term, and the weak-FP source signs. -/ theorem generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfLawConstantTestMassAndSourceSignsWithLogAction {Ω State Test : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] (hatX : Real → Ω → State) (Admissible : Test → Prop) (logRatioTest : Test) (dK massTerm targetTimeTerm : Real) (partialS driftDiv laplacian : Test → Real) (sigmaEta s0 : Real) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hlog : Admissible logRatioTest) (hklRaw : dK = partialS logRatioTest + massTerm - targetTimeTerm) (hmassDeriv : HasDerivAt (fun s => ∫ _, (1 : Real) ∂MeasureTheory.Measure.map (hatX s) P) massTerm s0) (hsourceSigns : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) : partialS logRatioTest = -(driftDiv logRatioTest) + (sigmaEta ^ 2 / 2) * laplacian logRatioTest ∧ dK = (-(driftDiv logRatioTest) + (sigmaEta ^ 2 / 2) * laplacian logRatioTest) - targetTimeTerm := by have hmass : massTerm = 0 := generalMovingTargetDiscreteKlMassTermZeroOfLawConstantTestDerivative P hatX s0 massTerm hhatX hmassDeriv exact generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction Admissible logRatioTest dK massTerm targetTimeTerm partialS driftDiv laplacian sigmaEta hlog hklRaw hmass hsourceSigns /-- Mathlib log-ratio convention for the discrete general KL boundary. The paper writes the weak test as `log(hat rho_s / tilde pi_s)`. In the Lean-facing backend this is represented by Mathlib's log-likelihood ratio `llr hatRho tildePi`, namely `log ((d hatRho / d tildePi).toReal)`. This records the zero-density convention used before proving weak-test admissibility or differentiating KL. -/ theorem generalMovingTargetDiscreteKlLogRatioLlrDef {State : Type*} [MeasurableSpace State] (hatRho tildePi : MeasureTheory.Measure State) : MeasureTheory.llr hatRho tildePi = fun x => Real.log (hatRho.rnDeriv tildePi x).toReal := by exact MeasureTheory.llr_def hatRho tildePi /-- Log-ratio measurability and integrability from finite KL. This cycle-87 lower theorem discharges the log-ratio measurability and integrability side hypotheses in `appendix.tex:1358-1366`, provided the local KL backend supplies finite `KL(hatRho || tildePi)`. Mathlib's finite-KL criterion also gives the absolute-continuity hypothesis needed to view the paper's `log(hat rho_s / tilde pi_s)` as the log-likelihood-ratio weak test. It does not prove weak-test admissibility, differentiability under the KL integral, mass conservation, or the target-time derivative identity. -/ theorem generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl {State : Type*} [MeasurableSpace State] (hatRho tildePi : MeasureTheory.Measure State) (hfiniteKl : InformationTheory.klDiv hatRho tildePi ≠ ⊤) : MeasureTheory.Measure.AbsolutelyContinuous hatRho tildePi ∧ MeasureTheory.AEStronglyMeasurable (MeasureTheory.llr hatRho tildePi) hatRho ∧ MeasureTheory.Integrable (MeasureTheory.llr hatRho tildePi) hatRho := by have hkl := InformationTheory.klDiv_ne_top_iff.mp hfiniteKl exact ⟨hkl.1, (MeasureTheory.stronglyMeasurable_llr hatRho tildePi).aestronglyMeasurable, hkl.2⟩ /-- Narrow weak-test closure boundary for the discrete general KL log-ratio. After cycle 87, finite KL supplies the absolute-continuity, measurability, and integrability side of the Mathlib `llr hatRho tildePi` representative. The remaining analytic theorem in `appendix.tex:1358-1366` is not another abstract `hlog`; it is the closure of the weak-Fokker--Planck admissible test class under the source log-ratio representative, including smoothing/Sobolev approximation, boundary/no-flux control, and closure of the drift-divergence and Laplacian actions. This structure names those exact ingredients without proving them. -/ structure GeneralMovingTargetDiscreteKlLogRatioAdmissibilityClosure {State : Type*} [MeasurableSpace State] (Admissible : (State → Real) → Prop) (hatRho tildePi : MeasureTheory.Measure State) where densityTimeRegular : Prop zeroSetConvention : Prop smoothSobolevApproximation : Prop boundaryNoFlux : Prop driftDivActionClosure : Prop laplacianActionClosure : Prop targetTimeIntegrability : Prop prerequisites : densityTimeRegular ∧ zeroSetConvention ∧ smoothSobolevApproximation ∧ boundaryNoFlux ∧ driftDivActionClosure ∧ laplacianActionClosure ∧ targetTimeIntegrability admissible_of_finite_kl_regularity : MeasureTheory.Measure.AbsolutelyContinuous hatRho tildePi → MeasureTheory.AEStronglyMeasurable (MeasureTheory.llr hatRho tildePi) hatRho → MeasureTheory.Integrable (MeasureTheory.llr hatRho tildePi) hatRho → densityTimeRegular → zeroSetConvention → smoothSobolevApproximation → boundaryNoFlux → driftDivActionClosure → laplacianActionClosure → targetTimeIntegrability → Admissible (MeasureTheory.llr hatRho tildePi) /-- Finite-KL handoff into the narrowed log-ratio admissibility boundary. This cycle-88 lower theorem removes the old broad supplied `hlog : Admissible logRatioTest` when the log-ratio test is the Mathlib `llr hatRho tildePi`. It proves the measure-regularity part from finite KL using `generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl`; the only remaining supplied analytic boundary is the named closure package above. It does not prove the smoothing/Sobolev approximation, boundary condition, weak-FP action closure, KL differentiability, mass conservation, integration by parts, or Fisher-information identification. -/ theorem generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure {State : Type*} [MeasurableSpace State] (Admissible : (State → Real) → Prop) (hatRho tildePi : MeasureTheory.Measure State) (hfiniteKl : InformationTheory.klDiv hatRho tildePi ≠ ⊤) (hclosure : GeneralMovingTargetDiscreteKlLogRatioAdmissibilityClosure Admissible hatRho tildePi) : Admissible (MeasureTheory.llr hatRho tildePi) := by have hreg := generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl hatRho tildePi hfiniteKl have hpre := hclosure.prerequisites exact hclosure.admissible_of_finite_kl_regularity hreg.1 hreg.2.1 hreg.2.2 hpre.1 hpre.2.1 hpre.2.2.1 hpre.2.2.2.1 hpre.2.2.2.2.1 hpre.2.2.2.2.2.1 hpre.2.2.2.2.2.2 /-- Raw KL weak-FP handoff for the Mathlib `llr` test and mapped-law mass. This cycle-93 lower refinement composes the two accepted KL/log-ratio backfills for `appendix.tex:1358-1366`: finite KL plus the named cycle-88 closure package supplies admissibility of the exact Mathlib `llr hatRho tildePi` test, and the mapped-law constant-test derivative supplies the mass drop for `hat rho_s = Law(hat X_s)`. The raw KL differentiability display, target-time term, weak-FP source signs, integration by parts, and Fisher identification remain explicit upstream boundaries. -/ theorem generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfFiniteKlLlrLawConstantTestMassWithLogAction {Ω State : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] (hatX : Real → Ω → State) (Admissible : (State → Real) → Prop) (hatRho tildePi : MeasureTheory.Measure State) (dK massTerm targetTimeTerm : Real) (partialS driftDiv laplacian : (State → Real) → Real) (sigmaEta s0 : Real) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hfiniteKl : InformationTheory.klDiv hatRho tildePi ≠ ⊤) (hclosure : GeneralMovingTargetDiscreteKlLogRatioAdmissibilityClosure Admissible hatRho tildePi) (hklRaw : dK = partialS (MeasureTheory.llr hatRho tildePi) + massTerm - targetTimeTerm) (hmassDeriv : HasDerivAt (fun s => ∫ _, (1 : Real) ∂MeasureTheory.Measure.map (hatX s) P) massTerm s0) (hsourceSigns : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) : partialS (MeasureTheory.llr hatRho tildePi) = -(driftDiv (MeasureTheory.llr hatRho tildePi)) + (sigmaEta ^ 2 / 2) * laplacian (MeasureTheory.llr hatRho tildePi) ∧ dK = (-(driftDiv (MeasureTheory.llr hatRho tildePi)) + (sigmaEta ^ 2 / 2) * laplacian (MeasureTheory.llr hatRho tildePi)) - targetTimeTerm := by have hlog : Admissible (MeasureTheory.llr hatRho tildePi) := generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure Admissible hatRho tildePi hfiniteKl hclosure exact generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfLawConstantTestMassAndSourceSignsWithLogAction P hatX Admissible (MeasureTheory.llr hatRho tildePi) dK massTerm targetTimeTerm partialS driftDiv laplacian sigmaEta s0 hhatX hlog hklRaw hmassDeriv hsourceSigns /-- Source-cited raw KL differentiability package at the finite-KL `llr` test. This is the cycle-99 boundary for `appendix.tex:1358-1366`. It is narrower than a primitive scalar `hklRaw`: finite KL fixes the Mathlib log-likelihood-ratio representative and supplies its measurability and integrability, while the remaining source-cited analysis is endpoint-safe differentiation of the KL integral, target-time derivative integrability and identification, and the mapped-law constant-test mass derivative. The package is deliberately not a proof of KL differentiability. It names the exact hypotheses a future lower proof must discharge before the existing weak-FP-to-`dK` handoff can run without a primitive `hklRaw` premise. -/ structure GeneralMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlr {Ω State : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] (hatX : Real → Ω → State) (hatRho tildePi : MeasureTheory.Measure State) (dK massTerm targetTimeTerm : Real) (partialS : (State → Real) → Real) (s0 : Real) where finiteKl : InformationTheory.klDiv hatRho tildePi ≠ ⊤ hatX_aemeasurable : ∀ s, AEMeasurable (hatX s) P densityPathRegular : Prop endpointSafeKlDifferentiation : Prop llrWeakActionIntegrable : Prop targetTimeTermIntegrable : Prop targetTimeDerivativeFormula : Prop prerequisites : densityPathRegular ∧ endpointSafeKlDifferentiation ∧ llrWeakActionIntegrable ∧ targetTimeTermIntegrable ∧ targetTimeDerivativeFormula massTermDerivative : HasDerivAt (fun s => ∫ _, (1 : Real) ∂MeasureTheory.Measure.map (hatX s) P) massTerm s0 rawKlDerivative_of_regularity : MeasureTheory.Measure.AbsolutelyContinuous hatRho tildePi → MeasureTheory.AEStronglyMeasurable (MeasureTheory.llr hatRho tildePi) hatRho → MeasureTheory.Integrable (MeasureTheory.llr hatRho tildePi) hatRho → densityPathRegular → endpointSafeKlDifferentiation → llrWeakActionIntegrable → targetTimeTermIntegrable → targetTimeDerivativeFormula → dK = partialS (MeasureTheory.llr hatRho tildePi) + massTerm - targetTimeTerm /-- Extract the raw KL display from the source-cited finite-KL `llr` package. The only proof done here is the Mathlib finite-KL handoff to absolute continuity, measurability, and integrability of `llr`; the analytic parametric-integral/KL differentiation content remains in the package fields. -/ theorem generalMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlrHklRaw {Ω State : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] (hatX : Real → Ω → State) (hatRho tildePi : MeasureTheory.Measure State) (dK massTerm targetTimeTerm : Real) (partialS : (State → Real) → Real) (s0 : Real) (hraw : GeneralMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlr P hatX hatRho tildePi dK massTerm targetTimeTerm partialS s0) : dK = partialS (MeasureTheory.llr hatRho tildePi) + massTerm - targetTimeTerm := by have hreg := generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl hatRho tildePi hraw.finiteKl have hpre := hraw.prerequisites exact hraw.rawKlDerivative_of_regularity hreg.1 hreg.2.1 hreg.2.2 hpre.1 hpre.2.1 hpre.2.2.1 hpre.2.2.2.1 hpre.2.2.2.2 /-- KL weak-FP handoff from the narrowed raw-KL finite-KL `llr` boundary. This cycle-99 theorem removes the primitive `hklRaw` and `hmassDeriv` inputs from the exact `llr` route by consuming `GeneralMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlr`. The package still has source-cited analytic fields for KL differentiability and target-time identification; the theorem only composes that narrowed boundary with the already compiled finite-KL admissibility and mapped-law mass handoff. -/ theorem generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlBoundaryAtFiniteKlLlrWithLogAction {Ω State : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] (hatX : Real → Ω → State) (Admissible : (State → Real) → Prop) (hatRho tildePi : MeasureTheory.Measure State) (dK massTerm targetTimeTerm : Real) (partialS driftDiv laplacian : (State → Real) → Real) (sigmaEta s0 : Real) (hraw : GeneralMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlr P hatX hatRho tildePi dK massTerm targetTimeTerm partialS s0) (hclosure : GeneralMovingTargetDiscreteKlLogRatioAdmissibilityClosure Admissible hatRho tildePi) (hsourceSigns : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) : partialS (MeasureTheory.llr hatRho tildePi) = -(driftDiv (MeasureTheory.llr hatRho tildePi)) + (sigmaEta ^ 2 / 2) * laplacian (MeasureTheory.llr hatRho tildePi) ∧ dK = (-(driftDiv (MeasureTheory.llr hatRho tildePi)) + (sigmaEta ^ 2 / 2) * laplacian (MeasureTheory.llr hatRho tildePi)) - targetTimeTerm := by have hklRaw := generalMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlrHklRaw P hatX hatRho tildePi dK massTerm targetTimeTerm partialS s0 hraw exact generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfFiniteKlLlrLawConstantTestMassWithLogAction P hatX Admissible hatRho tildePi dK massTerm targetTimeTerm partialS driftDiv laplacian sigmaEta s0 hraw.hatX_aemeasurable hraw.finiteKl hclosure hklRaw hraw.massTermDerivative hsourceSigns /-- Source-cited no-mass raw KL package at the finite-KL `llr` test. This lower refinement uses the already formalized mapped-law constant-test calculus instead of keeping the mass derivative inside the raw-KL boundary. The remaining source-cited analytic theorem is therefore the no-mass differentiation display from `appendix.tex:1358-1366`: `dK = partialS (llr hatRho tildePi) - targetTimeTerm`. -/ structure GeneralMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlr {Ω State : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] (hatX : Real → Ω → State) (hatRho tildePi : MeasureTheory.Measure State) (dK targetTimeTerm : Real) (partialS : (State → Real) → Real) (s0 : Real) where finiteKl : InformationTheory.klDiv hatRho tildePi ≠ ⊤ hatX_aemeasurable : ∀ s, AEMeasurable (hatX s) P densityPathRegular : Prop endpointSafeKlDifferentiation : Prop llrWeakActionIntegrable : Prop targetTimeTermIntegrable : Prop targetTimeDerivativeFormula : Prop prerequisites : densityPathRegular ∧ endpointSafeKlDifferentiation ∧ llrWeakActionIntegrable ∧ targetTimeTermIntegrable ∧ targetTimeDerivativeFormula rawKlDerivativeNoMass_of_regularity : MeasureTheory.Measure.AbsolutelyContinuous hatRho tildePi → MeasureTheory.AEStronglyMeasurable (MeasureTheory.llr hatRho tildePi) hatRho → MeasureTheory.Integrable (MeasureTheory.llr hatRho tildePi) hatRho → densityPathRegular → endpointSafeKlDifferentiation → llrWeakActionIntegrable → targetTimeTermIntegrable → targetTimeDerivativeFormula → dK = partialS (MeasureTheory.llr hatRho tildePi) - targetTimeTerm /-- Extract the no-mass raw KL display from the finite-KL `llr` package. -/ theorem generalMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlrHkl {Ω State : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] (hatX : Real → Ω → State) (hatRho tildePi : MeasureTheory.Measure State) (dK targetTimeTerm : Real) (partialS : (State → Real) → Real) (s0 : Real) (hraw : GeneralMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlr P hatX hatRho tildePi dK targetTimeTerm partialS s0) : dK = partialS (MeasureTheory.llr hatRho tildePi) - targetTimeTerm := by have hreg := generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl hatRho tildePi hraw.finiteKl have hpre := hraw.prerequisites exact hraw.rawKlDerivativeNoMass_of_regularity hreg.1 hreg.2.1 hreg.2.2 hpre.1 hpre.2.1 hpre.2.2.1 hpre.2.2.2.1 hpre.2.2.2.2 /-- KL weak-FP handoff from the no-mass raw-KL finite-KL `llr` boundary. This theorem removes the `massTermDerivative` field from the exact finite-KL `llr` route. The derivative of the mapped-law constant weak-test pairing is proved locally from `lawMapIntegralHasDerivAtOfSample`; the only remaining source-cited KL boundary is the no-mass raw derivative display above. -/ theorem generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfNoMassRawKlBoundaryAtFiniteKlLlrWithLogAction {Ω State : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] (hatX : Real → Ω → State) (Admissible : (State → Real) → Prop) (hatRho tildePi : MeasureTheory.Measure State) (dK targetTimeTerm : Real) (partialS driftDiv laplacian : (State → Real) → Real) (sigmaEta s0 : Real) (hraw : GeneralMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlr P hatX hatRho tildePi dK targetTimeTerm partialS s0) (hclosure : GeneralMovingTargetDiscreteKlLogRatioAdmissibilityClosure Admissible hatRho tildePi) (hsourceSigns : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) : partialS (MeasureTheory.llr hatRho tildePi) = -(driftDiv (MeasureTheory.llr hatRho tildePi)) + (sigmaEta ^ 2 / 2) * laplacian (MeasureTheory.llr hatRho tildePi) ∧ dK = (-(driftDiv (MeasureTheory.llr hatRho tildePi)) + (sigmaEta ^ 2 / 2) * laplacian (MeasureTheory.llr hatRho tildePi)) - targetTimeTerm := by have hklNoMass := generalMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlrHkl P hatX hatRho tildePi dK targetTimeTerm partialS s0 hraw have hklRaw : dK = partialS (MeasureTheory.llr hatRho tildePi) + (0 : Real) - targetTimeTerm := by rw [hklNoMass] ring have hmassDeriv : HasDerivAt (fun s => ∫ _, (1 : Real) ∂MeasureTheory.Measure.map (hatX s) P) (0 : Real) s0 := by refine lawMapIntegralHasDerivAtOfSample (P := P) (X := hatX) (φ := fun _ : State => (1 : Real)) hraw.hatX_aemeasurable ?_ ?_ · intro s exact MeasureTheory.aestronglyMeasurable_const · simpa using hasDerivAt_const s0 (∫ _, (1 : Real) ∂P) exact generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfFiniteKlLlrLawConstantTestMassWithLogAction P hatX Admissible hatRho tildePi dK (0 : Real) targetTimeTerm partialS driftDiv laplacian sigmaEta s0 hraw.hatX_aemeasurable hraw.finiteKl hclosure hklRaw hmassDeriv hsourceSigns /-- Pure source-cited no-mass KL differentiability package at the finite-KL `llr` test. Cycle 105 removes the sample-space and mapped-law mass data from the no-mass KL boundary. At this point the remaining theorem is purely a measure-path statement: finite KL fixes the Mathlib log-likelihood-ratio representative, and the source-cited analysis must justify endpoint-safe differentiation of `KL(hatRho || tildePi)` directly in the no-mass form `dK = partialS (llr hatRho tildePi) - targetTimeTerm`. -/ structure GeneralMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlr {State : Type*} [MeasurableSpace State] (hatRho tildePi : MeasureTheory.Measure State) (dK targetTimeTerm : Real) (partialS : (State → Real) → Real) where finiteKl : InformationTheory.klDiv hatRho tildePi ≠ ⊤ densityPathRegular : Prop endpointSafeKlDifferentiation : Prop llrWeakActionIntegrable : Prop targetTimeTermIntegrable : Prop targetTimeDerivativeFormula : Prop prerequisites : densityPathRegular ∧ endpointSafeKlDifferentiation ∧ llrWeakActionIntegrable ∧ targetTimeTermIntegrable ∧ targetTimeDerivativeFormula rawKlDerivativeNoMass_of_regularity : MeasureTheory.Measure.AbsolutelyContinuous hatRho tildePi → MeasureTheory.AEStronglyMeasurable (MeasureTheory.llr hatRho tildePi) hatRho → MeasureTheory.Integrable (MeasureTheory.llr hatRho tildePi) hatRho → densityPathRegular → endpointSafeKlDifferentiation → llrWeakActionIntegrable → targetTimeTermIntegrable → targetTimeDerivativeFormula → dK = partialS (MeasureTheory.llr hatRho tildePi) - targetTimeTerm /-- Extract the no-mass KL display from the pure finite-KL `llr` package. -/ theorem generalMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlrHkl {State : Type*} [MeasurableSpace State] (hatRho tildePi : MeasureTheory.Measure State) (dK targetTimeTerm : Real) (partialS : (State → Real) → Real) (hraw : GeneralMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlr hatRho tildePi dK targetTimeTerm partialS) : dK = partialS (MeasureTheory.llr hatRho tildePi) - targetTimeTerm := by have hreg := generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl hatRho tildePi hraw.finiteKl have hpre := hraw.prerequisites exact hraw.rawKlDerivativeNoMass_of_regularity hreg.1 hreg.2.1 hreg.2.2 hpre.1 hpre.2.1 hpre.2.2.1 hpre.2.2.2.1 hpre.2.2.2.2 /-- KL weak-FP handoff from the pure no-mass raw-KL finite-KL `llr` boundary. Unlike the cycle-99 no-mass handoff, this theorem does not route through a zero mass derivative or a sample-space law. It consumes the pure no-mass KL boundary, the finite-KL admissibility closure for the exact `llr` test, and the normalized weak-FP source signs, then returns the source-signed `dK` display. -/ theorem generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfPureNoMassRawKlBoundaryAtFiniteKlLlrWithLogAction {State : Type*} [MeasurableSpace State] (Admissible : (State → Real) → Prop) (hatRho tildePi : MeasureTheory.Measure State) (dK targetTimeTerm : Real) (partialS driftDiv laplacian : (State → Real) → Real) (sigmaEta : Real) (hraw : GeneralMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlr hatRho tildePi dK targetTimeTerm partialS) (hclosure : GeneralMovingTargetDiscreteKlLogRatioAdmissibilityClosure Admissible hatRho tildePi) (hsourceSigns : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ) : partialS (MeasureTheory.llr hatRho tildePi) = -(driftDiv (MeasureTheory.llr hatRho tildePi)) + (sigmaEta ^ 2 / 2) * laplacian (MeasureTheory.llr hatRho tildePi) ∧ dK = (-(driftDiv (MeasureTheory.llr hatRho tildePi)) + (sigmaEta ^ 2 / 2) * laplacian (MeasureTheory.llr hatRho tildePi)) - targetTimeTerm := by have hkl := generalMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlrHkl hatRho tildePi dK targetTimeTerm partialS hraw have hlog : Admissible (MeasureTheory.llr hatRho tildePi) := generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure Admissible hatRho tildePi hraw.finiteKl hclosure exact generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction Admissible (MeasureTheory.llr hatRho tildePi) dK targetTimeTerm partialS driftDiv laplacian sigmaEta hlog hkl hsourceSigns /-- Dominated target-time derivative for the general discrete KL boundary. In `appendix.tex:1358-1366`, the target-time term is `int (hat rho_s / tilde pi_s) * partial_s tilde pi_s dx`. This theorem is the Mathlib parametric-integral substep for that term: a fixed density-ratio weight, a target-density path, pointwise `HasDerivAt`, and a local integrable dominating function give both integrability of the weighted target derivative and the derivative of the weighted target integral. -/ theorem generalMovingTargetDiscreteTargetTimeDerivativeOfDominated {State : Type*} [MeasurableSpace State] (baseMeasure : MeasureTheory.Measure State) (densityRatio : State → Real) (targetDensity targetDensityDeriv : Real → State → Real) (s0 targetTimeTerm : Real) (neighborhood : Set Real) (bound : State → Real) (hterm : targetTimeTerm = ∫ x, densityRatio x * targetDensityDeriv s0 x ∂baseMeasure) (hneighborhood : neighborhood ∈ 𝓝 s0) (hFMeas : ∀ᶠ s in 𝓝 s0, MeasureTheory.AEStronglyMeasurable (fun x => densityRatio x * targetDensity s x) baseMeasure) (hFInt : MeasureTheory.Integrable (fun x => densityRatio x * targetDensity s0 x) baseMeasure) (hDerivMeas : MeasureTheory.AEStronglyMeasurable (fun x => densityRatio x * targetDensityDeriv s0 x) baseMeasure) (hDerivBound : ∀ᵐ x ∂baseMeasure, ∀ s ∈ neighborhood, ‖densityRatio x * targetDensityDeriv s x‖ ≤ bound x) (hBoundInt : MeasureTheory.Integrable bound baseMeasure) (hTargetDensityDeriv : ∀ᵐ x ∂baseMeasure, ∀ s ∈ neighborhood, HasDerivAt (fun t => targetDensity t x) (targetDensityDeriv s x) s) : MeasureTheory.Integrable (fun x => densityRatio x * targetDensityDeriv s0 x) baseMeasure ∧ HasDerivAt (fun s => ∫ x, densityRatio x * targetDensity s x ∂baseMeasure) targetTimeTerm s0 := by have hWeightedDeriv : ∀ᵐ x ∂baseMeasure, ∀ s ∈ neighborhood, HasDerivAt (fun t => densityRatio x * targetDensity t x) (densityRatio x * targetDensityDeriv s x) s := by filter_upwards [hTargetDensityDeriv] with x hx intro s hs exact (hx s hs).const_mul (densityRatio x) have hmain := hasDerivAt_integral_of_dominated_loc_of_deriv_le (F := fun s x => densityRatio x * targetDensity s x) (F' := fun s x => densityRatio x * targetDensityDeriv s x) (x₀ := s0) (s := neighborhood) (bound := bound) (μ := baseMeasure) hneighborhood hFMeas hFInt hDerivMeas hDerivBound hBoundInt hWeightedDeriv refine ⟨hmain.1, ?_⟩ rw [hterm] exact hmain.2 /-- Transfer the target-time theorem from a chosen weight to the paper's source density-ratio representative. The dominated theorem above intentionally keeps the fixed weight abstract. This lower bridge isolates the remaining source identification: once that weight agrees almost everywhere with the paper's `hat rho_s / tilde pi_s` representative, the target-time integral, integrability, and weighted target-integral derivative can be stated with the source ratio instead. -/ theorem generalMovingTargetDiscreteTargetTimeDerivativeSourceRatioCongr {State : Type*} [MeasurableSpace State] (baseMeasure : MeasureTheory.Measure State) (densityRatio sourceRatio : State → Real) (targetDensity targetDensityDeriv : Real → State → Real) (s0 targetTimeTerm : Real) (hratio : densityRatio =ᵐ[baseMeasure] sourceRatio) (hterm : targetTimeTerm = ∫ x, densityRatio x * targetDensityDeriv s0 x ∂baseMeasure) (hIntegrable : MeasureTheory.Integrable (fun x => densityRatio x * targetDensityDeriv s0 x) baseMeasure) (hDerivative : HasDerivAt (fun s => ∫ x, densityRatio x * targetDensity s x ∂baseMeasure) targetTimeTerm s0) : targetTimeTerm = ∫ x, sourceRatio x * targetDensityDeriv s0 x ∂baseMeasure ∧ MeasureTheory.Integrable (fun x => sourceRatio x * targetDensityDeriv s0 x) baseMeasure ∧ HasDerivAt (fun s => ∫ x, sourceRatio x * targetDensity s x ∂baseMeasure) targetTimeTerm s0 := by have hDerivIntegrand : (fun x => densityRatio x * targetDensityDeriv s0 x) =ᵐ[baseMeasure] fun x => sourceRatio x * targetDensityDeriv s0 x := by filter_upwards [hratio] with x hx rw [hx] have hTargetIntegrand : ∀ s, (fun x => densityRatio x * targetDensity s x) =ᵐ[baseMeasure] fun x => sourceRatio x * targetDensity s x := by intro s filter_upwards [hratio] with x hx rw [hx] refine ⟨?_, ?_, ?_⟩ · rw [hterm] exact MeasureTheory.integral_congr_ae hDerivIntegrand · exact hIntegrable.congr hDerivIntegrand · have hfun : (fun s => ∫ x, sourceRatio x * targetDensity s x ∂baseMeasure) = fun s => ∫ x, densityRatio x * targetDensity s x ∂baseMeasure := by funext s exact (MeasureTheory.integral_congr_ae (hTargetIntegrand s)).symm simpa [hfun] using hDerivative /-- Feed the dominated target-time theorem into the pure finite-KL `llr` KL-differentiability package fields. This narrows the remaining cycle-105 boundary without adding sample-space law data. Finite KL still supplies the Mathlib `llr` regularity, while the target-time fields are reduced to the concrete dominated parametric-integral statement above plus source-specific bridges from that concrete statement to the package's two abstract target-time propositions. -/ theorem generalMovingTargetDiscretePureRawKlTargetTimeFieldsOfDominated {State : Type*} [MeasurableSpace State] (hatRho tildePi baseMeasure : MeasureTheory.Measure State) (densityRatio : State → Real) (targetDensity targetDensityDeriv : Real → State → Real) (s0 targetTimeTerm : Real) (targetTimeTermIntegrable targetTimeDerivativeFormula : Prop) (neighborhood : Set Real) (bound : State → Real) (hfiniteKl : InformationTheory.klDiv hatRho tildePi ≠ ⊤) (hterm : targetTimeTerm = ∫ x, densityRatio x * targetDensityDeriv s0 x ∂baseMeasure) (hneighborhood : neighborhood ∈ 𝓝 s0) (hFMeas : ∀ᶠ s in 𝓝 s0, MeasureTheory.AEStronglyMeasurable (fun x => densityRatio x * targetDensity s x) baseMeasure) (hFInt : MeasureTheory.Integrable (fun x => densityRatio x * targetDensity s0 x) baseMeasure) (hDerivMeas : MeasureTheory.AEStronglyMeasurable (fun x => densityRatio x * targetDensityDeriv s0 x) baseMeasure) (hDerivBound : ∀ᵐ x ∂baseMeasure, ∀ s ∈ neighborhood, ‖densityRatio x * targetDensityDeriv s x‖ ≤ bound x) (hBoundInt : MeasureTheory.Integrable bound baseMeasure) (hTargetDensityDeriv : ∀ᵐ x ∂baseMeasure, ∀ s ∈ neighborhood, HasDerivAt (fun t => targetDensity t x) (targetDensityDeriv s x) s) (hIntegrableField : MeasureTheory.Measure.AbsolutelyContinuous hatRho tildePi → MeasureTheory.AEStronglyMeasurable (MeasureTheory.llr hatRho tildePi) hatRho → MeasureTheory.Integrable (MeasureTheory.llr hatRho tildePi) hatRho → MeasureTheory.Integrable (fun x => densityRatio x * targetDensityDeriv s0 x) baseMeasure → targetTimeTermIntegrable) (hFormulaField : MeasureTheory.Measure.AbsolutelyContinuous hatRho tildePi → MeasureTheory.AEStronglyMeasurable (MeasureTheory.llr hatRho tildePi) hatRho → MeasureTheory.Integrable (MeasureTheory.llr hatRho tildePi) hatRho → HasDerivAt (fun s => ∫ x, densityRatio x * targetDensity s x ∂baseMeasure) targetTimeTerm s0 → targetTimeDerivativeFormula) : targetTimeTermIntegrable ∧ targetTimeDerivativeFormula := by have hreg := generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl hatRho tildePi hfiniteKl have htarget := generalMovingTargetDiscreteTargetTimeDerivativeOfDominated baseMeasure densityRatio targetDensity targetDensityDeriv s0 targetTimeTerm neighborhood bound hterm hneighborhood hFMeas hFInt hDerivMeas hDerivBound hBoundInt hTargetDensityDeriv exact ⟨hIntegrableField hreg.1 hreg.2.1 hreg.2.2 htarget.1, hFormulaField hreg.1 hreg.2.1 hreg.2.2 htarget.2⟩ /-- KL-derivative handoff composed through the admissible weak-FP source signs. This lower wrapper makes the cycle-73 dependency on the cycle-72 admissible weak-test source-sign theorem explicit. It first normalizes the supplied weak conditional Fokker--Planck identity to the paper coefficient `sigma_eta^2/2`, then substitutes the normalized identity into the differentiated KL formula at the log-density-ratio test. It is still only Real equality bookkeeping under explicit analytic hypotheses. -/ theorem generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns {Test : Type*} (Admissible : Test → Prop) (logRatioTest : Test) (dK targetTimeTerm : Real) (partialS driftDiv laplacian : Test → Real) (sigmaEta sigmaCoeff : Real) (hlog : Admissible logRatioTest) (hkl : dK = partialS logRatioTest - targetTimeTerm) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hweak : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + sigmaCoeff * laplacian φ) : dK = (-(driftDiv logRatioTest) + (sigmaEta ^ 2 / 2) * laplacian logRatioTest) - targetTimeTerm := by have hweakSmul : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + sigmaCoeff • laplacian φ := by intro φ hφ simpa [smul_eq_mul] using hweak φ hφ have hsourceSigns : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff Admissible partialS driftDiv laplacian sigmaEta sigmaCoeff hcoeff hweakSmul exact generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns Admissible logRatioTest dK targetTimeTerm partialS driftDiv laplacian sigmaEta hlog hkl hsourceSigns /-- KL-derivative handoff composed through the generator-level weak-FP pieces. This cycle-78 wrapper connects the cycle-77 generator/source-sign refinement directly to the cycle-73 KL derivative substitution. The analytic generator theorem, conditional law, density, admissible log-ratio test, and boundary facts are still supplied as hypotheses; the proof only composes those supplied facts and preserves the paper's source signs and `sigma_eta^2/2` coefficient. -/ theorem generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces {Test : Type*} (Admissible : Test → Prop) (logRatioTest : Test) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (dK targetTimeTerm : Real) (partialS generatorAction driftAction diffusionAction driftDiv laplacian : Test → Real) (sigmaEta sigmaCoeff : Real) (hlog : Admissible logRatioTest) (hcommon : commonSpace) (hkernel : conditionalKernel) (hdrift : driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hkl : dK = partialS logRatioTest - targetTimeTerm) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hgenerator : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → partialS φ = generatorAction φ) (hgeneratorSplit : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → generatorAction φ = driftAction φ + diffusionAction φ) (hdriftSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction φ = -(driftDiv φ)) (hdiffusionSource : ∀ φ, Admissible φ → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction φ = sigmaCoeff • laplacian φ) : dK = (-(driftDiv logRatioTest) + (sigmaEta ^ 2 / 2) * laplacian logRatioTest) - targetTimeTerm := by have hsourceSigns : ∀ φ, Admissible φ → partialS φ = -(driftDiv φ) + (sigmaEta ^ 2 / 2) • laplacian φ := generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior partialS generatorAction driftAction diffusionAction driftDiv laplacian sigmaEta sigmaCoeff hcommon hkernel hdrift hdensity htests hboundary hcoeff hgenerator hgeneratorSplit hdriftSource hdiffusionSource exact generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns Admissible logRatioTest dK targetTimeTerm partialS driftDiv laplacian sigmaEta hlog hkl hsourceSigns /-- Cycle-83 handoff from endpoint/conditional weak-FP source signs to the KL-derivative display. This wrapper composes the cycle-82 endpoint/conditional source-sign handoff with the normalized weak-FP-to-KL substitution. It is still a supplied- hypothesis bridge: the endpoint law, conditional kernel, density/time regularity, generator theorem, admissible log-ratio test, KL differentiation, boundary behavior, and integration-by-parts steps all remain upstream obligations. -/ theorem generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoff {Omega State Vec Kernel Test : Type*} [MeasurableSpace Omega] [MeasurableSpace State] [AddCommGroup State] [Module Real State] [AddCommGroup Vec] [Module Real Vec] (P : MeasureTheory.Measure Omega) (hatX : Real → Omega → State) (Xk XNext frozenDrift W noise : Omega → State) (rhoK rhoNext hatRhoS : MeasureTheory.Measure State) (kernel : Kernel) (KernelCompatibleSwapped : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (KernelCompatibleOriginal : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (KernelIntegralField : Kernel → (State → Vec) → Prop) (FieldMeasurable FieldIntegrable : (State → Vec) → Prop) (WeakFpPrereq : MeasureTheory.Measure State → Kernel → (State → Vec) → Prop) (barB condC condScore : State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (dK targetTimeTerm : Real) (partialS generatorAction driftAction diffusionAction driftDiv laplacian : Test → Real) (logRatioTest : Test) (s sK sNext eta sigmaEta dotTk barCoeff fpCoeff : Real) (hlog : Admissible logRatioTest) (hkl : dK = partialS logRatioTest - targetTimeTerm) (hXk : Measurable Xk) (hhatAtS : Measurable (hatX s)) (hhatLeft : hatX sK =ᵐ[P] fun omega => Xk omega + (sK - sK) • frozenDrift omega + sigmaEta • (W omega - W omega)) (hhatRight : hatX sNext =ᵐ[P] fun omega => Xk omega + (sNext - sK) • frozenDrift omega + sigmaEta • noise omega) (hstep : sNext - sK = eta) (hupdate : XNext =ᵐ[P] fun omega => Xk omega + eta • frozenDrift omega + sigmaEta • noise omega) (hrhoK : rhoK = MeasureTheory.Measure.map Xk P) (hrhoNext : rhoNext = MeasureTheory.Measure.map XNext P) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map (hatX s) P) (hcompatSwapped : KernelCompatibleSwapped (MeasureTheory.Measure.map (fun omega => (hatX s omega, Xk omega)) P) hatRhoS kernel) (hcompatBridge : ∀ {joint jointSwapped : MeasureTheory.Measure (State × State)} {mu : MeasureTheory.Measure State}, MeasureTheory.Measure.map Prod.swap joint = jointSwapped → KernelCompatibleSwapped jointSwapped mu kernel → KernelCompatibleOriginal joint mu kernel) (hcondC : KernelIntegralField kernel condC) (hcondScore : KernelIntegralField kernel condScore) (hmeasOfKernel : ∀ {f : State → Vec}, KernelIntegralField kernel f → FieldMeasurable f) (hintOfKernel : ∀ {f : State → Vec}, KernelIntegralField kernel f → FieldIntegrable f) (hbar : ∀ x, barB x = dotTk • condC x + barCoeff • condScore x) (hmeasAdd : ∀ {f g : State → Vec}, FieldMeasurable f → FieldMeasurable g → FieldMeasurable (fun x => f x + g x)) (hmeasSmul : ∀ (a : Real) {f : State → Vec}, FieldMeasurable f → FieldMeasurable (fun x => a • f x)) (hintAdd : ∀ {f g : State → Vec}, FieldIntegrable f → FieldIntegrable g → FieldIntegrable (fun x => f x + g x)) (hintSmul : ∀ (a : Real) {f : State → Vec}, FieldIntegrable f → FieldIntegrable (fun x => a • f x)) (hmeasCongr : ∀ {f g : State → Vec}, (∀ x, f x = g x) → FieldMeasurable f → FieldMeasurable g) (hintCongr : ∀ {f g : State → Vec}, (∀ x, f x = g x) → FieldIntegrable f → FieldIntegrable g) (hweakReady : MeasureTheory.Measure.map (hatX sK) P = rhoK → MeasureTheory.Measure.map (hatX sNext) P = rhoNext → MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun omega => (Xk omega, hatX s omega)) P) = hatRhoS → MeasureTheory.Measure.map Prod.fst (MeasureTheory.Measure.map (fun omega => (hatX s omega, Xk omega)) P) = hatRhoS → MeasureTheory.Measure.map Prod.swap (MeasureTheory.Measure.map (fun omega => (Xk omega, hatX s omega)) P) = MeasureTheory.Measure.map (fun omega => (hatX s omega, Xk omega)) P → KernelCompatibleOriginal (MeasureTheory.Measure.map (fun omega => (Xk omega, hatX s omega)) P) hatRhoS kernel → FieldMeasurable barB → FieldIntegrable barB → WeakFpPrereq hatRhoS kernel barB) (hcommonOfReady : WeakFpPrereq hatRhoS kernel barB → commonSpace) (hkernelOfReady : WeakFpPrereq hatRhoS kernel barB → conditionalKernel) (hdriftOfReady : WeakFpPrereq hatRhoS kernel barB → driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : fpCoeff = sigmaEta ^ 2 / 2) (hgenerator : ∀ phi, Admissible phi → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → partialS phi = generatorAction phi) (hgeneratorSplit : ∀ phi, Admissible phi → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → generatorAction phi = driftAction phi + diffusionAction phi) (hdriftSource : ∀ phi, Admissible phi → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction phi = -(driftDiv phi)) (hdiffusionSource : ∀ phi, Admissible phi → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction phi = fpCoeff • laplacian phi) : dK = (-(driftDiv logRatioTest) + (sigmaEta ^ 2 / 2) * laplacian logRatioTest) - targetTimeTerm := by have hsourceSigns : ∀ phi, Admissible phi → partialS phi = -(driftDiv phi) + (sigmaEta ^ 2 / 2) • laplacian phi := generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff P hatX Xk XNext frozenDrift W noise rhoK rhoNext hatRhoS kernel KernelCompatibleSwapped KernelCompatibleOriginal KernelIntegralField FieldMeasurable FieldIntegrable WeakFpPrereq barB condC condScore Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior partialS generatorAction driftAction diffusionAction driftDiv laplacian s sK sNext eta sigmaEta dotTk barCoeff fpCoeff hXk hhatAtS hhatLeft hhatRight hstep hupdate hrhoK hrhoNext hhatRhoS hcompatSwapped hcompatBridge hcondC hcondScore hmeasOfKernel hintOfKernel hbar hmeasAdd hmeasSmul hintAdd hintSmul hmeasCongr hintCongr hweakReady hcommonOfReady hkernelOfReady hdriftOfReady hdensity htests hboundary hcoeff hgenerator hgeneratorSplit hdriftSource hdiffusionSource exact generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns Admissible logRatioTest dK targetTimeTerm partialS driftDiv laplacian sigmaEta hlog hkl hsourceSigns /-- Endpoint/conditional KL handoff retaining the log-ratio weak-FP action. Cycle 84 keeps the active EM backend on the same source block `appendix.tex:1358-1387`, but asks lower work to consume the accepted endpoint-readiness, source-sign, and KL-substitution wrappers before opening a new measure-theory fallback. This theorem composes those wrappers and returns both the weak-FP action evaluated at the log-density-ratio test and the resulting KL derivative display. All analytic content remains supplied by the explicit hypotheses. -/ theorem generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction {Omega State Vec Kernel Test : Type*} [MeasurableSpace Omega] [MeasurableSpace State] [AddCommGroup State] [Module Real State] [AddCommGroup Vec] [Module Real Vec] (P : MeasureTheory.Measure Omega) (hatX : Real → Omega → State) (Xk XNext frozenDrift W noise : Omega → State) (rhoK rhoNext hatRhoS : MeasureTheory.Measure State) (kernel : Kernel) (KernelCompatibleSwapped : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (KernelCompatibleOriginal : MeasureTheory.Measure (State × State) → MeasureTheory.Measure State → Kernel → Prop) (KernelIntegralField : Kernel → (State → Vec) → Prop) (FieldMeasurable FieldIntegrable : (State → Vec) → Prop) (WeakFpPrereq : MeasureTheory.Measure State → Kernel → (State → Vec) → Prop) (barB condC condScore : State → Vec) (Admissible : Test → Prop) (commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior : Prop) (dK targetTimeTerm : Real) (partialS generatorAction driftAction diffusionAction driftDiv laplacian : Test → Real) (logRatioTest : Test) (s sK sNext eta sigmaEta dotTk barCoeff fpCoeff : Real) (hlog : Admissible logRatioTest) (hkl : dK = partialS logRatioTest - targetTimeTerm) (hXk : Measurable Xk) (hhatAtS : Measurable (hatX s)) (hhatLeft : hatX sK =ᵐ[P] fun omega => Xk omega + (sK - sK) • frozenDrift omega + sigmaEta • (W omega - W omega)) (hhatRight : hatX sNext =ᵐ[P] fun omega => Xk omega + (sNext - sK) • frozenDrift omega + sigmaEta • noise omega) (hstep : sNext - sK = eta) (hupdate : XNext =ᵐ[P] fun omega => Xk omega + eta • frozenDrift omega + sigmaEta • noise omega) (hrhoK : rhoK = MeasureTheory.Measure.map Xk P) (hrhoNext : rhoNext = MeasureTheory.Measure.map XNext P) (hhatRhoS : hatRhoS = MeasureTheory.Measure.map (hatX s) P) (hcompatSwapped : KernelCompatibleSwapped (MeasureTheory.Measure.map (fun omega => (hatX s omega, Xk omega)) P) hatRhoS kernel) (hcompatBridge : ∀ {joint jointSwapped : MeasureTheory.Measure (State × State)} {mu : MeasureTheory.Measure State}, MeasureTheory.Measure.map Prod.swap joint = jointSwapped → KernelCompatibleSwapped jointSwapped mu kernel → KernelCompatibleOriginal joint mu kernel) (hcondC : KernelIntegralField kernel condC) (hcondScore : KernelIntegralField kernel condScore) (hmeasOfKernel : ∀ {f : State → Vec}, KernelIntegralField kernel f → FieldMeasurable f) (hintOfKernel : ∀ {f : State → Vec}, KernelIntegralField kernel f → FieldIntegrable f) (hbar : ∀ x, barB x = dotTk • condC x + barCoeff • condScore x) (hmeasAdd : ∀ {f g : State → Vec}, FieldMeasurable f → FieldMeasurable g → FieldMeasurable (fun x => f x + g x)) (hmeasSmul : ∀ (a : Real) {f : State → Vec}, FieldMeasurable f → FieldMeasurable (fun x => a • f x)) (hintAdd : ∀ {f g : State → Vec}, FieldIntegrable f → FieldIntegrable g → FieldIntegrable (fun x => f x + g x)) (hintSmul : ∀ (a : Real) {f : State → Vec}, FieldIntegrable f → FieldIntegrable (fun x => a • f x)) (hmeasCongr : ∀ {f g : State → Vec}, (∀ x, f x = g x) → FieldMeasurable f → FieldMeasurable g) (hintCongr : ∀ {f g : State → Vec}, (∀ x, f x = g x) → FieldIntegrable f → FieldIntegrable g) (hweakReady : MeasureTheory.Measure.map (hatX sK) P = rhoK → MeasureTheory.Measure.map (hatX sNext) P = rhoNext → MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.map (fun omega => (Xk omega, hatX s omega)) P) = hatRhoS → MeasureTheory.Measure.map Prod.fst (MeasureTheory.Measure.map (fun omega => (hatX s omega, Xk omega)) P) = hatRhoS → MeasureTheory.Measure.map Prod.swap (MeasureTheory.Measure.map (fun omega => (Xk omega, hatX s omega)) P) = MeasureTheory.Measure.map (fun omega => (hatX s omega, Xk omega)) P → KernelCompatibleOriginal (MeasureTheory.Measure.map (fun omega => (Xk omega, hatX s omega)) P) hatRhoS kernel → FieldMeasurable barB → FieldIntegrable barB → WeakFpPrereq hatRhoS kernel barB) (hcommonOfReady : WeakFpPrereq hatRhoS kernel barB → commonSpace) (hkernelOfReady : WeakFpPrereq hatRhoS kernel barB → conditionalKernel) (hdriftOfReady : WeakFpPrereq hatRhoS kernel barB → driftRegular) (hdensity : densityRegular) (htests : testRegular) (hboundary : boundaryBehavior) (hcoeff : fpCoeff = sigmaEta ^ 2 / 2) (hgenerator : ∀ phi, Admissible phi → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → partialS phi = generatorAction phi) (hgeneratorSplit : ∀ phi, Admissible phi → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → generatorAction phi = driftAction phi + diffusionAction phi) (hdriftSource : ∀ phi, Admissible phi → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → driftAction phi = -(driftDiv phi)) (hdiffusionSource : ∀ phi, Admissible phi → commonSpace → conditionalKernel → driftRegular → densityRegular → testRegular → boundaryBehavior → diffusionAction phi = fpCoeff • laplacian phi) : partialS logRatioTest = -(driftDiv logRatioTest) + (sigmaEta ^ 2 / 2) * laplacian logRatioTest ∧ dK = (-(driftDiv logRatioTest) + (sigmaEta ^ 2 / 2) * laplacian logRatioTest) - targetTimeTerm := by have hsourceSigns : ∀ phi, Admissible phi → partialS phi = -(driftDiv phi) + (sigmaEta ^ 2 / 2) • laplacian phi := generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff P hatX Xk XNext frozenDrift W noise rhoK rhoNext hatRhoS kernel KernelCompatibleSwapped KernelCompatibleOriginal KernelIntegralField FieldMeasurable FieldIntegrable WeakFpPrereq barB condC condScore Admissible commonSpace conditionalKernel driftRegular densityRegular testRegular boundaryBehavior partialS generatorAction driftAction diffusionAction driftDiv laplacian s sK sNext eta sigmaEta dotTk barCoeff fpCoeff hXk hhatAtS hhatLeft hhatRight hstep hupdate hrhoK hrhoNext hhatRhoS hcompatSwapped hcompatBridge hcondC hcondScore hmeasOfKernel hintOfKernel hbar hmeasAdd hmeasSmul hintAdd hintSmul hmeasCongr hintCongr hweakReady hcommonOfReady hkernelOfReady hdriftOfReady hdensity htests hboundary hcoeff hgenerator hgeneratorSplit hdriftSource hdiffusionSource exact generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction Admissible logRatioTest dK targetTimeTerm partialS driftDiv laplacian sigmaEta hlog hkl hsourceSigns /-- Source-sign handoff for the discrete general EM Fokker--Planck equation. In `appendix.tex:1379-1387`, the source writes the conditional-drift Fokker--Planck equation with drift sign `-div(hat rho_s*bar b_{k,s})` and diffusion coefficient `sigma_eta^2/2`. This theorem only rewrites an abstract weak FP identity whose diffusion coefficient has already been supplied as a named scalar. It does not construct the conditional law, density, absolute-continuity, weak FP theorem, or KL derivative backend. -/ theorem generalMovingTargetDiscreteConditionalFpSourceSignsHandoff {E F : Type*} [AddCommGroup E] [AddCommGroup F] [Module Real F] (D : E → F) (partialS laplacian : F) (sigmaEta sigmaCoeff : Real) (frozen : E) (hcoeff : sigmaCoeff = sigmaEta ^ 2 / 2) (hfp : partialS = -D frozen + sigmaCoeff • laplacian) : partialS = -D frozen + (sigmaEta ^ 2 / 2) • laplacian := by rw [hfp, hcoeff] /-- Lower handoff algebra for the sigma-weighted conditional Fokker--Planck split in the discrete general VA-SALD proof. This is the proof-producing part of `appendix.tex:1380-1387`: once the weak conditional-drift Fokker--Planck equation, the Laplacian split, and linearity of divergence are supplied, the source regrouping has the displayed `(sigma_eta^2/2)` coefficient. It does not construct the conditional drift, density, absolute-continuity, or integration-by-parts backend. -/ theorem generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff {E F : Type*} [AddCommGroup E] [AddCommGroup F] [Module Real E] [Module Real F] (D : E → F) (hadd : ∀ x y, D (x + y) = D x + D y) (hneg : ∀ x, D (-x) = -D x) (hsmul : ∀ (a : Real) (x : E), D (a • x) = a • D x) (partialS laplacian : F) (sigmaCoeff : Real) (frozen rel target : E) (hfp : partialS = -D frozen + sigmaCoeff • laplacian) (hlap : laplacian = D rel + D target) : partialS = sigmaCoeff • D rel + D (sigmaCoeff • target - frozen) := by have hdivTarget : D (sigmaCoeff • target - frozen) = sigmaCoeff • D target - D frozen := by rw [sub_eq_add_neg, hadd, hsmul, hneg] simp [sub_eq_add_neg] calc partialS = -D frozen + sigmaCoeff • (D rel + D target) := by rw [hfp, hlap] _ = sigmaCoeff • D rel + (sigmaCoeff • D target - D frozen) := by rw [smul_add] abel _ = sigmaCoeff • D rel + D (sigmaCoeff • target - frozen) := by rw [hdivTarget] /-- Scalar coefficient rewrite for the doubled residual term in the discrete general VA-SALD Gronwall bridge. This formalizes the paper algebra turning `dot{s}(t) * (2*sigma_eta(t)^(-2)*dot t(s(t))^2*alpha^(-1))` into `2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*alpha^(-1)`. -/ theorem generalMovingTargetDiscreteResidualCoefficientRewriteScalar (dotS dotT sigmaInvSq alphaInv : Real) (hdotT : dotT = dotS⁻¹) (hdotS : dotS ≠ 0) : dotS * (2 * sigmaInvSq * dotT ^ 2 * alphaInv) = 2 * sigmaInvSq * dotS⁻¹ * alphaInv := by calc dotS * (2 * sigmaInvSq * dotT ^ 2 * alphaInv) = (2 * sigmaInvSq * alphaInv) * (dotS * dotT ^ 2) := by ring _ = (2 * sigmaInvSq * alphaInv) * dotS⁻¹ := by rw [generalMovingTargetDiscreteConstantScheduleSquareScalar dotS dotT hdotT hdotS] _ = 2 * sigmaInvSq * dotS⁻¹ * alphaInv := by ring /-- Scalar coefficient rearrangement for the frozen `Gamma` term after the `s`-to-`t` time change in the discrete general VA-SALD proof. -/ theorem generalMovingTargetDiscreteGammaCoefficientRewriteScalar (dotS gamma etaSq alphaPrimeInv : Real) : dotS * (2 * gamma * etaSq * alphaPrimeInv) = 2 * dotS * etaSq * alphaPrimeInv * gamma := by ring /-- Scalar coefficient rearrangement for the frozen `Delta` term after the `s`-to-`t` time change in the discrete general VA-SALD proof. -/ theorem generalMovingTargetDiscreteDeltaCoefficientRewriteScalar (dotS delta eta : Real) : dotS * (2 * delta * eta) = 2 * dotS * eta * delta := by ring /-- Module-level algebra for the frozen/residual decomposition. In appendix lines 1469-1478, after the analytic identifications `delta = dotT • c + score - frozen`, `tildeV = dotT • v`, and `m = v - c` have been supplied, the source cross field rewrites as `delta + dotT • m`. This proves only that local vector-field algebra; the conditional drift, score, slowed transport, and pointwise field identifications remain side-condition obligations. -/ theorem generalMovingTargetDiscreteFrozenResidualAlgebraVector {E : Type*} [AddCommGroup E] [Module Real E] (dotT : Real) (score frozen c v m delta tildeV : E) (hm : m = v - c) (hdelta : delta = dotT • c + score - frozen) (htildeV : tildeV = dotT • v) : score - frozen + tildeV = delta + dotT • m := by subst m subst delta subst tildeV simp [sub_eq_add_neg, smul_add, smul_neg] abel /-- Source coefficient identity for one `sigma_eta^2/8` Young share. In appendix lines 1493-1511 the two cross terms each consume one quarter of the available Fisher dissipation `(sigma_eta^2/2)*FI`. This lemma closes only the real arithmetic identifying that quarter with `(sigma_eta^2/8)*FI`; the inner-product Young inequality and the analytic FI identification remain separate obligations. -/ theorem generalMovingTargetDiscreteYoungFisherShareScalar (sigmaSq fi : Real) : ((sigmaSq / 2) * fi) / 4 = (sigmaSq / 8) * fi := by ring /-- Scalar budget after the two `sigma_eta^2/8` Young splits. Instantiating `fisherDissipation` with `(sigma_eta^2/2)*FI`, the two Young cross-term bounds leave exactly one half of that dissipation, namely the source coefficient `-(sigma_eta^2/4)*FI` before the LSI step. This does not prove the cross-term bounds or the frozen-delta lemma. -/ theorem generalMovingTargetDiscreteTwoYoungFisherBudgetScalar (fisherDissipation residualTerm frozenKlTerm frozenDeltaTerm mCross deltaCross : Real) (hm : mCross ≤ fisherDissipation / 4 + residualTerm) (hdelta : deltaCross ≤ fisherDissipation / 4 + frozenKlTerm + frozenDeltaTerm) : -fisherDissipation + deltaCross + mCross ≤ -(fisherDissipation / 2) + residualTerm + frozenKlTerm + frozenDeltaTerm := by linarith /-- Scalar residual coefficient produced by Young with `epsilon = sigma_eta^2/4`. The source's residual cross term has `b^2=dot t(s)^2*||m||^2`. Once the analytic Young inequality has supplied the coefficient `1/(2*epsilon)` and `sigmaInvSq` is identified with `sigma_eta^{-2}`, this lemma rewrites it to the displayed `2*sigma_eta^{-2}*dot t(s)^2*||m||^2` coefficient. -/ theorem generalMovingTargetDiscreteResidualYoungCoefficientScalar (sigmaSq sigmaInvSq dotT energy : Real) (hsigmaInvSq : sigmaInvSq = sigmaSq⁻¹) (hsigmaSq : sigmaSq ≠ 0) : (1 / (2 * (sigmaSq / 4))) * (dotT ^ 2 * energy) = 2 * sigmaInvSq * dotT ^ 2 * energy := by rw [hsigmaInvSq] field_simp [hsigmaSq] ring /-- Scalar post-Young handoff for the discrete general VA-SALD derivative. This packages appendix lines 1469-1517 after the analytic KL derivative, frozen/residual decomposition, residual Young inequality, and frozen-delta bound are supplied. It proves only the real/order bookkeeping that two `sigma_eta^2/8` Fisher shares leave the source coefficient `-(sigma_eta^2/4)*FI`. -/ theorem generalMovingTargetDiscretePostYoungDerivativeBoundScalar (dKds fisher mCross deltaCross residualCoeff frozenKlTerm frozenDeltaTerm sigmaSq : Real) (hderiv : dKds = -(sigmaSq / 2) * fisher + deltaCross + mCross) (hm : mCross ≤ (sigmaSq / 8) * fisher + residualCoeff) (hdelta : deltaCross ≤ (sigmaSq / 8) * fisher + frozenKlTerm + frozenDeltaTerm) : dKds ≤ -(sigmaSq / 4) * fisher + residualCoeff + frozenKlTerm + frozenDeltaTerm := by calc dKds = -(sigmaSq / 2) * fisher + deltaCross + mCross := hderiv _ ≤ -(sigmaSq / 2) * fisher + ((sigmaSq / 8) * fisher + frozenKlTerm + frozenDeltaTerm) + ((sigmaSq / 8) * fisher + residualCoeff) := by linarith _ = -(sigmaSq / 4) * fisher + residualCoeff + frozenKlTerm + frozenDeltaTerm := by ring /-- Scalar LSI handoff for the discrete general VA-SALD derivative. Once `eq:LSI-KL-FI` supplies `C_LSI*K <= (1/2)*FI`, this converts the post-Young term `-(sigma_eta^2/4)*FI` into the source `-(sigma_eta^2/2)*C_LSI*K` damping. The density-test and FI backends remain separate obligations. -/ theorem generalMovingTargetDiscretePostLsiDerivativeBoundScalar (dKds fisher kl cLSI residualCoeff sigmaSq : Real) (hsigmaSq : 0 ≤ sigmaSq) (hpostYoung : dKds ≤ -(sigmaSq / 4) * fisher + residualCoeff) (hlsiHalf : cLSI * kl ≤ (1 / 2) * fisher) : dKds ≤ -(((sigmaSq / 2) * cLSI) * kl) + residualCoeff := by have hcoeffNonneg : 0 ≤ sigmaSq / 2 := by positivity have hscaled : (sigmaSq / 2) * (cLSI * kl) ≤ (sigmaSq / 2) * ((1 / 2) * fisher) := by exact mul_le_mul_of_nonneg_left hlsiHalf hcoeffNonneg have htarget : ((sigmaSq / 2) * cLSI) * kl ≤ (sigmaSq / 4) * fisher := by calc ((sigmaSq / 2) * cLSI) * kl = (sigmaSq / 2) * (cLSI * kl) := by ring _ ≤ (sigmaSq / 2) * ((1 / 2) * fisher) := hscaled _ = (sigmaSq / 4) * fisher := by ring have hneg : -(sigmaSq / 4) * fisher ≤ -(((sigmaSq / 2) * cLSI) * kl) := by simpa only [neg_mul] using neg_le_neg htarget calc dKds ≤ -(sigmaSq / 4) * fisher + residualCoeff := hpostYoung _ ≤ -(((sigmaSq / 2) * cLSI) * kl) + residualCoeff := by simpa [add_comm, add_left_comm, add_assoc] using add_le_add_right hneg residualCoeff /-- Scalar post-DV handoff for the discrete general VA-SALD derivative. After DV supplies `||m||^2 <= alphaInv*K + E_alpha`, this rewrites the residual energy term into the exact `s`-time damping and residual coefficients in appendix lines 1544-1570. -/ theorem generalMovingTargetDiscretePostDvDerivativeBoundScalar (dKds kl cLSI sigmaSq sigmaInvSq dotT alphaInv residualSq eAlpha gamma etaSq alphaPrimeInv delta eta : Real) (hpre : dKds ≤ -(((sigmaSq / 2) * cLSI) * kl) + 2 * sigmaInvSq * dotT ^ 2 * residualSq + 2 * gamma * etaSq * alphaPrimeInv * kl + 2 * delta * eta) (hcoeff : 0 ≤ 2 * sigmaInvSq * dotT ^ 2) (hdv : residualSq ≤ alphaInv * kl + eAlpha) : dKds ≤ -((((sigmaSq / 2) * cLSI) - 2 * sigmaInvSq * dotT ^ 2 * alphaInv - 2 * gamma * etaSq * alphaPrimeInv) * kl) + 2 * sigmaInvSq * dotT ^ 2 * eAlpha + 2 * delta * eta := by have hres : 2 * sigmaInvSq * dotT ^ 2 * residualSq ≤ 2 * sigmaInvSq * dotT ^ 2 * (alphaInv * kl + eAlpha) := by exact mul_le_mul_of_nonneg_left hdv hcoeff calc dKds ≤ -(((sigmaSq / 2) * cLSI) * kl) + 2 * sigmaInvSq * dotT ^ 2 * residualSq + 2 * gamma * etaSq * alphaPrimeInv * kl + 2 * delta * eta := hpre _ ≤ -(((sigmaSq / 2) * cLSI) * kl) + 2 * sigmaInvSq * dotT ^ 2 * (alphaInv * kl + eAlpha) + 2 * gamma * etaSq * alphaPrimeInv * kl + 2 * delta * eta := by linarith _ = -((((sigmaSq / 2) * cLSI) - 2 * sigmaInvSq * dotT ^ 2 * alphaInv - 2 * gamma * etaSq * alphaPrimeInv) * kl) + 2 * sigmaInvSq * dotT ^ 2 * eAlpha + 2 * delta * eta := by ring /-- Scalar time-change handoff for the discrete general VA-SALD derivative. This is the real/order part of appendix lines 1573-1583: multiply the `s`-time inequality by `dot{s}(t)`, use `dot t(s(t)) = dot{s}(t)^(-1)`, and preserve the doubled residual and frozen-delta coefficients used by the final Gronwall step. -/ theorem generalMovingTargetDiscreteTimeChangedDerivativeBoundScalar (dKds dKdt kl cLSI sigmaSq sigmaInvSq dotS dotT alphaInv eAlpha gamma etaSq alphaPrimeInv delta eta : Real) (hdotSNonneg : 0 ≤ dotS) (hdKdt : dKdt = dotS * dKds) (hsBound : dKds ≤ -((((sigmaSq / 2) * cLSI) - 2 * sigmaInvSq * dotT ^ 2 * alphaInv - 2 * gamma * etaSq * alphaPrimeInv) * kl) + 2 * sigmaInvSq * dotT ^ 2 * eAlpha + 2 * delta * eta) (hdotT : dotT = dotS⁻¹) (hdotS : dotS ≠ 0) : dKdt ≤ -((((sigmaSq / 2) * dotS * cLSI) - 2 * sigmaInvSq * dotS⁻¹ * alphaInv - 2 * dotS * etaSq * alphaPrimeInv * gamma) * kl) + 2 * sigmaInvSq * dotS⁻¹ * eAlpha + 2 * dotS * eta * delta := by rw [hdKdt] have hmul : dotS * dKds ≤ dotS * (-((((sigmaSq / 2) * cLSI) - 2 * sigmaInvSq * dotT ^ 2 * alphaInv - 2 * gamma * etaSq * alphaPrimeInv) * kl) + 2 * sigmaInvSq * dotT ^ 2 * eAlpha + 2 * delta * eta) := by exact mul_le_mul_of_nonneg_left hsBound hdotSNonneg calc dotS * dKds ≤ dotS * (-((((sigmaSq / 2) * cLSI) - 2 * sigmaInvSq * dotT ^ 2 * alphaInv - 2 * gamma * etaSq * alphaPrimeInv) * kl) + 2 * sigmaInvSq * dotT ^ 2 * eAlpha + 2 * delta * eta) := hmul _ = -((((sigmaSq / 2) * dotS * cLSI) - 2 * sigmaInvSq * dotS⁻¹ * alphaInv - 2 * dotS * etaSq * alphaPrimeInv * gamma) * kl) + 2 * sigmaInvSq * dotS⁻¹ * eAlpha + 2 * dotS * eta * delta := by rw [hdotT] field_simp [hdotS] /-- Source-shaped scalar handoff for the discrete general VA-SALD derivative. This composes the compiled post-Young, LSI, DV, and time-change scalar steps for appendix lines 1469-1583. All analytic inputs remain explicit: EM Fokker--Planck/KL differentiation, integration by parts, frozen-delta estimate, LSI density test, DV common-space/finite-log-mgf witness, and schedule calculus are not proved here. -/ theorem generalMovingTargetDiscreteDerivativeDvTimeChangedScalar (dKds dKdt fisher mCross deltaCross residualSq kl cLSI sigmaSq sigmaInvSq dotS dotT alphaInv eAlpha gamma etaSq alphaPrimeInv delta eta : Real) (hsigmaSq : 0 ≤ sigmaSq) (hdotSNonneg : 0 ≤ dotS) (hcoeff : 0 ≤ 2 * sigmaInvSq * dotT ^ 2) (hderiv : dKds = -(sigmaSq / 2) * fisher + deltaCross + mCross) (hm : mCross ≤ (sigmaSq / 8) * fisher + 2 * sigmaInvSq * dotT ^ 2 * residualSq) (hdelta : deltaCross ≤ (sigmaSq / 8) * fisher + 2 * gamma * etaSq * alphaPrimeInv * kl + 2 * delta * eta) (hlsiHalf : cLSI * kl ≤ (1 / 2) * fisher) (hdv : residualSq ≤ alphaInv * kl + eAlpha) (hdKdt : dKdt = dotS * dKds) (hdotT : dotT = dotS⁻¹) (hdotS : dotS ≠ 0) : dKdt ≤ -((((sigmaSq / 2) * dotS * cLSI) - 2 * sigmaInvSq * dotS⁻¹ * alphaInv - 2 * dotS * etaSq * alphaPrimeInv * gamma) * kl) + 2 * sigmaInvSq * dotS⁻¹ * eAlpha + 2 * dotS * eta * delta := by have hpostYoung : dKds ≤ -(sigmaSq / 4) * fisher + 2 * sigmaInvSq * dotT ^ 2 * residualSq + 2 * gamma * etaSq * alphaPrimeInv * kl + 2 * delta * eta := generalMovingTargetDiscretePostYoungDerivativeBoundScalar dKds fisher mCross deltaCross (2 * sigmaInvSq * dotT ^ 2 * residualSq) (2 * gamma * etaSq * alphaPrimeInv * kl) (2 * delta * eta) sigmaSq hderiv hm hdelta have hlsi : dKds ≤ -(((sigmaSq / 2) * cLSI) * kl) + (2 * sigmaInvSq * dotT ^ 2 * residualSq + 2 * gamma * etaSq * alphaPrimeInv * kl + 2 * delta * eta) := generalMovingTargetDiscretePostLsiDerivativeBoundScalar dKds fisher kl cLSI (2 * sigmaInvSq * dotT ^ 2 * residualSq + 2 * gamma * etaSq * alphaPrimeInv * kl + 2 * delta * eta) sigmaSq hsigmaSq (by simpa [add_assoc] using hpostYoung) hlsiHalf have hpostDv : dKds ≤ -((((sigmaSq / 2) * cLSI) - 2 * sigmaInvSq * dotT ^ 2 * alphaInv - 2 * gamma * etaSq * alphaPrimeInv) * kl) + 2 * sigmaInvSq * dotT ^ 2 * eAlpha + 2 * delta * eta := generalMovingTargetDiscretePostDvDerivativeBoundScalar dKds kl cLSI sigmaSq sigmaInvSq dotT alphaInv residualSq eAlpha gamma etaSq alphaPrimeInv delta eta (by simpa [add_assoc] using hlsi) hcoeff hdv exact generalMovingTargetDiscreteTimeChangedDerivativeBoundScalar dKds dKdt kl cLSI sigmaSq sigmaInvSq dotS dotT alphaInv eAlpha gamma etaSq alphaPrimeInv delta eta hdotSNonneg hdKdt hpostDv hdotT hdotS /-- Pointwise Gronwall-input wrapper for the discrete general VA-SALD time change. The scalar theorem above handles one fixed time after the EM/KL derivative, LSI, residual DV, and constant inverse-schedule inputs have supplied the `s`-time inequality. This wrapper gives the lower Gronwall side-condition packet the pointwise `K'(t) <= -a(t) K(t) + b(t)` shape used in appendix lines 1573-1600, without proving stitched regularity, endpoint identifications, or `lem:gronwall`. -/ theorem generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged (K dKds dKdt cLSI sigmaSq sigmaInvSq dotS dotT eAlpha gamma delta : Real → Real) (alphaInv etaSq alphaPrimeInv eta : Real) (hdotSNonneg : ∀ t, 0 ≤ dotS t) (hdKdt : ∀ t, dKdt t = dotS t * dKds t) (hsBound : ∀ t, dKds t ≤ -((((sigmaSq t / 2) * cLSI t) - 2 * sigmaInvSq t * dotT t ^ 2 * alphaInv - 2 * gamma t * etaSq * alphaPrimeInv) * K t) + 2 * sigmaInvSq t * dotT t ^ 2 * eAlpha t + 2 * delta t * eta) (hdotT : ∀ t, dotT t = (dotS t)⁻¹) (hdotS : ∀ t, dotS t ≠ 0) : ∀ t, dKdt t ≤ -((((sigmaSq t / 2) * dotS t * cLSI t) - 2 * sigmaInvSq t * (dotS t)⁻¹ * alphaInv - 2 * dotS t * etaSq * alphaPrimeInv * gamma t) * K t) + 2 * sigmaInvSq t * (dotS t)⁻¹ * eAlpha t + 2 * dotS t * eta * delta t := by intro t exact generalMovingTargetDiscreteTimeChangedDerivativeBoundScalar (dKds t) (dKdt t) (K t) (cLSI t) (sigmaSq t) (sigmaInvSq t) (dotS t) (dotT t) alphaInv (eAlpha t) (gamma t) etaSq alphaPrimeInv (delta t) eta (hdotSNonneg t) (hdKdt t) (hsBound t) (hdotT t) (hdotS t) /-- Named-coefficient handoff for the final discrete general VA-SALD Gronwall step. Cycle 58 supplies the pointwise derivative inequality with the source coefficient expression. This wrapper lets the final side-condition packet introduce named Gronwall functions `a` and `b` and prove that the source inequality has exactly the `K'(t) <= -a(t)K(t)+b(t)` shape needed by `lem:gronwall`. It is only display-matching algebra; coefficient regularity, endpoint stitching, and Gronwall itself remain obligations. -/ theorem generalMovingTargetDiscreteGronwallNamedCoefficientInput (K dKdt cLSI sigmaSq sigmaInvSq dotS eAlpha gamma delta a b : Real → Real) (alphaInv etaSq alphaPrimeInv eta : Real) (hinput : ∀ t, dKdt t ≤ -((((sigmaSq t / 2) * dotS t * cLSI t) - 2 * sigmaInvSq t * (dotS t)⁻¹ * alphaInv - 2 * dotS t * etaSq * alphaPrimeInv * gamma t) * K t) + 2 * sigmaInvSq t * (dotS t)⁻¹ * eAlpha t + 2 * dotS t * eta * delta t) (ha : ∀ t, a t = ((sigmaSq t / 2) * dotS t * cLSI t) - 2 * sigmaInvSq t * (dotS t)⁻¹ * alphaInv - 2 * dotS t * etaSq * alphaPrimeInv * gamma t) (hb : ∀ t, b t = 2 * sigmaInvSq t * (dotS t)⁻¹ * eAlpha t + 2 * dotS t * eta * delta t) : ∀ t, dKdt t ≤ -(a t * K t) + b t := by intro t rw [ha t, hb t] simpa [add_assoc] using hinput t /-- Endpoint rewrite for the final discrete general VA-SALD Gronwall bound. After `lem:gronwall` is applied to the stitched function `K`, this closes the pure endpoint-rewrite step from `K(T)` and `K(0)` to the theorem endpoints `KL(rho_K^eta || pi_T)` and `KL(rho_0 || pi_0)`. The endpoint law identifications themselves remain explicit hypotheses supplied by the side-condition backend. -/ theorem generalMovingTargetDiscreteGronwallEndpointRewriteScalar (K : Real → Real) (T initialKl terminalKl initialCoeff residualIntegral : Real) (hgronwall : K T ≤ initialCoeff * K 0 + residualIntegral) (hterminal : K T = terminalKl) (hinitial : K 0 = initialKl) : terminalKl ≤ initialCoeff * initialKl + residualIntegral := by rw [← hterminal, ← hinitial] exact hgronwall /-- Final scalar/display bridge for the discrete general VA-SALD theorem. This is the cycle-68 lower proof-producing wrapper for the selected `sald.unified_discrete_general.cycle68_discrete_general_bridge` packet. It starts after the discrete derivative/DV/time-change route has supplied the source pointwise differential inequality, uses the named-coefficient wrapper to present it to the Gronwall backend, and then rewrites the endpoint values to the theorem display. The theorem takes the Gronwall result itself as an explicit hypothesis. It therefore does not prove endpoint stitching, coefficient regularity, endpoint-safe Gronwall, EM/Fokker--Planck, LSI/KL/FI, DV, frozen-delta analysis, or the discrete theorem. -/ theorem generalMovingTargetDiscreteGronwallDisplayBridgeScalar (K dKdt cLSI sigmaSq sigmaInvSq dotS eAlpha gamma delta a b : Real → Real) (alphaInv etaSq alphaPrimeInv eta T initialKl terminalKl initialCoeff residualIntegral : Real) (hinput : ∀ t, dKdt t ≤ -((((sigmaSq t / 2) * dotS t * cLSI t) - 2 * sigmaInvSq t * (dotS t)⁻¹ * alphaInv - 2 * dotS t * etaSq * alphaPrimeInv * gamma t) * K t) + 2 * sigmaInvSq t * (dotS t)⁻¹ * eAlpha t + 2 * dotS t * eta * delta t) (ha : ∀ t, a t = ((sigmaSq t / 2) * dotS t * cLSI t) - 2 * sigmaInvSq t * (dotS t)⁻¹ * alphaInv - 2 * dotS t * etaSq * alphaPrimeInv * gamma t) (hb : ∀ t, b t = 2 * sigmaInvSq t * (dotS t)⁻¹ * eAlpha t + 2 * dotS t * eta * delta t) (hgronwall : (∀ t, dKdt t ≤ -(a t * K t) + b t) → K T ≤ initialCoeff * K 0 + residualIntegral) (hterminal : K T = terminalKl) (hinitial : K 0 = initialKl) : terminalKl ≤ initialCoeff * initialKl + residualIntegral := by have hnamed : ∀ t, dKdt t ≤ -(a t * K t) + b t := generalMovingTargetDiscreteGronwallNamedCoefficientInput K dKdt cLSI sigmaSq sigmaInvSq dotS eAlpha gamma delta a b alphaInv etaSq alphaPrimeInv eta hinput ha hb exact generalMovingTargetDiscreteGronwallEndpointRewriteScalar K T initialKl terminalKl initialCoeff residualIntegral (hgronwall hnamed) hterminal hinitial /-- Scalar upper-bound core for the PI norm-equivalence step. In `appendix.tex:104-112`, the analytic obligations identify `l2Sq` with the mean-zero variance term and `dotSq` with the gradient norm squared. Once PI has supplied `l2Sq <= invC * dotSq`, this closes only the real algebra in the displayed weighted `H^1(mu)` upper bound. -/ theorem piVelocityNormMeanZeroH1UpperScalar (l2Sq dotSq invC : Real) (hl2 : l2Sq ≤ invC * dotSq) : l2Sq + dotSq ≤ (1 + invC) * dotSq := by have hsum : l2Sq + dotSq ≤ invC * dotSq + dotSq := by linarith calc l2Sq + dotSq ≤ invC * dotSq + dotSq := hsum _ = (1 + invC) * dotSq := by ring /-- Scalar bounded-functional core for the first PI velocity-norm lower slice. The source obtains `T_mu(psi) <= ||psi||_L2 ||g||_L2` by Cauchy--Schwarz and then uses PI to replace `||psi||_L2` by `C_PI^{-1/2}||psi||_{dot H^1}`. This lemma formalizes only that order propagation; measurability, integrability, absolute values, and the weighted Sobolev backend remain obligations. -/ theorem piVelocityNormBoundedFunctionalScalar (functionalAbs psiL2 psiDot gL2 piScale : Real) (hcs : functionalAbs ≤ psiL2 * gL2) (hpi : psiL2 ≤ piScale * psiDot) (hg : 0 ≤ gL2) : functionalAbs ≤ piScale * psiDot * gL2 := by exact hcs.trans (mul_le_mul_of_nonneg_right hpi hg) /-- Scalar coefficient audit for the source LSI-to-KL/FI display. After the analytic obligations identify the LSI Dirichlet term with `(1/4) * FI(rho||pi)`, this lemma preserves the paper's constant `1/(2*C_LSI)`. It does not prove the density, admissibility, entropy, or Fisher-information chain-rule backends. -/ theorem lsiKlFiCoefficientAuditScalar (kl fi dirichlet cLSI : Real) (hlsi : kl ≤ (2 / cLSI) * dirichlet) (hchain : dirichlet = (1 / 4) * fi) (hcLSI : cLSI ≠ 0) : kl ≤ fi / (2 * cLSI) := by calc kl ≤ (2 / cLSI) * dirichlet := hlsi _ = fi / (2 * cLSI) := by rw [hchain] field_simp [hcLSI] ring /-- Scalar bridge for applying the normalized LSI test `phi=sqrt(rho/pi)`. This packages the source handoff in `main_body.tex:208-215` after the analytic backend has supplied the LSI test normalization, entropy-to-KL identity, and Fisher-information chain rule. It does not prove those analytic inputs. -/ theorem lsiKlFiDensityTestBridgeScalar (testMass entropy kl fi dirichlet cLSI : Real) (hnormalize : testMass = 1) (hlsi : testMass = 1 → entropy ≤ (2 / cLSI) * dirichlet) (hentropy : entropy = kl) (hchain : dirichlet = (1 / 4) * fi) (hcLSI : cLSI ≠ 0) : kl ≤ fi / (2 * cLSI) := by have hlsiApplied : entropy ≤ (2 / cLSI) * dirichlet := hlsi hnormalize have hlsiKl : kl ≤ (2 / cLSI) * dirichlet := by rw [← hentropy] exact hlsiApplied exact lsiKlFiCoefficientAuditScalar kl fi dirichlet cLSI hlsiKl hchain hcLSI /-- Convert the displayed KL/FI comparison into the half-Fisher form used later. The forward-KL proof consumes the LSI output as `C_LSI*K <= (1/2)*FI` before substituting it into the derivative inequality. This lemma proves only that scalar coefficient handoff from the paper's displayed `KL <= FI/(2*C_LSI)` under the source assumption `C_LSI>0`. -/ theorem lsiKlFiHalfFisherScalar (kl fi cLSI : Real) (hcLSI : 0 < cLSI) (hklfi : kl ≤ fi / (2 * cLSI)) : cLSI * kl ≤ (1 / 2) * fi := by have hmul : cLSI * kl ≤ cLSI * (fi / (2 * cLSI)) := mul_le_mul_of_nonneg_left hklfi (le_of_lt hcLSI) calc cLSI * kl ≤ cLSI * (fi / (2 * cLSI)) := hmul _ = (1 / 2) * fi := by field_simp [ne_of_gt hcLSI] /-- Normalized density-test bridge directly in the half-Fisher derivative form. After the analytic density-test backend supplies normalization, the LSI test inequality, entropy-to-KL, and Dirichlet-to-FI identities, this combines the cycle-33 scalar bridge with `lsiKlFiHalfFisherScalar`. It still does not prove the Radon-Nikodym construction, admissibility of `sqrt(r)`, or the Fisher-information chain rule. -/ theorem lsiKlFiDensityTestHalfFisherScalar (testMass entropy kl fi dirichlet cLSI : Real) (hnormalize : testMass = 1) (hlsi : testMass = 1 → entropy ≤ (2 / cLSI) * dirichlet) (hentropy : entropy = kl) (hchain : dirichlet = (1 / 4) * fi) (hcLSI : 0 < cLSI) : cLSI * kl ≤ (1 / 2) * fi := by exact lsiKlFiHalfFisherScalar kl fi cLSI hcLSI (lsiKlFiDensityTestBridgeScalar testMass entropy kl fi dirichlet cLSI hnormalize hlsi hentropy hchain (ne_of_gt hcLSI)) /-- Scalar raw-derivative split for discrete forward-KL. This is the cycle-89 lower core for the first blocker found by the `thm:forward-KL-discrete` pressure test. It replaces the older opaque input `dK = -FI + frozenCross + movingCross` by the source-shaped raw KL derivative split from `eq:KL-derivative-0-discrete`: a mass term dropped by conservation, the weak-FP/integration-by-parts/Fisher identity for `eq:KL-derivative-1-discrete`, and the target-transport integration-by-parts identity for `eq:KL-derivative-2-discrete`. -/ theorem discreteForwardKlDerivativeSplitOfRawIbpsScalar (dK firstTerm massTerm targetTerm fisher frozenCross movingCross : Real) (hklRaw : dK = firstTerm + massTerm + targetTerm) (hmass : massTerm = 0) (hfirst : firstTerm = -fisher + frozenCross) (htarget : targetTerm = movingCross) : dK = -fisher + frozenCross + movingCross := by rw [hklRaw, hmass, hfirst, htarget] ring /-- Mass term is zero when it is the derivative of a locally constant total mass. For `eq:KL-derivative-0-discrete`, the paper uses `int partial_s hat rho_s dx = 0`. This lemma isolates the local calculus part: once the analytic backend identifies `massTerm` as the derivative of the total-mass pairing and the interpolated law has total mass one near `s0`, the mass term is zero. It does not prove density-to-law mass preservation or differentiation under the integral. -/ theorem discreteForwardKlMassTermZeroOfTotalMassDerivative (totalMass : Real → Real) (s0 massTerm : Real) (hmassDeriv : HasDerivAt totalMass massTerm s0) (hprobMass : ∀ᶠ s in nhds s0, totalMass s = 1) : massTerm = 0 := by have hconstDeriv : HasDerivAt (fun _ : Real => (1 : Real)) 0 s0 := hasDerivAt_const s0 (1 : Real) have hzeroDeriv : HasDerivAt totalMass 0 s0 := hconstDeriv.congr_of_eventuallyEq hprobMass exact hmassDeriv.unique hzeroDeriv /-- Discrete derivative split with mass conservation derived from total mass. This removes the primitive `hmass : massTerm = 0` input from the cycle-89 raw IBP scalar route. The remaining analytic boundary is the source-cited identification of `massTerm` as the derivative of the total-mass integral for the EM interpolation density. -/ theorem discreteForwardKlDerivativeSplitOfMassDerivativeScalar (dK firstTerm massTerm targetTerm fisher frozenCross movingCross s0 : Real) (totalMass : Real → Real) (hklRaw : dK = firstTerm + massTerm + targetTerm) (hmassDeriv : HasDerivAt totalMass massTerm s0) (hprobMass : ∀ᶠ s in nhds s0, totalMass s = 1) (hfirst : firstTerm = -fisher + frozenCross) (htarget : targetTerm = movingCross) : dK = -fisher + frozenCross + movingCross := by have hmass : massTerm = 0 := discreteForwardKlMassTermZeroOfTotalMassDerivative totalMass s0 massTerm hmassDeriv hprobMass exact discreteForwardKlDerivativeSplitOfRawIbpsScalar dK firstTerm massTerm targetTerm fisher frozenCross movingCross hklRaw hmass hfirst htarget /-- Constant weak-test mass of a mapped probability law. For `eq:KL-derivative-0-discrete`, this is the law-normalization part of `int partial_s hat rho_s dx = 0`: if `hat rho_s` is represented as `Measure.map (hatX s) P` and `P` is a probability measure, the integral of the constant weak test `1` against that law is one. The theorem does not identify the KL-display mass term with a time derivative. -/ theorem discreteForwardKlLawConstantTestTotalMassOne {Ω State : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] (hatXAtS : Ω → State) (hhatX : AEMeasurable hatXAtS P) : (∫ _, (1 : Real) ∂MeasureTheory.Measure.map hatXAtS P) = 1 := by have hprobMap : MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.map hatXAtS P) := MeasureTheory.Measure.isProbabilityMeasure_map hhatX rw [MeasureTheory.integral_const] simp /-- The mapped-law constant weak test has zero derivative. This closes the elementary derivative side of the mass-conservation sentence in `eq:KL-derivative-0-discrete`: after rewriting the law integral to the sample-space integral by `lawMapIntegralHasDerivAtOfSample`, the constant weak test is a constant function of `s`. The remaining source-cited analytic step is to identify the paper's scalar `massTerm` with this derivative. -/ theorem discreteForwardKlLawConstantTestHasDerivAtZero {Ω State : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] (hatX : Real → Ω → State) (s0 : Real) (hhatX : ∀ s, AEMeasurable (hatX s) P) : HasDerivAt (fun s => ∫ _, (1 : Real) ∂MeasureTheory.Measure.map (hatX s) P) 0 s0 := by refine lawMapIntegralHasDerivAtOfSample (P := P) (X := hatX) (φ := fun _ : State => (1 : Real)) hhatX ?_ ?_ · intro s exact MeasureTheory.aestronglyMeasurable_const · simpa using hasDerivAt_const s0 (∫ _, (1 : Real) ∂P) /-- Mass term is zero for the concrete mapped-law constant weak test. This lower-cycle refinement removes the abstract `totalMass`/local-normalization inputs from the cycle-90 middle handoff. It still requires the source-cited identification that the raw KL scalar `massTerm` is the derivative of the constant weak-test pairing for `hat rho_s = Law(hat X_s)`. -/ theorem discreteForwardKlMassTermZeroOfLawConstantTestDerivative {Ω State : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] (hatX : Real → Ω → State) (s0 massTerm : Real) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hmassDeriv : HasDerivAt (fun s => ∫ _, (1 : Real) ∂MeasureTheory.Measure.map (hatX s) P) massTerm s0) : massTerm = 0 := by have hzeroDeriv : HasDerivAt (fun s => ∫ _, (1 : Real) ∂MeasureTheory.Measure.map (hatX s) P) 0 s0 := discreteForwardKlLawConstantTestHasDerivAtZero P hatX s0 hhatX exact hmassDeriv.unique hzeroDeriv /-- Discrete derivative split with mass conservation from the mapped law. Compared with `discreteForwardKlDerivativeSplitOfMassDerivativeScalar`, this specializes the total-mass function to the source law `hat rho_s = Measure.map (hatX s) P` and proves the constant-test derivative is zero locally. The only remaining mass input is the analytic derivative identification for the raw KL display. -/ theorem discreteForwardKlDerivativeSplitOfLawConstantTestMassScalar {Ω State : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] (hatX : Real → Ω → State) (dK firstTerm massTerm targetTerm fisher frozenCross movingCross s0 : Real) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hklRaw : dK = firstTerm + massTerm + targetTerm) (hmassDeriv : HasDerivAt (fun s => ∫ _, (1 : Real) ∂MeasureTheory.Measure.map (hatX s) P) massTerm s0) (hfirst : firstTerm = -fisher + frozenCross) (htarget : targetTerm = movingCross) : dK = -fisher + frozenCross + movingCross := by have hmass : massTerm = 0 := discreteForwardKlMassTermZeroOfLawConstantTestDerivative P hatX s0 massTerm hhatX hmassDeriv exact discreteForwardKlDerivativeSplitOfRawIbpsScalar dK firstTerm massTerm targetTerm fisher frozenCross movingCross hklRaw hmass hfirst htarget /-- Scalar derivative handoff for discrete forward-KL before the DV step. This is the lower-cycle proof-producing core for `appendix.tex:388-491`. It starts after the EM conditional Fokker--Planck and integration-by-parts backend has supplied the KL derivative display `dK = -FI + frozenCross + movingCross`, after `lem:frozen_delta_cross_lip_sald` has supplied the frozen-defect bound, and after Young's inequality has supplied the moving-target velocity bound. The lemma then consumes the LSI half-Fisher comparison to obtain the exact pre-DV `s`-time inequality used in Eq. `eq:KL-derivative-5-discrete`. -/ theorem discreteForwardKlPostLsiDerivativeBoundScalar (dK frozenCross movingCross fisher kl cLSI tildeVelocitySq eta alphaPrimeInv gamma delta : Real) (hderiv : dK = -fisher + frozenCross + movingCross) (hfrozen : frozenCross ≤ (1 / 4) * fisher + (2 * eta ^ 2 * alphaPrimeInv * gamma) * kl + 2 * eta * delta) (hmoving : movingCross ≤ (1 / 4) * fisher + tildeVelocitySq) (hlsiHalf : cLSI * kl ≤ (1 / 2) * fisher) : dK ≤ -(cLSI - 2 * eta ^ 2 * alphaPrimeInv * gamma) * kl + tildeVelocitySq + 2 * eta * delta := by rw [hderiv] calc -fisher + frozenCross + movingCross ≤ -fisher + ((1 / 4) * fisher + (2 * eta ^ 2 * alphaPrimeInv * gamma) * kl + 2 * eta * delta) + ((1 / 4) * fisher + tildeVelocitySq) := by linarith _ ≤ -cLSI * kl + (2 * eta ^ 2 * alphaPrimeInv * gamma) * kl + tildeVelocitySq + 2 * eta * delta := by linarith _ = -(cLSI - 2 * eta ^ 2 * alphaPrimeInv * gamma) * kl + tildeVelocitySq + 2 * eta * delta := by ring /-- Discrete forward-KL derivative handoff using the source KL/FI comparison. This packages the `eq:LSI-KL-FI` scalar half-Fisher bridge into `discreteForwardKlPostLsiDerivativeBoundScalar`. The density-test proof of `KL <= FI/(2*C_LSI)`, the EM Fokker--Planck derivative identity, the frozen defect lemma, and Young/Cauchy analytic inputs remain explicit obligations. -/ theorem discreteForwardKlPostLsiDerivativeBoundOfKlFiScalar (dK frozenCross movingCross fisher kl cLSI tildeVelocitySq eta alphaPrimeInv gamma delta : Real) (hcLSI : 0 < cLSI) (hderiv : dK = -fisher + frozenCross + movingCross) (hfrozen : frozenCross ≤ (1 / 4) * fisher + (2 * eta ^ 2 * alphaPrimeInv * gamma) * kl + 2 * eta * delta) (hmoving : movingCross ≤ (1 / 4) * fisher + tildeVelocitySq) (hklfi : kl ≤ fisher / (2 * cLSI)) : dK ≤ -(cLSI - 2 * eta ^ 2 * alphaPrimeInv * gamma) * kl + tildeVelocitySq + 2 * eta * delta := by exact discreteForwardKlPostLsiDerivativeBoundScalar dK frozenCross movingCross fisher kl cLSI tildeVelocitySq eta alphaPrimeInv gamma delta hderiv hfrozen hmoving (lsiKlFiHalfFisherScalar kl fisher cLSI hcLSI hklfi) /-- Discrete forward-KL derivative handoff from raw KL and named IBP pieces. This composes the cycle-89 raw derivative split with the existing LSI scalar handoff. The theorem proves only Real/order bookkeeping once the analytic backend supplies mass conservation, `eq:KL-derivative-1-discrete`, and `eq:KL-derivative-2-discrete`; those integration-by-parts, density, boundary, and Fisher-information facts remain obligations. -/ theorem discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar (dK firstTerm massTerm targetTerm frozenCross movingCross fisher kl cLSI tildeVelocitySq eta alphaPrimeInv gamma delta : Real) (hcLSI : 0 < cLSI) (hklRaw : dK = firstTerm + massTerm + targetTerm) (hmass : massTerm = 0) (hfirst : firstTerm = -fisher + frozenCross) (htarget : targetTerm = movingCross) (hfrozen : frozenCross ≤ (1 / 4) * fisher + (2 * eta ^ 2 * alphaPrimeInv * gamma) * kl + 2 * eta * delta) (hmoving : movingCross ≤ (1 / 4) * fisher + tildeVelocitySq) (hklfi : kl ≤ fisher / (2 * cLSI)) : dK ≤ -(cLSI - 2 * eta ^ 2 * alphaPrimeInv * gamma) * kl + tildeVelocitySq + 2 * eta * delta := by have hderiv : dK = -fisher + frozenCross + movingCross := discreteForwardKlDerivativeSplitOfRawIbpsScalar dK firstTerm massTerm targetTerm fisher frozenCross movingCross hklRaw hmass hfirst htarget exact discreteForwardKlPostLsiDerivativeBoundOfKlFiScalar dK frozenCross movingCross fisher kl cLSI tildeVelocitySq eta alphaPrimeInv gamma delta hcLSI hderiv hfrozen hmoving hklfi /-- Discrete post-LSI handoff with mass conservation derived from total mass. This is the cycle-90 middle route for `eq:KL-derivative-0-discrete`: it feeds the total-mass derivative lemma into the cycle-89 raw IBP route, so lower work can target the smaller analytic theorem "differentiate the probability-law normalization with the constant weak test" rather than a bare `hmass` scalar. -/ theorem discreteForwardKlPostLsiDerivativeBoundOfMassDerivativeScalar (dK firstTerm massTerm targetTerm frozenCross movingCross fisher kl cLSI tildeVelocitySq eta alphaPrimeInv gamma delta s0 : Real) (totalMass : Real → Real) (hcLSI : 0 < cLSI) (hklRaw : dK = firstTerm + massTerm + targetTerm) (hmassDeriv : HasDerivAt totalMass massTerm s0) (hprobMass : ∀ᶠ s in nhds s0, totalMass s = 1) (hfirst : firstTerm = -fisher + frozenCross) (htarget : targetTerm = movingCross) (hfrozen : frozenCross ≤ (1 / 4) * fisher + (2 * eta ^ 2 * alphaPrimeInv * gamma) * kl + 2 * eta * delta) (hmoving : movingCross ≤ (1 / 4) * fisher + tildeVelocitySq) (hklfi : kl ≤ fisher / (2 * cLSI)) : dK ≤ -(cLSI - 2 * eta ^ 2 * alphaPrimeInv * gamma) * kl + tildeVelocitySq + 2 * eta * delta := by have hmass : massTerm = 0 := discreteForwardKlMassTermZeroOfTotalMassDerivative totalMass s0 massTerm hmassDeriv hprobMass exact discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar dK firstTerm massTerm targetTerm frozenCross movingCross fisher kl cLSI tildeVelocitySq eta alphaPrimeInv gamma delta hcLSI hklRaw hmass hfirst htarget hfrozen hmoving hklfi /-- Discrete post-LSI handoff with mapped-law constant-test mass conservation. This is the cycle-90 lower route for `eq:KL-derivative-0-discrete`: the raw derivative split no longer needs a standalone `hmass` hypothesis or an abstract locally constant `totalMass`; it uses the concrete law `Measure.map (hatX s) P` and the constant weak-test derivative. The remaining mass-side analytic boundary is the derivative identification hypothesis `hmassDeriv`. -/ theorem discreteForwardKlPostLsiDerivativeBoundOfLawConstantTestMassScalar {Ω State : Type*} [MeasurableSpace Ω] [MeasurableSpace State] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] (hatX : Real → Ω → State) (dK firstTerm massTerm targetTerm frozenCross movingCross fisher kl cLSI tildeVelocitySq eta alphaPrimeInv gamma delta s0 : Real) (hhatX : ∀ s, AEMeasurable (hatX s) P) (hcLSI : 0 < cLSI) (hklRaw : dK = firstTerm + massTerm + targetTerm) (hmassDeriv : HasDerivAt (fun s => ∫ _, (1 : Real) ∂MeasureTheory.Measure.map (hatX s) P) massTerm s0) (hfirst : firstTerm = -fisher + frozenCross) (htarget : targetTerm = movingCross) (hfrozen : frozenCross ≤ (1 / 4) * fisher + (2 * eta ^ 2 * alphaPrimeInv * gamma) * kl + 2 * eta * delta) (hmoving : movingCross ≤ (1 / 4) * fisher + tildeVelocitySq) (hklfi : kl ≤ fisher / (2 * cLSI)) : dK ≤ -(cLSI - 2 * eta ^ 2 * alphaPrimeInv * gamma) * kl + tildeVelocitySq + 2 * eta * delta := by have hmass : massTerm = 0 := discreteForwardKlMassTermZeroOfLawConstantTestDerivative P hatX s0 massTerm hhatX hmassDeriv exact discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar dK firstTerm massTerm targetTerm frozenCross movingCross fisher kl cLSI tildeVelocitySq eta alphaPrimeInv gamma delta hcLSI hklRaw hmass hfirst htarget hfrozen hmoving hklfi /-- Scalar substitution for the first derivative term in continuous forward-KL. In appendix lines 168-185, the analytic obligations first produce the KL derivative identity and then identify the SALD Fokker--Planck/integration-by- parts contribution with `-FI`. This lemma only substitutes that supplied identity into the scalar derivative display; it does not prove mass conservation, differentiation under the integral, Fokker--Planck, boundary decay, or the Fisher-information identity. -/ theorem forwardKlFirstTermFisherSubstitutionScalar (dK firstTerm targetTerm fisher : Real) (hkl : dK = firstTerm + targetTerm) (hfirst : firstTerm = -fisher) : dK = -fisher + targetTerm := by rw [hkl, hfirst] /-- Scalar mass-conservation drop in the continuous forward-KL derivative. In `appendix.tex:168-174`, differentiating the KL integrand first produces the extra scalar term corresponding to `int partial_s rho_s dx`. The source then drops it using mass conservation. This lemma only records that real-algebra handoff after the analytic mass-conservation identity has been supplied. -/ theorem forwardKlMassConservationDropScalar (dK firstTerm massTerm targetTerm : Real) (hkl : dK = firstTerm + massTerm + targetTerm) (hmass : massTerm = 0) : dK = firstTerm + targetTerm := by rw [hkl, hmass] ring /-- First continuous forward-KL derivative scalar handoff after mass conservation. This composes the source mass-conservation drop with the already isolated `-FI` first-term substitution from `appendix.tex:176-185`. It does not prove mass conservation, differentiation under the integral, Fokker--Planck, integration by parts, or the Fisher-information identity. -/ theorem forwardKlMassConservationFirstTermFisherScalar (dK firstTerm massTerm targetTerm fisher : Real) (hkl : dK = firstTerm + massTerm + targetTerm) (hmass : massTerm = 0) (hfirst : firstTerm = -fisher) : dK = -fisher + targetTerm := by exact forwardKlFirstTermFisherSubstitutionScalar dK firstTerm targetTerm fisher (forwardKlMassConservationDropScalar dK firstTerm massTerm targetTerm hkl hmass) hfirst /-- Scalar Young bound for the target-side transport term in forward-KL. In appendix lines 199-208, Cauchy--Schwarz first gives the target-side term bounded by `sqrt(FI) * ||tilde v_s||`. This lemma formalizes only the final Real Young step with the source coefficients `1/2` and `1/2`; target transport, integration by parts, Cauchy--Schwarz, and L2/FI identifications remain analytic obligations. -/ theorem forwardKlTargetTransportYoungBoundScalar (targetTerm fisher velocitySq : Real) (hfisher : 0 ≤ fisher) (hvelocitySq : 0 ≤ velocitySq) (hcauchy : targetTerm ≤ Real.sqrt fisher * Real.sqrt velocitySq) : targetTerm ≤ (1 / 2) * fisher + (1 / 2) * velocitySq := by have hyoung : Real.sqrt fisher * Real.sqrt velocitySq ≤ (1 / 2) * fisher + (1 / 2) * velocitySq := by have hsq : 0 ≤ (Real.sqrt fisher - Real.sqrt velocitySq) ^ 2 := sq_nonneg _ have hf : (Real.sqrt fisher) ^ 2 = fisher := Real.sq_sqrt hfisher have hv : (Real.sqrt velocitySq) ^ 2 = velocitySq := Real.sq_sqrt hvelocitySq nlinarith exact hcauchy.trans hyoung /-- Scalar post-Young derivative bound for continuous forward-KL. This is the theorem-independent arithmetic after the analytic source steps in `appendix.tex:168-208` have supplied the KL derivative display, the first-term Fisher identity, and the target-side Cauchy--Schwarz/Young bound. It does not prove differentiation under the integral, Fokker--Planck, integration by parts, or the target transport identity. -/ theorem forwardKlPostYoungDerivativeBoundScalar (dK firstTerm targetTerm fisher velocitySq : Real) (hkl : dK = firstTerm + targetTerm) (hfirst : firstTerm = -fisher) (htarget : targetTerm ≤ (1 / 2) * fisher + (1 / 2) * velocitySq) : dK ≤ -(1 / 2) * fisher + (1 / 2) * velocitySq := by rw [hkl, hfirst] linarith /-- Post-Young derivative bound using the target-side Cauchy input directly. This composes `forwardKlTargetTransportYoungBoundScalar` with the existing post-Young derivative bookkeeping. It still starts after the source analytic steps have supplied the KL derivative display, the first-term Fisher identity, and the Cauchy--Schwarz target-side estimate. -/ theorem forwardKlPostYoungDerivativeBoundOfCauchyScalar (dK firstTerm targetTerm fisher velocitySq : Real) (hfisher : 0 ≤ fisher) (hvelocitySq : 0 ≤ velocitySq) (hkl : dK = firstTerm + targetTerm) (hfirst : firstTerm = -fisher) (hcauchy : targetTerm ≤ Real.sqrt fisher * Real.sqrt velocitySq) : dK ≤ -(1 / 2) * fisher + (1 / 2) * velocitySq := by exact forwardKlPostYoungDerivativeBoundScalar dK firstTerm targetTerm fisher velocitySq hkl hfirst (forwardKlTargetTransportYoungBoundScalar targetTerm fisher velocitySq hfisher hvelocitySq hcauchy) /-- Scalar LSI substitution for the continuous forward-KL derivative bound. After `appendix.tex:199-208` gives the post-Young bound, the source applies LSI in `appendix.tex:210-217`. This lemma records only the real-order handoff from a supplied comparison `C_LSI*K <= (1/2)*FI`; the density-test LSI backend remains `probability.lsi_to_kl_fi`. -/ theorem forwardKlLsiDerivativeBoundScalar (dK fisher velocitySq kl cLSI : Real) (hpostYoung : dK ≤ -(1 / 2) * fisher + (1 / 2) * velocitySq) (hlsiHalf : cLSI * kl ≤ (1 / 2) * fisher) : dK ≤ -cLSI * kl + (1 / 2) * velocitySq := by linarith /-- LSI substitution using the source KL/FI comparison directly. The paper cites `eq:LSI-KL-FI` in the form `KL <= FI/(2*C_LSI)` and then uses it as `C_LSI*KL <= (1/2)*FI` in the derivative estimate. This lemma closes only that scalar handoff for the forward-KL derivative block; the density-test backend proving the KL/FI comparison remains `probability.lsi_to_kl_fi`. -/ theorem forwardKlLsiDerivativeBoundOfKlFiScalar (dK fisher velocitySq kl cLSI : Real) (hpostYoung : dK ≤ -(1 / 2) * fisher + (1 / 2) * velocitySq) (hcLSI : 0 < cLSI) (hklfi : kl ≤ fisher / (2 * cLSI)) : dK ≤ -cLSI * kl + (1 / 2) * velocitySq := by exact forwardKlLsiDerivativeBoundScalar dK fisher velocitySq kl cLSI hpostYoung (lsiKlFiHalfFisherScalar kl fisher cLSI hcLSI hklfi) /-- Scalar inverse-schedule handoff for the continuous forward-KL derivative. After `appendix.tex:210-217` gives the `s`-time LSI derivative inequality, `appendix.tex:218-228` changes variables from `s` to `t`. This lemma proves only the real-order bookkeeping after the analytic chain rule, velocity scaling, and inverse-derivative identity have been supplied. -/ theorem forwardKlTimeChangedDerivativeBoundScalar (dKds dKdt kl cLSI tildeVelocitySq velocitySq dotS dotT : Real) (hdotSNonneg : 0 ≤ dotS) (hdKdt : dKdt = dotS * dKds) (hsBound : dKds ≤ -cLSI * kl + (1 / 2) * tildeVelocitySq) (hvelocity : tildeVelocitySq = dotT ^ 2 * velocitySq) (hdotT : dotT = dotS⁻¹) (hdotS : dotS ≠ 0) : dKdt ≤ -(dotS * cLSI) * kl + (1 / 2) * dotS⁻¹ * velocitySq := by rw [hdKdt] have hmul : dotS * dKds ≤ dotS * (-cLSI * kl + (1 / 2) * tildeVelocitySq) := by exact mul_le_mul_of_nonneg_left hsBound hdotSNonneg calc dotS * dKds ≤ dotS * (-cLSI * kl + (1 / 2) * tildeVelocitySq) := hmul _ = dotS * (-cLSI * kl + (1 / 2) * (dotT ^ 2 * velocitySq)) := by rw [hvelocity] _ = -(dotS * cLSI) * kl + (1 / 2) * (dotS * (dotT ^ 2 * velocitySq)) := by ring _ = -(dotS * cLSI) * kl + (1 / 2) * (dotS⁻¹ * velocitySq) := by rw [discreteForwardKlTimeChangeSquareCoefficientRewriteScalar dotS dotT velocitySq hdotT hdotS] _ = -(dotS * cLSI) * kl + (1 / 2) * dotS⁻¹ * velocitySq := by ring /-- Scalar inverse-derivative handoff for the continuous forward-KL schedule. The analytic inverse-function theorem is still part of `sald.forward_kl.schedule_time_change`. This lemma only proves the Real algebra used after that backend supplies the product identity `dot{s}(t) * dot{t}(s(t)) = 1`. -/ theorem forwardKlInverseScheduleDerivativeScalar (dotS dotT : Real) (hinv : dotS * dotT = 1) : dotT = dotS⁻¹ ∧ dotS ≠ 0 := by have hdotS : dotS ≠ 0 := by intro hzero rw [hzero] at hinv norm_num at hinv constructor · calc dotT = 1 * dotT := by ring _ = (dotS * dotS⁻¹) * dotT := by rw [mul_inv_cancel₀ hdotS] _ = dotS⁻¹ * (dotS * dotT) := by ring _ = dotS⁻¹ * 1 := by rw [hinv] _ = dotS⁻¹ := by ring · exact hdotS /-- Source-shaped square-coefficient rewrite for the forward-KL time change. This version starts from the inverse-derivative product identity rather than a pre-rewritten `dotT = dotS⁻¹`. It is pure scalar algebra for `appendix.tex:218-228`; the inverse-function calculus remains an obligation. -/ theorem forwardKlTimeChangeSquareCoefficientRewriteOfProductScalar (dotS dotT coeff : Real) (hinv : dotS * dotT = 1) : dotS * (dotT ^ 2 * coeff) = dotS⁻¹ * coeff := by have hinvData := forwardKlInverseScheduleDerivativeScalar dotS dotT hinv exact discreteForwardKlTimeChangeSquareCoefficientRewriteScalar dotS dotT coeff hinvData.1 hinvData.2 /-- Scalar velocity-square scaling used by the slowed target. In `appendix.tex:191-197`, the paper defines `\tilde v_s = dot{t}(s) v_{t(s)}`. Once the analytic L2 backend has reduced that identity to scalar norm-square inputs, this lemma produces `||\tilde v_s||^2 = dot{t}(s)^2 ||v_t||^2`. -/ theorem forwardKlVelocitySquareScalingScalar (tildeVelocitySq velocitySq tildeVelocityNorm velocityNorm dotT : Real) (htildeSq : tildeVelocitySq = tildeVelocityNorm ^ 2) (hvelocitySq : velocitySq = velocityNorm ^ 2) (hscale : tildeVelocityNorm = dotT * velocityNorm) : tildeVelocitySq = dotT ^ 2 * velocitySq := by rw [htildeSq, hvelocitySq, hscale] ring /-- Time-changed forward-KL derivative bound from source-shaped schedule data. This composes `forwardKlTimeChangedDerivativeBoundScalar` with the scalar inverse-derivative handoff from `dotS * dotT = 1`. It still assumes the analytic chain rule and velocity-square scaling as explicit inputs. -/ theorem forwardKlTimeChangedDerivativeBoundOfProductScalar (dKds dKdt kl cLSI tildeVelocitySq velocitySq dotS dotT : Real) (hdotSNonneg : 0 ≤ dotS) (hdKdt : dKdt = dotS * dKds) (hsBound : dKds ≤ -cLSI * kl + (1 / 2) * tildeVelocitySq) (hvelocity : tildeVelocitySq = dotT ^ 2 * velocitySq) (hinv : dotS * dotT = 1) : dKdt ≤ -(dotS * cLSI) * kl + (1 / 2) * dotS⁻¹ * velocitySq := by have hinvData := forwardKlInverseScheduleDerivativeScalar dotS dotT hinv exact forwardKlTimeChangedDerivativeBoundScalar dKds dKdt kl cLSI tildeVelocitySq velocitySq dotS dotT hdotSNonneg hdKdt hsBound hvelocity hinvData.1 hinvData.2 /-- Scalar pipeline for the continuous forward-KL pre-DV derivative bound. This theorem composes the already-formalized scalar steps for `appendix.tex:168-228`: first-term Fisher substitution, target-side Cauchy/Young, LSI half-Fisher substitution, and inverse-schedule coefficient bookkeeping. It assumes the analytic KL differentiation, Fokker--Planck, integration-by-parts, LSI, and schedule identities explicitly; those backends remain proof obligations. -/ theorem forwardKlPreDvDerivativeBoundScalar (dKds dKdt firstTerm targetTerm fisher tildeVelocitySq velocitySq kl cLSI dotS dotT : Real) (hfisher : 0 ≤ fisher) (htildeVelocitySq : 0 ≤ tildeVelocitySq) (hdotSNonneg : 0 ≤ dotS) (hkl : dKds = firstTerm + targetTerm) (hfirst : firstTerm = -fisher) (hcauchy : targetTerm ≤ Real.sqrt fisher * Real.sqrt tildeVelocitySq) (hlsiHalf : cLSI * kl ≤ (1 / 2) * fisher) (hdKdt : dKdt = dotS * dKds) (hvelocity : tildeVelocitySq = dotT ^ 2 * velocitySq) (hdotT : dotT = dotS⁻¹) (hdotS : dotS ≠ 0) : dKdt ≤ -(dotS * cLSI) * kl + (1 / 2) * dotS⁻¹ * velocitySq := by have hpost : dKds ≤ -(1 / 2) * fisher + (1 / 2) * tildeVelocitySq := by exact forwardKlPostYoungDerivativeBoundOfCauchyScalar dKds firstTerm targetTerm fisher tildeVelocitySq hfisher htildeVelocitySq hkl hfirst hcauchy have hlsi : dKds ≤ -cLSI * kl + (1 / 2) * tildeVelocitySq := by exact forwardKlLsiDerivativeBoundScalar dKds fisher tildeVelocitySq kl cLSI hpost hlsiHalf exact forwardKlTimeChangedDerivativeBoundScalar dKds dKdt kl cLSI tildeVelocitySq velocitySq dotS dotT hdotSNonneg hdKdt hlsi hvelocity hdotT hdotS /-- Pre-DV derivative pipeline with source-shaped inverse-schedule input. This is the same scalar pipeline as `forwardKlPreDvDerivativeBoundScalar`, but the schedule side starts from the product identity `dot{s}(t) * dot{t}(s(t)) = 1` used in the source proof. -/ theorem forwardKlPreDvDerivativeBoundOfProductScalar (dKds dKdt firstTerm targetTerm fisher tildeVelocitySq velocitySq kl cLSI dotS dotT : Real) (hfisher : 0 ≤ fisher) (htildeVelocitySq : 0 ≤ tildeVelocitySq) (hdotSNonneg : 0 ≤ dotS) (hkl : dKds = firstTerm + targetTerm) (hfirst : firstTerm = -fisher) (hcauchy : targetTerm ≤ Real.sqrt fisher * Real.sqrt tildeVelocitySq) (hlsiHalf : cLSI * kl ≤ (1 / 2) * fisher) (hdKdt : dKdt = dotS * dKds) (hvelocity : tildeVelocitySq = dotT ^ 2 * velocitySq) (hinv : dotS * dotT = 1) : dKdt ≤ -(dotS * cLSI) * kl + (1 / 2) * dotS⁻¹ * velocitySq := by have hpost : dKds ≤ -(1 / 2) * fisher + (1 / 2) * tildeVelocitySq := by exact forwardKlPostYoungDerivativeBoundOfCauchyScalar dKds firstTerm targetTerm fisher tildeVelocitySq hfisher htildeVelocitySq hkl hfirst hcauchy have hlsi : dKds ≤ -cLSI * kl + (1 / 2) * tildeVelocitySq := by exact forwardKlLsiDerivativeBoundScalar dKds fisher tildeVelocitySq kl cLSI hpost hlsiHalf exact forwardKlTimeChangedDerivativeBoundOfProductScalar dKds dKdt kl cLSI tildeVelocitySq velocitySq dotS dotT hdotSNonneg hdKdt hlsi hvelocity hinv /-- Pre-DV derivative pipeline with the slowed-velocity square scaling exposed. This theorem matches the source `appendix.tex:191-228` bookkeeping most closely among the scalar lemmas: it derives the nonnegativity and square scaling of `||\tilde v_s||^2` from supplied norm-square identities, and derives the inverse coefficient rewrite from `dotS * dotT = 1`. -/ theorem forwardKlPreDvDerivativeBoundOfVelocityScalingScalar (dKds dKdt firstTerm targetTerm fisher tildeVelocitySq velocitySq tildeVelocityNorm velocityNorm kl cLSI dotS dotT : Real) (hfisher : 0 ≤ fisher) (hdotSNonneg : 0 ≤ dotS) (hkl : dKds = firstTerm + targetTerm) (hfirst : firstTerm = -fisher) (hcauchy : targetTerm ≤ Real.sqrt fisher * Real.sqrt tildeVelocitySq) (hlsiHalf : cLSI * kl ≤ (1 / 2) * fisher) (hdKdt : dKdt = dotS * dKds) (htildeSq : tildeVelocitySq = tildeVelocityNorm ^ 2) (hvelocitySq : velocitySq = velocityNorm ^ 2) (hscale : tildeVelocityNorm = dotT * velocityNorm) (hinv : dotS * dotT = 1) : dKdt ≤ -(dotS * cLSI) * kl + (1 / 2) * dotS⁻¹ * velocitySq := by have htildeVelocitySq : 0 ≤ tildeVelocitySq := by rw [htildeSq] exact sq_nonneg tildeVelocityNorm have hvelocity : tildeVelocitySq = dotT ^ 2 * velocitySq := forwardKlVelocitySquareScalingScalar tildeVelocitySq velocitySq tildeVelocityNorm velocityNorm dotT htildeSq hvelocitySq hscale exact forwardKlPreDvDerivativeBoundOfProductScalar dKds dKdt firstTerm targetTerm fisher tildeVelocitySq velocitySq kl cLSI dotS dotT hfisher htildeVelocitySq hdotSNonneg hkl hfirst hcauchy hlsiHalf hdKdt hvelocity hinv /-- Pre-DV derivative pipeline using the source KL/FI comparison. This is the lower-cycle theorem-specific bridge for `appendix.tex:168-228`. It threads the supplied KL derivative display, first-term Fisher identity, target Cauchy estimate, source `KL <= FI/(2*C_LSI)` comparison, velocity norm-square scaling, and inverse-schedule product identity into the t-time pre-DV inequality. The measure-theoretic KL derivative, Fokker--Planck, integration-by-parts, LSI density-test, and inverse-function backends remain separate obligations. -/ theorem forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar (dKds dKdt firstTerm targetTerm fisher tildeVelocitySq velocitySq tildeVelocityNorm velocityNorm kl cLSI dotS dotT : Real) (hfisher : 0 ≤ fisher) (hcLSI : 0 < cLSI) (hdotSNonneg : 0 ≤ dotS) (hkl : dKds = firstTerm + targetTerm) (hfirst : firstTerm = -fisher) (hcauchy : targetTerm ≤ Real.sqrt fisher * Real.sqrt tildeVelocitySq) (hklfi : kl ≤ fisher / (2 * cLSI)) (hdKdt : dKdt = dotS * dKds) (htildeSq : tildeVelocitySq = tildeVelocityNorm ^ 2) (hvelocitySq : velocitySq = velocityNorm ^ 2) (hscale : tildeVelocityNorm = dotT * velocityNorm) (hinv : dotS * dotT = 1) : dKdt ≤ -(dotS * cLSI) * kl + (1 / 2) * dotS⁻¹ * velocitySq := by have hlsiHalf : cLSI * kl ≤ (1 / 2) * fisher := lsiKlFiHalfFisherScalar kl fisher cLSI hcLSI hklfi exact forwardKlPreDvDerivativeBoundOfVelocityScalingScalar dKds dKdt firstTerm targetTerm fisher tildeVelocitySq velocitySq tildeVelocityNorm velocityNorm kl cLSI dotS dotT hfisher hdotSNonneg hkl hfirst hcauchy hlsiHalf hdKdt htildeSq hvelocitySq hscale hinv /-- Pre-DV derivative pipeline from the raw KL derivative split. This is the cycle-60 lower scalar wrapper for `appendix.tex:168-228`. It starts from the source derivative display before the mass-conservation term is dropped, uses the supplied `int partial_s rho_s = 0` input to remove that term, then reuses the existing first-term/FI, target Young, LSI, slowed-velocity, and inverse-schedule scalar pipeline. It does not prove differentiation under the integral, mass conservation, Fokker--Planck, integration by parts, LSI, or the inverse-function theorem. -/ theorem forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar (dKds dKdt firstTerm massTerm targetTerm fisher tildeVelocitySq velocitySq tildeVelocityNorm velocityNorm kl cLSI dotS dotT : Real) (hfisher : 0 ≤ fisher) (hcLSI : 0 < cLSI) (hdotSNonneg : 0 ≤ dotS) (hklRaw : dKds = firstTerm + massTerm + targetTerm) (hmass : massTerm = 0) (hfirst : firstTerm = -fisher) (hcauchy : targetTerm ≤ Real.sqrt fisher * Real.sqrt tildeVelocitySq) (hklfi : kl ≤ fisher / (2 * cLSI)) (hdKdt : dKdt = dotS * dKds) (htildeSq : tildeVelocitySq = tildeVelocityNorm ^ 2) (hvelocitySq : velocitySq = velocityNorm ^ 2) (hscale : tildeVelocityNorm = dotT * velocityNorm) (hinv : dotS * dotT = 1) : dKdt ≤ -(dotS * cLSI) * kl + (1 / 2) * dotS⁻¹ * velocitySq := by have hklReduced : dKds = firstTerm + targetTerm := forwardKlMassConservationDropScalar dKds firstTerm massTerm targetTerm hklRaw hmass exact forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar dKds dKdt firstTerm targetTerm fisher tildeVelocitySq velocitySq tildeVelocityNorm velocityNorm kl cLSI dotS dotT hfisher hcLSI hdotSNonneg hklReduced hfirst hcauchy hklfi hdKdt htildeSq hvelocitySq hscale hinv /-- Pointwise continuous forward-KL pre-DV derivative handoff. This is the cycle-65 lower wrapper for `appendix.tex:168-228`. The scalar lemma above handles one fixed time after the analytic KL derivative, mass-conservation, Fokker--Planck/integration-by-parts, LSI, target-transport, and inverse-schedule inputs have been supplied. This wrapper exposes the same result in the pointwise `t`-indexed shape consumed by the faithful `thm:forward-KL` skeleton before the separate DV and Gronwall obligations. -/ theorem forwardKlPointwisePreDvDerivativeBoundOfRawKlFiVelocityScaling (dKds dKdt firstTerm massTerm targetTerm fisher tildeVelocitySq velocitySq tildeVelocityNorm velocityNorm kl cLSI dotS dotT : Real → Real) (hfisher : ∀ t, 0 ≤ fisher t) (hcLSI : ∀ t, 0 < cLSI t) (hdotSNonneg : ∀ t, 0 ≤ dotS t) (hklRaw : ∀ t, dKds t = firstTerm t + massTerm t + targetTerm t) (hmass : ∀ t, massTerm t = 0) (hfirst : ∀ t, firstTerm t = -fisher t) (hcauchy : ∀ t, targetTerm t ≤ Real.sqrt (fisher t) * Real.sqrt (tildeVelocitySq t)) (hklfi : ∀ t, kl t ≤ fisher t / (2 * cLSI t)) (hdKdt : ∀ t, dKdt t = dotS t * dKds t) (htildeSq : ∀ t, tildeVelocitySq t = tildeVelocityNorm t ^ 2) (hvelocitySq : ∀ t, velocitySq t = velocityNorm t ^ 2) (hscale : ∀ t, tildeVelocityNorm t = dotT t * velocityNorm t) (hinv : ∀ t, dotS t * dotT t = 1) : ∀ t, dKdt t ≤ -(dotS t * cLSI t) * kl t + (1 / 2) * (dotS t)⁻¹ * velocitySq t := by intro t exact forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar (dKds t) (dKdt t) (firstTerm t) (massTerm t) (targetTerm t) (fisher t) (tildeVelocitySq t) (velocitySq t) (tildeVelocityNorm t) (velocityNorm t) (kl t) (cLSI t) (dotS t) (dotT t) (hfisher t) (hcLSI t) (hdotSNonneg t) (hklRaw t) (hmass t) (hfirst t) (hcauchy t) (hklfi t) (hdKdt t) (htildeSq t) (hvelocitySq t) (hscale t) (hinv t) /-- Source-shaped handoff from the KL derivative backend and DV to Gronwall. This is the cycle-50 lower scalar bridge for `appendix.tex:168-241`. It starts from the explicit analytic inputs owned by `sald.forward_kl.kl_derivative`, `probability.lsi_to_kl_fi`, and the theorem-specific DV witness, then collects the source coefficient `dot{s}(t)*C_LSI(t) - (1/2)*dot{s}(t)^(-1)*alpha^(-1)` used by Gronwall. It does not prove the Fokker--Planck/KL derivative identity, the LSI density-test theorem, the finite-log-mgf witness, or the inverse-schedule calculus. -/ theorem forwardKlDerivativeDvGronwallCoefficientOfKlFiVelocityScalingScalar (dKds dKdt firstTerm targetTerm fisher tildeVelocitySq velocitySq tildeVelocityNorm velocityNorm kl cLSI dotS dotT alpha logMgf eAlpha : Real) (hfisher : 0 ≤ fisher) (hcLSI : 0 < cLSI) (hdotSNonneg : 0 ≤ dotS) (halpha : 0 < alpha) (hkl : dKds = firstTerm + targetTerm) (hfirst : firstTerm = -fisher) (hcauchy : targetTerm ≤ Real.sqrt fisher * Real.sqrt tildeVelocitySq) (hklfi : kl ≤ fisher / (2 * cLSI)) (hdKdt : dKdt = dotS * dKds) (htildeSq : tildeVelocitySq = tildeVelocityNorm ^ 2) (hvelocitySq : velocitySq = velocityNorm ^ 2) (hscale : tildeVelocityNorm = dotT * velocityNorm) (hinv : dotS * dotT = 1) (hdv : alpha * velocitySq ≤ kl + logMgf) (heAlpha : eAlpha = alpha⁻¹ * logMgf) : dKdt ≤ -((dotS * cLSI) - (1 / 2) * dotS⁻¹ * alpha⁻¹) * kl + (1 / 2) * dotS⁻¹ * eAlpha := by have hpre : dKdt ≤ -(dotS * cLSI) * kl + (1 / 2) * dotS⁻¹ * velocitySq := forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar dKds dKdt firstTerm targetTerm fisher tildeVelocitySq velocitySq tildeVelocityNorm velocityNorm kl cLSI dotS dotT hfisher hcLSI hdotSNonneg hkl hfirst hcauchy hklfi hdKdt htildeSq hvelocitySq hscale hinv have hcoeff : 0 ≤ (1 / 2 : Real) * dotS⁻¹ := by exact mul_nonneg (by norm_num) (inv_nonneg.mpr hdotSNonneg) exact forwardKlPostDvGronwallCoefficientOfScheduleScalar dKdt kl cLSI dotS alpha velocitySq logMgf eAlpha hpre halpha hcoeff hdv heAlpha /-- Scalar residual split for the continuous general VA-SALD KL derivative. This is the cycle-57 lower proof-producing core for `appendix.tex:765-835`. After the analytic backend supplies the raw KL derivative split, the mass term as zero, the general Fokker--Planck/integration-by-parts evaluation of the `c_t` drift term, and the target-transport evaluation of the `v_t` term, this lemma rewrites the display into the residual form consumed by the existing Young/LSI/time-change pipeline. It does not prove mass conservation, differentiation under the integral, the Fokker--Planck equation, target transport, boundary decay, or any Fisher-information identity. -/ theorem generalMovingTargetKlDerivativeResidualSplitScalar (dKds firstTerm massTerm targetTerm cDrift vDrift fisher residualCross sigmaSq : Real) (hklRaw : dKds = firstTerm + massTerm + targetTerm) (hmass : massTerm = 0) (hfirst : firstTerm = cDrift - (sigmaSq / 2) * fisher) (htarget : targetTerm = -vDrift) (hresidual : residualCross = cDrift - vDrift) : dKds = -(sigmaSq / 2) * fisher + residualCross := by rw [hklRaw, hmass, hfirst, htarget, hresidual] ring /-- Scaled residual display for the continuous general VA-SALD KL derivative. This is a cycle-62 lower scalar core for `appendix.tex:813-835`. Once the analytic backend has supplied the target-transport contribution with `\tilde v_s=\dot t(s)v_{t(s)}` and the residual pairing `m_t=v_t-c_t`, this lemma records the source sign and scaling: the `c_t` drift minus the target `v_t` contribution is `-\dot t(s)` times the residual pairing. It does not prove target transport, integration by parts, or any vector/L2 identity. -/ theorem generalMovingTargetKlDerivativeScaledResidualDisplayScalar (dKds firstTerm massTerm targetTerm cDrift vDrift cPair vPair mPair fisher dotT sigmaSq : Real) (hklRaw : dKds = firstTerm + massTerm + targetTerm) (hmass : massTerm = 0) (hfirst : firstTerm = cDrift - (sigmaSq / 2) * fisher) (htarget : targetTerm = -vDrift) (hcDrift : cDrift = dotT * cPair) (hvDrift : vDrift = dotT * vPair) (hmPair : mPair = vPair - cPair) : dKds = -(sigmaSq / 2) * fisher - dotT * mPair := by rw [hklRaw, hmass, hfirst, htarget, hcDrift, hvDrift, hmPair] ring /-- Scalar post-Young derivative bound for continuous general VA-SALD. In `appendix.tex:835-864`, after the Fokker--Planck and target-transport identities have combined the `c_t` and `v_t` terms into the residual `m_t=v_t-c_t`, Young's inequality leaves exactly `-(sigma_t^2/4)*FI + sigma_t^(-2)*dot t(s)^2*||m_t||^2`. This lemma proves only that real/order handoff from a supplied residual Young bound; the Fokker--Planck, integration-by-parts, Holder/Young, and FI identifications stay as obligations. -/ theorem generalMovingTargetPostYoungDerivativeBoundScalar (dKds fisher residualCross residualCoeff sigmaSq : Real) (hkl : dKds = -(sigmaSq / 2) * fisher + residualCross) (hyoung : residualCross ≤ (sigmaSq / 4) * fisher + residualCoeff) : dKds ≤ -(sigmaSq / 4) * fisher + residualCoeff := by calc dKds = -(sigmaSq / 2) * fisher + residualCross := hkl _ ≤ -(sigmaSq / 2) * fisher + ((sigmaSq / 4) * fisher + residualCoeff) := by simpa [add_comm, add_left_comm, add_assoc] using add_le_add_left hyoung (-(sigmaSq / 2) * fisher) _ = -(sigmaSq / 4) * fisher + residualCoeff := by ring /-- Scalar LSI handoff for continuous general VA-SALD. The source uses `eq:LSI-KL-FI` after the residual Young step. Once the LSI backend has supplied `C_LSI*K <= (1/2)*FI`, this lemma converts `-(sigma_t^2/4)*FI` into the paper's `-(sigma_t^2/2)*C_LSI*K` term. It does not prove the density-test LSI bridge or any sigma regularity beyond the explicit nonnegativity input. -/ theorem generalMovingTargetLsiDerivativeBoundScalar (dKds fisher kl cLSI residualCoeff sigmaSq : Real) (hsigmaSq : 0 ≤ sigmaSq) (hpostYoung : dKds ≤ -(sigmaSq / 4) * fisher + residualCoeff) (hlsiHalf : cLSI * kl ≤ (1 / 2) * fisher) : dKds ≤ -(((sigmaSq / 2) * cLSI) * kl) + residualCoeff := by have hcoeffNonneg : 0 ≤ sigmaSq / 2 := by positivity have hscaled : (sigmaSq / 2) * (cLSI * kl) ≤ (sigmaSq / 2) * ((1 / 2) * fisher) := by exact mul_le_mul_of_nonneg_left hlsiHalf hcoeffNonneg have htarget : ((sigmaSq / 2) * cLSI) * kl ≤ (sigmaSq / 4) * fisher := by calc ((sigmaSq / 2) * cLSI) * kl = (sigmaSq / 2) * (cLSI * kl) := by ring _ ≤ (sigmaSq / 2) * ((1 / 2) * fisher) := hscaled _ = (sigmaSq / 4) * fisher := by ring have hneg : -(sigmaSq / 4) * fisher ≤ -(((sigmaSq / 2) * cLSI) * kl) := by simpa only [neg_mul] using neg_le_neg htarget calc dKds ≤ -(sigmaSq / 4) * fisher + residualCoeff := hpostYoung _ ≤ -(((sigmaSq / 2) * cLSI) * kl) + residualCoeff := by simpa [add_comm, add_left_comm, add_assoc] using add_le_add_right hneg residualCoeff /-- Scalar time-change handoff for the continuous general VA-SALD derivative. This is the `appendix.tex:865-884` real/order step after the analytic schedule backend has supplied `dK/dt=dot{s}(t)*dK/ds` and `dot t(s(t))=dot{s}(t)^(-1)`. It preserves the source coefficient `sigma_t^(-2)*dot{s}(t)^(-1)` on the residual energy. -/ theorem generalMovingTargetTimeChangedDerivativeBoundScalar (dKds dKdt kl cLSI residualSq sigmaSq sigmaInvSq dotS dotT : Real) (hdotSNonneg : 0 ≤ dotS) (hdKdt : dKdt = dotS * dKds) (hsBound : dKds ≤ -(((sigmaSq / 2) * cLSI) * kl) + sigmaInvSq * dotT ^ 2 * residualSq) (hdotT : dotT = dotS⁻¹) (hdotS : dotS ≠ 0) : dKdt ≤ -(((sigmaSq / 2) * dotS * cLSI) * kl) + sigmaInvSq * dotS⁻¹ * residualSq := by rw [hdKdt] have hmul : dotS * dKds ≤ dotS * (-(((sigmaSq / 2) * cLSI) * kl) + sigmaInvSq * dotT ^ 2 * residualSq) := by exact mul_le_mul_of_nonneg_left hsBound hdotSNonneg calc dotS * dKds ≤ dotS * (-(((sigmaSq / 2) * cLSI) * kl) + sigmaInvSq * dotT ^ 2 * residualSq) := hmul _ = -(((sigmaSq / 2) * dotS * cLSI) * kl) + dotS * (dotT ^ 2 * (sigmaInvSq * residualSq)) := by ring _ = -(((sigmaSq / 2) * dotS * cLSI) * kl) + dotS⁻¹ * (sigmaInvSq * residualSq) := by rw [discreteForwardKlTimeChangeSquareCoefficientRewriteScalar dotS dotT (sigmaInvSq * residualSq) hdotT hdotS] _ = -(((sigmaSq / 2) * dotS * cLSI) * kl) + sigmaInvSq * dotS⁻¹ * residualSq := by ring /-- Source-shaped scalar pre-DV derivative pipeline for continuous general VA-SALD. This composes the compiled scalar pieces for `appendix.tex:835-884`: residual Young bookkeeping, LSI half-Fisher substitution, and inverse-schedule coefficient rewriting. The measure-theoretic KL derivative, general Fokker--Planck equation, integration by parts, LSI density test, and schedule calculus remain explicit upstream obligations. -/ theorem generalMovingTargetPreDvDerivativeBoundScalar (dKds dKdt fisher residualCross residualSq kl cLSI sigmaSq sigmaInvSq dotS dotT : Real) (hsigmaSq : 0 ≤ sigmaSq) (hdotSNonneg : 0 ≤ dotS) (hkl : dKds = -(sigmaSq / 2) * fisher + residualCross) (hyoung : residualCross ≤ (sigmaSq / 4) * fisher + sigmaInvSq * dotT ^ 2 * residualSq) (hlsiHalf : cLSI * kl ≤ (1 / 2) * fisher) (hdKdt : dKdt = dotS * dKds) (hdotT : dotT = dotS⁻¹) (hdotS : dotS ≠ 0) : dKdt ≤ -(((sigmaSq / 2) * dotS * cLSI) * kl) + sigmaInvSq * dotS⁻¹ * residualSq := by have hpost : dKds ≤ -(sigmaSq / 4) * fisher + sigmaInvSq * dotT ^ 2 * residualSq := generalMovingTargetPostYoungDerivativeBoundScalar dKds fisher residualCross (sigmaInvSq * dotT ^ 2 * residualSq) sigmaSq hkl hyoung have hlsi : dKds ≤ -(((sigmaSq / 2) * cLSI) * kl) + sigmaInvSq * dotT ^ 2 * residualSq := generalMovingTargetLsiDerivativeBoundScalar dKds fisher kl cLSI (sigmaInvSq * dotT ^ 2 * residualSq) sigmaSq hsigmaSq hpost hlsiHalf exact generalMovingTargetTimeChangedDerivativeBoundScalar dKds dKdt kl cLSI residualSq sigmaSq sigmaInvSq dotS dotT hdotSNonneg hdKdt hlsi hdotT hdotS /-- Source-shaped pre-DV derivative handoff from the raw KL split. This composes the cycle-57 residual split for `appendix.tex:765-835` with the existing scalar pipeline for `appendix.tex:835-884`. All analytic inputs remain explicit hypotheses: law regularity, mass conservation, KL differentiation under the integral, Fokker--Planck, integration by parts, target transport, residual Young, LSI, sigma positivity, and inverse-schedule calculus are still obligations. -/ theorem generalMovingTargetKlDerivativePreDvBoundOfSplitScalar (dKds dKdt firstTerm massTerm targetTerm cDrift vDrift fisher residualCross residualSq kl cLSI sigmaSq sigmaInvSq dotS dotT : Real) (hsigmaSq : 0 ≤ sigmaSq) (hdotSNonneg : 0 ≤ dotS) (hklRaw : dKds = firstTerm + massTerm + targetTerm) (hmass : massTerm = 0) (hfirst : firstTerm = cDrift - (sigmaSq / 2) * fisher) (htarget : targetTerm = -vDrift) (hresidual : residualCross = cDrift - vDrift) (hyoung : residualCross ≤ (sigmaSq / 4) * fisher + sigmaInvSq * dotT ^ 2 * residualSq) (hlsiHalf : cLSI * kl ≤ (1 / 2) * fisher) (hdKdt : dKdt = dotS * dKds) (hdotT : dotT = dotS⁻¹) (hdotS : dotS ≠ 0) : dKdt ≤ -(((sigmaSq / 2) * dotS * cLSI) * kl) + sigmaInvSq * dotS⁻¹ * residualSq := by have hderiv : dKds = -(sigmaSq / 2) * fisher + residualCross := generalMovingTargetKlDerivativeResidualSplitScalar dKds firstTerm massTerm targetTerm cDrift vDrift fisher residualCross sigmaSq hklRaw hmass hfirst htarget hresidual exact generalMovingTargetPreDvDerivativeBoundScalar dKds dKdt fisher residualCross residualSq kl cLSI sigmaSq sigmaInvSq dotS dotT hsigmaSq hdotSNonneg hderiv hyoung hlsiHalf hdKdt hdotT hdotS /-- Post-DV scalar handoff for the continuous general VA-SALD coefficient. This is the `appendix.tex:885-907` real/order step after the analytic DV backend has supplied `alpha * energy <= K + log E_pi exp(alpha * energy)`. It rewrites the pre-DV residual-energy term into the exact Gronwall coefficient `damping - coeff * alpha^(-1)` and residual `coeff * E_alpha`. It does not prove DV, finite log-mgf, common-space, measurability, or the theorem-specific alpha-complexity witness. -/ theorem generalMovingTargetPostDvGronwallCoefficientScalar (dKdt kl damping alpha energy logMgf eAlpha coeff : Real) (hpre : dKdt ≤ -(damping * kl) + coeff * energy) (halpha : 0 < alpha) (hcoeff : 0 ≤ coeff) (hdv : alpha * energy ≤ kl + logMgf) (heAlpha : eAlpha = alpha⁻¹ * logMgf) : dKdt ≤ -((damping - coeff * alpha⁻¹) * kl) + coeff * eAlpha := by have hdvCoeff := forwardKlDvPositiveAlphaCoefficientScalar alpha energy kl logMgf eAlpha coeff halpha hcoeff hdv heAlpha calc dKdt ≤ -(damping * kl) + coeff * energy := hpre _ ≤ -(damping * kl) + ((coeff * alpha⁻¹) * kl + coeff * eAlpha) := by simpa [add_comm, add_left_comm, add_assoc] using add_le_add_left hdvCoeff (-(damping * kl)) _ = -((damping - coeff * alpha⁻¹) * kl) + coeff * eAlpha := by ring /-- Source-shaped post-DV handoff for `thm:general-moving-target-SALD`. This specializes `generalMovingTargetPostDvGronwallCoefficientScalar` to the sigma-weighted damping and residual prefactor in `appendix.tex:897-907`: `damping=(sigma_t^2/2)*dot{s}(t)*C_LSI(t)` and `coeff=sigma_t^(-2)*dot{s}(t)^(-1)`. -/ theorem generalMovingTargetPostDvGronwallCoefficientOfSigmaScheduleScalar (dKdt kl cLSI sigmaSq sigmaInvSq dotS alpha residualSq logMgf eAlpha : Real) (hpre : dKdt ≤ -(((sigmaSq / 2) * dotS * cLSI) * kl) + sigmaInvSq * dotS⁻¹ * residualSq) (halpha : 0 < alpha) (hcoeff : 0 ≤ sigmaInvSq * dotS⁻¹) (hdv : alpha * residualSq ≤ kl + logMgf) (heAlpha : eAlpha = alpha⁻¹ * logMgf) : dKdt ≤ -((((sigmaSq / 2) * dotS * cLSI) - (sigmaInvSq * dotS⁻¹) * alpha⁻¹) * kl) + (sigmaInvSq * dotS⁻¹) * eAlpha := by exact generalMovingTargetPostDvGronwallCoefficientScalar dKdt kl ((sigmaSq / 2) * dotS * cLSI) alpha residualSq logMgf eAlpha (sigmaInvSq * dotS⁻¹) hpre halpha hcoeff hdv heAlpha /-- Source-shaped scalar handoff from the general moving-target derivative and residual DV inputs to the Gronwall differential inequality. This composes the compiled pre-DV derivative pipeline for `appendix.tex:765-884` with the post-DV coefficient rewrite for `appendix.tex:885-907`. All analytic inputs remain explicit hypotheses: Fokker--Planck/KL differentiation, integration by parts, residual Young, LSI, schedule calculus, finite log-mgf, and the cited DV inequality are not proved here. -/ theorem generalMovingTargetDerivativeDvGronwallCoefficientScalar (dKds dKdt fisher residualCross residualSq kl cLSI sigmaSq sigmaInvSq dotS dotT alpha logMgf eAlpha : Real) (hsigmaSq : 0 ≤ sigmaSq) (hdotSNonneg : 0 ≤ dotS) (halpha : 0 < alpha) (hcoeff : 0 ≤ sigmaInvSq * dotS⁻¹) (hkl : dKds = -(sigmaSq / 2) * fisher + residualCross) (hyoung : residualCross ≤ (sigmaSq / 4) * fisher + sigmaInvSq * dotT ^ 2 * residualSq) (hlsiHalf : cLSI * kl ≤ (1 / 2) * fisher) (hdKdt : dKdt = dotS * dKds) (hdotT : dotT = dotS⁻¹) (hdotS : dotS ≠ 0) (hdv : alpha * residualSq ≤ kl + logMgf) (heAlpha : eAlpha = alpha⁻¹ * logMgf) : dKdt ≤ -((((sigmaSq / 2) * dotS * cLSI) - (sigmaInvSq * dotS⁻¹) * alpha⁻¹) * kl) + (sigmaInvSq * dotS⁻¹) * eAlpha := by have hpre : dKdt ≤ -(((sigmaSq / 2) * dotS * cLSI) * kl) + sigmaInvSq * dotS⁻¹ * residualSq := generalMovingTargetPreDvDerivativeBoundScalar dKds dKdt fisher residualCross residualSq kl cLSI sigmaSq sigmaInvSq dotS dotT hsigmaSq hdotSNonneg hkl hyoung hlsiHalf hdKdt hdotT hdotS exact generalMovingTargetPostDvGronwallCoefficientOfSigmaScheduleScalar dKdt kl cLSI sigmaSq sigmaInvSq dotS alpha residualSq logMgf eAlpha hpre halpha hcoeff hdv heAlpha /-- Cycle-67 scalar bridge from the residual KL split to the Gronwall input. This is the proof-producing lower wrapper for `appendix.tex:765-907` inside the selected `sald.general_moving_target.cycle67_residual_to_gronwall_bridge` packet. It starts from the source-shaped raw KL derivative split, consumes the existing residual Young/LSI/time-change scalar pipeline and the residual DV coefficient rewrite, and outputs exactly the differential inequality whose coefficient is later passed to the source-cited Gronwall interface: `(sigma_t^2/2)*dot{s}(t)*C_LSI(t) - sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1)`. It does not prove the Fokker--Planck/KL derivative identity, integration by parts, target transport, LSI/KL/FI density-test backend, finite log-mgf witness, DV formula, endpoint-safe Gronwall theorem, or pure-contraction normalization. -/ theorem generalMovingTargetResidualToGronwallBridgeScalar (dKds dKdt firstTerm massTerm targetTerm cDrift vDrift fisher residualCross residualSq kl cLSI sigmaSq sigmaInvSq dotS dotT alpha logMgf eAlpha : Real) (hsigmaSq : 0 ≤ sigmaSq) (hdotSNonneg : 0 ≤ dotS) (halpha : 0 < alpha) (hcoeff : 0 ≤ sigmaInvSq * dotS⁻¹) (hklRaw : dKds = firstTerm + massTerm + targetTerm) (hmass : massTerm = 0) (hfirst : firstTerm = cDrift - (sigmaSq / 2) * fisher) (htarget : targetTerm = -vDrift) (hresidual : residualCross = cDrift - vDrift) (hyoung : residualCross ≤ (sigmaSq / 4) * fisher + sigmaInvSq * dotT ^ 2 * residualSq) (hlsiHalf : cLSI * kl ≤ (1 / 2) * fisher) (hdKdt : dKdt = dotS * dKds) (hdotT : dotT = dotS⁻¹) (hdotS : dotS ≠ 0) (hdv : alpha * residualSq ≤ kl + logMgf) (heAlpha : eAlpha = alpha⁻¹ * logMgf) : dKdt ≤ -((((sigmaSq / 2) * dotS * cLSI) - (sigmaInvSq * dotS⁻¹) * alpha⁻¹) * kl) + (sigmaInvSq * dotS⁻¹) * eAlpha := by have hderiv : dKds = -(sigmaSq / 2) * fisher + residualCross := generalMovingTargetKlDerivativeResidualSplitScalar dKds firstTerm massTerm targetTerm cDrift vDrift fisher residualCross sigmaSq hklRaw hmass hfirst htarget hresidual exact generalMovingTargetDerivativeDvGronwallCoefficientScalar dKds dKdt fisher residualCross residualSq kl cLSI sigmaSq sigmaInvSq dotS dotT alpha logMgf eAlpha hsigmaSq hdotSNonneg halpha hcoeff hderiv hyoung hlsiHalf hdKdt hdotT hdotS hdv heAlpha /-- Lean-facing bridge for the source step `LSI + phi=sqrt(rho/pi)`. This is contract data for `eq:LSI-KL-FI`, not a proof. It keeps the exact paper route from LSI to the KL/FI comparison visible before theorem-specific forward-KL blocks reuse it. -/ structure LsiKlFiBridgeContract where sourceBlock : SourceAnchor absoluteContinuityInterface : String densityRatioInterface : String testFunctionInterface : String normalizationStep : String entropyIdentity : String fisherChainRule : String finiteQuantityInterfaces : List String := [] sourceGaps : List String := [] dependencies : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Narrow proof-obligation interface for the LSI test function `phi=sqrt(rho/pi)`. This record keeps the density, finite-quantity, and smooth-test-function requirements explicit instead of adding them silently to theorem statements. -/ structure LsiKlFiDensityTestContract where sourceBlock : SourceAnchor densityObject : String absoluteContinuity : String normalization : String positivityDomain : String finiteKlInterface : String finiteFiInterface : String sqrtTestFunction : String testAdmissibility : String entropyRewrite : String fisherChainRule : String coefficientAudit : String approximationFallback : String dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Middle-layer audit for applying the cited DV variational formula. The source lemma is cited from Boucheron et al.; this record does not prove it. It names the local interfaces required before the SALD theorem blocks can instantiate the formula with squared-velocity test functions. -/ structure DvFiniteLogMgfContract where sourceBlock : SourceAnchor formula : String commonSpaceInterface : String testFunctionClass : String finiteLogMgfCondition : String saldInstantiations : List String := [] alphaComplexityWitness : String dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Middle-layer audit for the PI vocabulary as used by the appendix velocity-norm lemma. The PI definition itself is contract data. The subsequent Sobolev, weak-PDE, and Riesz-representation route remains an analytic backend obligation. -/ structure PiVelocityNormDependencyContract where sourceBlock : SourceAnchor piDefinition : String weightedSobolevSpace : String meanZeroInterface : String normEquivalence : String weakPdeStatement : String boundedFunctionalStep : String rieszRepresentationStep : String velocityBound : String downstreamComplexityBounds : List String := [] dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Upper-role packet for the first appendix/vocabulary re-audit. This is workflow data, not mathematical proof content. It records the chosen faithful-paper objective, lower packet, and reviewer checklist for returning to `lem:gronwall`, `lem:dv_variation`, `def:PI`, and the KL/FI/LSI vocabulary without changing any theorem statement. -/ structure FirstAppendixVocabularyPacket where objective : String sourceLabels : List String := [] modeDiscipline : List String := [] nonGoals : List String := [] lowerPacket : List String := [] reviewerChecklist : List String := [] status : ProofStatus := ProofStatus.planned deriving Repr, DecidableEq /-- Source-index audit for the first appendix/vocabulary layer. This is upper-role workflow data. It keeps `SALD_original.jsonl`, the first proof-DAG labels, and the Lean-facing contracts synchronized without changing any mathematical theorem statement or proof status. -/ structure FirstAppendixSourceIndexAuditContract where sourceIndexPath : String sourceRoot : String excludedFiles : List String := [] indexedLabels : List String := [] sourceLineMap : List String := [] leanContractMap : List String := [] obligationMap : List String := [] lowerPacket : List String := [] reviewerChecklist : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Middle-role source-to-Lean audit for the first appendix/vocabulary layer. This workflow contract refines the upper source-index packet into a lower-ready map: every focused source step is classified as a Lean contract, cited result, or named proof obligation before proof search starts. -/ structure FirstAppendixMiddleAuditContract where sourceIndexPath : String focusLabels : List String := [] sourceReadWindows : List String := [] sourceStepMap : List String := [] leanStepMap : List String := [] citedResultMap : List String := [] obligationMap : List String := [] lowerPacket : List String := [] reviewerChecklist : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Upper-role packet for returning to the continuous forward-KL proof route. This is workflow data for `thm:forward-KL`. It records the chosen faithful objective and review constraints for the moving-target, LSI, DV, and Gronwall dependency chain without changing the theorem statement or claiming a proof of any analytic backend. -/ structure ForwardKlUpperPacket where objective : String sourceLabels : List String := [] modeDiscipline : List String := [] nonGoals : List String := [] lowerPacket : List String := [] reviewerChecklist : List String := [] status : ProofStatus := ProofStatus.planned deriving Repr, DecidableEq /-- Upper-role packet for returning to the discrete forward-KL proof route. This is workflow data for `thm:forward-KL-discrete`. It records one faithful-paper objective and lower packet while keeping the theorem statement and all analytic proof backends as contracts or obligations. -/ structure DiscreteForwardKlUpperPacket where objective : String sourceLabels : List String := [] modeDiscipline : List String := [] nonGoals : List String := [] lowerPacket : List String := [] reviewerChecklist : List String := [] status : ProofStatus := ProofStatus.planned deriving Repr, DecidableEq /-- Upper-role packet for guided/general VA-SALD proof routing. This is workflow data for the guided residual proposition, continuous general VA-SALD theorem, unified specialization, and discrete general theorem. It records one faithful-paper objective and review constraints without changing any theorem statement or promoting analytic obligations. -/ structure GeneralVaSaldUpperPacket where objective : String sourceLabels : List String := [] modeDiscipline : List String := [] nonGoals : List String := [] lowerPacket : List String := [] reviewerChecklist : List String := [] status : ProofStatus := ProofStatus.planned deriving Repr, DecidableEq /-- Middle-role source-to-Lean packet for the guided/general VA-SALD path. This workflow contract keeps the cycle focus synchronized across the guided residual proposition, continuous general theorem, unified specialization, and discrete general theorem. It does not change any theorem statement or close any analytic proof obligation. -/ structure GeneralVaSaldGuidedPathMiddleContract where guidedResidualSource : SourceAnchor generalContinuousSource : SourceAnchor unifiedSource : SourceAnchor discreteSource : SourceAnchor objective : String sourceStepMap : List String := [] leanStepMap : List String := [] citedResultInterfaces : List String := [] obligations : List String := [] lowerPacket : List String := [] reviewerChecklist : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Faithful data record for the source theorem `thm:forward-KL`. This pins the statement and appendix proof shape without claiming any of the measure-theoretic or calculus steps as formalized. -/ structure ForwardKlStatementContract where theoremLabel : String sourceStatement : SourceAnchor sourceProof : SourceAnchor movingTarget : String saldLaw : String lsiAssumption : String alphaComplexityAssumption : String alphaRange : String initialErrorFactor : String residualIntegral : String proofQuantity : String differentialInequality : String proofSteps : List String := [] dependencies : List String := [] status : ProofStatus := ProofStatus.contractOnly deriving Repr, DecidableEq /-- Contract for the paper's alpha-complexity vocabulary. The statement is definitional data. Finiteness, measurability, and monotonicity facts needed by theorem proofs remain obligations of the relevant theorem blocks. -/ structure AlphaComplexityContract where pathName : String velocityName : String alphaRange : String pointwiseDensity : String integratedComplexity : String finitenessAssumption : String source : SourceAnchor status : ProofStatus := ProofStatus.contractOnly deriving Repr, DecidableEq /-- Lean-facing interface for the derivative part of `thm:forward-KL`. This records the exact analytic route used in the appendix before DV and Gronwall enter the proof. -/ structure ForwardKlDerivativeCandidateContract where sourceBlock : SourceAnchor densityAndLawInterface : String scheduleInterface : String saldFokkerPlanck : String targetVelocity : String klDerivativeIdentity : String firstTermEvaluation : String secondTermBound : String lsiStep : String timeChangedInequality : String requiredRegularity : List String := [] sourceGaps : List String := [] dependencies : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Explicit side-condition interface for the derivative block of `thm:forward-KL`. The source proof uses these conditions in the KL derivative and time-change steps, but the theorem statement does not state them as standalone hypotheses. They remain obligations rather than hidden assumptions. -/ structure ForwardKlDerivativeSideConditionContract where sourceBlock : SourceAnchor densityPath : String massConservation : String derivativeUnderIntegral : String saldIntegrationByParts : String targetIntegrationByParts : String l2CauchyYoungStep : String inverseSchedule : String timeChangeIdentity : String outputInequality : String obligations : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Lean-facing interface for the DV velocity-energy step in `thm:forward-KL`. -/ structure ForwardKlDvEnergyCandidateContract where sourceBlock : SourceAnchor dvMeasure : String dvTestFunction : String integrabilityRoute : String bound : String dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Theorem-specific finite-log-mgf witness needed before applying DV in `thm:forward-KL`. The source proof applies `lem:dv_variation` directly at `appendix.tex:230-241`. This record isolates the missing Lean interface: the test function must be measurable on the same state space, the alpha0-complexity assumption must supply a finite log-mgf for every smaller `alpha`, and the division by `alpha` must preserve the paper's coefficient. -/ structure ForwardKlDvFiniteLogMgfWitnessContract where sourceBlock : SourceAnchor theoremStatement : SourceAnchor dvMeasures : String testFunction : String finiteAlpha0Assumption : String alphaMonotonicityBridge : String commonSpaceAndAbsoluteContinuity : String measurabilityInterface : String scalingStep : String outputBound : String coefficientUse : String dependencies : List String := [] sourceGaps : List String := [] lowerPacket : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Narrow alpha-monotonicity interface for the forward-KL DV test function. The source theorem assumes finite alpha0-complexity and then applies DV for every `0 < alpha <= alpha0`. This record isolates the needed exponential moment monotonicity instead of adding a second finite-mgf hypothesis to `thm:forward-KL`. -/ structure ForwardKlDvAlphaMonotonicityContract where sourceBlock : SourceAnchor theoremStatement : SourceAnchor sourceAssumption : String alphaRange : String nonnegativeEnergy : String pointwiseDomination : String expectationBridge : String logMgfBridge : String alphaComplexityRewrite : String downstreamCoefficientUse : String dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Lean-facing interface for the final Gronwall instantiation in `thm:forward-KL`. -/ structure ForwardKlGronwallInstantiationContract where sourceBlock : SourceAnchor quantityK : String gronwallA : String gronwallB : String preSplitBound : String splitBound : String requiredRegularity : List String := [] sourceGaps : List String := [] dependencies : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Source-facing audit of the moving-target assumptions used by `thm:forward-KL`. This contract keeps the theorem statement fixed while identifying which assumptions are stated in the main body, which interfaces are imported from the appendix proof, and which analytic regularity facts remain obligations. -/ structure ForwardKlMovingTargetDependencyContract where sourceStatement : SourceAnchor sourceProof : SourceAnchor slowedTargetPath : String transportVelocityInterface : String saldLawInterface : String lsiBridge : String dvBridge : String gronwallBridge : String terminalIdentification : String lowerFormalizationTarget : String dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Line-by-line coefficient audit for the LSI/DV/Gronwall chain in `thm:forward-KL`. This is narrower than `ForwardKlMovingTargetDependencyContract`: it records how the source proof transforms the derivative inequality into the exact scalar Gronwall coefficients and theorem exponents. It prevents later proof work from silently changing the Young, LSI, DV, or exponent-splitting constants. -/ structure ForwardKlDependencyChainAuditContract where sourceBlock : SourceAnchor postYoungInequality : String lsiCoefficientStep : String timeChangeStep : String dvInstantiation : String scalarDifferentialInequality : String gronwallA : String gronwallB : String exponentSplit : String residualExponentSimplification : String terminalEndpointBridge : String sourceLineLedger : List String := [] scalarSideConditions : List String := [] sourceDependencyClassification : List String := [] dependencies : List String := [] sourceGaps : List String := [] lowerPacket : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Endpoint and exponent side conditions for the final `thm:forward-KL` Gronwall display. This is a narrow ledger for the last source step: identifying the endpoints of `K(t)` with the theorem statement and justifying the residual-exponent simplification. These facts are required by the paper proof but are not added as hidden hypotheses of the theorem contract. -/ structure ForwardKlGronwallSideConditionContract where sourceBlock : SourceAnchor theoremStatement : SourceAnchor endpointScheduleIdentities : String terminalKlIdentification : String initialKlIdentification : String coefficientRegularity : String signFactsForResidualDrop : String exponentSplitAlgebra : String residualExponentBound : String dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Narrow endpoint-schedule ledger for `thm:forward-KL`. This is the cycle-14 lower slice. It isolates the source's inverse-schedule endpoint rewrites from the derivative, DV, and Gronwall analytic backends. -/ structure ForwardKlEndpointScheduleContract where sourceStatement : SourceAnchor sourceProof : SourceAnchor slowdownInterface : String inverseEndpointIdentities : String slowedTargetIdentity : String klPathDefinition : String terminalRewrite : String initialRewrite : String gronwallUse : String dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Middle-role source-to-Lean map for the continuous `thm:forward-KL` proof. This workflow contract classifies each source step in `appendix.tex:168-252` as an existing Lean-facing contract, source-cited result, or named obligation. It does not change the theorem statement and does not close any analytic backend. -/ structure ForwardKlMiddleSourceToLeanContract where sourceStatement : SourceAnchor sourceProof : SourceAnchor derivativeSource : SourceAnchor dvSource : SourceAnchor gronwallSource : SourceAnchor objective : String sourceStepMap : List String := [] leanStepMap : List String := [] citedResultInterfaces : List String := [] obligations : List String := [] lowerPacket : List String := [] reviewerChecklist : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Faithful data record for the source theorem `thm:forward-KL-discrete`. The record pins the main-body theorem, its EM implementation, and the exact discrete error terms. It does not assert that the analytic estimates are formalized. -/ structure DiscreteForwardKlStatementContract where theoremLabel : String sourceStatement : SourceAnchor sourceProof : SourceAnchor emUpdate : String interpolation : String linearSlowdown : String lipschitzAssumptions : String complexityAssumptions : String stepSizeCondition : String alphaRange : String gammaDeltaDefinitions : String terminalBound : String proofSteps : List String := [] dependencies : List String := [] status : ProofStatus := ProofStatus.contractOnly deriving Repr, DecidableEq /-- Side-condition interface for the EM interpolation used in `thm:forward-KL-discrete`. The source proof uses three facts at different points: endpoint law matching for the interpolation, a conditional-drift Fokker--Planck equation on each step, and enough stitched-interval regularity to apply the scalar Gronwall lemma. This record keeps those facts visible as obligations instead of folding them into a single hidden assumption. -/ structure DiscreteForwardKlEmInterpolationSideConditionContract where sourceBlock : SourceAnchor endpointLawMatching : String conditionalFrozenDrift : String interpolationFokkerPlanck : String densityRegularity : String stitchedIntervalInterface : String obligations : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Lower-ready line ledger for the conditional Fokker--Planck slice. The cycle-15 middle packet selects `appendix.tex:347-385` as the first lower slice. This record keeps that slice narrower than the whole discrete theorem: it covers the conditional drift, the law/density interface for the frozen EM interpolation, the Fokker--Planck equation, and the Laplacian split needed by the KL derivative block. -/ structure DiscreteForwardKlEmConditionalFpLowerContract where sourceBlock : SourceAnchor parentPacket : String targetObligation : String conditionalDriftDefinition : String conditionalLawInterface : String fokkerPlanckEquation : String laplacianSplit : String klDerivativeBridge : String exclusions : List String := [] dependencies : List String := [] sourceGaps : List String := [] lowerPacket : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Narrow lower interface for defining the frozen conditional drift. Before the interpolation Fokker--Planck equation can be stated in Lean, the conditional expectation in `bar b_{k,s}` has to be represented as a measurable drift field against the law of `hat X_s`. This record isolates that measure/disintegration backend from the later divergence and Laplacian algebra. -/ structure DiscreteForwardKlConditionalDriftDensityContract where sourceBlock : SourceAnchor parentPacket : String targetObligation : String randomVariables : String conditionalLawKernel : String densityInterface : String driftMeasurability : String driftIntegrability : String fokkerPlanckInput : String exclusions : List String := [] dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Lean-facing interface for the omitted SALD frozen-defect lemma. The source states that this lemma follows from the later general frozen-defect lemma by taking c identically zero and sigma_eta(t)=sqrt(2). Until that specialization is formalized, the exact bound remains an obligation. -/ structure FrozenDeltaCrossLipSaldContract where sourceBlock : SourceAnchor specializationRoute : String frozenError : String assumptions : List String := [] bound : String gammaDefinition : String deltaDefinition : String dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Lean-facing interface for the discrete forward-KL derivative block. -/ structure DiscreteForwardKlDerivativeCandidateContract where sourceBlock : SourceAnchor interpolationLaw : String frozenConditionalDrift : String fokkerPlanck : String klDerivativeIdentity : String firstTermEvaluation : String targetVelocityTerm : String frozenDefectBound : String movingVelocityDvBound : String outputSInequality : String timeChangedInequality : String requiredRegularity : List String := [] sourceGaps : List String := [] dependencies : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Theorem-specific finite-log-mgf witness for the discrete forward-KL DV step. The discrete proof applies `lem:dv_variation` with `nu=hat rho_s` and `mu=tilde pi_s`. This record isolates the extra EM-interpolation interfaces needed before reusing the continuous alpha-complexity DV argument. -/ structure DiscreteForwardKlDvFiniteLogMgfWitnessContract where sourceBlock : SourceAnchor theoremStatement : SourceAnchor dvMeasures : String testFunction : String finiteAlpha0Assumption : String alphaMonotonicityBridge : String interpolationLawInterface : String commonSpaceAndAbsoluteContinuity : String measurabilityInterface : String scalingStep : String dotTScalingStep : String outputBound : String coefficientUse : String dependencies : List String := [] sourceGaps : List String := [] lowerPacket : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Lean-facing interface for the final Gronwall step in `thm:forward-KL-discrete`. -/ structure DiscreteForwardKlGronwallInstantiationContract where sourceBlock : SourceAnchor statementBlock : SourceAnchor quantityK : String gronwallA : String gronwallB : String preSpecializationBound : String linearSlowdownSpecialization : String accumulatedError : String requiredRegularity : List String := [] sourceGaps : List String := [] dependencies : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Final bridge from the appendix discrete Gronwall display to the theorem statement. The appendix ends with a general-schedule bound. The main body states the linear-slowdown theorem with accumulated `barGamma` and `barDelta` constants. This record keeps the endpoint rewrites, exponent split, and integral collection as a separate obligation. -/ structure DiscreteForwardKlAccumulatedErrorBridgeContract where sourceProof : SourceAnchor theoremStatement : SourceAnchor endpointBridge : String gronwallOutput : String linearSlowdownSubstitution : String initialExponentSplit : String residualExponentBound : String alphaComplexityCollection : String gammaAccumulation : String deltaAccumulation : String dependencies : List String := [] sourceGaps : List String := [] lowerPacket : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Middle-role source-to-Lean packet for the discrete forward-KL route. This contract does not add a new theorem statement. It records how the cycle focus spans the EM interpolation, one-step frozen defect, DV velocity estimate, Gronwall accumulation, and final accumulated-error collection, so lower work can target one existing backend without losing the paper correspondence. -/ structure DiscreteForwardKlEmDefectAccumulationMiddleContract where sourceStatement : SourceAnchor interpolationSource : SourceAnchor derivativeSource : SourceAnchor gronwallSource : SourceAnchor accumulatedSource : SourceAnchor objective : String sourceStepMap : List String := [] leanStepMap : List String := [] citedResultInterfaces : List String := [] obligations : List String := [] lowerPacket : List String := [] reviewerChecklist : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Coefficient audit for the discrete forward-KL proof. This keeps the one-step `Gamma`/`Delta` coefficients synchronized from the frozen defect lemma through the derivative inequality, the `s` to `t` time change, the Gronwall exponent, and the main-body linear-slowdown collection. -/ structure DiscreteForwardKlCoefficientChainAuditContract where sourceProof : SourceAnchor theoremStatement : SourceAnchor frozenCrossCoefficient : String movingCrossCoefficient : String lsiStep : String dvVelocityCoefficient : String timeChangeStep : String gronwallA : String gronwallB : String linearSlowdownExponent : String accumulatedErrorCollection : String endpointBridge : String sourceLineLedger : List String := [] scalarSideConditions : List String := [] dependencies : List String := [] sourceGaps : List String := [] lowerPacket : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Lean-facing interface for the guided-path residual proposition. This is algebraic contract data for the appendix computation. The derivative of the normalizer, integration by parts, and mean-zero statement stay as obligations until a density/regularity backend is available. -/ structure GuidedResidualIdentityContract where sourceStatement : SourceAnchor sourceProof : SourceAnchor basePath : String guideTilt : String guideTransportDerivative : String normalizerDerivative : String residualIdentity : String meanZeroStatement : String requiredRegularity : List String := [] sourceGaps : List String := [] dependencies : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Faithful data record for `thm:general-moving-target-SALD`. The source theorem is the continuous general VA-SALD bound. It differs from `thm:forward-KL` by using an implementable velocity `c_t`, diffusion scale `sigma_t`, and residual velocity `m_t=v_t-c_t`. -/ structure GeneralMovingTargetStatementContract where theoremLabel : String sourceStatement : SourceAnchor sourceProof : SourceAnchor dynamics : String residualField : String lsiAssumption : String alphaComplexityAssumption : String alphaRange : String terminalBound : String pureContractionCase : String proofQuantity : String differentialInequality : String proofSteps : List String := [] dependencies : List String := [] status : ProofStatus := ProofStatus.contractOnly deriving Repr, DecidableEq /-- Lean-facing interface for the derivative block of the continuous general VA-SALD theorem. -/ structure GeneralMovingTargetDerivativeCandidateContract where sourceBlock : SourceAnchor densityAndLawInterface : String dynamicsFokkerPlanck : String targetVelocity : String residualVelocity : String klDerivativeIdentity : String firstTermEvaluation : String secondTermEvaluation : String youngStep : String lsiStep : String timeChangedInequality : String requiredRegularity : List String := [] sourceGaps : List String := [] dependencies : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Lean-facing interface for the DV residual-energy step in the continuous general VA-SALD theorem. -/ structure GeneralMovingTargetDvEnergyCandidateContract where sourceBlock : SourceAnchor dvMeasure : String dvTestFunction : String integrabilityRoute : String bound : String dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Theorem-specific finite-log-mgf witness for the residual DV step in `thm:general-moving-target-SALD`. The source proof applies `lem:dv_variation` directly with `Z=alpha*||m_t||^2`. This record isolates the common-space, measurability, alpha0-to-alpha finite-log-mgf, and positive-alpha scaling obligations before the residual-energy inequality is reused by the unified theorem. -/ structure GeneralMovingTargetDvFiniteLogMgfWitnessContract where sourceBlock : SourceAnchor theoremStatement : SourceAnchor dvMeasures : String testFunction : String finiteAlpha0Assumption : String alphaMonotonicityBridge : String commonSpaceAndAbsoluteContinuity : String measurabilityInterface : String scalingStep : String outputBound : String coefficientUse : String unifiedSpecializationUse : String dependencies : List String := [] sourceGaps : List String := [] lowerPacket : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Lower-level backend for the positive-alpha scaling step in the residual DV bound for `thm:general-moving-target-SALD`. After the cited DV formula is instantiated with `Z=alpha*||m_t||^2`, the appendix divides by `alpha>0` and rewrites the log-mgf term as `mathfrak E_alpha(pi_t,m_t)`. This record isolates that scalar/order rewrite from the finite-log-mgf and common-space witnesses. -/ structure GeneralMovingTargetDvPositiveAlphaScalingContract where sourceBlock : SourceAnchor theoremStatement : SourceAnchor dvTestFunction : String positivityInput : String dvInequalityBeforeScaling : String divisionStep : String logMgfRewrite : String coefficientAudit : String unifiedSpecializationUse : String dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Lean-facing interface for the Gronwall step in the continuous general VA-SALD theorem. -/ structure GeneralMovingTargetGronwallInstantiationContract where sourceBlock : SourceAnchor quantityK : String gronwallA : String gronwallB : String preSplitBound : String theoremBound : String pureContractionSpecialization : String requiredRegularity : List String := [] sourceGaps : List String := [] dependencies : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Endpoint, exponent, and pure-contraction side conditions for the final `thm:general-moving-target-SALD` Gronwall display. The appendix applies Gronwall and then states that the displayed theorem bound follows. This contract keeps the endpoint rewrites, sigma-weighted exponent split, residual-exponent drop, and zero-residual specialization visible as obligations instead of turning them into hidden theorem assumptions. -/ structure GeneralMovingTargetGronwallSideConditionContract where sourceBlock : SourceAnchor theoremStatement : SourceAnchor endpointScheduleIdentities : String terminalKlIdentification : String initialKlIdentification : String coefficientRegularity : String signFactsForResidualDrop : String exponentSplitAlgebra : String residualExponentBound : String pureContractionResidualZero : String dependencies : List String := [] sourceGaps : List String := [] lowerPacket : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Source-to-general-theorem bridge for `thm:unified-forward-KL`. The paper proves the unified VA-SALD theorem by one specialization line: set `c_t <- u_t` in the general moving-target theorem. This contract expands only the source bookkeeping needed for that specialization. It does not replace the source proof route with a direct VA-SALD KL proof. -/ structure UnifiedForwardKlSpecializationContract where sourceStatement : SourceAnchor sourceProof : SourceAnchor vaSaldDynamics : SourceAnchor residualEquation : SourceAnchor correctionEquation : SourceAnchor generalTheorem : SourceAnchor specialization : String correctionFieldTransportBridge : String residualFieldIdentification : String assumptionsBridge : String terminalBoundMatch : String proofRoute : List String := [] dependencies : List String := [] sourceGaps : List String := [] lowerPacket : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Middle-role packet for the cycle-16 unified transport bridge. This contract narrows `thm:unified-forward-KL` to the paper's transport algebra before any lower proof search: combine the centered guided residual identity with the correction-field divergence equation, then hand the resulting transport velocity to the general moving-target theorem specialization. -/ structure UnifiedForwardKlTransportBridgeMiddleContract where residualIdentitySource : SourceAnchor correctionEquationSource : SourceAnchor transportBridgeSource : SourceAnchor specializationProofSource : SourceAnchor objective : String sourceAlgebraLedger : List String := [] leanInterfaceMap : List String := [] citedResultInterfaces : List String := [] obligations : List String := [] lowerPacket : List String := [] reviewerChecklist : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Lower interface for the cycle-16 unified transport bridge. This record isolates the only algebra selected for lower work: the signed cancellation between the guided residual identity and the correction-field divergence equation. The analytic interpretation of divergence and the existence/regularity of `w_t` stay as obligations. -/ structure UnifiedForwardKlTransportBridgeLowerContract where sourceBlock : SourceAnchor parentPacket : String targetObligation : String residualEquation : String correctionEquation : String divergenceLinearity : String cancellationStep : String transportVelocityConclusion : String specializationIdentifications : String exclusions : List String := [] dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Faithful data record for `thm:general-moving-target-SALD-discrete`. The source theorem is the discrete-time general VA-SALD bound. It reuses the continuous general theorem hypotheses and the general frozen-delta lemma, then adds EM interpolation and Gronwall accumulation with the paper's doubled residual-energy coefficients. -/ structure GeneralMovingTargetDiscreteStatementContract where theoremLabel : String sourceStatement : SourceAnchor sourceProof : SourceAnchor emUpdate : String interpolation : String frozenError : String constantSchedule : String assumptions : String alphaRange : String terminalBound : String proofQuantity : String differentialInequality : String proofSteps : List String := [] dependencies : List String := [] status : ProofStatus := ProofStatus.contractOnly deriving Repr, DecidableEq /-- Lean-facing interface for the general VA-SALD frozen-delta lemma. -/ structure GeneralFrozenDeltaCrossLipContract where sourceBlock : SourceAnchor frozenError : String assumptions : List String := [] stepSizeCondition : String bound : String gammaDefinition : String deltaDefinition : String proofRoute : List String := [] dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Lean-facing interface for the derivative block of the discrete general VA-SALD theorem. -/ structure GeneralMovingTargetDiscreteDerivativeCandidateContract where sourceBlock : SourceAnchor interpolationLaw : String frozenConditionalDrift : String fokkerPlanck : String klDerivativeIdentity : String frozenResidualDecomposition : String mYoungStep : String frozenDeltaStep : String lsiStep : String dvStep : String outputSInequality : String timeChangedInequality : String requiredRegularity : List String := [] sourceGaps : List String := [] dependencies : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Side-condition ledger for the discrete general VA-SALD derivative block. This keeps the source's interval-wise EM law, conditional drift, frozen/residual algebra, Young coefficient bookkeeping, and final time-change interfaces visible before any attempt to prove the derivative inequality. -/ structure GeneralMovingTargetDiscreteDerivativeSideConditionContract where sourceBlock : SourceAnchor endpointLawInterface : String conditionalDriftInterface : String fokkerPlanckSplitInterface : String transportVelocityInterface : String frozenResidualAlgebra : String youngCoefficientBookkeeping : String lsiBookkeeping : String dvFiniteMgfInterface : String timeChangeAndStitchingInterface : String requiredRegularity : List String := [] sourceGaps : List String := [] dependencies : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Regular conditional-drift interface for the discrete general VA-SALD Euler--Maruyama interpolation. This isolates the source line defining `bar b_{k,s}(x)` from the later weak Fokker--Planck identity. It records the regular conditional law, selected conditional expectation, measurability, and integrability obligations needed before the drift can be inserted into `-div(hat rho_s * bar b_{k,s})`. -/ structure GeneralMovingTargetDiscreteConditionalDriftContract where sourceBlock : SourceAnchor parentInterface : String randomVariables : String conditioningMap : String conditionalLawKernel : String driftSummands : String conditionalExpectationLinearity : String selectedDriftField : String measurabilityInterface : String integrabilityInterface : String fokkerPlanckInput : String exclusions : List String := [] dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Conditional-law and regularity interface for the named frozen drift `bar b_{k,s}` in the discrete general VA-SALD EM interpolation. This is a source-facing ledger, not a construction of disintegration. It separates the regular conditional kernel, component conditional expectations, measurability, integrability, and weak-FP handoff expected before `bar b_{k,s}` can be used in divergence form. -/ structure GeneralMovingTargetDiscreteConditionalLawMeasurabilityContract where sourceBlock : SourceAnchor parentInterface : String commonSpace : String interpolationLaw : String conditionalKernel : String kernelCompatibility : String componentConditionalFields : String selectedDriftField : String measurabilitySideConditions : String integrabilitySideConditions : String weakFpHandoff : String exclusions : List String := [] dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Endpoint-to-conditional compatibility interface for the named EM law. This sits between endpoint/common-space `Measure.map` bookkeeping and the regular conditional kernel required for `bar b_{k,s}`. It records that the joint law of `(X_k^eta,hat X_s)` has second marginal equal to the named `\hat\rho_s=Law(hat X_s)`, and that any supplied conditional-kernel compatibility predicate must be transported to that named marginal before the weak Fokker--Planck statement consumes it. -/ structure GeneralMovingTargetDiscreteEndpointConditionalCompatibilityContract where sourceBlock : SourceAnchor parentInterface : String commonSpace : String jointLawRepresentation : String hatRhoMarginal : String conditionalKernelCompatibility : String compatibilityTransport : String weakFpUse : String exclusions : List String := [] dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Weak-test source-sign interface for the conditional Fokker--Planck line. This records the analytic statement invoked at `appendix.tex:1379-1387`: after the regular conditional drift `bar b_{k,s}` has been constructed, the frozen EM interpolation law satisfies a weak Fokker--Planck equation whose distributional right-hand side is `-div(hat rho_s*bar b_{k,s}) + (sigma_eta^2/2)*Delta hat rho_s`. The record is intentionally a contract, not a proof of the SDE law, density, or integration-by-parts backend. -/ structure GeneralMovingTargetDiscreteWeakConditionalFpSourceSignContract where sourceBlock : SourceAnchor parentInterface : String fixedInterval : String namedLaw : String conditionalDrift : String weakTestClass : String weakEquation : String driftSignConvention : String diffusionSignConvention : String explicitHypotheses : List String := [] downstreamHandoffs : List String := [] exclusions : List String := [] dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Handoff from weak conditional Fokker--Planck to the discrete KL derivative. This interface starts at the differentiated KL display `eq:general_KL_derivative_0_discrete` and records the single analytic bridge needed before the paper performs integration by parts: the weak conditional FP identity must be applied to the log-density-ratio test `log(hat rho_s/tilde pi_s)`. -/ structure GeneralMovingTargetDiscreteKlDerivativeWeakFpHandoffContract where sourceBlock : SourceAnchor parentInterface : String differentiatedKlFormula : String selectedWeakTest : String admissibilityInterface : String weakFpSubstitution : String sourceSignedDerivative : String integrationByPartsHandoff : String downstreamDerivativeInterface : String explicitHypotheses : List String := [] downstreamHandoffs : List String := [] exclusions : List String := [] dependencies : List String := [] sourceGaps : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Theorem-specific finite-log-mgf witness for the discrete general VA-SALD residual DV step. The discrete proof applies `lem:dv_variation` under the EM interpolation law with `nu=hat rho_s`, `mu=tilde pi_s`, and `Z=alpha*||m_{t(s)}||^2`. This record keeps the EM common-space and absolute-continuity side conditions separate from the continuous residual log-mgf witness. -/ structure GeneralMovingTargetDiscreteDvFiniteLogMgfWitnessContract where sourceBlock : SourceAnchor theoremStatement : SourceAnchor dvMeasures : String testFunction : String finiteAlpha0Assumption : String alphaMonotonicityBridge : String interpolationLawInterface : String commonSpaceAndAbsoluteContinuity : String measurabilityInterface : String scalingStep : String outputBound : String coefficientUse : String dependencies : List String := [] sourceGaps : List String := [] lowerPacket : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Lean-facing interface for the final Gronwall step in the discrete general VA-SALD theorem. -/ structure GeneralMovingTargetDiscreteGronwallInstantiationContract where sourceBlock : SourceAnchor quantityK : String gronwallA : String gronwallB : String theoremBound : String constantScheduleInterface : String requiredRegularity : List String := [] sourceGaps : List String := [] dependencies : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Side-condition ledger for the final discrete general VA-SALD Gronwall step. The appendix derives the `t`-time differential inequality on stitched Euler--Maruyama intervals and then says that applying Gronwall finishes the proof. This record isolates the endpoint laws, constant inverse-schedule interface, coefficient regularity, and theorem-display matching as obligations instead of adding hidden assumptions to `thm:general-moving-target-SALD-discrete`. -/ structure GeneralMovingTargetDiscreteGronwallSideConditionContract where sourceBlock : SourceAnchor theoremStatement : SourceAnchor endpointLawIdentities : String constantScheduleIdentities : String terminalKlIdentification : String initialKlIdentification : String stitchedRegularity : String coefficientRegularity : String gronwallDisplayMatch : String residualCoefficientAudit : String frozenDeltaCoefficientAudit : String dependencies : List String := [] sourceGaps : List String := [] lowerPacket : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq /-- Upper-level ledger for the SALD main proof-skeleton sprint. This is assignment and source-to-Lean route data, not a theorem. It records which slow analytic interfaces are allowed to remain source-cited or obligation-level while the faithful theorem skeletons are wired in paper order. -/ structure MainSkeletonAnalyticInterfaceLedger where sourceBlock : SourceAnchor objective : String sourceLabels : List String := [] analyticInterfaces : List String := [] theoremRoute : List String := [] modeDiscipline : List String := [] nonGoals : List String := [] lowerPacket : List String := [] reviewerChecklist : List String := [] dependencies : List String := [] status : ProofStatus := ProofStatus.obligation deriving Repr, DecidableEq def saldGronwallCandidateContract : GronwallCandidateContract where timeDomain := "0 <= t1, with all hypotheses over t in Set.Icc (0 : Real) t1." aRegularity := "a : Real -> Real is ContinuousOn a (Set.Icc 0 t1), matching the source continuous a_t." bRegularity := "b : Real -> Real is ContinuousOn b (Set.Icc 0 t1), matching the source continuous b_t." kRegularity := "K : Real -> Real is differentiable on [0,t1] in the source; the Lean backend should choose HasDerivWithinAt on Set.Icc 0 t1 or an equivalent endpoint-safe formulation." differentialInequality := "For t in [0,t1], dK/dt t <= -(a t) * K t + b t." boundAtT1 := "K t1 <= exp (-(int u in 0..t1, a u)) * K 0 + int t in 0..t1, exp (-(int u in t..t1, a u)) * b t." integratingFactorDerivative := "d/dt (exp (int u in 0..t, a u) * K t) <= exp (int u in 0..t, a u) * b t." integratedFactorInequality := "exp (int u in 0..t1, a u) * K t1 <= K 0 + int t in 0..t1, exp (int u in 0..t, a u) * b t." mathlibRoute := [ "MeasureTheory intervalIntegral over volume on Real intervals", "ContinuousOn.intervalIntegrable for a, b, and integrating-factor products", "HasDerivWithinAt or HasDerivAt plus Real.hasDerivAt_exp and product derivative rules", "SALD.gronwallIntegratingFactorProductDerivative for appendix.tex:58-60", "SALD.gronwallIntegratingFactorDerivativeInequalityScalar and SALD.gronwallIntegratingFactorDerivativeLeOfIntegral for appendix.tex:60-61", "SALD.gronwallIntegratingFactorBoundOfDerivatives, SALD.gronwallIntegratingFactorBoundOfIntegral, and SALD.gronwallIntegratingFactorBoundOfContinuousData assemble appendix.tex:58-69 under explicit global Mathlib side conditions", "SALD.gronwallIntegratingFactorBoundOfDifferentiable rewrites the paper derivative as deriv K, with one explicit product-derivative interval-integrability side condition; SALD.gronwallIntegratingFactorBoundOfC1 discharges that condition when deriv K is continuous", "SALD.gronwallOrderIntegrationOfHasDerivRight, SALD.gronwallIntegratingFactorBoundOfInteriorDerivatives, SALD.gronwallIntegratingFactorBoundOfInteriorContinuousData, and SALD.gronwallIntegratingFactorBoundOfInteriorC1 remove endpoint derivative hypotheses by using continuity on [0,t1] and right derivatives on (0,t1)", "intervalIntegral fundamental theorem of calculus / integration by parts for the integrating factor" ] source := saldGronwallSource status := ProofStatus.obligation def saldGronwallEndpointCalculusContract : GronwallEndpointCalculusContract where sourceBlock := saldGronwallSource intervalInterface := "The source works on the closed interval t in [0,t1]; a Lean theorem must choose either endpoint-safe HasDerivWithinAt hypotheses or an absolutely-continuous interval formulation." derivativeMode := "K is only stated differentiable on [0,t1]; the derivative inequality at endpoints and the FTC step must be phrased so endpoint values K 0 and K t1 are admissible." integratingFactor := "I(t)=exp(int u in 0..t, a u), exactly the factor in appendix.tex:58." productDerivativeStep := "Show d/dt (I(t)*K(t)) <= I(t)*b(t) by differentiating I, using d/dt int_0^t a=a(t), and substituting dK/dt <= -a(t)*K(t)+b(t); the pointwise HasDerivAt/product/scalar part now builds as SALD.gronwallIntegratingFactorDerivativeLeOfIntegral." orderIntegrationStep := "Integrate the derivative inequality from 0 to t1 and preserve the inequality direction using interval-integral monotonicity; the local HasDerivAt/FTC order step now builds as SALD.gronwallOrderIntegrationOfHasDerivAt once f' and g are interval-integrable." endpointEvaluation := "The integrated expression is I(t1)*K(t1) <= K(0)+int_0^t1 I(t)*b(t)dt, since I(0)=exp(0)=1; the scalar endpoint/multiplication algebra now builds as SALD.gronwallEndpointEvaluationScalar and SALD.gronwallEndpointMultiplyByExpNegScalar." exponentAlgebra := "After multiplying by exp(-int_0^t1 a), rewrite exp(-int_0^t1 a)*exp(int_0^t a) as exp(-int_t^t1 a), matching appendix.tex:65-69; SALD.gronwallEndpointIntegralRewrite now packages this endpoint integral rewrite." reusableInstantiations := [ "continuous forward-KL Gronwall application", "discrete forward-KL Gronwall accumulation", "continuous general moving-target VA-SALD Gronwall application", "discrete general moving-target VA-SALD Gronwall application" ] dependencies := [ "SALD.saldGronwallCandidateContract", "SALD.gronwallIntegratingFactorProductDerivative", "SALD.gronwallIntegratingFactorDerivativeInequalityScalar", "SALD.gronwallIntegratingFactorDerivativeLe", "SALD.gronwallIntegratingFactorDerivativeLeOfIntegral", "SALD.gronwallOrderIntegrationOfHasDerivAt", "SALD.gronwallOrderIntegrationOfHasDerivRight", "SALD.gronwallEndpointEvaluationScalar", "SALD.gronwallEndpointMultiplyByExpNegScalar", "SALD.gronwallEndpointIntegralRewrite", "SALD.gronwallIntegratingFactorBoundOfDerivatives", "SALD.gronwallIntegratingFactorBoundOfIntegral", "SALD.gronwallIntegratingFactorBoundOfInteriorDerivatives", "SALD.gronwallCoefficientSideConditionsOfContinuous", "SALD.gronwallIntegratingFactorBoundOfContinuousData", "SALD.gronwallIntegratingFactorBoundOfDifferentiable", "SALD.gronwallIntegratingFactorBoundOfC1", "SALD.gronwallIntegratingFactorBoundOfInteriorContinuousData", "SALD.gronwallIntegratingFactorBoundOfInteriorC1", "Real intervalIntegral FTC", "closed-interval derivative or absolute-continuity backend", "Real.exp positivity and interval-integral additivity" ] sourceGaps := [ "appendix.tex:47-71 does not specify whether differentiability on [0,t1] means endpoint one-sided derivatives or an open-neighborhood derivative with endpoint continuity", "the proof integrates a derivative inequality without spelling out the absolute-continuity or FTC hypotheses", "the final exponent rewrite uses interval-integral additivity and exp addition algebra that must be supplied locally" ] status := ProofStatus.obligation def saldGronwallExponentRewriteContract : GronwallExponentRewriteContract where sourceBlock := saldGronwallExponentRewriteSource startingFactor := "After appendix.tex:63, the remaining integral term is exp(-(int u in 0..t1, a u)) * int t in 0..t1, exp(int u in 0..t, a u) * b t dt." intervalAdditivity := "For each t in [0,t1], use interval-integral additivity int_0^t1 a = int_0^t a + int_t^t1 a, with orientation handled exactly by the source interval 0 <= t <= t1." negIntegralRewrite := "Rewrite -(int_0^t1 a) + int_0^t a as -(int_t^t1 a), without imposing sign assumptions on a; the scalar real identity is compiled as SALD.gronwallNegIntegralRewriteScalar once interval additivity supplies the three integral values." expProductRewrite := "Use exp(x)*exp(y)=exp(x+y) to identify exp(-(int_0^t1 a))*exp(int_0^t a) with exp(-(int_t^t1 a)); the scalar Real.exp product step is compiled as SALD.gronwallExpProductRewriteScalar." integralTermRewrite := "Push the pointwise exponential rewrite through the integral over t in [0,t1], preserving the multiplier b_t and the inequality direction from the previous line." noSignAssumptions := "The rewrite uses only interval additivity and exponential algebra; it must not add positivity or monotonicity assumptions on a or b." dependencies := [ "SALD.saldGronwallEndpointCalculusContract", "SALD.gronwallNegIntegralRewriteScalar", "SALD.gronwallExpProductRewriteScalar", "SALD.gronwallIntervalIntegralAdditivityScalar", "SALD.gronwallExpProductRewriteIntervalIntegral", "SALD.gronwallExpProductRewriteIntegralCongr", "SALD.gronwallEndpointIntegralRewrite", "interval-integrability on adjacent Real intervals" ] sourceGaps := [ "appendix.tex:65-69 performs the exponent rewrite in one display; SALD.gronwallIntervalIntegralAdditivityScalar now supplies the adjacent-interval scalar equality once interval-integrability is proved", "SALD.gronwallExpProductRewriteIntegralCongr now pushes the pointwise exponential rewrite through the t-integral once adjacent interval-integrability is supplied for each t in the source interval", "SALD.gronwallEndpointIntegralRewrite now moves the endpoint inverse factor through the b_t integral and applies the compiled congruence" ] status := ProofStatus.obligation def saldKLContract : KLContract where rho := "rho" pi := "pi" expression := "KL(rho || pi) = integral log((rho/pi)(x)) rho(dx), for rho absolutely continuous with respect to pi." source := saldKlFiLsiSource status := ProofStatus.contractOnly def saldFIContract : FIContract where rho := "rho" pi := "pi" expression := "FI(rho || pi) = integral ||nabla log((rho/pi)(x))||^2 rho(dx)." source := saldKlFiLsiSource status := ProofStatus.contractOnly def saldLSIContract : LSIContract where measureName := "pi" constantName := "C_LSI" statement := "For smooth phi with integral phi^2 dpi = 1, integral phi^2 log(phi^2) dpi <= (2/C_LSI) integral ||nabla phi||^2 dpi." source := saldKlFiLsiSource status := ProofStatus.obligation def saldLsiKlFiBridgeContract : LsiKlFiBridgeContract where sourceBlock := saldKlFiLsiSource absoluteContinuityInterface := "rho << pi; write r=d rho/d pi, with r >= 0 pi-a.e. and integral r d pi = 1." densityRatioInterface := "The paper's rho/pi ratio is represented by r; KL(rho||pi)=integral log(r) r d pi and FI(rho||pi)=integral ||nabla log r||^2 r d pi when finite." testFunctionInterface := "Use phi=sqrt(r) as the LSI test function; Lean must either require phi is smooth/admissible or supply an approximation/closure lemma." normalizationStep := "integral phi^2 d pi = integral r d pi = 1, matching the LSI normalization hypothesis." entropyIdentity := "integral phi^2 log(phi^2) d pi = KL(rho||pi)." fisherChainRule := "integral ||nabla phi||^2 d pi = (1/4)*FI(rho||pi), so the source LSI display gives KL(rho||pi) <= FI(rho||pi)/(2*C_LSI)." finiteQuantityInterfaces := [ "finite KL(rho||pi)", "finite FI(rho||pi)", "measurable nonnegative density ratio r", "integrability of log(r)*r and ||nabla log r||^2*r" ] sourceGaps := [ "main_body.tex states the substitution phi=sqrt(rho/pi) but does not spell out density smoothness or approximation hypotheses", "the source assumes the Fisher-information chain rule needed for the constant 1/(2*C_LSI)" ] dependencies := ["KLContract", "FIContract", "LSIContract", "probability.lsi_to_kl_fi"] status := ProofStatus.obligation def saldLsiKlFiDensityTestContract : LsiKlFiDensityTestContract where sourceBlock := saldKlFiLsiSource densityObject := "r = rho/pi is the Radon-Nikodym density d rho / d pi for rho << pi." absoluteContinuity := "rho << pi is required before the paper's ratio rho/pi, KL integral, and FI integral are meaningful." normalization := "integral r d pi = 1, hence phi=sqrt(r) satisfies integral phi^2 d pi = 1; cycle 43 formalizes this for Mathlib Radon-Nikodym densities via AutoSamplingTheory.lsiKlFiRnDerivDensityMassOne and AutoSamplingTheory.lsiKlFiSqrtRnDerivTestMassOne." positivityDomain := "r >= 0 pi-a.e.; any log(r) and nabla log(r) interface must handle the zero-density set or work under a positive smooth approximation." finiteKlInterface := "finite integral log(r) * r d pi is needed to identify the entropy side with KL(rho||pi)." finiteFiInterface := "finite integral ||nabla log r||^2 * r d pi is needed before the Fisher-information chain rule can be used." sqrtTestFunction := "phi = sqrt(r), exactly as stated after the LSI definition in main_body.tex." testAdmissibility := "phi must be an admissible smooth LSI test function, or the proof must pass through an approximation/closure theorem." entropyRewrite := "integral phi^2 * log(phi^2) d pi = integral r * log(r) d pi = KL(rho||pi); cycle 43 formalizes the Mathlib log-likelihood transport step through AutoSamplingTheory.lsiKlFiRnDerivEntropyIntegral and AutoSamplingTheory.lsiKlFiSqrtRnDerivEntropyIntegral." fisherChainRule := "nabla sqrt(r) = (1/2) sqrt(r) * nabla log(r), giving integral ||nabla phi||^2 d pi = (1/4) FI(rho||pi); cycle 43 lower formalizes the finite-coordinate a.e. integral handoff once coordinate derivative identities are supplied." coefficientAudit := "Combining the source LSI factor 2/C_LSI with the 1/4 Fisher chain-rule factor gives FI(rho||pi)/(2*C_LSI), matching eq:LSI-KL-FI; SALD.lsiKlFiCoefficientAuditScalar formalizes the coefficient algebra, SALD.lsiKlFiDensityTestBridgeScalar formalizes the normalized-test scalar handoff after the analytic inputs are supplied, cycle 38 adds the pointwise positive-density scalar Fisher coefficient plus the downstream half-Fisher coefficient bridge, and cycle 43 adds Mathlib-backed density normalization, entropy transport, and a finite-coordinate integral Fisher-chain handoff for the sqrt Radon-Nikodym test." approximationFallback := "If sqrt(r) is not smooth or positive, record a closure/approximation lemma as a separate obligation rather than adding hidden assumptions to forward-KL theorems." dependencies := [ "KLContract", "FIContract", "LSIContract", "AutoSamplingTheory.lsiKlFiRnDerivLIntegralMassOne", "AutoSamplingTheory.lsiKlFiRnDerivDensityMassOne", "AutoSamplingTheory.lsiKlFiSqrtRnDerivTestMassOne", "AutoSamplingTheory.lsiKlFiRnDerivEntropyIntegral", "AutoSamplingTheory.lsiKlFiSqrtRnDerivEntropyIntegral", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainOfDerivativesScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralFiniteSum", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralHandoffScalar", "SALD.lsiKlFiCoefficientAuditScalar", "SALD.lsiKlFiDensityTestBridgeScalar", "SALD.lsiKlFiHalfFisherScalar", "SALD.lsiKlFiDensityTestHalfFisherScalar", "probability.lsi_to_kl_fi" ] sourceGaps := [ "main_body.tex:208 states rho << pi but not the Radon-Nikodym backend or zero-density convention", "main_body.tex:208-215 does not state smoothness/admissibility hypotheses for sqrt(rho/pi)", "the Fisher-information chain rule and approximation/closure argument are used but not proved in the source" ] status := ProofStatus.obligation def saldDvFiniteLogMgfContract : DvFiniteLogMgfContract where sourceBlock := saldDvVariationSource formula := "KL(nu||mu)=sup_Z {E_nu[Z]-log E_mu[exp Z]}, with the supremum over random variables satisfying log E_mu[exp Z] < +infty." commonSpaceInterface := "mu and nu are probability distributions on the same measurable space; later SALD uses nu=rho_{s(t)} or hat rho_s and mu=pi_t or tilde pi_s." testFunctionClass := "Z is a measurable real random variable; theorem blocks instantiate Z with alpha*||v_t||^2, alpha*||m_t||^2, and frozen-defect squared norms." finiteLogMgfCondition := "Each instantiation must provide log E_mu[exp Z] < +infty before the cited formula can be applied." saldInstantiations := [ "forward-KL: Z=alpha*||v_t||^2 under mu=pi_t.", "discrete forward-KL: Z=alpha*||v_{t(s)}||^2 under mu=pi_{t(s)}.", "general moving-target VA-SALD: Z=alpha*||m_t||^2 under mu=pi_t.", "general discrete VA-SALD and frozen-delta bounds: residual or frozen squared-norm tests under the source alpha/alpha' complexity assumptions." ] alphaComplexityWitness := "The finite alpha0-complexity assumptions in the theorem statements are the intended source witness for finite log-mgf when 0 < alpha <= alpha0; Lean still needs the monotonicity/measurability bridge." dependencies := ["probability.dv_variational_formula", "dvVariationalFormulaInterface", "KLContract", "def:alpha-complexity"] sourceGaps := [ "appendix.tex cites the DV formula but does not state the measurable-space or random-variable class in Lean terms", "finite log-mgf monotonicity from alpha0 to smaller alpha is used later but is not proved in the source" ] status := ProofStatus.obligation def saldPiVelocityNormDependencyContract : PiVelocityNormDependencyContract where sourceBlock := saldPiVelocityNormSource piDefinition := "appendix.tex:86-94 defines PI by Var_mu(phi) <= C_PI^{-1} * integral ||nabla phi||^2 dmu for all smooth phi." weightedSobolevSpace := "appendix.tex:96-106 introduces dot H^1(mu) as the mean-zero weighted Sobolev space with gradient inner product." meanZeroInterface := "For psi in dot H^1(mu), the proof uses E_mu[psi]=0 so ||psi||_{L2(mu)}^2=Var_mu[psi]." normEquivalence := "PI gives ||psi||_{L2(mu)} <= C_PI^{-1/2} ||psi||_{dot H^1(mu)}, and hence equivalence with the weighted H^1 norm on the mean-zero subspace; SALD.piVelocityNormMeanZeroH1UpperScalar records only the resulting real upper-bound algebra." weakPdeStatement := "For g in dot H^1(mu), solve div(mu*v)=-g*mu with v=nabla phi through the weak form _{dot H^1}=int psi*g dmu." boundedFunctionalStep := "Cauchy-Schwarz plus PI bounds T_mu(psi)=int psi*g dmu by C_PI^{-1/2}||psi||_{dot H^1(mu)}||g||_{L2(mu)}; SALD.piVelocityNormBoundedFunctionalScalar formalizes only this scalar order propagation after the analytic hypotheses are supplied." rieszRepresentationStep := "Riesz representation on dot H^1(mu) yields phi solving the weak PDE; Lean must provide the Hilbert-space and quotient/mean-zero backend." velocityBound := "The source conclusion is ||v||_{L2(mu)}=||nabla phi||_{L2(mu)} <= C_PI^{-1/2}||g||_{L2(mu)}." downstreamComplexityBounds := [ "For moving targets, use g=partial_t log pi_t to bound A_0(pi,v).", "For guided paths, use g_t=partial_t f_t+nabla f_t^T u_t and the centered term g_t-E_{pi_t}[g_t] to bound A_0(pi,w)." ] dependencies := ["PIContract", "variance vocabulary", "weighted Sobolev space", "SALD.piVelocityNormMeanZeroH1UpperScalar", "SALD.piVelocityNormBoundedFunctionalScalar", "Riesz representation theorem"] sourceGaps := [ "the source does not spell out the Hilbert-space completion and mean-zero quotient details for dot H^1(mu)", "the cycle 25 lower scalar helpers still require Lean instances of the mean-zero variance identity, L2 pairing/Cauchy-Schwarz bound, nonnegative L2 norms, and the PI square-root norm bound", "the weak PDE existence step requires boundary/regularity assumptions for div(mu*nabla phi)", "the displayed proof says v=nabla phi but initially writes psi in the solution ansatz; Lean should normalize this notation before proof search" ] status := ProofStatus.obligation def cycle9FirstAppendixVocabularyPacket : FirstAppendixVocabularyPacket where objective := "Re-audit the source-index and first appendix/vocabulary contracts for lem:gronwall, lem:dv_variation, def:PI, and eq:LSI-KL-FI before assigning lower proof search." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "def:PI", "eq:LSI-KL-FI" ] modeDiscipline := [ "faithfulPaper: use /home/nitanda_sub/mark/repos/sald/paper/main_body.tex and appendix.tex; sald_version_2.tex remains excluded.", "Preserve the source Gronwall inequality, DV supremum formula, PI variance bound, and LSI-to-KL/FI coefficient 1/(2*C_LSI).", "Keep Gronwall and LSI-to-KL/FI as obligations and DV as sourceCited until local Lean proofs replace them." ] nonGoals := [ "Do not restate or prove thm:forward-KL or later VA-SALD theorems in this packet.", "Do not replace DV or LSI with Pinsker, Talagrand, PI, Girsanov, or another proof route.", "Do not add density, endpoint, smoothness, or finite-mgf assumptions silently to theorem contracts; record them as obligations/source gaps." ] lowerPacket := [ "Target one first-layer interface: saldGronwallCandidateContract, saldGronwallEndpointCalculusContract, dvVariationalObligation saldDvVariationSource, saldPIContract, or saldLsiKlFiDensityTestContract.", "For Gronwall, refine endpoint-safe real calculus/integral assumptions only; keep the source signs and bound.", "For DV, expose finite log-mgf and common probability-space interfaces without marking the cited formula formalized.", "For PI/LSI/KL/FI, refine variance/Sobolev or density-test obligations without changing theorem-level hypotheses." ] reviewerChecklist := [ "source-index contains lem:gronwall, lem:dv_variation, def:PI, and eq:LSI-KL-FI and excludes sald_version_2.tex.", "saldFirstProofDag preserves statuses: Gronwall obligation, DV sourceCited, PI contractOnly, LSI/KL/FI obligation.", "No first-layer analytic fact is closed by axiom, sorry, admit, Prop := True, or := trivial.", "Forward-KL and VA-SALD theorem statements remain unchanged." ] status := ProofStatus.obligation def saldAlphaComplexityContract : AlphaComplexityContract where pathName := "pi=(pi_t)_{t in [0,T]}" velocityName := "v=(v_t)_{t in [0,T]}" alphaRange := "alpha > 0" pointwiseDensity := "E_alpha(pi_t,v_t) = alpha^{-1} * log E_{pi_t}[exp(alpha * ||v_t||^2)]." integratedComplexity := "A_alpha(pi,v) = int_0^T E_alpha(pi_t,v_t) dt." finitenessAssumption := "thm:forward-KL assumes some alpha0 > 0 with E_{alpha0}(pi_t,v_t) < +infty for every t, then uses any alpha in (0, alpha0]." source := saldAlphaComplexitySource status := ProofStatus.contractOnly def continuousForwardKlStatementContract : ForwardKlStatementContract where theoremLabel := "thm:forward-KL" sourceStatement := saldForwardKlSource sourceProof := saldForwardKlProofSource movingTarget := "pi_t on t in [0,T], with transport velocity field v_t generating pi through the continuity equation." saldLaw := "rho_s is the probability law of SALD eq:SALD for s in [0,S], with inverse time schedule s=s(t)." lsiAssumption := "For every t in [0,T], pi_t satisfies LSI with constant C_LSI(t) >= 0; C_LSI(t)=0 is allowed when no positive LSI is available." alphaComplexityAssumption := "There exists alpha0 > 0 such that E_{alpha0}(pi_t,v_t) < +infty for every t in [0,T]." alphaRange := "For any alpha in (0, alpha0]." initialErrorFactor := "exp (-(int_0^T dot{s}(t) C_LSI(t) dt)) * exp (int_0^T (1/2) dot{s}(t)^(-1) alpha^(-1) dt) * KL(rho_0 || pi_0)." residualIntegral := "int_0^T exp (int_t^T (1/2) dot{s}(u)^(-1) alpha^(-1) du) * (1/2) dot{s}(t)^(-1) * E_alpha(pi_t,v_t) dt." proofQuantity := "K(t) := KL(rho_{s(t)} || pi_t) = KL(rho_{s(t)} || tilde_pi_{s(t)})." differentialInequality := "dK/dt <= -(dot{s}(t) C_LSI(t) - (1/2) dot{s}(t)^(-1) alpha^(-1)) * K(t) + (1/2) dot{s}(t)^(-1) * E_alpha(pi_t,v_t)." proofSteps := [ "appendix.tex:166-189 differentiates KL(rho_s || tilde_pi_s) and substitutes the SALD Fokker-Planck equation.", "appendix.tex:190-225 changes from s to t and applies LSI to the Fisher information term.", "appendix.tex:229-241 applies Donsker--Varadhan to bound ||v_t||_{L2(rho_{s(t)})}^2 by alpha^{-1}K(t)+E_alpha(pi_t,v_t).", "appendix.tex:242-252 applies lem:gronwall with the exact exponent signs, then separates the LSI and alpha-complexity exponent terms." ] dependencies := [ "eq:LSI-KL-FI", "def:alpha-complexity", "lem:dv_variation", "lem:gronwall", "sald.forward_kl.moving_target_dependency_chain", "sald.forward_kl.gronwall_side_conditions", "sald.forward_kl.kl_derivative", "sald.forward_kl.dv_energy_bound", "sald.forward_kl.gronwall_application" ] status := ProofStatus.contractOnly def forwardKlDerivativeCandidateContract : ForwardKlDerivativeCandidateContract where sourceBlock := saldForwardKlDerivativeSource densityAndLawInterface := "rho_s is Law(X_s) for SALD, tilde_pi_s = pi_{t(s)}, rho_s << tilde_pi_s, and KL/FI are finite along the proof interval." scheduleInterface := "t=t(s) is smooth monotone increasing with inverse s=s(t), dot{s}(t)>0, and dot{t}(s(t)) = dot{s}(t)^(-1)." saldFokkerPlanck := "partial_s rho_s = div(rho_s * nabla log(rho_s / tilde_pi_s)), from eq:FP-eq." targetVelocity := "v_t satisfies partial_t pi_t + div(v_t*pi_t)=0; tilde_v_s := dot{t}(s)*v_{t(s)} generates tilde_pi_s." klDerivativeIdentity := "d/ds KL(rho_s || tilde_pi_s) = int partial_s rho_s * log(rho_s/tilde_pi_s) dx - int (rho_s/tilde_pi_s) * partial_s tilde_pi_s dx." firstTermEvaluation := "The SALD Fokker--Planck equation and integration by parts give -FI(rho_s || tilde_pi_s)." secondTermBound := "The target-velocity term is bounded by sqrt(FI(rho_s||tilde_pi_s))*||tilde_v_s||_{L2(rho_s)} <= (1/2)*FI + (1/2)*||tilde_v_s||^2." lsiStep := "Using LSI for pi_{t(s)}=tilde_pi_s gives FI(rho_s||tilde_pi_s) >= 2*C_LSI(t(s))*KL(rho_s||tilde_pi_s)." timeChangedInequality := "d/dt KL(rho_{s(t)} || pi_t) <= -dot{s}(t)*C_LSI(t)*KL(rho_{s(t)}||pi_t) + (1/2)*dot{s}(t)^(-1)*||v_t||_{L2(rho_{s(t)})}^2." requiredRegularity := [ "differentiability in s and t of the density paths and KL quantity", "smooth positive densities for rho_s, tilde_pi_s, and pi_t on the common state space", "integration-by-parts hypotheses or boundary decay eliminating boundary terms", "continuity/integrability of C_LSI(t), dot{s}(t), and the velocity L2 term" ] sourceGaps := [ "main_body.tex states the slowdown is smooth monotone but does not spell out the full inverse-function interface needed for dot{t}(s(t)) = dot{s}(t)^(-1)", "appendix.tex uses differentiation under the integral and integration by parts without explicit dominated-convergence or boundary assumptions", "the theorem statement does not explicitly state continuity of t -> C_LSI(t) or t -> E_alpha(pi_t,v_t), which Gronwall/integration later require" ] dependencies := [ "eq:SALD", "eq:FP-eq", "eq:LSI-KL-FI", "TransportVelocityContract", "KLContract", "FIContract" ] status := ProofStatus.obligation def forwardKlDerivativeSideConditionContract : ForwardKlDerivativeSideConditionContract where sourceBlock := saldForwardKlDerivativeSource densityPath := "rho_s and tilde_pi_s=pi_{t(s)} are smooth positive densities on a common state space, with rho_s absolutely continuous with respect to tilde_pi_s." massConservation := "appendix.tex:168-174 uses int partial_s rho_s dx = 0 when differentiating KL(rho_s || tilde_pi_s)." derivativeUnderIntegral := "The map s -> int log(rho_s/tilde_pi_s) rho_s dx is differentiable, and the derivative may pass through the integral." saldIntegrationByParts := "appendix.tex:176-185 integrates div(rho_s nabla log(rho_s/tilde_pi_s)) against log(rho_s/tilde_pi_s) with no boundary contribution." targetIntegrationByParts := "appendix.tex:199-208 integrates div(tilde_v_s tilde_pi_s) against rho_s/tilde_pi_s with no boundary contribution, then applies Cauchy--Schwarz and Young." l2CauchyYoungStep := "sqrt(FI(rho_s||tilde_pi_s))*||tilde_v_s||_{L2(rho_s)} <= (1/2)*FI(rho_s||tilde_pi_s)+(1/2)*||tilde_v_s||_{L2(rho_s)}^2." inverseSchedule := "t=t(s) is smooth monotone with inverse s=s(t), dot{s}(t)>0, and dot{t}(s(t))=dot{s}(t)^(-1)." timeChangeIdentity := "appendix.tex:218-228 uses d/dt K(s(t)) = dot{s}(t)*dK/ds and dot{s}(t)*dot{t}(s(t))^2 = dot{s}(t)^(-1)." outputInequality := "dK/dt <= -dot{s}(t)*C_LSI(t)*K(t) + (1/2)*dot{s}(t)^(-1)*||v_t||_{L2(rho_{s(t)})}^2." obligations := [ "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change" ] sourceGaps := [ "the source proof invokes differentiating under the integral and integration by parts without spelling out domination or boundary decay hypotheses", "the source theorem names the inverse slowdown s=s(t), but not the inverse-function derivative identities needed by the time-change calculation" ] status := ProofStatus.obligation def forwardKlDvEnergyCandidateContract : ForwardKlDvEnergyCandidateContract where sourceBlock := saldForwardKlDvEnergySource dvMeasure := "Apply lem:dv_variation with nu=rho_{s(t)} and mu=pi_t." dvTestFunction := "Z(x)=alpha*||v_t(x)||^2." integrabilityRoute := "The theorem assumes E_{alpha0}(pi_t,v_t)<+infty and alpha in (0,alpha0]; a local monotonicity/finiteness lemma for exp(alpha*||v_t||^2) is needed before invoking DV." bound := "||v_t||_{L2(rho_{s(t)})}^2 <= alpha^{-1}*KL(rho_{s(t)}||pi_t) + E_alpha(pi_t,v_t)." dependencies := [ "lem:dv_variation", "def:alpha-complexity", "KLContract", "sald.forward_kl.dv_alpha_mgf_monotonicity", "sald.forward_kl.dv_finite_log_mgf_witness", "SALD.forwardKlDvPositiveAlphaScalingScalar", "SALD.forwardKlDvPositiveAlphaCoefficientScalar" ] sourceGaps := [ "the appendix directly applies DV; the Lean route must expose the finite log-mgf witness for Z from the alpha0 assumption" ] status := ProofStatus.obligation def forwardKlDvFiniteLogMgfWitnessContract : ForwardKlDvFiniteLogMgfWitnessContract where sourceBlock := saldForwardKlDvEnergySource theoremStatement := saldForwardKlSource dvMeasures := "Use nu=rho_{s(t)} and mu=pi_t in the common Euclidean state space of the SALD law and moving target." testFunction := "Z_t(x)=alpha*||v_t(x)||^2, exactly the source choice in appendix.tex:230-235." finiteAlpha0Assumption := "main_body.tex:240-241 assumes E_{alpha0}(pi_t,v_t)<+infty for every t in [0,T]." alphaMonotonicityBridge := "Use forwardKlDvAlphaMonotonicityContract to prove log E_{pi_t}[exp(alpha*||v_t||^2)]<+infty for 00 and alpha<=alpha0 are required for alpha^(-1) and for the finite log-mgf witness from the source alpha0-complexity assumption.", "dot{s}(t)>0 and dot{t}(s(t))=dot{s}(t)^(-1) are required by the inverse-schedule time-change and coefficient rewrite.", "C_LSI(t)>=0 is used only in the residual-exponent drop, not to change the differential inequality or theorem statement.", "K(t), a(t), and b(t) need the differentiability/continuity or interval-integrability required by lem:gronwall.", "Endpoint identities s(0)=0, S=s(T), and tilde_pi_{s(t)}=pi_t are required before rewriting K(0) and K(T).", "rho_s, tilde_pi_s, and pi_t must live on a common state space with enough density and boundary regularity for the KL derivative and DV instantiation." ] sourceDependencyClassification := [ "eq:LSI-KL-FI: local density-test obligation using phi=sqrt(rho/pi), not a theorem-level assumption.", "lem:dv_variation: external-cited result with local finite-log-mgf/common-space instantiation; SLT entropy-duality remains reference-only.", "lem:gronwall: local real-analysis obligation over interval integrals and endpoint-safe derivative semantics.", "KL derivative and Fokker--Planck identity: local lemma plus source-contract gap for density, boundary, and differentiation-under-integral assumptions.", "Final exponent split and endpoint bridge: local scalar/integral algebra, tracked by the gronwall side-condition obligation." ] dependencies := [ "eq:LSI-KL-FI", "def:alpha-complexity", "lem:dv_variation", "lem:gronwall", "sald.forward_kl.moving_target_dependency_chain", "sald.forward_kl.kl_derivative", "sald.forward_kl.dv_energy_bound", "sald.forward_kl.gronwall_application" ] sourceGaps := [ "the paper does not isolate the nonnegativity/positivity facts used in the residual exponent simplification", "the theorem statement does not state the endpoint identities S=s(T) and s(0)=0 as standalone hypotheses", "the finite-log-mgf monotonicity from alpha <= alpha0 to the DV test function is used implicitly" ] lowerPacket := [ "First formalize the scalar coefficient algebra from the derivative and DV obligations before attempting a theorem-level proof.", "Keep the Young coefficient 1/2, the LSI coefficient C_LSI, and the DV contribution (1/2)*dot{s}^(-1)*alpha^(-1) unchanged.", "Do not replace the final residual exponent simplification with a different Gronwall normalization." ] status := ProofStatus.obligation def cycle10ForwardKlUpperPacket : ForwardKlUpperPacket where objective := "Keep thm:forward-KL fixed and sharpen the source dependency audit from the post-Young derivative inequality through LSI, inverse-schedule time change, DV, and Gronwall to the terminal theorem display." sourceLabels := [ "thm:forward-KL", "eq:LSI-KL-FI", "lem:dv_variation", "lem:gronwall", "def:alpha-complexity", "eq:SALD", "eq:FP-eq" ] modeDiscipline := [ "faithfulPaper: use main_body.tex:238-247 and appendix.tex:164-252 from the original source root, with sald_version_2.tex excluded.", "Preserve the source theorem statement, the two initial-error exponent factors, and the residual alpha-complexity integral exactly.", "Treat the KL derivative, LSI-to-KL/FI bridge, DV formula, inverse-schedule calculus, and Gronwall lemma as obligations or source-cited facts until local Lean proofs replace them." ] nonGoals := [ "Do not restate or prove thm:forward-KL in this cycle.", "Do not replace the paper route derivative -> LSI -> DV -> Gronwall with Pinsker, Talagrand, Girsanov, or another entropy method.", "Do not add endpoint, positivity, regularity, finite-log-mgf, or integrability assumptions silently to the theorem statement.", "Do not merge away the source's two exponent factors or alter the factor (1/2)*dot{s}(t)^(-1)*alpha^(-1)." ] lowerPacket := [ "Target exactly one interface: SALD.forwardKlDependencyChainAuditContract, SALD.forwardKlGronwallSideConditionContract, or the named obligation sald.forward_kl.coefficient_chain_audit.", "Refine one slice only: LSI coefficient bookkeeping, inverse-schedule scalar rewrite, DV finite-log-mgf witness, Gronwall coefficient regularity, or endpoint/exponent algebra.", "Use the new sourceLineLedger and scalarSideConditions entries as the checklist; unresolved analytic facts remain proof obligations.", "Keep thm:forward-KL and continuousForwardKlStatementContract unchanged." ] reviewerChecklist := [ "SALD.forwardKlDependencyChainAuditContract has sourceLineLedger, scalarSideConditions, and sourceDependencyClassification entries for appendix.tex:210-252.", "SALD.continuousSaldContract still lists the moving-target, coefficient-chain, gronwall-side-condition, derivative, DV, and Gronwall obligations.", "SALD.forwardKlProofDag still routes thm:forward-KL through moving_target_dependencies, coefficient_chain_audit, derivative, dv_energy, and gronwall_application blocks.", "The source index contains thm:forward-KL, eq:LSI-KL-FI, lem:dv_variation, and lem:gronwall, and sald_version_2.tex remains excluded.", "No analytic dependency is marked formalized and no fake proof closure is introduced." ] status := ProofStatus.obligation def forwardKlGronwallSideConditionContract : ForwardKlGronwallSideConditionContract where sourceBlock := saldForwardKlGronwallSource theoremStatement := saldForwardKlSource endpointScheduleIdentities := "The final bound is stated at t=T and t=0; the Lean route must provide the endpoint-schedule interface s(0)=0, S=s(T), t(s(T))=T, and tilde_pi_{s(t)}=pi_t on [0,T]." terminalKlIdentification := "The Gronwall output K(T) is rewritten as KL(rho_S||pi_T) using S=s(T) and tilde_pi_{s(T)}=pi_T." initialKlIdentification := "The initial term K(0) is rewritten as KL(rho_0||pi_0) using s(0)=0 and tilde_pi_{s(0)}=pi_0." coefficientRegularity := "a(t)=dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1) and b(t)=(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t) must satisfy the continuity or interval-integrability hypotheses required by lem:gronwall." signFactsForResidualDrop := "The residual-exponent simplification uses C_LSI(u)>=0 and dot{s}(u)>0, so -int_t^T dot{s}(u)*C_LSI(u) du <= 0." exponentSplitAlgebra := "Split exp(-int a) into exp(-int dot{s}*C_LSI)*exp(int (1/2)*dot{s}^(-1)*alpha^(-1)) for the initial term, and use SALD.forwardKlGronwallExpProductRewriteIntegralCongrOfPieces to assemble the LSI/alpha-piece interval-integrability hypotheses before applying the reusable Gronwall congruence." residualExponentBound := "For the residual integral, bound exp(-int_t^T a(u)du) by exp(int_t^T (1/2)*dot{s}(u)^(-1)*alpha^(-1)du) by dropping the nonpositive LSI contribution." dependencies := [ "sald.forward_kl.moving_target_dependency_chain", "sald.forward_kl.endpoint_schedule_identities", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "sald.forward_kl.dv_energy_bound", "sald.gronwall.integrating_factor", "SALD.gronwallIntervalIntegralAdditivityScalar", "SALD.gronwallExpProductRewriteIntervalIntegral", "SALD.gronwallExpProductRewriteIntegralCongr", "SALD.forwardKlGronwallCoeffIntervalIntegrable", "SALD.forwardKlGronwallCoeffAdjacentIntervalIntegrable", "SALD.forwardKlGronwallExpProductRewriteIntegralCongrOfPieces", "SALD.forwardKlGronwallCoeffIntegralSub", "SALD.forwardKlGronwallInitialExponentSplitScalar", "SALD.forwardKlGronwallInitialExponentSplitOfPieces", "SALD.forwardKlGronwallResidualExponentDropScalar", "SALD.forwardKlGronwallResidualExponentDropIntegral" ] sourceGaps := [ "the source theorem names the inverse slowdown but does not isolate s(0)=0, S=s(T), or t(s(T))=T as a separate endpoint lemma", "the proof applies Gronwall without separately proving continuity or interval-integrability for a(t) and b(t)", "the residual exponent drop still needs theorem-specific proof that int_t^T dot{s}(u)*C_LSI(u) du is nonnegative and that b(t) is nonnegative/integrable" ] status := ProofStatus.obligation def forwardKlEndpointScheduleContract : ForwardKlEndpointScheduleContract where sourceStatement := saldForwardKlSource sourceProof := saldForwardKlEndpointScheduleSource slowdownInterface := "main_body.tex:9-13 introduces t=t(s) on s in [0,S] as a smooth monotone increasing slowdown and main_body.tex:238 names the inverse s=s(t)." inverseEndpointIdentities := "For the theorem endpoint display, the Lean route must isolate s(0)=0, S=s(T), t(s(T))=T, and t(s(0))=0 as inverse-schedule endpoint identities rather than adding them to thm:forward-KL." slowedTargetIdentity := "appendix.tex:218-228 rewrites KL(rho_{s(t)}||tilde_pi_{s(t)}) as KL(rho_{s(t)}||pi_t), using tilde_pi_s=pi_{t(s)} and the inverse identity t(s(t))=t." klPathDefinition := "The scalar Gronwall quantity is K(t)=KL(rho_{s(t)}||pi_t) after the slowed-target rewrite, with K defined on the original t-interval [0,T]." terminalRewrite := "The terminal Gronwall endpoint K(T) becomes KL(rho_{s(T)}||pi_T), then KL(rho_S||pi_T) using S=s(T), exactly matching main_body.tex:243." initialRewrite := "The initial Gronwall endpoint K(0) becomes KL(rho_{s(0)}||pi_0), then KL(rho_0||pi_0) using s(0)=0, exactly matching main_body.tex:245." gronwallUse := "appendix.tex:244-252 uses these endpoint rewrites only after the differential inequality and DV step; they do not change the source a(t), b(t), or terminal theorem display." dependencies := [ "SALD.forwardKlMovingTargetDependencyContract", "SALD.forwardKlGronwallSideConditionContract", "sald.forward_kl.moving_target_dependency_chain", "sald.forward_kl.schedule_time_change", "sald.forward_kl.gronwall_side_conditions" ] sourceGaps := [ "the source states the slowdown and inverse schedule but does not prove a closed-interval inverse endpoint lemma", "the paper identifies S with s(T) in the theorem endpoint display without a named assumption or lemma", "the slowed-target equality tilde_pi_{s(t)}=pi_t depends on the inverse identity t(s(t))=t on [0,T]" ] status := ProofStatus.obligation def cycle14ForwardKlUpperPacket : ForwardKlUpperPacket where objective := "Keep thm:forward-KL fixed and select the theorem-level moving-target side conditions as the next lower target: endpoint schedule identities, transport/Fokker--Planck interfaces, DV finite-log-mgf witness, and Gronwall coefficient regularity along the original derivative -> LSI -> DV -> Gronwall route." sourceLabels := [ "thm:forward-KL", "eq:SALD", "eq:FP-eq", "eq:LSI-KL-FI", "def:alpha-complexity", "lem:dv_variation", "lem:gronwall" ] modeDiscipline := [ "faithfulPaper: use main_body.tex:238-247 and appendix.tex:164-252 from the original source root, with sald_version_2.tex excluded.", "Preserve the source theorem statement, the two initial-error exponent factors, the residual alpha-complexity integral, and the scalar coefficients a(t)=dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1) and b(t)=(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t).", "Treat endpoint schedule identities, density/boundary regularity, transport/Fokker--Planck backends, LSI-to-KL/FI, DV, finite-log-mgf monotonicity, and Gronwall regularity as obligations or source-cited facts until local Lean proofs replace them." ] nonGoals := [ "Do not restate or prove thm:forward-KL in this packet.", "Do not replace the paper route derivative -> LSI -> DV -> Gronwall with Pinsker, Talagrand, Girsanov, path-space, or another entropy method.", "Do not add endpoint, positivity, regularity, finite-log-mgf, absolute-continuity, or integrability assumptions silently to the theorem statement.", "Do not change the discrete or general moving-target theorem statements while refining this continuous forward-KL packet." ] lowerPacket := [ "Target exactly one interface: SALD.forwardKlEndpointScheduleContract, SALD.forwardKlMovingTargetDependencyContract, SALD.forwardKlGronwallSideConditionContract, SALD.forwardKlDerivativeSideConditionContract, or the named obligations sald.forward_kl.endpoint_schedule_identities, sald.forward_kl.moving_target_dependency_chain, and sald.forward_kl.gronwall_side_conditions.", "Preferred lower slice: isolate endpoint schedule identities s(0)=0, S=s(T), t(s(T))=T, and tilde_pi_{s(t)}=pi_t in SALD.forwardKlEndpointScheduleContract without adding them as theorem hypotheses.", "Alternative lower slices: transport velocity for the slowed target, density/boundary side conditions for the KL derivative, finite log-mgf/common-space witness for DV, or continuity/integrability of the Gronwall coefficients a(t) and b(t).", "Keep the differential inequality from appendix.tex:239-241 and the terminal theorem display in main_body.tex:243-246 unchanged." ] reviewerChecklist := [ "SALD.continuousSaldContract still lists the moving-target, derivative side-condition, DV witness, coefficient-chain, Gronwall-side-condition, derivative, DV-energy, and Gronwall obligations.", "SALD.forwardKlProofDag routes thm:forward-KL through moving_target_dependencies before derivative, DV-energy, and Gronwall proof search.", "SALD.forwardKlEndpointScheduleContract, SALD.forwardKlMovingTargetDependencyContract, and SALD.forwardKlGronwallSideConditionContract record endpoint identities and Gronwall regularity as obligations, not hidden theorem assumptions.", "research-wiki/source-index/SALD_original.jsonl contains thm:forward-KL, eq:LSI-KL-FI, lem:dv_variation, and lem:gronwall while excluding sald_version_2.tex.", "No analytic dependency is marked formalized and no fake proof closure is introduced." ] status := ProofStatus.obligation def cycle14ForwardKlMiddleContract : ForwardKlMiddleSourceToLeanContract where sourceStatement := saldForwardKlSource sourceProof := saldForwardKlProofSource derivativeSource := saldForwardKlDerivativeSource dvSource := saldForwardKlDvEnergySource gronwallSource := saldForwardKlGronwallSource objective := "Map the continuous forward-KL proof route from source lines to Lean-facing contracts and lower obligations, while keeping thm:forward-KL and its displayed bound fixed." sourceStepMap := [ "main_body.tex:238-247 states the inverse slowdown setup, LSI constants C_LSI(t)>=0, finite alpha0-complexity for the transport velocity v_t, alpha in (0,alpha0], and the terminal KL display.", "appendix.tex:168-185 differentiates KL(rho_s||tilde_pi_s), uses mass conservation and the SALD Fokker--Planck equation, and identifies the first term as -FI(rho_s||tilde_pi_s).", "appendix.tex:187-208 uses the transport velocity v_t for pi_t, defines tilde_v_s=dot{t}(s)*v_{t(s)}, proves it transports tilde_pi_s, and bounds the second derivative term by Young.", "appendix.tex:210-228 combines the derivative pieces with eq:LSI-KL-FI and changes from s to t, producing the pre-DV inequality with coefficient (1/2)*dot{s}(t)^(-1).", "appendix.tex:230-241 applies lem:dv_variation with Z=alpha*||v_t||^2 and rewrites the log-mgf contribution as E_alpha(pi_t,v_t).", "appendix.tex:244-252 applies lem:gronwall, splits the initial exponent, drops the nonpositive LSI contribution from the residual exponent, and matches main_body.tex:243-246." ] leanStepMap := [ "The theorem statement remains represented by SALD.continuousForwardKlStatementContract and SALD.continuousSaldContract.", "Density regularity, mass conservation, differentiation under the integral, and integration by parts are tracked by SALD.forwardKlDerivativeSideConditionContract and sald.forward_kl.density_boundary_regular.", "Slowed-target transport, inverse-schedule calculus, and endpoint rewrites are tracked by SALD.forwardKlEndpointScheduleContract, SALD.forwardKlMovingTargetDependencyContract, SALD.forwardKlDerivativeSideConditionContract, and sald.forward_kl.schedule_time_change.", "The LSI step is routed through SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi; no LSI proof is promoted.", "The DV step is split between SALD.forwardKlDvAlphaMonotonicityContract, SALD.forwardKlDvFiniteLogMgfWitnessContract, and SALD.forwardKlDvEnergyCandidateContract.", "The scalar coefficient and endpoint algebra are tracked by SALD.forwardKlDependencyChainAuditContract, SALD.forwardKlGronwallInstantiationContract, and SALD.forwardKlGronwallSideConditionContract." ] citedResultInterfaces := [ "lem:dv_variation remains source-cited through Boucheron Corollary 4.15 or the SLT entropy_duality pattern; no SLT theorem is imported.", "eq:LSI-KL-FI remains a local density-test obligation through probability.lsi_to_kl_fi.", "lem:gronwall remains a local real-analysis obligation through sald.gronwall.integrating_factor, sald.gronwall.endpoint_calculus, and sald.gronwall.exponent_rewrite.", "Transport-velocity existence/minimality is source background from main_body.tex:230-232; the proof only requires a stated transport velocity interface." ] obligations := [ "sald.forward_kl.middle_source_to_lean_map", "sald.forward_kl.endpoint_schedule_identities", "sald.forward_kl.moving_target_dependency_chain", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "probability.lsi_to_kl_fi", "sald.forward_kl.dv_alpha_mgf_monotonicity", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "sald.forward_kl.coefficient_chain_audit", "sald.forward_kl.gronwall_application", "sald.forward_kl.gronwall_side_conditions" ] lowerPacket := [ "Preferred lower target: refine SALD.forwardKlEndpointScheduleContract and sald.forward_kl.endpoint_schedule_identities for s(0)=0, S=s(T), t(s(T))=T, and tilde_pi_{s(t)}=pi_t.", "Keep the source proof route derivative -> LSI -> DV -> Gronwall; do not introduce path-space, Girsanov, Pinsker, Talagrand, or a different Gronwall normalization.", "If the endpoint slice is blocked, refine exactly one alternative interface: slowed-target transport, density/boundary side conditions, DV finite-log-mgf/common-space witness, or Gronwall coefficient regularity.", "Preserve the differential inequality from appendix.tex:239-241 and the terminal theorem display in main_body.tex:243-246 byte-for-byte at the contract level." ] reviewerChecklist := [ "Every source step in appendix.tex:168-252 is classified as a Lean contract, cited result, or named proof obligation.", "SALD.continuousSaldContract includes sald.forward_kl.middle_source_to_lean_map and still keeps all analytic backends at obligation or source-cited status.", "SALD.forwardKlProofDag exposes ASTIS.SALD.forward_KL.middle_source_to_lean_map before lower proof-search blocks.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes SALD.cycle14ForwardKlMiddleContract, sald.forward_kl.middle_source_to_lean_map, and sald.forward_kl.endpoint_schedule_identities.", "The source index still excludes sald_version_2.tex and no analytic dependency is marked formalized." ] status := ProofStatus.obligation def cycle11DiscreteForwardKlUpperPacket : DiscreteForwardKlUpperPacket where objective := "Keep thm:forward-KL-discrete fixed and isolate the discrete Donsker--Varadhan finite-log-mgf witness used for the EM interpolation velocity term in appendix.tex:493-523." sourceLabels := [ "thm:forward-KL-discrete", "lem:dv_variation", "def:alpha-complexity", "eq:frozen_interp_terminal_disc_prop_additive_final", "lem:gronwall", "eq:LSI-KL-FI" ] modeDiscipline := [ "faithfulPaper: use main_body.tex:301-323 and appendix.tex:493-523 for this refinement, with sald_version_2.tex excluded.", "Preserve the EM interpolation law hat rho_s, the DV choice nu=hat rho_s, mu=tilde pi_s, and Z=alpha*||v_{t(s)}||^2 exactly.", "Reuse the continuous alpha0-to-alpha monotonicity obligation for the pi_t log-mgf, but keep the discrete common-space and absolute-continuity interfaces separate because nu is now hat rho_s.", "Treat DV, EM endpoint laws, density regularity, finite log-mgf, and positive-alpha scaling as obligations or source-cited dependencies until local Lean proofs replace them." ] nonGoals := [ "Do not restate or prove thm:forward-KL-discrete in this cycle.", "Do not replace the source DV step by path-space, Girsanov, Pinsker, Talagrand, or another entropy method.", "Do not add a new finite-mgf or absolute-continuity hypothesis to the theorem statement.", "Do not change the coefficient dot t(s)^2*(alpha^(-1)*K_s+E_alpha(pi_{t(s)},v_{t(s)})) or the later dot{s}(t)^(-1)*alpha^(-1) term." ] lowerPacket := [ "Target exactly one interface: SALD.discreteForwardKlDvFiniteLogMgfWitnessContract or SALD.discreteForwardKlDvFiniteLogMgfWitnessObligation.", "Refine one backend only: EM-interpolation common-space/absolute-continuity, measurability of ||v_{t(s)}||^2, alpha0-to-alpha finite-log-mgf reuse, positive-alpha scaling, or dot t(s)^2 coefficient preservation.", "Use appendix.tex:493-523 as the source line ledger and keep lem:dv_variation source-cited.", "Keep SALD.discreteForwardKlStatementContract and the theorem bound in main_body.tex:309-323 unchanged." ] reviewerChecklist := [ "SALD.discreteSaldContract lists SALD.discreteForwardKlDvFiniteLogMgfWitnessObligation before SALD.discreteForwardKlDvVelocityObligation.", "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.dv_finite_log_mgf_witness before the Gronwall accumulation block.", "SALD.discreteForwardKlDvVelocityObligation depends on sald.discrete_forward_kl.dv_finite_log_mgf_witness and keeps the appendix.tex:493-523 coefficient.", "The conversion window and proof-obligation ledger mention the discrete DV witness under thm:forward-KL-discrete.", "No analytic dependency is marked formalized and no fake proof closure is introduced." ] status := ProofStatus.obligation def cycle15DiscreteForwardKlUpperPacket : DiscreteForwardKlUpperPacket where objective := "Keep thm:forward-KL-discrete fixed and select the Euler--Maruyama interpolation side-condition spine as the next lower target: endpoint laws, conditional-drift Fokker--Planck equation, and stitched KL regularity before the one-step Gamma/Delta defects are accumulated." sourceLabels := [ "thm:forward-KL-discrete", "eq:frozen_interp_terminal_disc_prop_additive_final", "lem:frozen_delta_cross_lip_sald", "lem:dv_variation", "lem:gronwall", "eq:LSI-KL-FI" ] modeDiscipline := [ "faithfulPaper: use main_body.tex:299-323 and appendix.tex:260-590 from the original source root, with sald_version_2.tex excluded.", "Preserve the source theorem statement, the linear slowdown t(s)=s/r, the EM interpolation law hat rho_s, the frozen one-step Gamma and Delta coefficients, and the final barGamma/barDelta accumulated-error display.", "Keep endpoint laws, conditional drift, EM Fokker--Planck, density/boundary regularity, stitched-interval Gronwall regularity, DV, and Gronwall as obligations or source-cited facts until local Lean proofs replace them.", "Treat SLT one_step_discretization only as a route reference for the EM backend; it is not an imported dependency and is not marked formalized." ] nonGoals := [ "Do not restate or prove thm:forward-KL-discrete in this packet.", "Do not bundle the EM interpolation, frozen-defect lemma, DV velocity estimate, and accumulated-error bridge into one hidden theorem assumption.", "Do not change Gamma, Delta, barGamma, barDelta, alpha, alpha', eta, r, or the source step-size condition.", "Do not replace the paper route EM interpolation -> frozen defect -> LSI -> DV -> Gronwall -> linear-slowdown collection with path-space, Girsanov, Pinsker, Talagrand, or another entropy route." ] lowerPacket := [ "Target exactly one interface: SALD.discreteForwardKlEmInterpolationSideConditionContract, with the first lower slice sald.discrete_forward_kl.em_conditional_fokker_planck.", "Refine appendix.tex:347-385 only: define bar b_{k,s}, derive partial_s hat rho_s = -div(hat rho_s*bar b_{k,s}) + Delta hat rho_s, and record the Laplacian split relative to tilde pi_s.", "Keep sald.discrete_forward_kl.em_endpoint_laws and sald.discrete_forward_kl.stitched_interval_regularity as separate obligations; do not discharge them by adding endpoint or smoothness hypotheses to the theorem.", "When the conditional Fokker--Planck backend is blocked, record the precise missing conditional-expectation, density, or integration-by-parts interface instead of weakening the discrete theorem." ] reviewerChecklist := [ "SALD.discreteSaldContract still lists the EM endpoint, conditional Fokker--Planck, stitched-interval, frozen defect, DV witness, Gronwall, and accumulated-error obligations.", "SALD.discreteForwardKlProofDag routes through ASTIS.SALD.forward_KL_discrete.em_interpolation_side_conditions before derivative, DV velocity, and Gronwall accumulation blocks.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle15DiscreteForwardKlUpperPacket and the EM endpoint, conditional Fokker--Planck, and stitched-interval obligations.", "The conversion window, proof-obligation ledger, SLT reuse audit, and source index still point to main_body.tex:301 and appendix.tex:260-590, while sald_version_2.tex remains excluded.", "No analytic dependency is marked formalized and no fake proof closure is introduced." ] status := ProofStatus.obligation def cycle15DiscreteForwardKlMiddleContract : DiscreteForwardKlEmDefectAccumulationMiddleContract where sourceStatement := saldForwardKlDiscreteSource interpolationSource := saldForwardKlDiscreteInterpolationSource derivativeSource := saldForwardKlDiscreteDerivativeSource gronwallSource := saldForwardKlDiscreteGronwallSource accumulatedSource := saldForwardKlDiscreteAccumulatedErrorSource objective := "Refine the cycle-15 upper EM side-condition packet into a lower-ready source-to-Lean map for thm:forward-KL-discrete, with sald.discrete_forward_kl.em_conditional_fokker_planck as the first lower slice and all one-step and accumulated-error constants left unchanged." sourceStepMap := [ "main_body.tex:299-323 fixes the linear slowdown t(s)=s/r and the terminal bound with Gamma, Delta, barGamma, barDelta, alpha, alpha', eta, and r.", "appendix.tex:260-266 and 334-335 define the frozen EM interpolation hat X_s and use endpoint laws hat rho_{s_k}=rho_k^eta and hat rho_{s_{k+1}}=rho_{k+1}^eta.", "appendix.tex:347-385 defines bar b_{k,s}, invokes partial_s hat rho_s = -div(hat rho_s*bar b_{k,s}) + Delta hat rho_s, and splits Delta hat rho_s relative to tilde pi_s.", "appendix.tex:454-491 uses lem:frozen_delta_cross_lip_sald, Young, and LSI only after the EM conditional Fokker--Planck identity has produced the KL derivative block.", "appendix.tex:493-523 applies the discrete DV finite-log-mgf witness for nu=hat rho_s, mu=tilde pi_s, and Z=alpha*||v_{t(s)}||^2 while preserving the dot t(s)^2 coefficient.", "appendix.tex:526-590 changes variables from s to t, applies lem:gronwall, and leaves the main-body linear-slowdown collection to the accumulated-error bridge." ] leanStepMap := [ "SALD.cycle15DiscreteForwardKlUpperPacket chooses the EM side-condition spine; this middle contract records the ordered source-to-Lean map for lower work.", "SALD.discreteForwardKlEmInterpolationSideConditionContract splits the EM backend into sald.discrete_forward_kl.em_endpoint_laws, sald.discrete_forward_kl.em_conditional_fokker_planck, and sald.discrete_forward_kl.stitched_interval_regularity.", "SALD.discreteForwardKlDerivativeCandidateContract depends on the conditional Fokker--Planck slice before it can use the frozen-defect lemma, LSI bridge, and DV velocity witness.", "SALD.discreteForwardKlGronwallInstantiationContract and SALD.discreteForwardKlAccumulatedErrorBridgeContract stay separate from the EM backend so endpoint, exponent, barGamma, and barDelta algebra is not hidden.", "SALD.discreteForwardKlCoefficientChainAuditContract remains the reviewer-facing coefficient ledger for the one-step Gamma/Delta and accumulated-error constants." ] citedResultInterfaces := [ "SLT one_step_discretization remains a route reference for the EM interpolation backend only; no SLT theorem is imported or marked formalized.", "lem:dv_variation remains source-cited through Boucheron Corollary 4.15 or the SLT entropy_duality pattern for the later velocity estimate.", "lem:gronwall remains a local real-analysis obligation through sald.gronwall.integrating_factor and the stitched-interval interface.", "eq:LSI-KL-FI remains the local density-test obligation probability.lsi_to_kl_fi." ] obligations := [ "sald.discrete_forward_kl.cycle15_middle_em_spine", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.frozen_delta_cross_lip", "sald.discrete_forward_kl.kl_derivative", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.coefficient_chain_audit" ] lowerPacket := [ "Lower target exactly one interface first: SALD.discreteForwardKlEmConditionalFpObligation, not the full theorem and not the accumulated-error bridge.", "Use SALD.cycle15DiscreteForwardKlEmConditionalFpLowerContract as the line ledger for appendix.tex:347-385 before attempting any derivative or Gronwall proof search.", "Inside that ledger, refine SALD.cycle15DiscreteForwardKlConditionalDriftDensityContract first: the regular conditional law, density, measurability, and integrability needed to define bar b_{k,s}.", "For appendix.tex:347-385, expose the conditional expectation defining bar b_{k,s}, the density/law interface for hat rho_s, the frozen-interpolation Fokker--Planck equation, and the Laplacian split relative to tilde pi_s.", "Keep endpoint law matching and stitched-interval Gronwall regularity as named sibling obligations; do not discharge the conditional Fokker--Planck slice by adding theorem-level smoothness assumptions.", "Once the conditional Fokker--Planck backend is refined, the next lower slices remain the endpoint/stitching interfaces or the accumulated-error bridge, not a restatement of thm:forward-KL-discrete." ] reviewerChecklist := [ "Every source step from appendix.tex:260-590 that is used for thm:forward-KL-discrete is classified as a Lean contract, cited result, or named proof obligation.", "SALD.discreteSaldContract lists SALD.cycle15DiscreteForwardKlMiddleEmSpineObligation and SALD.cycle15DiscreteForwardKlConditionalDriftDensityObligation alongside the existing EM endpoint, conditional Fokker--Planck, stitched-interval, frozen-defect, DV, Gronwall, and accumulated-error obligations.", "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.conditional_drift_density before the conditional Fokker--Planck lower packet.", "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle15_conditional_fp_lower_packet before the derivative block.", "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle15_middle_em_spine after the upper packet and before derivative, DV, and Gronwall blocks.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle15DiscreteForwardKlMiddleContract, sald.discrete_forward_kl.cycle15_middle_em_spine, and sald.discrete_forward_kl.conditional_drift_density.", "No theorem statement, source constant, source file selection, or analytic dependency status is changed." ] status := ProofStatus.obligation def cycle15DiscreteForwardKlEmConditionalFpLowerContract : DiscreteForwardKlEmConditionalFpLowerContract where sourceBlock := saldForwardKlDiscreteConditionalFpSource parentPacket := "SALD.cycle15DiscreteForwardKlMiddleContract" targetObligation := "sald.discrete_forward_kl.em_conditional_fokker_planck" conditionalDriftDefinition := "appendix.tex:347-354 defines bar b_{k,s}(x)=E[nabla log pi_{t_k}(X_k^eta) | hat X_s=x]." conditionalLawInterface := "The frozen interpolation hat X_s must provide a law hat rho_s with enough density and conditional-expectation structure for bar b_{k,s} to be a measurable drift field." fokkerPlanckEquation := "appendix.tex:357-364 invokes partial_s hat rho_s = -div(hat rho_s*bar b_{k,s}) + Delta hat rho_s on each interval [s_k,s_{k+1}]." laplacianSplit := "appendix.tex:365-385 rewrites Delta hat rho_s as div(hat rho_s*nabla log(hat rho_s/tilde pi_s)) + div(hat rho_s*nabla log tilde pi_s), then groups the frozen drift defect." klDerivativeBridge := "The output feeds appendix.tex:388-413, where integration by parts turns the divergence form into -FI(hat rho_s||tilde pi_s) and the frozen cross term." exclusions := [ "Do not include endpoint law matching beyond the input needed to identify the interval [s_k,s_{k+1}].", "Do not include the later frozen Gamma/Delta bound, DV velocity estimate, Gronwall accumulation, or linear-slowdown barGamma/barDelta collection.", "Do not add theorem-level smoothness or absolute-continuity assumptions; record missing density and boundary interfaces as obligations." ] dependencies := [ "SALD.cycle15DiscreteForwardKlMiddleContract", "SALD.cycle15DiscreteForwardKlConditionalDriftDensityContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.em_conditional_fokker_planck", "FokkerPlanckContract", "KLContract", "FIContract" ] sourceGaps := [ "The source names the Fokker--Planck equation associated with the frozen interpolation but does not prove it.", "The source does not spell out the conditional-expectation regularity that makes bar b_{k,s} a drift in divergence form.", "The Laplacian split and following integration by parts require density positivity, differentiability, and boundary decay not separately stated in the theorem." ] lowerPacket := [ "First formalize sald.discrete_forward_kl.conditional_drift_density: the law/density interface for hat X_s and bar b_{k,s} on a fixed interval.", "Then prove or cite the conditional-drift Fokker--Planck equation for the frozen EM interpolation.", "Finally expose the Laplacian split relative to tilde pi_s as the input to the KL derivative identity.", "Leave endpoint stitching and accumulated-error algebra to their sibling obligations." ] status := ProofStatus.obligation def cycle15DiscreteForwardKlConditionalDriftDensityContract : DiscreteForwardKlConditionalDriftDensityContract where sourceBlock := saldForwardKlDiscreteConditionalFpSource parentPacket := "SALD.cycle15DiscreteForwardKlEmConditionalFpLowerContract" targetObligation := "sald.discrete_forward_kl.conditional_drift_density" randomVariables := "On a fixed interval [s_k,s_{k+1}], the pair (X_k^eta, hat X_s) is generated by the frozen EM interpolation in appendix.tex:260-266." conditionalLawKernel := "appendix.tex:347-354 requires a regular conditional law for X_k^eta given hat X_s=x so bar b_{k,s}(x)=E[nabla log pi_{t_k}(X_k^eta) | hat X_s=x] is defined pointwise or almost everywhere." densityInterface := "The law hat rho_s=Law(hat X_s) must admit a smooth positive density on the Euclidean state space before the divergence form in appendix.tex:357-364 is meaningful." driftMeasurability := "The conditional expectation must select a measurable vector field x |-> bar b_{k,s}(x) compatible with the density hat rho_s." driftIntegrability := "The product hat rho_s*bar b_{k,s} must have enough local integrability and weak differentiability for div(hat rho_s*bar b_{k,s}) to be used in the Fokker--Planck equation." fokkerPlanckInput := "This contract supplies only the drift/density input to partial_s hat rho_s = -div(hat rho_s*bar b_{k,s}) + Delta hat rho_s; the FP identity and Laplacian split remain separate obligations." exclusions := [ "Do not prove the Fokker--Planck equation in this interface.", "Do not perform the Laplacian split relative to tilde pi_s.", "Do not include endpoint stitching, frozen Gamma/Delta bounds, DV, Gronwall, or accumulated-error algebra." ] dependencies := [ "eq:frozen_interp_terminal_disc_prop_additive_final", "SALD.discreteSaldEulerMaruyamaContract", "sald.discrete_forward_kl.em_endpoint_laws" ] sourceGaps := [ "The source writes the conditional expectation defining bar b_{k,s} but does not state the regular conditional probability or measurability interface.", "The source uses hat rho_s as a density in divergence form without separately proving density existence, positivity, or smoothness for the EM interpolation law.", "The source does not isolate the integrability needed for hat rho_s*bar b_{k,s} before applying the Fokker--Planck equation." ] status := ProofStatus.obligation def cycle19DiscreteForwardKlUpperPacket : DiscreteForwardKlUpperPacket where objective := "Keep thm:forward-KL-discrete fixed and select the accumulated-error bridge as the next lower target: endpoint rewrites after the EM interpolation, the linear-slowdown exponent split, the residual exponent bound, and the A_alpha/barGamma/barDelta collection from appendix.tex:557-590 to main_body.tex:309-323." sourceLabels := [ "thm:forward-KL-discrete", "proof:thm:forward-KL-discrete:gronwall", "proof:thm:forward-KL-discrete:accumulated-error", "proof:thm:forward-KL-discrete:coefficient-chain", "lem:gronwall", "def:alpha-complexity", "eq:frozen_interp_terminal_disc_prop_additive_final" ] modeDiscipline := [ "faithfulPaper: use main_body.tex:299-323 and appendix.tex:526-592 from the original source root, with sald_version_2.tex excluded.", "Preserve the source theorem statement, the linear slowdown t(s)=s/r with r>=1, the appendix Gronwall coefficients a(t)=dot{s}(t)*C_LSI(t)-dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t) and b(t)=dot{s}(t)^(-1)*E_alpha(pi_t,v_t)+2*dot{s}(t)*eta*Delta(t), and the main-body constants T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'}.", "Treat EM endpoint laws, stitched-interval regularity, coefficient integrability, residual-exponent monotonicity, barGamma/barDelta integral identifications, and Gronwall as obligations or source-cited facts until local Lean proofs replace them.", "Classify the blocked lower scalar bridge as local real/integral algebra plus source-contract gaps; do not import or mark any SLT result formalized for this packet." ] nonGoals := [ "Do not restate or prove thm:forward-KL-discrete in this packet.", "Do not reopen the conditional Fokker--Planck, frozen one-step Gamma/Delta, LSI, or DV velocity subproofs except as dependencies of the accumulated-error bridge.", "Do not alter Gamma, Delta, barGamma, barDelta, alpha, alpha', eta, r, the source step-size condition, or the theorem's open alpha range.", "Do not hide endpoint rewrites, residual-exponent monotonicity, or A_alpha/Gamma/Delta integral collection as a new theorem assumption.", "Do not replace the source route EM interval inequalities -> Gronwall -> linear-slowdown collection with a path-space, Girsanov, Pinsker, Talagrand, or different entropy route." ] lowerPacket := [ "Target exactly one interface: SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.", "Preferred first lower sub-slice: SALD.discreteForwardKlResidualExponentBoundObligation, proving only the residual exponent drop and full-interval Gamma bound used inside the accumulated-error bridge.", "Keep the appendix Gronwall display from appendix.tex:557-590 separate from the main-body theorem display in main_body.tex:309-323; bridge them by explicit endpoint, exponent, and integral-collection steps.", "If endpoint law matching, stitched interval regularity, coefficient integrability, or interval-integral monotonicity is blocked, record the precise source-contract gap rather than weakening the theorem statement.", "Do not bundle the one-step frozen defect, DV witness, Gronwall accumulation, residual exponent bound, and final accumulated-error collection into one opaque assumption." ] reviewerChecklist := [ "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle19_upper_packet before the linear-slowdown, residual-exponent, and accumulated-error blocks.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle19DiscreteForwardKlUpperPacket while retaining the cycle-15 EM packets and all existing discrete obligations.", "SALD.discreteForwardKlAccumulatedErrorBridgeContract still uses appendix.tex:557-590 and main_body.tex:309-323, with the source a(t), b(t), barGamma, and barDelta constants unchanged.", "The conversion window and proof-obligation ledger classify the cycle-19 packet as workflow/obligation data, not a proof of thm:forward-KL-discrete.", "No analytic dependency is marked formalized and no fake proof closure is introduced." ] status := ProofStatus.obligation /-- Cycle-19 middle packet for the discrete forward-KL accumulated-error bridge. This translates the upper-selected accumulated-error target into a lower-ready source-to-Lean map. It keeps the final scalar bridge separate from the EM interpolation, one-step frozen defect, DV velocity estimate, and Gronwall accumulation backends. -/ def cycle19DiscreteForwardKlMiddleContract : DiscreteForwardKlEmDefectAccumulationMiddleContract where sourceStatement := saldForwardKlDiscreteSource interpolationSource := saldForwardKlDiscreteInterpolationSource derivativeSource := saldForwardKlDiscreteDerivativeSource gronwallSource := saldForwardKlDiscreteGronwallSource accumulatedSource := saldForwardKlDiscreteAccumulatedErrorSource objective := "Translate the cycle-19 accumulated-error bridge into a lower-ready map for sald.discrete_forward_kl.accumulated_error_bridge, with sald.discrete_forward_kl.residual_exponent_bound isolated as the first scalar sub-slice and all source constants preserved." sourceStepMap := [ "appendix.tex:557-571: Gronwall gives the initial-error term with a(t)=dot{s}(t)*C_LSI(t)-dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t).", "appendix.tex:573-589: the residual integral uses the same exponential kernel and b(t)=dot{s}(t)^(-1)*E_alpha(pi_t,v_t)+2*dot{s}(t)*eta*Delta(t).", "main_body.tex:299-323: the theorem specializes to linear slowdown t(s)=s/r, so dot{s}(t)=r and the positive exponent is T/(r*alpha)+2*r*eta^2*barGamma/alpha'.", "main_body.tex:310-323: the endpoint term keeps the LSI contraction factor exp(-r*int_0^T C_LSI(t)dt), while the residual integral is bounded by the common positive exponential factor.", "main_body.tex:316-323: the residual b(t) collection becomes (1/r)*A_alpha(pi,v)+2*r*eta*barDelta_{alpha'} using the source definitions of A_alpha, barGamma, and barDelta." ] leanStepMap := [ "The theorem statement remains SALD.discreteForwardKlStatementContract and SALD.discreteSaldContract; this middle packet changes neither.", "The source Gronwall output is tracked by SALD.discreteForwardKlGronwallInstantiationContract and sald.discrete_forward_kl.gronwall_accumulation.", "The final bridge is tracked by SALD.discreteForwardKlAccumulatedErrorBridgeContract and sald.discrete_forward_kl.accumulated_error_bridge.", "The first lower scalar target is SALD.discreteForwardKlResidualExponentBoundObligation / sald.discrete_forward_kl.residual_exponent_bound.", "Endpoint rewrites are inherited from sald.discrete_forward_kl.em_endpoint_laws and sald.discrete_forward_kl.stitched_interval_regularity; they are dependencies of the bridge, not new theorem assumptions.", "The coefficient audit remains SALD.discreteForwardKlCoefficientChainAuditContract, which checks the Gamma, Delta, barGamma, barDelta, alpha, alpha', eta, and r constants against the source displays." ] citedResultInterfaces := [ "lem:gronwall remains a local real-analysis obligation; this packet only maps the scalar output after Gronwall is invoked.", "lem:dv_variation remains source-cited through the existing discrete DV velocity witness and is not reopened by this accumulated-error packet.", "eq:LSI-KL-FI remains an inherited obligation; the only LSI use here is the sign C_LSI(t)>=0 needed to drop the nonpositive residual exponent contribution.", "No SLT theorem is used for endpoint rewrites, exponent splitting, residual exponent monotonicity, or barGamma/barDelta collection." ] obligations := [ "sald.discrete_forward_kl.cycle19_accumulated_error_middle", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.coefficient_chain_audit", "sald.gronwall.integrating_factor" ] lowerPacket := [ "Target exactly SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.", "First sub-slice: sald.discrete_forward_kl.residual_exponent_bound, proving only the residual exponent drop and the replacement of interval Gamma integrals by the full barGamma contribution.", "Do not alter the source a(t), b(t), theorem endpoint, alpha ranges, step-size condition, Gamma, Delta, barGamma, or barDelta definitions.", "Keep endpoint matching and stitched-interval regularity as explicit dependencies; if those are blocked, refine those obligations rather than adding assumptions to thm:forward-KL-discrete.", "Leave the frozen one-step defect, DV velocity witness, LSI bridge, and full Gronwall lemma outside this lower slice." ] reviewerChecklist := [ "SALD.discreteForwardKlProofDag contains the cycle-19 middle accumulated-error block between the upper packet and the linear-slowdown/residual-exponent/accumulated-error blocks.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle19DiscreteForwardKlMiddleContract and sald.discrete_forward_kl.cycle19_accumulated_error_middle.", "The conversion window, proof-obligation ledger, and SLT audit classify this packet as local real/integral algebra plus source-contract gaps, not as a proof of thm:forward-KL-discrete.", "No analytic dependency is promoted beyond its current obligation/source-cited status." ] status := ProofStatus.obligation /-- Cycle-23 upper packet for the discrete forward-KL proof spine. This returns to `thm:forward-KL-discrete` after the continuous forward-KL coefficient work. It keeps the full source route visible for middle, but chooses one lower-facing target so the next proof attempt audits constants and side conditions rather than proving the theorem. -/ def cycle23DiscreteForwardKlUpperPacket : DiscreteForwardKlUpperPacket where objective := "Rebaseline thm:forward-KL-discrete across the Euler-Maruyama interpolation, one-step frozen score defect, and accumulated-error bridge, then choose the coefficient-chain audit as the single lower target for this cycle." sourceLabels := [ "thm:forward-KL-discrete", "eq:frozen_interp_terminal_disc_prop_additive_final", "lem:frozen_delta_cross_lip_sald", "proof:thm:forward-KL-discrete:em-interpolation", "proof:thm:forward-KL-discrete:one-step-defects", "proof:thm:forward-KL-discrete:accumulated-error", "lem:dv_variation", "lem:gronwall" ] modeDiscipline := [ "faithfulPaper: use only main_body.tex:273-323 and appendix.tex:260-592 from the original source root, with sald_version_2.tex excluded.", "Preserve the source theorem statement, linear slowdown t(s)=s/r, step-size condition 4*eta^2*L_{pi,space}^2<1/2, alpha ranges, Gamma/Delta definitions, and barGamma/barDelta accumulated constants.", "Keep the proof route EM interval laws -> conditional Fokker-Planck -> frozen defect and Young/LSI -> DV velocity -> time change -> Gronwall -> linear-slowdown collection.", "Treat EM Fokker-Planck, omitted SALD frozen-defect proof, LSI-to-KL/FI, DV, Gronwall, endpoint stitching, coefficient integrability, and interval-integral monotonicity as obligations or source-cited facts until local Lean proofs replace them.", "Middle must keep AutoSamplingTheory/SALD.lean, conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, research-wiki/source-index/SALD_original.jsonl, and the cited TeX windows synchronized in both directions." ] nonGoals := [ "Do not prove or restate thm:forward-KL-discrete in this packet.", "Do not reopen the continuous forward-KL theorem except as an inherited dependency.", "Do not replace the paper's one-step frozen-defect route with a path-space, Girsanov, Pinsker, Talagrand, or alternative entropy route.", "Do not change Gamma, Delta, barGamma, barDelta, alpha, alpha', eta, r, or the source Lipschitz and exponential-moment assumptions.", "Do not promote lem:dv_variation, lem:gronwall, eq:LSI-KL-FI, the EM Fokker-Planck backend, or the omitted frozen-defect proof beyond their current statuses." ] lowerPacket := [ "Target exactly one interface: SALD.discreteForwardKlCoefficientChainAuditContract / SALD.discreteForwardKlCoefficientChainObligation / sald.discrete_forward_kl.coefficient_chain_audit.", "First lower sub-slice: audit appendix.tex:454-553, including the two 1/4*FI cross-term bounds, LSI conversion, DV coefficient dot{t}(s)^2*alpha^(-1), and the time-change rewrite to dot{s}(t)^(-1)*alpha^(-1).", "Second sub-slice only if the first is stable: connect appendix.tex:557-590 to main_body.tex:309-323 through endpoint stitching, residual exponent drop, and full-interval A_alpha, barGamma, and barDelta collection.", "Keep EM endpoint laws, conditional Fokker-Planck, frozen Gamma/Delta lemma, DV witness, and Gronwall as named dependencies; if one is missing, refine the corresponding ProofObligation instead of adding a theorem hypothesis.", "Do not bundle the EM interpolation, one-step defect, Gronwall accumulation, and final theorem display into one opaque assumption." ] reviewerChecklist := [ "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle23_upper_packet before the coefficient-chain audit block.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle23DiscreteForwardKlUpperPacket while retaining the cycle-15 and cycle-19 packets.", "The conversion window and proof-obligation ledger cite main_body.tex:273-323 and appendix.tex:260-592, and the source index still excludes sald_version_2.tex.", "SALD.discreteForwardKlCoefficientChainAuditContract preserves the source coefficients T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'}.", "No analytic dependency is marked formalized and no fake proof closure is introduced." ] status := ProofStatus.obligation /-- Cycle-23 middle packet for the discrete forward-KL coefficient chain. This translates the upper-selected coefficient audit into a lower-ready source-to-Lean map. It keeps the first lower slice on appendix lines 454-553 and leaves the endpoint/accumulated-error bridge as a separate follow-on slice. -/ def cycle23DiscreteForwardKlMiddleContract : DiscreteForwardKlEmDefectAccumulationMiddleContract where sourceStatement := saldForwardKlDiscreteSource interpolationSource := saldForwardKlDiscreteInterpolationSource derivativeSource := saldForwardKlDiscreteDerivativeSource gronwallSource := saldForwardKlDiscreteGronwallSource accumulatedSource := saldForwardKlDiscreteAccumulatedErrorSource objective := "Translate the cycle-23 coefficient-chain target into a lower-ready audit for sald.discrete_forward_kl.coefficient_chain_audit, with appendix.tex:454-553 as the first slice and appendix.tex:557-590 to main_body.tex:309-323 kept as the follow-on accumulated-error bridge." sourceStepMap := [ "main_body.tex:273-323 fixes the score Lipschitz assumptions, Gamma, Delta, barGamma, barDelta, t(s)=s/r, alpha ranges, step-size condition, and final theorem constants.", "appendix.tex:454-467 applies lem:frozen_delta_cross_lip_sald and contributes the first (1/4)*FI term plus 2*eta^2*alpha'^(-1)*Gamma(t(s))*K_s and 2*eta*Delta(t(s)).", "appendix.tex:469-479 applies Cauchy-Schwarz/Young to the moving velocity cross term and contributes the second (1/4)*FI term plus ||tilde v_s||_{L2(hat rho_s)}^2.", "appendix.tex:481-493 combines the two cross-term bounds, leaves -(1/2)*FI, and invokes the LSI-to-KL/FI comparison to obtain -C_LSI(t(s))*K_s.", "appendix.tex:496-523 applies DV with nu=hat rho_s, mu=tilde pi_s, and Z=alpha*||v_{t(s)}||^2, preserving the dot t(s)^2*alpha^(-1) coefficient.", "appendix.tex:526-553 changes variables from s to t and rewrites dot{s}(t)*dot t(s(t))^2 as dot{s}(t)^(-1), while preserving the Gamma and Delta coefficients.", "appendix.tex:557-590 and main_body.tex:309-323 are not part of the first lower slice; they remain the endpoint, residual-exponent, barGamma, and barDelta accumulated-error bridge." ] leanStepMap := [ "The theorem statement remains SALD.discreteForwardKlStatementContract and SALD.discreteSaldContract; this packet adds no hypothesis and proves no terminal KL bound.", "The lower-facing ledger is SALD.discreteForwardKlCoefficientChainAuditContract and the named obligation sald.discrete_forward_kl.coefficient_chain_audit.", "The first lower slice depends on sald.discrete_forward_kl.frozen_delta_cross_lip, sald.discrete_forward_kl.kl_derivative, sald.discrete_forward_kl.dv_finite_log_mgf_witness, sald.discrete_forward_kl.dv_velocity_bound, and probability.lsi_to_kl_fi.", "The time-change coefficient rewrite uses SALD.discreteForwardKlTimeChangeSquareCoefficientRewriteScalar once inverse-schedule regularity supplies dot t(s(t))=dot{s}(t)^(-1) and dot{s}(t)>0; those analytic facts remain obligations.", "The follow-on bridge is SALD.discreteForwardKlAccumulatedErrorBridgeContract / sald.discrete_forward_kl.accumulated_error_bridge, with SALD.discreteForwardKlResidualExponentBoundScalar and SALD.discreteForwardKlResidualExpBoundScalar as only the scalar cores already formalized." ] citedResultInterfaces := [ "eq:LSI-KL-FI remains the source-cited LSI-to-KL/FI comparison through SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi.", "lem:dv_variation remains source-cited through Boucheron Corollary 4.15 or the SLT entropy_duality pattern; this packet imports no SLT theorem.", "lem:gronwall remains a local real-analysis obligation through sald.gronwall.integrating_factor and is not reopened by the first coefficient slice.", "The omitted SALD frozen-defect proof remains a faithful specialization obligation from the later general frozen-defect lemma, not a completed local theorem.", "EM endpoint laws, conditional Fokker-Planck, stitched-interval regularity, coefficient integrability, and interval-integral monotonicity remain explicit obligations." ] obligations := [ "sald.discrete_forward_kl.cycle23_coefficient_chain_middle", "sald.discrete_forward_kl.coefficient_chain_audit", "sald.discrete_forward_kl.frozen_delta_cross_lip", "sald.discrete_forward_kl.kl_derivative", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.discreteForwardKlTimeChangeSquareCoefficientRewriteScalar", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.stitched_interval_regularity" ] lowerPacket := [ "Target exactly SALD.discreteForwardKlCoefficientChainAuditContract / SALD.discreteForwardKlCoefficientChainObligation / sald.discrete_forward_kl.coefficient_chain_audit.", "First sub-slice: appendix.tex:454-553 only, auditing the two (1/4)*FI cross-term coefficients, LSI conversion, DV dot t(s)^2*alpha^(-1) coefficient, and the time-change rewrite to dot{s}(t)^(-1)*alpha^(-1).", "Do not attempt the full theorem, the EM Fokker-Planck backend, the omitted frozen-defect proof, the DV theorem, the Gronwall theorem, or the accumulated-error bridge in the same lower slice.", "If the first slice is stable, the next slice is appendix.tex:557-590 to main_body.tex:309-323 through endpoint stitching, residual exponent drop, and full-interval A_alpha, barGamma, and barDelta collection.", "Report any missing endpoint, integrability, or inverse-schedule fact as the corresponding named ProofObligation rather than adding hypotheses to thm:forward-KL-discrete." ] reviewerChecklist := [ "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle23_middle_coefficient_chain between the cycle-23 upper packet and the coefficient-chain audit block.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle23DiscreteForwardKlMiddleContract and sald.discrete_forward_kl.cycle23_coefficient_chain_middle.", "The conversion window, proof-obligation ledger, and SLT audit classify this packet as workflow/obligation data, not as a proof of thm:forward-KL-discrete.", "The source constants T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'} are unchanged.", "No analytic dependency is marked formalized and sald_version_2.tex remains excluded." ] status := ProofStatus.obligation /-- Cycle-27 upper packet for the discrete forward-KL accumulated-error bridge. This returns to `thm:forward-KL-discrete` after the coefficient-chain audit and selects the next faithful lower slice inside the final Gronwall-to-theorem bridge. The packet does not change the theorem statement or promote any analytic backend; it only narrows the next lower target. -/ def cycle27DiscreteForwardKlUpperPacket : DiscreteForwardKlUpperPacket where objective := "Keep thm:forward-KL-discrete fixed and select the accumulated-error collection slice after the coefficient-chain audit: endpoint rewrites, the linear-slowdown exponent split, and the A_alpha/barGamma/barDelta collection from appendix.tex:557-590 to main_body.tex:309-323." sourceLabels := [ "thm:forward-KL-discrete", "proof:thm:forward-KL-discrete:gronwall", "proof:thm:forward-KL-discrete:accumulated-error", "proof:thm:forward-KL-discrete:coefficient-chain", "eq:frozen_interp_terminal_disc_prop_additive_final", "lem:gronwall", "def:alpha-complexity" ] modeDiscipline := [ "faithfulPaper: use only main_body.tex:299-323 and appendix.tex:557-590 from the original source root, with sald_version_2.tex excluded.", "Preserve the theorem statement, linear slowdown t(s)=s/r, r>=1, alpha ranges, the step-size condition, Gamma, Delta, barGamma, barDelta, and the constants T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'}.", "Treat endpoint laws, stitched-interval regularity, coefficient integrability, residual-exponent monotonicity, full-interval Gamma/Delta identifications, and Gronwall as obligations or source-cited facts until local Lean proofs replace them.", "Use the already-separated coefficient-chain audit only as a dependency; do not reopen the frozen-defect, LSI, DV, or time-change coefficient subproofs in this upper packet.", "Require middle to keep AutoSamplingTheory/SALD.lean, conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, and research-wiki/source-index/SALD_original.jsonl synchronized against these exact source windows." ] nonGoals := [ "Do not prove or restate thm:forward-KL-discrete in this packet.", "Do not introduce a new theorem-level smoothness, sign, integrability, or endpoint assumption; record missing facts as obligations.", "Do not change the source a(t), b(t), Gamma/Delta, barGamma/barDelta, alpha, alpha', eta, r, or the theorem's exponential factors.", "Do not replace the source route Gronwall output -> linear slowdown -> residual exponent bound -> integral collection with another entropy or path-space argument.", "Do not mark lem:gronwall, interval-integral monotonicity, EM endpoint stitching, or the accumulated-error bridge as formalized." ] lowerPacket := [ "Target exactly SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.", "First lower sub-slice: endpointBridge plus alphaComplexityCollection and deltaAccumulation in SALD.discreteForwardKlAccumulatedErrorBridgeContract, using appendix.tex:557-590 and main_body.tex:309-323.", "Use SALD.discreteForwardKlResidualExponentBoundScalar and SALD.discreteForwardKlResidualExpBoundScalar only as existing scalar cores; the remaining interval-integral monotonicity and barGamma identification stay obligations if not proved.", "Keep sald.discrete_forward_kl.coefficient_chain_audit as a dependency and reviewer ledger for constants, not as the lower target for this cycle.", "If endpoint matching, stitched regularity, or full-interval A_alpha/barDelta collection is blocked, refine sald.discrete_forward_kl.accumulated_error_bridge or a named sibling obligation instead of adding assumptions." ] reviewerChecklist := [ "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle27DiscreteForwardKlUpperPacket and sald.discrete_forward_kl.cycle27_accumulated_collection_upper while retaining cycle-15, cycle-19, and cycle-23 packets.", "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle27_upper_accumulated_collection linked to the accumulated-error bridge and coefficient-chain audit.", "SALD.discreteForwardKlAccumulatedErrorBridgeContract still preserves endpointBridge, initialExponentSplit, residualExponentBound, alphaComplexityCollection, gammaAccumulation, and deltaAccumulation as separate fields.", "The conversion window and proof-obligation ledger cite appendix.tex:557-590 and main_body.tex:309-323 for this packet, and source-index refresh keeps sald_version_2.tex excluded.", "No analytic dependency is marked formalized and no fake proof closure is introduced." ] status := ProofStatus.obligation /-- Cycle-27 middle packet for the discrete forward-KL accumulated collection. This translates the upper-selected accumulated-error slice into a lower-ready source-to-Lean map. It keeps the first lower target on endpoint matching plus the `A_alpha` and `barDelta` additive collections, while leaving the residual exponent and `barGamma` interval monotonicity as named dependencies. -/ def cycle27DiscreteForwardKlMiddleContract : DiscreteForwardKlEmDefectAccumulationMiddleContract where sourceStatement := saldForwardKlDiscreteSource interpolationSource := saldForwardKlDiscreteInterpolationSource derivativeSource := saldForwardKlDiscreteDerivativeSource gronwallSource := saldForwardKlDiscreteGronwallSource accumulatedSource := saldForwardKlDiscreteAccumulatedErrorSource objective := "Translate the cycle-27 accumulated-collection target into a lower-ready map for sald.discrete_forward_kl.accumulated_error_bridge, with endpointBridge plus alphaComplexityCollection and deltaAccumulation as the first sub-slice and all source constants preserved." sourceStepMap := [ "appendix.tex:557-590: start from the general-schedule Gronwall output for K(T), with a(t)=dot{s}(t)*C_LSI(t)-dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t).", "appendix.tex:560-571: the endpoint term must rewrite K(T) to KL(rho_K^eta||pi_T) and K(0) to KL(rho_0||pi_0) using EM endpoint laws and linear-slowdown endpoints.", "appendix.tex:586-589: the residual integrand is dot{s}(t)^(-1)*E_alpha(pi_t,v_t)+2*dot{s}(t)*eta*Delta(t).", "main_body.tex:299-323: under t(s)=s/r, dot{s}(t)=r and dot{s}(t)^(-1)=1/r, so the additive residual collection is (1/r)*A_alpha(pi,v)+2*r*eta*barDelta_{alpha'}.", "main_body.tex:310-323: the positive exponent still uses T/(r*alpha)+2*r*eta^2*barGamma/alpha'; residual-exponent monotonicity and barGamma identification remain separate obligations." ] leanStepMap := [ "The theorem statement remains SALD.discreteForwardKlStatementContract and SALD.discreteSaldContract; this packet adds no hypothesis and proves no terminal KL bound.", "The lower-facing target is SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.", "The first lower sub-slice is the contract fields endpointBridge, alphaComplexityCollection, and deltaAccumulation.", "Endpoint rewrites depend on sald.discrete_forward_kl.em_endpoint_laws, sald.discrete_forward_kl.stitched_interval_regularity, and sald.discrete_forward_kl.linear_slowdown_specialization.", "The A_alpha and barDelta collections use def:alpha-complexity and the main-body definitions of barGamma and barDelta; no new constants are introduced.", "SALD.discreteForwardKlResidualExponentBoundScalar and SALD.discreteForwardKlResidualExpBoundScalar remain only scalar cores for the residual exponent dependency, not a completed accumulated-error bridge." ] citedResultInterfaces := [ "lem:gronwall remains a local real-analysis obligation through sald.discrete_forward_kl.gronwall_accumulation and sald.gronwall.integrating_factor.", "lem:dv_variation remains inherited through the existing discrete DV velocity witness and is not reopened by this packet.", "eq:LSI-KL-FI remains inherited only through the nonnegative LSI contribution used by the residual exponent bound.", "No SLT theorem applies to endpoint rewrites, alpha-complexity collection, Delta accumulation, barGamma identification, or interval-integral monotonicity." ] obligations := [ "sald.discrete_forward_kl.cycle27_accumulated_collection_middle", "sald.discrete_forward_kl.cycle27_accumulated_collection_lower", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.coefficient_chain_audit", "def:alpha-complexity", "SALD.discreteForwardKlAlphaComplexityCollectionScalar", "SALD.discreteForwardKlDeltaAccumulationScalar", "SALD.discreteForwardKlAccumulatedErrorCollectionScalar", "sald.gronwall.integrating_factor" ] lowerPacket := [ "Target exactly SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.", "First sub-slice: endpointBridge plus alphaComplexityCollection and deltaAccumulation, using appendix.tex:557-590 and main_body.tex:309-323.", "Instantiate dot{s}=r and dot{s}^(-1)=1/r only through the existing linear-slowdown obligation; do not add those identities as theorem hypotheses.", "Keep sald.discrete_forward_kl.residual_exponent_bound and the barGamma full-interval identification as dependencies unless the lower slice explicitly proves them.", "Do not reopen the frozen-defect, LSI, DV, time-change coefficient, or full Gronwall subproofs." ] reviewerChecklist := [ "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle27_middle_accumulated_collection between the cycle-27 upper packet and the accumulated-error bridge.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle27DiscreteForwardKlMiddleContract and sald.discrete_forward_kl.cycle27_accumulated_collection_middle.", "The conversion window, proof-obligation ledger, and SLT audit classify this packet as workflow/source-to-Lean data, not as a proof of thm:forward-KL-discrete.", "The constants T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'} are unchanged.", "No analytic dependency is promoted beyond obligation/source-cited status." ] status := ProofStatus.obligation def discreteSaldEulerMaruyamaContract : EulerMaruyamaContract where id := "ASTIS.SALD.forward_KL_discrete.euler_maruyama" updateFormula := "X_{k+1}^eta = X_k^eta + eta*nabla log pi_{t_k}(X_k^eta) + sqrt(2*eta)*xi_k, with t_k=t(k*eta)." interpolationFormula := "For s in [s_k,s_{k+1}], hat X_s = X_k^eta + (s-s_k)*nabla log pi_{t_k}(X_k^eta) + sqrt(2)*(W_s-W_{s_k})." source := saldForwardKlDiscreteInterpolationSource status := ProofStatus.contractOnly def discreteForwardKlStatementContract : DiscreteForwardKlStatementContract where theoremLabel := "thm:forward-KL-discrete" sourceStatement := saldForwardKlDiscreteSource sourceProof := saldForwardKlDiscreteProofSource emUpdate := "Discrete SALD eq:sald_em_terminal_disc_prop_additive_final with rho_k^eta=Law(X_k^eta), K=S/eta, and t_k=t(k*eta)." interpolation := "The proof uses the continuous EM interpolation hat X_s from eq:frozen_interp_terminal_disc_prop_additive_final and hat rho_s=Law(hat X_s)." linearSlowdown := "The statement is for t(s)=s/r with r>=1, hence S=r*T and dot{s}(t)=r." lipschitzAssumptions := "Score space Lipschitz eq:lip_SALD_1 and time Lipschitz eq:lip_SALD_2 with constants L_{pi,space}, L_{pi,time} and measurable M." complexityAssumptions := "There exists alpha0'>0 with E_{alpha0'}(pi_t,nabla log pi_t)0" ] sourceGaps := [ "appendix.tex uses the interpolation Fokker--Planck equation and conditional drift without a standalone theorem", "the differentiability and boundary conditions are inherited from the continuous proof but not restated for the EM interpolation", "the Gronwall argument is applied across the stitched interpolation intervals without separately spelling out endpoint continuity of K" ] dependencies := [ "eq:frozen_interp_terminal_disc_prop_additive_final", "lem:frozen_delta_cross_lip_sald", "eq:LSI-KL-FI", "lem:dv_variation", "SALD.discreteForwardKlPostLsiDerivativeBoundScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfKlFiScalar", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.em_conditional_fokker_planck" ] status := ProofStatus.obligation def discreteForwardKlDvFiniteLogMgfWitnessContract : DiscreteForwardKlDvFiniteLogMgfWitnessContract where sourceBlock := saldForwardKlDiscreteDvVelocitySource theoremStatement := saldForwardKlDiscreteSource dvMeasures := "appendix.tex:496-507 applies lem:dv_variation with nu=hat rho_s and mu=tilde pi_s=pi_{t(s)} on the common Euclidean state space." testFunction := "Z_s(x)=alpha*||v_{t(s)}(x)||^2, the same squared-velocity test as the continuous theorem but evaluated along the EM interpolation time." finiteAlpha0Assumption := "The theorem assumes the same hypotheses as thm:forward-KL, including finite E_{alpha0}(pi_t,v_t) for every t in [0,T], and main_body.tex:306 takes alpha in (0,alpha0)." alphaMonotonicityBridge := "Reuse SALD.forwardKlDvAlphaMonotonicityContract to get finite log E_{pi_{t(s)}}[exp(alpha*||v_{t(s)}||^2)] for the smaller alpha before invoking DV." interpolationLawInterface := "The EM interpolation supplies the law hat rho_s and the slowed target tilde pi_s on each interval [s_k,s_{k+1}], with endpoint laws tracked separately by SALD.discreteForwardKlEmInterpolationSideConditionContract." commonSpaceAndAbsoluteContinuity := "The DV step requires hat rho_s and tilde pi_s on the same measurable space with the absolute-continuity/density interface needed for KL(hat rho_s||tilde pi_s)." measurabilityInterface := "v_{t(s)} and ||v_{t(s)}||^2 must be measurable under both hat rho_s and tilde pi_s; this is not supplied by the source as a standalone lemma." scalingStep := "After DV bounds E_{hat rho_s}[alpha*||v_{t(s)}||^2], divide by alpha>0 and rewrite the log-mgf term as E_alpha(pi_{t(s)},v_{t(s)})." dotTScalingStep := "Multiply the resulting bound by dot t(s)^2 because ||tilde v_s||^2=dot t(s)^2*||v_{t(s)}||^2 in appendix.tex:499-515." outputBound := "||tilde v_s||_{L2(hat rho_s)}^2 <= dot t(s)^2*(alpha^(-1)*KL(hat rho_s||tilde pi_s)+E_alpha(pi_{t(s)},v_{t(s)}))." coefficientUse := "This supplies the dot t(s)^2*alpha^(-1)*K_s coefficient in appendix.tex:520-531 and, after time change, dot{s}(t)^(-1)*alpha^(-1) in appendix.tex:541-553." dependencies := [ "lem:dv_variation", "def:alpha-complexity", "SALD.saldDvFiniteLogMgfContract", "sald.dv_variation.finite_log_mgf_interface", "sald.forward_kl.dv_alpha_mgf_monotonicity", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.em_interpolation_fp" ] sourceGaps := [ "the appendix directly applies DV under hat rho_s and tilde pi_s without isolating common-space or absolute-continuity hypotheses for the EM interpolation law", "the source reuses the finite alpha0-complexity assumption from thm:forward-KL but does not restate the alpha0-to-alpha finite-log-mgf bridge for the discrete proof", "measurability of v_{t(s)} and squared-norm integrability under the interpolated law are implicit in the paper's smoothness vocabulary" ] lowerPacket := [ "Formalize or refine the common-space and absolute-continuity side condition for hat rho_s and tilde pi_s before attempting the full DV velocity bound.", "Reuse the continuous alpha0-to-alpha monotonicity obligation for the pi_t log-mgf; do not add a new theorem assumption.", "Keep the dot t(s)^2 factor outside the DV bound so the later time-change coefficient remains dot{s}(t)^(-1)." ] status := ProofStatus.obligation def discreteForwardKlGronwallInstantiationContract : DiscreteForwardKlGronwallInstantiationContract where sourceBlock := saldForwardKlDiscreteGronwallSource statementBlock := saldForwardKlDiscreteSource quantityK := "K(t)=KL(hat rho_{s(t)}||pi_t), with endpoint K(T)=KL(rho_K^eta||pi_T)." gronwallA := "a(t)=dot{s}(t)*C_LSI(t) - dot{s}(t)^(-1)*alpha^(-1) - 2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t)." gronwallB := "b(t)=dot{s}(t)^(-1)*E_alpha(pi_t,v_t) + 2*dot{s}(t)*eta*Delta(t)." preSpecializationBound := "K(T) <= exp(-int_0^T a(t)dt)*K(0) + int_0^T exp(-int_t^T a(u)du)*b(t)dt." linearSlowdownSpecialization := "For t(s)=s/r, dot{s}(t)=r and dot{s}(t)^(-1)=1/r; substituting in the appendix Gronwall bound gives the main-body exponent terms -r*int_0^T C_LSI(t)dt, T/(r*alpha), and 2*r*eta^2*alpha'^(-1)*barGamma." accumulatedError := "The residual integral is bounded by the common positive exponential factor times ((1/r)*A_alpha(pi,v)+2*r*eta*barDelta_{alpha'}), after dropping nonpositive LSI contributions." requiredRegularity := [ "integrability of C_LSI, Gamma, Delta, and E_alpha(pi_t,v_t) on [0,T]", "piecewise differentiability or endpoint-safe absolute continuity of K(t) after stitching EM intervals", "nonnegativity of C_LSI to discard the LSI part from the residual exponential" ] sourceGaps := [ "the source does not state a separate stitched-interval Gronwall lemma for the EM interpolation", "continuity/integrability of Gamma and Delta is asserted informally as controlled by smoothness and complexities", "appendix.tex stops at the general-schedule Gronwall bound; the final linear-slowdown algebra is required to match main_body.tex lines 309-323" ] dependencies := [ "lem:gronwall", "sald.discrete_forward_kl.kl_derivative", "sald.gronwall.integrating_factor", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.linear_slowdown_specialization" ] status := ProofStatus.obligation def discreteForwardKlAccumulatedErrorBridgeContract : DiscreteForwardKlAccumulatedErrorBridgeContract where sourceProof := saldForwardKlDiscreteAccumulatedErrorSource theoremStatement := saldForwardKlDiscreteSource endpointBridge := "The Gronwall K(T) and K(0) rewrite to KL(rho_K^eta||pi_T) and KL(rho_0||pi_0) using EM endpoint laws, s(0)=0, S=s(T), and the linear slowdown t(s)=s/r." gronwallOutput := "Start from appendix.tex:557-590: K(T)<=exp(-int_0^T a)*K(0)+int_0^T exp(-int_t^T a)*((dot{s})^(-1)*E_alpha+2*dot{s}*eta*Delta)dt, with a(t)=dot{s}*C_LSI-dot{s}^(-1)*alpha^(-1)-2*dot{s}*eta^2*alpha'^(-1)*Gamma." linearSlowdownSubstitution := "For t(s)=s/r, dot{s}(t)=r and dot{s}(t)^(-1)=1/r, so a(t)=r*C_LSI(t)-(r*alpha)^(-1)-2*r*eta^2*alpha'^(-1)*Gamma(t)." initialExponentSplit := "Split exp(-int_0^T a) as exp(-r*int_0^T C_LSI(t)dt)*exp(T/(r*alpha)+2*r*eta^2*alpha'^(-1)*int_0^T Gamma(t)dt), matching main_body.tex lines 310-315 after barGamma identification." residualExponentBound := "For each t, exp(-int_t^T a) is bounded by exp(T/(r*alpha)+2*r*eta^2*barGamma/alpha') after discarding the nonpositive LSI contribution and bounding interval integrals by the full [0,T] integrals." alphaComplexityCollection := "The E_alpha term collects to (1/r)*A_alpha(pi,v), using A_alpha(pi,v)=int_0^T E_alpha(pi_t,v_t)dt." gammaAccumulation := "barGamma denotes int_0^T Gamma(t)dt, so the Gamma contribution in the positive exponent is 2*r*eta^2*barGamma/alpha'." deltaAccumulation := "barDelta_{alpha'} denotes int_0^T Delta(t)dt, so the residual Delta contribution collects to 2*r*eta*barDelta_{alpha'}." dependencies := [ "lem:gronwall", "def:alpha-complexity", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "SALD.discreteForwardKlGronwallCoeffIntervalIntegrable", "SALD.discreteForwardKlGronwallCoeffIntegralSubSub", "SALD.discreteForwardKlGronwallInitialExponentSplitScalar", "SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces", "SALD.discreteForwardKlResidualExponentBoundScalar", "SALD.discreteForwardKlResidualExpBoundScalar", "SALD.discreteForwardKlAlphaComplexityCollectionScalar", "SALD.discreteForwardKlDeltaAccumulationScalar", "SALD.discreteForwardKlAccumulatedErrorCollectionScalar", "SALD.discreteForwardKlResidualIntegralDisplayBoundScalar", "SALD.discreteForwardKlMainDisplayBoundScalar" ] sourceGaps := [ "the appendix proof stops at the general-schedule Gronwall display and does not spell out the main-body linear-slowdown collection", "the source states that barGamma and barDelta are controlled by smoothness and complexities but does not define the Lean integral objects or regularity assumptions", "the residual exponent bound uses sign and interval-integral monotonicity facts implicitly", "SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces now formalizes the local three-piece exponent split once the source LSI, alpha, and Gamma coefficient integrability hypotheses are supplied", "the compiled cycle-27 collection scalars only factor constants through interval integrals after A_alpha and barDelta have already been identified with the source full-interval integrals", "SALD.discreteForwardKlResidualIntegralDisplayBoundScalar now composes the supplied residual-kernel bound with that collection algebra; endpoint stitching and residual exponent monotonicity remain obligations", "SALD.discreteForwardKlMainDisplayBoundScalar now combines the supplied Gronwall initial term and supplied residual display into the exact two-term main-body bound, but does not prove the analytic source identifications" ] lowerPacket := [ "Prove the endpoint rewrites and exponent split before attempting the theorem-level discrete bound.", "Cycle 46 lower compiled scalar core: SALD.discreteForwardKlGronwallCoeffIntervalIntegrable, SALD.discreteForwardKlGronwallCoeffIntegralSubSub, SALD.discreteForwardKlGronwallInitialExponentSplitScalar, and SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces cover only the initial Gronwall exponent split under explicit interval-integrability inputs.", "Refine sald.discrete_forward_kl.residual_exponent_bound as the lower scalar bridge for the residual Gronwall exponent before collecting the residual integral.", "Use the source a(t) from appendix.tex:557-590 exactly; do not normalize away the Gamma term.", "Separate the alpha-complexity, Gamma, and Delta integral collections so each can be checked against main_body.tex:309-323.", "Cycle 27 lower target: after the coefficient-chain audit, focus on endpointBridge plus alphaComplexityCollection and deltaAccumulation, keeping residual-exponent monotonicity and barGamma identification as named obligations if they remain blocked.", "Cycle 27 lower compiled scalar core: SALD.discreteForwardKlAlphaComplexityCollectionScalar, SALD.discreteForwardKlDeltaAccumulationScalar, and SALD.discreteForwardKlAccumulatedErrorCollectionScalar cover only constant-factor extraction for the additive residual integrals.", "Cycle 61 lower compiled scalar wrapper: SALD.discreteForwardKlResidualIntegralDisplayBoundScalar connects a supplied common-exponential residual bound to the exact main-body additive display.", "Cycle 66 lower compiled scalar wrapper: SALD.discreteForwardKlMainDisplayBoundScalar combines the supplied initial exponent split and supplied residual display into the exact main-body two-term bound." ] status := ProofStatus.obligation def cycle11DiscreteForwardKlMiddleContract : DiscreteForwardKlEmDefectAccumulationMiddleContract where sourceStatement := saldForwardKlDiscreteSource interpolationSource := saldForwardKlDiscreteInterpolationSource derivativeSource := saldForwardKlDiscreteDerivativeSource gronwallSource := saldForwardKlDiscreteGronwallSource accumulatedSource := saldForwardKlDiscreteAccumulatedErrorSource objective := "Map the cycle-focus proof route for thm:forward-KL-discrete from EM interpolation through one-step defects and accumulated errors, while keeping the theorem statement and constants unchanged." sourceStepMap := [ "appendix.tex:260-266 and 334-335: define hat X_s and use endpoint laws for hat rho_{s_k}=rho_k^eta and hat rho_{s_{k+1}}=rho_{k+1}^eta.", "appendix.tex:347-385: define bar b_{k,s} and invoke the conditional-drift Fokker--Planck equation for the frozen EM interpolation.", "appendix.tex:454-467: apply lem:frozen_delta_cross_lip_sald to introduce the one-step Gamma and Delta defects.", "appendix.tex:469-523: apply Young to the moving velocity term and then the DV finite-log-mgf witness with the dot t(s)^2 coefficient.", "appendix.tex:526-553: change from s to t and preserve dot{s}(t)^(-1)*alpha^(-1), 2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma, and 2*dot{s}(t)*eta*Delta.", "appendix.tex:557-590 with main_body.tex:309-323: apply Gronwall, specialize t(s)=s/r, and collect A_alpha, barGamma, and barDelta_{alpha'}." ] leanStepMap := [ "EM endpoint laws and interpolation Fokker--Planck are tracked by SALD.discreteForwardKlEmInterpolationSideConditionContract and sald.discrete_forward_kl.em_interpolation_fp.", "The frozen one-step defect is tracked by SALD.frozenDeltaCrossLipSaldContract and sald.discrete_forward_kl.frozen_delta_cross_lip.", "The derivative inequality before Gronwall is tracked by SALD.discreteForwardKlDerivativeCandidateContract and sald.discrete_forward_kl.kl_derivative.", "The DV velocity estimate is split between SALD.discreteForwardKlDvFiniteLogMgfWitnessContract and sald.discrete_forward_kl.dv_velocity_bound.", "The global scalar step is split between SALD.discreteForwardKlGronwallInstantiationContract, SALD.discreteForwardKlAccumulatedErrorBridgeContract, sald.discrete_forward_kl.residual_exponent_bound, and SALD.discreteForwardKlCoefficientChainAuditContract." ] citedResultInterfaces := [ "lem:dv_variation remains source-cited through Boucheron Corollary 4.15 or the SLT entropy_duality pattern; no SLT theorem is imported.", "lem:gronwall remains a local real-analysis obligation through sald.gronwall.integrating_factor.", "eq:LSI-KL-FI remains an obligation through probability.lsi_to_kl_fi.", "SLT one_step_discretization is only a reference pattern for the EM interpolation backend." ] obligations := [ "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.frozen_delta_cross_lip", "sald.discrete_forward_kl.kl_derivative", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.coefficient_chain_audit" ] lowerPacket := [ "Preferred lower target for EM work: refine SALD.discreteForwardKlEmInterpolationSideConditionContract or sald.discrete_forward_kl.em_conditional_fokker_planck, keeping endpoint laws and conditional drift separate.", "Preferred lower target for accumulated-error work: refine SALD.discreteForwardKlAccumulatedErrorBridgeContract or sald.discrete_forward_kl.accumulated_error_bridge, keeping the appendix Gronwall display distinct from the main-body linear-slowdown theorem bound.", "Cycle 11 lower scalar target: refine sald.discrete_forward_kl.residual_exponent_bound for the residual exponent drop and full-interval Gamma bound.", "Do not restate thm:forward-KL-discrete, add hidden smoothness assumptions, or alter the Gamma, Delta, barGamma, barDelta, alpha, alpha', eta, or r coefficients.", "If touching the DV substep, depend on sald.discrete_forward_kl.dv_finite_log_mgf_witness rather than adding a new finite-mgf theorem assumption." ] reviewerChecklist := [ "Every source step from appendix.tex:260-590 maps to an existing Lean contract, cited result, or proof obligation.", "The EM interpolation backend, frozen-defect lemma, DV witness, Gronwall accumulation, and accumulated-error bridge remain separate lower targets.", "The final theorem display in main_body.tex:309-323 is unchanged.", "sald_version_2.tex remains excluded from source correspondence.", "No analytic dependency is marked formalized and no fake proof closure is introduced." ] status := ProofStatus.obligation def discreteForwardKlCoefficientChainAuditContract : DiscreteForwardKlCoefficientChainAuditContract where sourceProof := saldForwardKlDiscreteCoefficientChainSource theoremStatement := saldForwardKlDiscreteSource frozenCrossCoefficient := "appendix.tex:454-467 uses lem:frozen_delta_cross_lip_sald to contribute (1/4)*FI + 2*eta^2*alpha'^(-1)*Gamma(t(s))*K_s + 2*eta*Delta(t(s))." movingCrossCoefficient := "appendix.tex:469-479 applies Young to the moving-target velocity cross term, contributing another (1/4)*FI plus ||tilde v_s||_{L2(hat rho_s)}^2." lsiStep := "After the two quarter-FI cross-term bounds, appendix.tex:481-493 leaves -(1/2)*FI and uses eq:LSI-KL-FI to obtain -C_LSI(t(s))*K_s." dvVelocityCoefficient := "appendix.tex:496-515 applies lem:dv_variation with Z=alpha*||v_{t(s)}||^2 under the EM-interpolation witness, yielding dot t(s)^2*(alpha^(-1)*K_s + E_alpha(pi_{t(s)},v_{t(s)}))." timeChangeStep := "appendix.tex:534-553 multiplies by dot{s}(t), rewrites dot{s}(t)*dot t(s(t))^2 as dot{s}(t)^(-1), and produces the source t-inequality with +2*dot{s}(t)*eta*Delta(t)." gronwallA := "a(t)=dot{s}(t)*C_LSI(t)-dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t)." gronwallB := "b(t)=dot{s}(t)^(-1)*E_alpha(pi_t,v_t)+2*dot{s}(t)*eta*Delta(t)." linearSlowdownExponent := "For t(s)=s/r, main_body.tex:309-321 requires exp(-r*int_0^T C_LSI(t)dt)*exp(T/(r*alpha)+2*r*eta^2*barGamma/alpha') for the initial term and the positive exponent factor for the residual term." accumulatedErrorCollection := "The residual integral must collect to (1/r)*A_alpha(pi,v)+2*r*eta*barDelta_{alpha'} using barGamma=int_0^T Gamma(t)dt and barDelta_{alpha'}=int_0^T Delta(t)dt." endpointBridge := "The Gronwall output identifies K(T)=KL(rho_K^eta||pi_T) and K(0)=KL(rho_0||pi_0) through the EM endpoint laws and linear slowdown endpoints." sourceLineLedger := [ "appendix.tex:454-467: the frozen defect contributes the first 1/4*FI term plus 2*eta^2*alpha'^(-1)*Gamma*K and 2*eta*Delta.", "appendix.tex:469-479: the moving velocity Young bound contributes the second 1/4*FI term plus ||tilde v_s||^2.", "appendix.tex:481-493: the remaining -(1/2)*FI is converted by eq:LSI-KL-FI into -C_LSI(t(s))*K_s.", "appendix.tex:496-515: DV with Z=alpha*||v_{t(s)}||^2 under the EM-interpolation witness contributes dot t(s)^2*alpha^(-1)*K_s and dot t(s)^2*E_alpha(pi_t,v_t).", "appendix.tex:534-553: the time change multiplies by dot{s}(t), rewrites dot{s}*dot t^2 to dot{s}^(-1) using SALD.discreteForwardKlTimeChangeSquareCoefficientRewriteScalar once inverse-schedule side conditions are supplied, and preserves the Gamma and Delta coefficients.", "appendix.tex:557-590 and main_body.tex:309-323: Gronwall plus t(s)=s/r gives T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'}." ] scalarSideConditions := [ "alpha>0 and alpha'>0 are required for the inverse coefficients alpha^(-1) and alpha'^(-1) in the source a(t) and b(t).", "hat rho_s and tilde pi_s must satisfy the common-space, absolute-continuity, and finite-log-mgf interfaces required by the DV witness at appendix.tex:493-523.", "dot{s}(t)>0 and dot t(s(t))=dot{s}(t)^(-1) are required inputs for SALD.discreteForwardKlTimeChangeSquareCoefficientRewriteScalar; under linear slowdown the nonzero/positive derivative follows from r>=1.", "C_LSI(t)>=0 is used only to discard the nonpositive LSI contribution in residual exponents, not as a new theorem assumption.", "Gamma, Delta, E_alpha(pi_t,v_t), and C_LSI must be interval-integrable on [0,T] for the Gronwall and accumulation expressions.", "The residual exponent bound uses interval-integral monotonicity to replace int_t^T Gamma and int_t^T dot{s}^(-1)*alpha^(-1) by the full positive exponent T/(r*alpha)+2*r*eta^2*barGamma/alpha'.", "Endpoint continuity of the stitched EM KL path is needed before K(T) and K(0) can be rewritten as the theorem endpoints." ] dependencies := [ "eq:LSI-KL-FI", "def:alpha-complexity", "lem:dv_variation", "lem:gronwall", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.frozen_delta_cross_lip", "sald.discrete_forward_kl.kl_derivative", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.discreteForwardKlTimeChangeSquareCoefficientRewriteScalar", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge" ] sourceGaps := [ "the SALD-specific frozen-defect proof is omitted and must be supplied by a faithful specialization of the later general lemma", "the inverse-schedule identity dot t(s(t))=dot{s}(t)^(-1) and nonzero dot{s}(t) are still analytic side conditions; only the scalar coefficient algebra after those inputs is formalized", "the source applies Gronwall after interval-wise EM inequalities without a standalone stitched-interval lemma", "the appendix gives the general-schedule Gronwall display, while the main-body theorem supplies the final linear-slowdown collected constants" ] lowerPacket := [ "Target the scalar coefficient algebra from appendix.tex:454-553 before attempting the full discrete theorem.", "Keep the two 1/4 FI Young contributions, the full DV coefficient dot{s}^(-1)*alpha^(-1), and the one-step 2*dot{s}*eta^2*alpha'^(-1)*Gamma term unchanged; SALD.discreteForwardKlTimeChangeSquareCoefficientRewriteScalar covers only the scalar dot{s}*dot t^2 rewrite.", "Use sald.discrete_forward_kl.accumulated_error_bridge for the endpoint, exponent, barGamma, and barDelta collection tied to main_body.tex:309-323." ] status := ProofStatus.obligation def guidedResidualIdentityContract : GuidedResidualIdentityContract where sourceStatement := saldGuidedResidualSource sourceProof := saldGuidedResidualProofSource basePath := "p_t has transport velocity u_t: partial_t p_t + div(p_t*u_t)=0." guideTilt := "Z_t=int p_t(x)*exp(-f_t(x)) dx and pi_t(x)=Z_t^(-1)*p_t(x)*exp(-f_t(x))." guideTransportDerivative := "g_t(x)=partial_t f_t(x)+nabla f_t(x)^T*u_t(x)." normalizerDerivative := "dot Z_t/Z_t = -E_{pi_t}[g_t], obtained by differentiating Z_t and integrating div(p_t*u_t)*exp(-f_t) by parts." residualIdentity := "partial_t pi_t + div(pi_t*u_t) = -pi_t*(g_t-E_{pi_t}[g_t])." meanZeroStatement := "int (g_t-E_{pi_t}[g_t]) * pi_t dx = 0." requiredRegularity := [ "differentiability of t -> p_t, f_t, and Z_t", "positive finite normalizer Z_t", "transport equation for p_t with velocity u_t", "integration-by-parts or boundary decay for div(p_t*u_t)*exp(-f_t)", "integrability of g_t under pi_t" ] sourceGaps := [ "the source proof uses differentiation under the integral for Z_t and pi_t without an explicit dominated-convergence hypothesis", "boundary conditions for the integration-by-parts step are implicit", "positivity and finiteness of Z_t are used as part of the guided-density vocabulary" ] dependencies := [ "TransportVelocityContract", "GuidedTiltContract" ] status := ProofStatus.obligation def generalMovingTargetStatementContract : GeneralMovingTargetStatementContract where theoremLabel := "thm:general-moving-target-SALD" sourceStatement := saldGeneralMovingTargetSource sourceProof := saldGeneralMovingTargetSource dynamics := "dX_s=(dot{t}(s)*c_{t(s)}(X_s)+(sigma_{t(s)}^2/2)*nabla log pi_{t(s)}(X_s)) ds + sigma_{t(s)} dW_s." residualField := "v_t is a transport velocity for pi_t and m_t(x)=v_t(x)-c_t(x)." lsiAssumption := "pi_t satisfies LSI with constant C_LSI(t)>=0 for every t in [0,T]." alphaComplexityAssumption := "There exists alpha0>0 such that E_{alpha0}(pi_t,m_t)<+infty for every t in [0,T]." alphaRange := "For any alpha in (0,alpha0]." terminalBound := "KL(rho_S||pi_T) is bounded by the product of exp(-int_0^T (sigma_t^2/2)*dot{s}(t)*C_LSI(t)dt), exp(int_0^T sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1)dt), and KL(rho_0||pi_0), plus the residual integral with sigma_t^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t)." pureContractionCase := "If c_t=v_t for all t,x, then m_t=0, E_alpha(pi_t,m_t)=0, and the residual integral vanishes." proofQuantity := "K(t)=KL(rho_{s(t)}||pi_t)." differentialInequality := "dK/dt <= -((sigma_t^2/2)*dot{s}(t)*C_LSI(t)-sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1))*K(t)+sigma_t^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t)." proofSteps := [ "appendix.tex:765-835 differentiates KL(rho_s||pi_{t(s)}) and inserts the general VA-SALD Fokker--Planck equation.", "appendix.tex:835-864 combines the c_t drift with the target transport velocity v_t, exposing the residual m_t=v_t-c_t and applying Young's inequality with epsilon=2*dot{t}(s)/sigma_{t(s)}^2.", "appendix.tex:866-884 changes from s to t and applies LSI to convert FI into KL contraction.", "appendix.tex:886-907 applies Donsker--Varadhan with Z=alpha*||m_t||^2.", "appendix.tex:908-934 applies lem:gronwall and obtains the displayed theorem bound.", "appendix.tex:936-945 specializes c_t=v_t to obtain the pure contraction estimate." ] dependencies := [ "eq:general_moving_target_SALD", "eq:general_moving_target_FP", "eq:LSI-KL-FI", "def:alpha-complexity", "lem:dv_variation", "lem:gronwall", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction" ] status := ProofStatus.contractOnly def generalMovingTargetDerivativeCandidateContract : GeneralMovingTargetDerivativeCandidateContract where sourceBlock := saldGeneralMovingTargetDerivativeSource densityAndLawInterface := "rho_s is the law of eq:general_moving_target_SALD, pi_{t(s)} is the slowed target, rho_s << pi_{t(s)}, and KL/FI are finite along the interval." dynamicsFokkerPlanck := "partial_s rho_s = -div(dot{t}(s)*c_{t(s)}*rho_s)+(sigma_{t(s)}^2/2)*div(rho_s*nabla log(rho_s/pi_{t(s)}))." targetVelocity := "v_t generates pi_t, hence tilde v_s=dot{t}(s)*v_{t(s)} generates pi_{t(s)}." residualVelocity := "m_t=v_t-c_t; the c_t drift and target-velocity term combine as -dot{t}(s)*int rho_s ." klDerivativeIdentity := "d/ds KL(rho_s||pi_{t(s)}) = int partial_s rho_s log(rho_s/pi_{t(s)}) dx - int (rho_s/pi_{t(s)}) partial_s pi_{t(s)} dx." firstTermEvaluation := "The general VA-SALD Fokker--Planck equation and integration by parts give dot{t}(s)*int rho_s -(sigma_{t(s)}^2/2)*FI(rho_s||pi_{t(s)}). SALD.generalMovingTargetKlDerivativeResidualSplitScalar consumes this as a supplied scalar identity." secondTermEvaluation := "The target transport equation gives -int rho_s ; together with the supplied mass-conservation drop and m_t=v_t-c_t identification, SALD.generalMovingTargetKlDerivativeResidualSplitScalar derives the residual derivative display. SALD.generalMovingTargetKlDerivativeScaledResidualDisplayScalar additionally records the dot{t}(s) scaling and the source sign -dot{t}(s)*int rho_s ." youngStep := "With epsilon=2*dot{t}(s)/sigma_{t(s)}^2 as stated in the source route, the residual cross term is bounded by (sigma_{t(s)}^2/4)*FI plus sigma_{t(s)}^(-2)*dot{t}(s)^2*||m_{t(s)}||_{L2(rho_s)}^2; SALD.generalMovingTargetPostYoungDerivativeBoundScalar compiles only the subsequent real-order subtraction." lsiStep := "LSI gives FI(rho_s||pi_{t(s)}) >= 2*C_LSI(t(s))*KL(rho_s||pi_{t(s)}); SALD.generalMovingTargetLsiDerivativeBoundScalar consumes the supplied half-Fisher comparison without proving the density-test backend." timeChangedInequality := "dK/dt <= -(sigma_t^2/2)*dot{s}(t)*C_LSI(t)*K(t)+sigma_t^(-2)*dot{s}(t)^(-1)*||m_t||_{L2(rho_{s(t)})}^2; SALD.generalMovingTargetTimeChangedDerivativeBoundScalar and SALD.generalMovingTargetPreDvDerivativeBoundScalar compile the scalar schedule handoff after inverse-schedule inputs are supplied." requiredRegularity := [ "smooth positive densities for rho_s and pi_{t(s)}", "valid Fokker--Planck equation for eq:general_moving_target_SALD", "integration-by-parts hypotheses for both c_t and v_t terms", "positive sigma_t and dot{s}(t), with inverse-schedule identity dot{t}(s(t))=dot{s}(t)^(-1)", "finite KL/FI and L2 residual-energy terms" ] sourceGaps := [ "the source theorem does not isolate the regularity assumptions needed for the general Fokker--Planck and integration-by-parts steps", "the inverse-schedule calculus is reused from the SALD proof but not restated for the sigma-weighted general theorem", "positivity and regularity of sigma_t are used by sigma_t^(-2) and Young's inequality but are not packaged as a standalone assumption" ] dependencies := [ "eq:general_moving_target_SALD", "TransportVelocityContract", "KLContract", "FIContract", "eq:LSI-KL-FI", "SALD.generalMovingTargetKlDerivativeResidualSplitScalar", "SALD.generalMovingTargetKlDerivativeScaledResidualDisplayScalar", "SALD.generalMovingTargetPostYoungDerivativeBoundScalar", "SALD.generalMovingTargetLsiDerivativeBoundScalar", "SALD.generalMovingTargetTimeChangedDerivativeBoundScalar", "SALD.generalMovingTargetPreDvDerivativeBoundScalar", "SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change" ] status := ProofStatus.obligation def generalMovingTargetDvEnergyCandidateContract : GeneralMovingTargetDvEnergyCandidateContract where sourceBlock := saldGeneralMovingTargetDvGronwallSource dvMeasure := "Apply lem:dv_variation with nu=rho_{s(t)} and mu=pi_t." dvTestFunction := "Z(x)=alpha*||m_t(x)||^2." integrabilityRoute := "The theorem assumes E_{alpha0}(pi_t,m_t)<+infty and alpha in (0,alpha0]; a finite log-mgf monotonicity lemma is needed before invoking DV." bound := "||m_t||_{L2(rho_{s(t)})}^2 <= alpha^(-1)*K(t)+E_alpha(pi_t,m_t)." dependencies := [ "lem:dv_variation", "def:alpha-complexity", "KLContract", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling" ] sourceGaps := [ "the appendix applies DV directly; Lean must expose the finite log-mgf witness for m_t" ] status := ProofStatus.obligation def generalMovingTargetDvFiniteLogMgfWitnessContract : GeneralMovingTargetDvFiniteLogMgfWitnessContract where sourceBlock := saldGeneralMovingTargetResidualDvSource theoremStatement := saldGeneralMovingTargetSource dvMeasures := "Use nu=rho_{s(t)} and mu=pi_t in the common Euclidean state space of the general VA-SALD law and moving target." testFunction := "Z_t(x)=alpha*||m_t(x)||^2, with m_t=v_t-c_t exactly as in appendix.tex:724-726 and 885-895." finiteAlpha0Assumption := "appendix.tex:724-727 assumes E_{alpha0}(pi_t,m_t)<+infty for every t in [0,T]." alphaMonotonicityBridge := "From 00 and the rewrite to E_alpha(pi_t,m_t) are displayed in one equality but require separate Lean order and scalar algebra", "the unified theorem inherits the residual log-mgf witness after m_t=w_t but the appendix proof only states the specialization in one line" ] lowerPacket := [ "Target this contract or sald.general_moving_target.dv_finite_log_mgf_witness only.", "Refine one backend: alpha0-to-alpha monotonicity for m_t, common-space/absolute-continuity, measurability of ||m_t||^2, or positive-alpha scaling.", "Keep lem:dv_variation source-cited and do not add a new finite-mgf assumption to thm:general-moving-target-SALD or thm:unified-forward-KL.", "Preserve the downstream sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1) coefficient." ] status := ProofStatus.obligation def generalMovingTargetDvPositiveAlphaScalingContract : GeneralMovingTargetDvPositiveAlphaScalingContract where sourceBlock := saldGeneralMovingTargetResidualDvSource theoremStatement := saldGeneralMovingTargetSource dvTestFunction := "Use the already-admissible DV test Z_t(x)=alpha*||m_t(x)||^2 from appendix.tex:885-893." positivityInput := "The theorem range alpha in (0,alpha0] supplies alpha>0; this is the only scalar fact used to divide the DV inequality." dvInequalityBeforeScaling := "After DV, alpha*E_{rho_{s(t)}}[||m_t||^2] <= KL(rho_{s(t)}||pi_t)+log E_{pi_t}[exp(alpha*||m_t||^2)]." divisionStep := "Divide the inequality by alpha>0 to preserve the inequality direction and get alpha^(-1)*K(t)+alpha^(-1)*log E_{pi_t}[exp(alpha*||m_t||^2)]." logMgfRewrite := "Rewrite alpha^(-1)*log E_{pi_t}[exp(alpha*||m_t||^2)] as mathfrak E_alpha(pi_t,m_t), exactly matching appendix.tex:891-895." coefficientAudit := "Substitution into appendix.tex:899-907 must keep sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1) on K(t) and sigma_t^(-2)*dot{s}(t)^(-1)*mathfrak E_alpha(pi_t,m_t) on the residual term." unifiedSpecializationUse := "Under the unified specialization c_t<-u_t, the same scaling rewrite applies after m_t=w_t and preserves the main_body.tex:381 and 390 coefficients." dependencies := [ "lem:dv_variation", "def:alpha-complexity", "SALD.saldDvFiniteLogMgfContract", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "KLContract", "Real positive scalar division/order algebra" ] sourceGaps := [ "appendix.tex:887-895 writes the divided inequality directly and does not isolate the alpha>0 order argument", "the definition of mathfrak E_alpha must be unfolded with the same alpha and residual field m_t before the residual coefficient is multiplied in", "main_body.tex:372-390 inherits this rewrite through m_t=w_t, but the source proof only says to specialize c_t<-u_t" ] status := ProofStatus.obligation def generalMovingTargetGronwallInstantiationContract : GeneralMovingTargetGronwallInstantiationContract where sourceBlock := saldGeneralMovingTargetDvGronwallSource quantityK := "K(t)=KL(rho_{s(t)}||pi_t)." gronwallA := "a(t)=(sigma_t^2/2)*dot{s}(t)*C_LSI(t)-sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1)." gronwallB := "b(t)=sigma_t^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t)." preSplitBound := "K(T) <= exp(-int_0^T a(t)dt)*K(0)+int_0^T exp(-int_t^T a(u)du)*b(t)dt." theoremBound := "Separate the exponent into the negative LSI term and positive sigma-weighted alpha term, then drop the nonpositive LSI part in the residual exponential to match eq:general_moving_target_KL_bound." pureContractionSpecialization := "If c_t=v_t, then m_t=0 and E_alpha(pi_t,m_t)=alpha^(-1)*log E_{pi_t}[1]=0, leaving only the initial-error contraction factor." requiredRegularity := [ "continuity or interval-integrability of a(t), b(t), sigma_t, dot{s}(t), C_LSI(t), and E_alpha(pi_t,m_t)", "differentiability of K(t) in the sense required by lem:gronwall", "positivity of sigma_t and dot{s}(t)" ] sourceGaps := [ "the source theorem does not separately state Gronwall regularity for the sigma-weighted coefficients", "the proof drops the LSI contribution in the residual exponential using nonnegativity of C_LSI and positivity of sigma_t^2*dot{s}(t)", "the pure-contraction clause uses the constant-zero residual energy identity, which still requires the alpha-complexity definition" ] dependencies := [ "lem:gronwall", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_m_energy", "def:alpha-complexity" ] status := ProofStatus.obligation def generalMovingTargetGronwallSideConditionContract : GeneralMovingTargetGronwallSideConditionContract where sourceBlock := saldGeneralMovingTargetDvGronwallSource theoremStatement := saldGeneralMovingTargetSource endpointScheduleIdentities := "The theorem bound is stated at t=T and t=0; the Lean route must provide s(0)=0, S=s(T), t(s(T))=T, and the slowed-target identity tilde_pi_{s(t)}=pi_t." terminalKlIdentification := "The Gronwall output K(T) is rewritten as KL(rho_S||pi_T) using S=s(T) and K(t)=KL(rho_{s(t)}||pi_t)." initialKlIdentification := "The initial term K(0) is rewritten as KL(rho_0||pi_0) using s(0)=0 and the source initial law X_0~rho_0." coefficientRegularity := "a(t)=(sigma_t^2/2)*dot{s}(t)*C_LSI(t)-sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1) and b(t)=sigma_t^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t) must satisfy the continuity or interval-integrability hypotheses required by lem:gronwall." signFactsForResidualDrop := "The residual-exponent simplification uses C_LSI(u)>=0, sigma_u^2>=0, and dot{s}(u)>0, so -int_t^T (sigma_u^2/2)*dot{s}(u)*C_LSI(u) du <= 0." exponentSplitAlgebra := "Split exp(-int_0^T a) into exp(-int_0^T (sigma_t^2/2)*dot{s}(t)*C_LSI(t)dt)*exp(int_0^T sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1)dt)." residualExponentBound := "For the residual integral, bound exp(-int_t^T a(u)du) by exp(int_t^T sigma_u^(-2)*dot{s}(u)^(-1)*alpha^(-1)du) after dropping the nonpositive LSI contribution." pureContractionResidualZero := "Under c_t=v_t, m_t=0; hence exp(alpha*||m_t||^2)=1, E_alpha(pi_t,m_t)=alpha^(-1)*log 1=0, and the residual integral in eq:general_moving_target_KL_bound vanishes." dependencies := [ "SALD.cycle24GeneralVaSaldUpperPacket", "SALD.cycle24GeneralVaSaldMiddleContract", "sald.general_moving_target.cycle24_gronwall_middle", "sald.forward_kl.schedule_time_change", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_application", "sald.gronwall.integrating_factor", "SALD.generalMovingTargetGronwallCoeffAdjacentIntervalIntegrable", "SALD.generalMovingTargetGronwallExpProductRewriteIntegralCongrOfPieces", "def:alpha-complexity" ] sourceGaps := [ "the source theorem does not isolate s(0)=0, S=s(T), and slowed-target endpoint identities as a separate lemma", "the proof applies Gronwall without separately proving continuity or interval-integrability of the sigma-weighted coefficients; SALD.generalMovingTargetGronwallCoeffAdjacentIntervalIntegrable packages only the local interval-integrability closure once those hypotheses are supplied", "the residual exponent drop relies on nonnegativity of C_LSI, sigma_t^2, and dot{s}, which must be exposed before formal proof search", "the zero-residual alpha-complexity calculation uses normalization of pi_t and log(1)=0, which are not formalized locally" ] lowerPacket := [ "Target this contract or the named gronwall-side-condition obligation only.", "Preserve the sigma-weighted coefficients and the exponent split from appendix.tex:908-934 and theorem lines 727-743.", "Preferred first lower sub-slice after cycle 24 middle: theorem-specific coefficient regularity and adjacent interval-integrability for the LSI coefficient, alpha coefficient, and residual b(t) term.", "SALD.generalMovingTargetGronwallCoeffAdjacentIntervalIntegrable and SALD.generalMovingTargetGronwallExpProductRewriteIntegralCongrOfPieces are compiled local algebra wrappers only after those source regularity hypotheses are supplied.", "Record endpoint rewrites, coefficient regularity, residual-exponent monotonicity, and zero-residual alpha-complexity as obligations unless compiled Lean lemmas prove them.", "Do not add these side conditions to the theorem statement as hidden assumptions." ] status := ProofStatus.obligation def unifiedForwardKlSpecializationContract : UnifiedForwardKlSpecializationContract where sourceStatement := saldUnifiedForwardKlSource sourceProof := saldUnifiedForwardKlProofSource vaSaldDynamics := saldVaSaldItoSource residualEquation := saldGuidedResidualMainSource correctionEquation := saldCorrectionFieldSource generalTheorem := saldGeneralMovingTargetSource specialization := "Set c_t <- u_t in eq:general_moving_target_SALD; then the general SDE becomes VA-SALD eq:SALD_Ito because its score term is (sigma_t^2/2)*nabla log pi_t." correctionFieldTransportBridge := "main_body.tex:359-368 combines prop:guided_path_residual with div(pi_t*w_t)=pi_t*(g_t-E_pi_t[g_t]) to infer partial_t pi_t+div(pi_t*(u_t+w_t))=0." residualFieldIdentification := "Use v_t=u_t+w_t as the transport velocity for pi_t and c_t=u_t as the implementable velocity; the general residual m_t=v_t-c_t is w_t." assumptionsBridge := "The unified theorem's LSI assumption and finite E_alpha0(pi_t,w_t) assumption are exactly the general theorem assumptions after m_t=w_t." terminalBoundMatch := "Substituting m_t=w_t into eq:general_moving_target_KL_bound gives main_body.tex lines 374-390, with the same sigma-weighted exponents and residual complexity term." proofRoute := [ "main_body.tex:359-363 quotes prop:guided_path_residual for the guided path residual.", "main_body.tex:364-368 introduces w_t by the Poisson/divergence equation and states u_t+w_t is a transport velocity for pi_t.", "main_body.tex:372-390 states the unified theorem with residual complexity E_alpha(pi_t,w_t).", "appendix.tex:949-951 proves the theorem by specializing thm:general-moving-target-SALD with c_t <- u_t." ] dependencies := [ "prop:guided_path_residual", "eq:poisson-eq", "eq:SALD_Ito", "thm:general-moving-target-SALD", "sald.guided_path_residual.identity", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions" ] sourceGaps := [ "the source introduces w_t through a divergence equation but does not formalize existence, regularity, or boundary conditions for a solution", "the transport-velocity bridge from the residual identity plus the correction equation to u_t+w_t generating pi_t is stated in prose and remains a local obligation", "the equality between VA-SALD eq:SALD_Ito and the general moving-target SDE under c_t=u_t requires the guided-density score identity for pi_t", "no direct theorem proof is supplied for thm:unified-forward-KL; the only faithful route is the specialization of thm:general-moving-target-SALD" ] lowerPacket := [ "Target the transport-velocity bridge u_t+w_t for pi_t before theorem-level proof search.", "Keep the specialization c_t <- u_t and residual m_t=w_t exactly; do not introduce a direct VA-SALD proof.", "Record existence and regularity of the correction field w_t as obligations unless a compiled Poisson/divergence backend is added." ] status := ProofStatus.obligation def cycle12GeneralVaSaldUpperPacket : GeneralVaSaldUpperPacket where objective := "Keep the guided/general VA-SALD theorem statements fixed and isolate the residual-field Donsker--Varadhan finite-log-mgf witness for m_t, including the unified specialization m_t=w_t and the discrete EM-interpolation reuse." sourceLabels := [ "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "lem:dv_variation", "def:alpha-complexity" ] modeDiscipline := [ "faithfulPaper: use main_body.tex:359-395 and appendix.tex:724-951 plus appendix.tex:1544-1603, with sald_version_2.tex excluded.", "Preserve m_t=v_t-c_t, the unified specialization c_t<-u_t and m_t=w_t, and the discrete residual coefficient 2*sigma_eta(t)^(-2)*dot{s}(t)^(-1).", "Treat DV, alpha0-to-alpha finite-log-mgf monotonicity, common-space/absolute-continuity, EM interpolation laws, and Gronwall as obligations or source-cited facts until compiled Lean proofs replace them." ] nonGoals := [ "Do not restate or prove thm:general-moving-target-SALD, thm:unified-forward-KL, or thm:general-moving-target-SALD-discrete.", "Do not replace the source residual DV step with Pinsker, Talagrand, path-space, Girsanov, or a direct VA-SALD proof.", "Do not add finite-log-mgf, measurability, absolute-continuity, endpoint, or schedule hypotheses silently to theorem statements.", "Do not simplify away the sigma-weighted or doubled discrete residual coefficients." ] lowerPacket := [ "Target exactly one interface: SALD.generalMovingTargetDvFiniteLogMgfWitnessContract, SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract, or their named obligations.", "For the continuous theorem, refine alpha0-to-alpha monotonicity for Z=alpha*||m_t||^2, common-space/absolute-continuity, measurability, or positive-alpha scaling.", "For the unified theorem, only check the specialization bridge m_t=w_t after c_t<-u_t; do not introduce a separate direct proof.", "For the discrete theorem, keep nu=hat rho_s, mu=tilde pi_s, and the coefficient 2*sigma_eta^(-2)*dot t(s)^2 before time change." ] reviewerChecklist := [ "SALD.generalVaSaldContract and SALD.unifiedForwardKlContract list SALD.generalMovingTargetDvFiniteLogMgfWitnessObligation before the residual DV-energy obligation.", "SALD.generalVaSaldDiscreteContract lists SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessObligation before SALD.generalMovingTargetDiscreteDvMEnergyObligation.", "SALD.generalVaSaldProofDag and SALD.generalVaSaldDiscreteProofDag contain residual DV witness blocks before the DV-energy blocks.", "The conversion window, proof-obligation ledger, SLT audit, and source index remain synchronized and sald_version_2.tex remains excluded.", "No analytic dependency is marked formalized and no fake proof closure is introduced." ] status := ProofStatus.obligation def cycle12GeneralVaSaldMiddleContract : GeneralVaSaldGuidedPathMiddleContract where guidedResidualSource := saldGuidedResidualSource generalContinuousSource := saldGeneralMovingTargetSource unifiedSource := saldUnifiedForwardKlSource discreteSource := saldGeneralMovingTargetDiscreteSource objective := "Map the cycle-focus guided/general VA-SALD proof route from the guided residual identity through the continuous general theorem, the unified c_t<-u_t specialization, and the discrete general theorem while keeping all theorem statements and coefficients unchanged." sourceStepMap := [ "appendix.tex:619-704 proves prop:guided_path_residual by differentiating Z_t, differentiating pi_t=Z_t^(-1)*p_t*exp(-f_t), canceling div(p_t*u_t), and centering g_t.", "main_body.tex:359-368 uses the residual identity and div(pi_t*w_t)=pi_t*(g_t-E_pi_t[g_t]) to state that u_t+w_t is a transport velocity for pi_t.", "appendix.tex:765-884 derives the continuous general VA-SALD sigma-weighted KL differential inequality with residual m_t=v_t-c_t before DV.", "appendix.tex:885-934 applies DV to Z=alpha*||m_t||^2 and then Gronwall with the sigma-weighted a(t), b(t).", "appendix.tex:936-951 gives the pure-contraction c_t=v_t case and the one-line unified theorem proof by setting c_t<-u_t.", "appendix.tex:1354-1600 repeats the route under the general EM interpolation, splits delta_pi^VA and dot t(s)*m_t, applies the residual DV witness, changes to t, and applies Gronwall.", "appendix.tex:1603 specializes the discrete general theorem to discrete VA-SALD by replacing c with u." ] leanStepMap := [ "Guided residual normalization and centering are tracked by SALD.guidedResidualIdentityContract, sald.guided_path_residual.normalizer_derivative, and sald.guided_path_residual.identity.", "The continuous derivative route is tracked by SALD.generalMovingTargetDerivativeCandidateContract and sald.general_moving_target.kl_derivative.", "The continuous residual DV step is split between SALD.generalMovingTargetDvFiniteLogMgfWitnessContract, SALD.generalMovingTargetDvPositiveAlphaScalingContract, SALD.generalMovingTargetDvEnergyCandidateContract, sald.general_moving_target.dv_finite_log_mgf_witness, sald.general_moving_target.dv_positive_alpha_scaling, and sald.general_moving_target.dv_m_energy.", "The continuous Gronwall and pure-contraction bookkeeping is tracked by SALD.generalMovingTargetGronwallInstantiationContract, SALD.generalMovingTargetGronwallSideConditionContract, sald.general_moving_target.gronwall_application, sald.general_moving_target.gronwall_side_conditions, and sald.general_moving_target.pure_contraction.", "The unified theorem bridge is tracked by SALD.unifiedForwardKlSpecializationContract and sald.unified_forward_kl.specialization.", "The discrete route is tracked by SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract, SALD.generalMovingTargetDiscreteGronwallSideConditionContract, and the matching sald.general_moving_target_discrete.* obligations.", "The discrete guided specialization is tracked by sald.general_moving_target_discrete.unified_specialization." ] citedResultInterfaces := [ "lem:dv_variation remains source-cited; residual finite-log-mgf witnesses expose theorem-specific side conditions before each DV-energy obligation.", "eq:LSI-KL-FI remains an obligation through probability.lsi_to_kl_fi.", "lem:gronwall remains a local real-analysis obligation through sald.gronwall.integrating_factor and the Gronwall side-condition contracts.", "lem:frozen_delta_cross_lip is an internal appendix lemma whose proof still depends on local EM estimates and source-cited DV substeps." ] obligations := [ "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "sald.unified_forward_kl.specialization", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions", "sald.general_moving_target_discrete.unified_specialization" ] lowerPacket := [ "Target exactly one existing backend obligation from this map; preferred next targets are sald.unified_forward_kl.specialization or sald.general_moving_target_discrete.derivative_side_conditions.", "If working on the unified specialization, prove only the transport-velocity bridge from prop:guided_path_residual plus eq:poisson-eq to v_t=u_t+w_t and m_t=w_t.", "If working on the discrete derivative side conditions, preserve the two sigma_eta^2/8 Young splits, the residual coefficient 2*sigma_eta^(-2)*dot t(s)^2 before time change, and the frozen-delta Gamma/Delta coefficients.", "Do not add correction-field existence, density regularity, finite-log-mgf, endpoint, or schedule assumptions silently to any theorem statement." ] reviewerChecklist := [ "The guided residual, general continuous, unified, and discrete theorem contracts all remain contract-only with named obligations.", "The residual DV witness obligations remain before the residual DV-energy obligations in both continuous and discrete DAGs.", "The unified theorem uses only the source specialization c_t<-u_t and does not introduce a direct VA-SALD proof route.", "The discrete theorem display keeps the doubled residual coefficient and the Gamma/Delta terms from appendix.tex:1316-1347.", "No analytic dependency is marked formalized and no fake proof closure is introduced." ] status := ProofStatus.obligation def cycle16GeneralVaSaldUpperPacket : GeneralVaSaldUpperPacket where objective := "Keep the guided/general VA-SALD theorem statements fixed and select the unified transport-velocity bridge from prop:guided_path_residual plus eq:poisson-eq as the next lower target." sourceLabels := [ "prop:guided_path_residual", "eq:poisson-eq", "eq:SALD_Ito", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] modeDiscipline := [ "faithfulPaper: use main_body.tex:359-395, appendix.tex:619-704, appendix.tex:724-951, and appendix.tex:1603 for source correspondence; sald_version_2.tex remains out of scope.", "Preserve the signs in partial_t pi_t+div(pi_t*u_t)=-pi_t*(g_t-E_pi_t[g_t]) and div(pi_t*w_t)=pi_t*(g_t-E_pi_t[g_t]), so the cancellation yields u_t+w_t as the transport velocity for pi_t.", "Keep thm:unified-forward-KL as the source specialization of thm:general-moving-target-SALD with c_t<-u_t, v_t=u_t+w_t, and m_t=w_t; do not introduce a direct VA-SALD proof route.", "Leave the continuous sigma-weighted coefficients and the discrete doubled residual coefficients under their existing obligations." ] nonGoals := [ "Do not prove or restate thm:general-moving-target-SALD, thm:unified-forward-KL, or thm:general-moving-target-SALD-discrete.", "Do not solve existence, uniqueness, regularity, or boundary conditions for the correction field w_t unless a compiled backend is explicitly supplied.", "Do not replace the correction-field transport bridge with Girsanov, Pinsker, Talagrand, or path-space reasoning.", "Do not add correction-field, density, finite-log-mgf, endpoint, or schedule assumptions silently to any theorem statement." ] lowerPacket := [ "Target exactly one interface: sald.unified_forward_kl.transport_velocity_bridge, using SALD.unifiedForwardKlSpecializationContract as the line ledger.", "Show only the source algebra from main_body.tex:359-368: combine the centered residual identity with div(pi_t*w_t)=pi_t*(g_t-E_pi_t[g_t]) to obtain partial_t pi_t+div(pi_t*(u_t+w_t))=0.", "After the transport bridge, record the general-theorem specialization v_t=u_t+w_t, c_t=u_t, and m_t=w_t; stop before DV, Gronwall, or discrete EM proof search.", "If the transport proof backend is not ready, refine SALD.unifiedForwardKlTransportBridgeObligation rather than strengthening the theorem assumptions." ] reviewerChecklist := [ "SALD.unifiedForwardKlTransportBridgeObligation is present, sourced to main_body.tex:359-368, and remains an obligation.", "SALD.unifiedForwardKlSpecializationObligation depends on sald.unified_forward_kl.transport_velocity_bridge and still follows appendix.tex:949-951.", "SALD.saldDependenciesForLabel \"thm:unified-forward-KL\" includes SALD.cycle16GeneralVaSaldUpperPacket and sald.unified_forward_kl.transport_velocity_bridge.", "The conversion window, proof-obligation ledger, SLT audit, source index, and dialogue handoff are synchronized, with sald_version_2.tex excluded.", "No analytic dependency is marked formalized and no fake proof closure is introduced." ] status := ProofStatus.obligation /-- Cycle-20 upper packet returning to the guided/general VA-SALD path. The selected lower target is the final Gronwall/display bridge for `thm:general-moving-target-SALD-discrete`. This packet is workflow data only: it keeps the theorem statement, the continuous general theorem, and the unified specialization fixed, and it does not promote any analytic dependency. -/ def cycle20GeneralVaSaldUpperPacket : GeneralVaSaldUpperPacket where objective := "Return to the guided/general VA-SALD path and select the discrete general Gronwall side-condition/display bridge as the next lower target: stitched endpoint laws, constant-schedule coefficient rewrites, coefficient regularity, and exact matching of appendix.tex:1586-1600 to the theorem display appendix.tex:1316-1347." sourceLabels := [ "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "eq:SALD_general_EM", "lem:frozen_delta_cross_lip", "lem:gronwall" ] modeDiscipline := [ "faithfulPaper: use appendix.tex:1313-1603 for the discrete general theorem, with appendix.tex:724-951 and main_body.tex:359-395 only as fixed upstream guided/general dependencies; sald_version_2.tex remains out of scope.", "Preserve the exact theorem-display coefficients: (sigma_eta(t)^2/2)*dot{s}(t)*C_LSI(t), 2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*alpha^(-1), 2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t), and b(t)=2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t)+2*dot{s}(t)*eta*Delta(t).", "Keep the source route interval EM derivative -> residual/frozen split -> DV/LSI -> s-to-t time change -> lem:gronwall; continuous general VA-SALD and unified VA-SALD remain dependencies, not alternate proof routes.", "Treat endpoint stitching, constant inverse-schedule algebra, coefficient integrability, piecewise differentiability of K(t), and Gronwall as obligations until compiled Lean proofs replace them." ] nonGoals := [ "Do not restate or prove thm:general-moving-target-SALD-discrete in this packet.", "Do not reopen the frozen-delta lemma, residual DV finite-log-mgf witness, LSI-to-KL/FI bridge, or KL derivative proof except as dependencies of the final Gronwall side-condition ledger.", "Do not turn the discrete guided specialization into a direct VA-SALD proof; it remains the source replacement c<-u after the general theorem.", "Do not simplify away the doubled residual coefficient, the Gamma/Delta terms, alpha, alpha', eta, sigma_eta, or the constant-schedule hypothesis." ] lowerPacket := [ "Target exactly SALD.generalMovingTargetDiscreteGronwallSideConditionContract / SALD.generalMovingTargetDiscreteGronwallSideConditionObligation / sald.general_moving_target_discrete.gronwall_side_conditions.", "Refine one lower sub-slice only: endpoint stitching for K(0)/K(T), constant-schedule coefficient rewrites, coefficient regularity for a(t), b(t), or exact Gronwall-display matching.", "Preserve appendix.tex:1586-1597 and theorem display appendix.tex:1316-1347 verbatim at the contract level; bridge them by explicit endpoint, schedule, and coefficient lemmas.", "If stitched EM regularity, endpoint laws, interval-integrability, or schedule algebra is blocked, record the precise source-contract gap rather than adding theorem hypotheses." ] reviewerChecklist := [ "SALD.generalVaSaldDiscreteProofDag contains ASTIS.SALD.general_moving_target_discrete.cycle20_upper_packet before the gronwall_side_conditions block.", "SALD.saldDependenciesForLabel \"thm:general-moving-target-SALD-discrete\" includes SALD.cycle20GeneralVaSaldUpperPacket while retaining the existing discrete EM, frozen-delta, DV, derivative, and Gronwall obligations.", "SALD.generalVaSaldDiscreteContract remains contractOnly and the named gronwall-side-condition obligation remains an obligation.", "The conversion window, proof-obligation ledger, SLT audit, and source index classify this as workflow/local algebra work and keep sald_version_2.tex excluded.", "No analytic dependency is marked formalized and no fake proof closure is introduced." ] status := ProofStatus.obligation /-- Cycle-20 middle packet for the discrete general VA-SALD Gronwall bridge. This translates the upper-selected target into a lower-ready source-to-Lean map for `sald.general_moving_target_discrete.gronwall_side_conditions`. It keeps the theorem statement fixed and separates endpoint stitching, constant-schedule coefficient rewrites, Gronwall regularity, and display matching from the upstream EM, frozen-delta, DV, LSI, and derivative obligations. -/ def cycle20GeneralVaSaldMiddleContract : GeneralVaSaldGuidedPathMiddleContract where guidedResidualSource := saldGuidedResidualSource generalContinuousSource := saldGeneralMovingTargetSource unifiedSource := saldUnifiedForwardKlSource discreteSource := saldGeneralMovingTargetDiscreteSource objective := "Translate the cycle-20 discrete general Gronwall/display bridge into a lower-ready map for sald.general_moving_target_discrete.gronwall_side_conditions, with endpoint stitching, constant-schedule coefficient rewrites, coefficient regularity, and theorem-display matching kept as explicit obligations." sourceStepMap := [ "appendix.tex:1316-1347 fixes the theorem display for KL(rho_K^eta||pi_T), including the exact a(t) and b(t) coefficients.", "appendix.tex:1558-1570 gives the pre-time-change differential inequality with the doubled residual term 2*sigma_eta^{-2}*dot t(s)^2 and the frozen-delta terms 2*Gamma(t(s))*eta^2*alpha'^{-1} and 2*Delta(t(s))*eta.", "appendix.tex:1573-1583 defines K(t)=KL(hat rho_{s(t)}||pi_t)=KL(hat rho_{s(t)}||tilde pi_{s(t)}) and changes derivatives by dK/dt=dot{s}(t)*d/ds KL|_{s=s(t)}.", "appendix.tex:1583 uses the constant inverse-schedule identity dot t(s(t))=dot{s}(t)^(-1) before rewriting the residual and frozen-delta coefficients.", "appendix.tex:1586-1597 is the final t-time differential inequality; the lower proof must preserve the two residual coefficients and the dot{s}(t) multipliers on Gamma and Delta.", "appendix.tex:1600 invokes lem:gronwall, whose endpoint laws, coefficient regularity, and exact display matching are not expanded in the source proof." ] leanStepMap := [ "The theorem statement remains SALD.generalMovingTargetDiscreteStatementContract and SALD.generalVaSaldDiscreteContract; this packet changes neither.", "SALD.generalMovingTargetDiscreteGronwallInstantiationContract records the source a(t), b(t), and Gronwall target before theorem-display side conditions.", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract records endpoint stitching, constant-schedule identities, coefficient regularity, residual/frozen coefficient audits, and display matching.", "SALD.generalMovingTargetDiscreteConstantScheduleObligation supplies schedule and stitched-interval interfaces as obligations rather than new theorem assumptions.", "Cycle-20 lower scalar helpers formalize only the real coefficient algebra after the inverse-schedule identity: SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar, SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar, SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar, and SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar.", "SALD.cycle20GeneralVaSaldDiscreteGronwallMiddleObligation keeps this middle source map synchronized with the lower target sald.general_moving_target_discrete.gronwall_side_conditions.", "The cycle-17 and cycle-18 Gronwall scalar helpers are reusable only after theorem-specific interval-integrability and endpoint equalities are supplied; they do not prove this discrete general Gronwall step." ] citedResultInterfaces := [ "lem:gronwall remains a local real-analysis obligation through sald.gronwall.integrating_factor and theorem-specific regularity/display side conditions.", "lem:dv_variation remains source-cited only through the existing residual DV-energy obligations; this middle packet does not reopen or formalize DV.", "eq:LSI-KL-FI remains the inherited density-test obligation used before the final Gronwall inequality.", "No SLT theorem applies to endpoint stitching, constant-schedule coefficient rewrites, coefficient regularity, or exact display matching." ] obligations := [ "sald.general_moving_target_discrete.cycle20_gronwall_middle", "sald.general_moving_target_discrete.gronwall_side_conditions", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.constant_schedule_stitching", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "sald.gronwall.integrating_factor" ] lowerPacket := [ "Target exactly SALD.generalMovingTargetDiscreteGronwallSideConditionContract / SALD.generalMovingTargetDiscreteGronwallSideConditionObligation / sald.general_moving_target_discrete.gronwall_side_conditions.", "Preferred first sub-slice: constant-schedule coefficient rewrite from appendix.tex:1579-1597, especially dot{s}(t)*dot t(s(t))^2 = dot{s}(t)^(-1) and the unchanged Gamma/Delta multipliers.", "Alternative one-slice targets are endpoint stitching for K(0)/K(T), coefficient regularity for a(t), b(t), or exact Gronwall-display matching against appendix.tex:1316-1347.", "Keep the EM interpolation, frozen-delta lemma, residual DV finite-log-mgf witness, LSI bridge, KL derivative, and full Gronwall lemma as separate dependencies.", "Do not add schedule, endpoint, regularity, or integrability assumptions to thm:general-moving-target-SALD-discrete; record blocked interfaces as obligations." ] reviewerChecklist := [ "Every source step from appendix.tex:1573-1600 and the theorem display appendix.tex:1316-1347 is classified as Lean contract, cited result, or named obligation.", "SALD.generalVaSaldDiscreteContract lists SALD.cycle20GeneralVaSaldDiscreteGronwallMiddleObligation and still keeps the theorem contractOnly.", "SALD.generalVaSaldDiscreteProofDag contains ASTIS.SALD.general_moving_target_discrete.cycle20_middle_gronwall_bridge between the upper packet and gronwall_side_conditions blocks.", "SALD.saldDependenciesForLabel \"thm:general-moving-target-SALD-discrete\" includes SALD.cycle20GeneralVaSaldMiddleContract and sald.general_moving_target_discrete.cycle20_gronwall_middle.", "No theorem statement, source coefficient, source file selection, or analytic dependency status is changed." ] status := ProofStatus.obligation /-- Cycle-24 upper packet for the continuous general VA-SALD Gronwall bridge. This returns to `thm:general-moving-target-SALD` after the discrete and forward-KL coefficient audits. It selects only the endpoint/exponent side-condition ledger behind the final Gronwall display, leaving the theorem statement, unified specialization, discrete theorem, and analytic backends unchanged. -/ def cycle24GeneralVaSaldUpperPacket : GeneralVaSaldUpperPacket where objective := "Return to the guided/general VA-SALD path and select the continuous general moving-target Gronwall endpoint/exponent/pure-contraction side-condition bridge as the next lower target: K(0)/K(T) endpoint rewrites, sigma-weighted coefficient regularity, exponent splitting, residual-exponent monotonicity, and the zero-residual alpha-complexity specialization in appendix.tex:909-945." sourceLabels := [ "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "lem:dv_variation", "lem:gronwall", "eq:LSI-KL-FI", "def:alpha-complexity" ] modeDiscipline := [ "faithfulPaper: use appendix.tex:724-951 for the continuous general theorem, main_body.tex:359-395 for the unified specialization, and appendix.tex:1313-1603 only as downstream discrete reuse; sald_version_2.tex remains out of scope.", "Preserve the exact continuous coefficients a(t)=(sigma_t^2/2)*dot{s}(t)*C_LSI(t)-sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1) and b(t)=sigma_t^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t).", "Keep the source route derivative -> LSI -> residual DV -> Gronwall -> theorem display -> pure contraction; unified VA-SALD remains only the c_t<-u_t specialization with m_t=w_t.", "Treat endpoint schedule identities, coefficient regularity, sign facts for dropping the LSI contribution, zero-residual alpha-complexity, DV, LSI, and Gronwall as obligations until compiled Lean proofs replace them." ] nonGoals := [ "Do not prove or restate thm:general-moving-target-SALD, thm:unified-forward-KL, or thm:general-moving-target-SALD-discrete in this packet.", "Do not replace the sigma-weighted Gronwall route with the forward-KL proof, a discrete argument, Girsanov, Pinsker, Talagrand, or any path-space proof.", "Do not add hidden endpoint, schedule, coefficient-integrability, density, boundary, finite-log-mgf, or sign assumptions to theorem statements.", "Do not simplify away sigma_t, dot{s}(t), alpha, the residual alpha-complexity, the pure-contraction exponent factors, or the discrete doubled residual/Gamma/Delta coefficients." ] lowerPacket := [ "Target exactly SALD.generalMovingTargetGronwallSideConditionContract / SALD.generalMovingTargetGronwallSideConditionObligation / sald.general_moving_target.gronwall_side_conditions.", "First sub-slice for middle: classify appendix.tex:909-934 against theorem display appendix.tex:727-743, separating endpoint K(T)/K(0) rewrites, coefficient regularity for a(t), b(t), exponent split, and residual-exponent drop.", "Alternative one-slice targets are endpoint schedule identities, theorem-specific interval-integrability of the sigma/LSI and alpha coefficient pieces, residual-exponent monotonicity, or appendix.tex:936-945 zero-residual alpha-complexity.", "If any endpoint, regularity, sign, or alpha-complexity backend is blocked, refine the named proof obligation instead of strengthening the theorem or marking an analytic result formalized." ] reviewerChecklist := [ "SALD.generalVaSaldProofDag contains ASTIS.SALD.general_moving_target.cycle24_upper_packet before ASTIS.SALD.general_moving_target.gronwall_side_conditions.", "SALD.saldDependenciesForLabel \"thm:general-moving-target-SALD\" and \"thm:unified-forward-KL\" include SALD.cycle24GeneralVaSaldUpperPacket while retaining the existing guided residual, DV, LSI, Gronwall, and transport-bridge obligations.", "SALD.generalMovingTargetGronwallSideConditionObligation remains an obligation; no endpoint, coefficient regularity, residual exponent drop, pure-contraction, DV, LSI, or full Gronwall backend is promoted.", "The unified theorem still follows only by setting c_t<-u_t with v_t=u_t+w_t and m_t=w_t; the discrete theorem remains under the cycle-20 Gronwall-side-condition packet.", "The conversion window, proof-obligation ledger, source index, and dialogue handoff stay synchronized, sald_version_2.tex is excluded, and python3 tools/astis.py check passes." ] status := ProofStatus.obligation /-- Cycle-24 middle packet for the continuous general VA-SALD Gronwall bridge. This translates the upper-selected target into a lower-ready source-to-Lean map for `sald.general_moving_target.gronwall_side_conditions`. It keeps the continuous general theorem, unified specialization, and downstream discrete reuse fixed while separating endpoint rewrites, coefficient regularity, exponent splitting, residual-exponent monotonicity, and pure-contraction alpha-complexity from the upstream derivative, LSI, DV, and Gronwall obligations. -/ def cycle24GeneralVaSaldMiddleContract : GeneralVaSaldGuidedPathMiddleContract where guidedResidualSource := saldGuidedResidualSource generalContinuousSource := saldGeneralMovingTargetSource unifiedSource := saldUnifiedForwardKlSource discreteSource := saldGeneralMovingTargetDiscreteSource objective := "Translate appendix.tex:909-945 into a lower-ready middle ledger for sald.general_moving_target.gronwall_side_conditions, with theorem-display endpoint rewrites, sigma-weighted coefficient regularity, exponent splitting, residual-exponent monotonicity, and zero-residual alpha-complexity kept as explicit obligations." sourceStepMap := [ "appendix.tex:908-910 gives the scalar differential inequality after residual DV with a(t)=(sigma_t^2/2)*dot{s}(t)*C_LSI(t)-sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1) and b(t)=sigma_t^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t).", "appendix.tex:911-920 applies lem:gronwall and identifies the Gronwall endpoint K(T) with KL(rho_S||pi_T) and K(0) with KL(rho_0||pi_0).", "appendix.tex:913-920 splits exp(-int_0^T a) into the theorem's LSI contraction factor and positive alpha factor without changing sigma_t, dot{s}(t), or alpha.", "appendix.tex:921-932 rewrites the residual integral and drops only the nonpositive LSI contribution from exp(-int_t^T a), using C_LSI>=0, sigma_u^2>=0, and dot{s}(u)>0 as side conditions.", "appendix.tex:936-945 specializes c_t=v_t, so m_t=0 and E_alpha(pi_t,m_t)=alpha^(-1)*log E_{pi_t}[1]=0, leaving the pure-contraction display.", "appendix.tex:949-951 and main_body.tex:372-395 reuse the same bridge only after the unified specialization c_t<-u_t, v_t=u_t+w_t, and m_t=w_t." ] leanStepMap := [ "SALD.generalMovingTargetGronwallInstantiationContract records the source a(t), b(t), and pre-split Gronwall output.", "SALD.generalMovingTargetGronwallSideConditionContract records endpointScheduleIdentities, terminalKlIdentification, initialKlIdentification, coefficientRegularity, exponentSplitAlgebra, residualExponentBound, and pureContractionResidualZero.", "SALD.cycle24GeneralVaSaldGronwallMiddleObligation keeps this middle source map synchronized with the lower target sald.general_moving_target.gronwall_side_conditions.", "SALD.generalMovingTargetGronwallCoeffAdjacentIntervalIntegrable and SALD.generalMovingTargetGronwallExpProductRewriteIntegralCongrOfPieces package the cycle-24 lower coefficient slice only after theorem-specific interval-integrability for the sigma/LSI, alpha, and residual pieces is supplied.", "The reusable scalar Gronwall exponent helpers may be used only after theorem-specific endpoint equalities, coefficient regularity, interval-integrability, and sign facts are supplied.", "The unified theorem remains SALD.unifiedForwardKlSpecializationContract; this middle packet does not introduce a direct VA-SALD KL proof.", "The discrete general theorem remains under the cycle-20 discrete Gronwall-side-condition packet and is only listed as downstream reuse." ] citedResultInterfaces := [ "lem:gronwall remains a local real-analysis obligation through sald.gronwall.integrating_factor and theorem-specific side conditions.", "lem:dv_variation remains source-cited through sald.general_moving_target.dv_m_energy; this middle packet starts after the residual DV inequality is available.", "eq:LSI-KL-FI remains the inherited density-test obligation that supplies C_LSI>=0 and the FI-to-KL contraction step.", "No SLT theorem applies to endpoint rewrites, sigma-weighted coefficient regularity, residual-exponent monotonicity, or zero-residual alpha-complexity." ] obligations := [ "sald.general_moving_target.cycle24_gronwall_middle", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.kl_derivative", "sald.forward_kl.schedule_time_change", "sald.gronwall.integrating_factor", "sald.gronwall.exponent_rewrite", "probability.lsi_to_kl_fi" ] lowerPacket := [ "Target exactly SALD.generalMovingTargetGronwallSideConditionContract / SALD.generalMovingTargetGronwallSideConditionObligation / sald.general_moving_target.gronwall_side_conditions.", "Preferred first sub-slice: theorem-specific coefficient regularity and adjacent interval-integrability for the LSI coefficient (sigma_t^2/2)*dot{s}(t)*C_LSI(t), the alpha coefficient sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1), and b(t)=sigma_t^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t).", "Use SALD.generalMovingTargetGronwallCoeffAdjacentIntervalIntegrable only as the local assembly step; it does not prove the source regularity hypotheses.", "Alternative one-slice targets are endpoint K(0)/K(T) rewrites, exponent-split algebra against appendix.tex:727-743, residual-exponent monotonicity on [t,T], or zero-residual alpha-complexity for appendix.tex:936-945.", "Keep the derivative, LSI-to-KL/FI, residual DV witness, full Gronwall theorem, unified transport bridge, and discrete general theorem as separate dependencies.", "Do not add endpoint, sign, sigma positivity, coefficient-integrability, normalization, or finite-log-mgf assumptions to theorem statements; record blocked interfaces as obligations." ] reviewerChecklist := [ "Every source step from appendix.tex:909-945 and theorem display appendix.tex:727-743 is classified as Lean contract, cited result, or named obligation.", "SALD.generalVaSaldContract and SALD.unifiedForwardKlContract list SALD.cycle24GeneralVaSaldGronwallMiddleObligation while preserving contractOnly status.", "SALD.generalVaSaldProofDag contains ASTIS.SALD.general_moving_target.cycle24_middle_gronwall_bridge between the upper packet and gronwall_side_conditions blocks.", "SALD.saldDependenciesForLabel \"thm:general-moving-target-SALD\" and \"thm:unified-forward-KL\" include SALD.cycle24GeneralVaSaldMiddleContract and sald.general_moving_target.cycle24_gronwall_middle.", "No theorem statement, source coefficient, source file selection, or analytic dependency status is changed." ] status := ProofStatus.obligation /-- Cycle-28 upper packet for the discrete general VA-SALD derivative side conditions. This returns to the guided/general path after the discrete forward-KL accumulated collection work. It selects the pre-Gronwall derivative side-condition ledger for `thm:general-moving-target-SALD-discrete`, especially the frozen/residual algebra and the two Young coefficient splits that produce the doubled residual coefficient and the Gamma/Delta terms. -/ def cycle28GeneralVaSaldUpperPacket : GeneralVaSaldUpperPacket where objective := "Return to the guided/general VA-SALD path and select the discrete general derivative side-condition bridge as the next lower target: EM endpoint laws, conditional-drift Fokker--Planck split, slowed transport velocity, frozen/residual algebra, the two sigma_eta^2/8 Young splits, LSI bookkeeping, residual DV witness handoff, and stitched time-change interfaces from appendix.tex:1354-1598." sourceLabels := [ "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "eq:general_discrete_delta_def", "lem:frozen_delta_cross_lip", "eq:LSI-KL-FI", "lem:dv_variation" ] modeDiscipline := [ "faithfulPaper: use appendix.tex:1354-1598 for the discrete derivative route, appendix.tex:1313-1347 for the fixed theorem display, and appendix.tex:724-951 plus main_body.tex:359-395 only as upstream guided/general dependencies; sald_version_2.tex remains out of scope.", "Preserve the source decomposition (sigma_eta^2/2)*nabla log tilde pi_s - bar b_{k,s} + tilde v_s = delta_pi^VA + dot{t}(s)*m_{t(s)} and the two Young splits with sigma_eta^2/8 each.", "Keep the resulting coefficients exactly: 2*sigma_eta^(-2)*dot{t}(s)^2 on the residual energy, 2*Gamma(t(s))*eta^2*alpha'^(-1) on KL, and 2*Delta(t(s))*eta as the additive frozen-delta term before time change.", "Treat EM endpoint laws, conditional drift/disintegration, Fokker--Planck, integration by parts, LSI-to-KL/FI, residual DV finite-log-mgf, stitched time change, and later Gronwall as obligations until compiled Lean proofs replace them." ] nonGoals := [ "Do not restate or prove thm:general-moving-target-SALD-discrete, thm:general-moving-target-SALD, or thm:unified-forward-KL in this packet.", "Do not reopen the final Gronwall display bridge selected in cycle 20 except as a downstream dependency of the derivative inequality.", "Do not replace the paper's EM interval derivative route with a path-space, Girsanov, Pinsker, Talagrand, or direct guided VA-SALD proof.", "Do not simplify away the doubled residual coefficient, the Gamma/Delta terms, alpha, alpha', eta, sigma_eta, dot{t}, dot{s}, or the constant-schedule hypothesis." ] lowerPacket := [ "Target exactly SALD.generalMovingTargetDiscreteDerivativeSideConditionContract / SALD.generalMovingTargetDiscreteDerivativeSideConditionObligation / sald.general_moving_target_discrete.derivative_side_conditions.", "Preferred first lower sub-slice: appendix.tex:1469-1511, proving or refining only the frozen/residual algebra and Young coefficient bookkeeping that yields the residual coefficient 2*sigma_eta^(-2)*dot{t}(s)^2 and frozen coefficients 2*Gamma*eta^2*alpha'^(-1), 2*Delta*eta.", "Keep endpoint laws, conditional Fokker--Planck, LSI-to-KL/FI, residual DV finite-log-mgf, s-to-t time change, and Gronwall as named dependencies unless a separate compiled proof handles exactly one of them.", "If conditional drift, divergence/integration-by-parts, slowed transport velocity, or stitched regularity is blocked, refine the corresponding source-contract gap rather than adding assumptions to the theorem statement." ] reviewerChecklist := [ "SALD.generalVaSaldDiscreteProofDag contains ASTIS.SALD.general_moving_target_discrete.cycle28_upper_derivative_side_conditions before the derivative_side_conditions block.", "SALD.saldDependenciesForLabel \"thm:general-moving-target-SALD-discrete\" includes SALD.cycle28GeneralVaSaldUpperPacket while retaining cycle-20 Gronwall and all existing EM, frozen-delta, DV, derivative, and Gronwall obligations.", "SALD.generalMovingTargetDiscreteDerivativeSideConditionObligation remains an obligation and preserves appendix.tex:1469-1511 source coefficients exactly.", "The unified theorem remains only the source specialization c_t<-u_t after the correction-field transport bridge; no direct VA-SALD proof route is introduced.", "The conversion window, proof-obligation ledger, SLT audit, source index, and dialogue handoff stay synchronized, sald_version_2.tex is excluded, and python3 tools/astis.py check passes." ] status := ProofStatus.obligation /-- Cycle-28 middle packet for the discrete general VA-SALD derivative side. This translates the upper-selected source slice `appendix.tex:1469-1511` into a lower-ready source-to-Lean map. It keeps the theorem display fixed and narrows lower work to the frozen/residual decomposition plus the two Young coefficient splits before LSI, DV, time change, or Gronwall proof search. -/ def cycle28GeneralVaSaldMiddleContract : GeneralVaSaldGuidedPathMiddleContract where guidedResidualSource := saldGuidedResidualSource generalContinuousSource := saldGeneralMovingTargetSource unifiedSource := saldUnifiedForwardKlSource discreteSource := saldGeneralMovingTargetDiscreteSource objective := "Translate appendix.tex:1469-1511 into a lower-ready middle ledger for sald.general_moving_target_discrete.derivative_side_conditions, preserving the frozen/residual algebra, the two sigma_eta^2/8 Young splits, and the resulting residual/Gamma/Delta coefficients while leaving EM/Fokker--Planck, LSI, DV, time-change, and Gronwall backends as named obligations." sourceStepMap := [ "appendix.tex:1469-1478 rewrites (sigma_eta^2/2)*nabla log tilde pi_s - bar b_{k,s} + tilde v_s as delta_pi^VA + dot{t}(s)*m_{t(s)} using eq:general_discrete_delta_def and m_t=v_t-c_t.", "appendix.tex:1481-1488 substitutes that decomposition into the KL derivative, splitting the cross term into frozen delta_pi^VA and residual dot{t}(s)*m_{t(s)} pieces.", "appendix.tex:1493-1501 applies Young to the residual cross term with one sigma_eta^2/8 FI share and produces 2*sigma_eta^(-2)*dot{t}(s)^2*||m_{t(s)}||_{L2(hat rho_s)}^2.", "appendix.tex:1503-1511 applies the frozen-delta lemma to the delta_pi^VA cross term, consuming the second sigma_eta^2/8 FI share and yielding 2*Gamma(t(s))*eta^2*alpha'^(-1)*KL plus 2*Delta(t(s))*eta.", "appendix.tex:1513-1524 combines both Young outputs with the original -(sigma_eta^2/2)*FI dissipation, leaving -(sigma_eta^2/4)*FI before the LSI handoff." ] leanStepMap := [ "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract.frozenResidualAlgebra records the vector-field rewrite; SALD.generalMovingTargetDiscreteFrozenResidualAlgebraVector formalizes the module-level algebra once delta_pi^VA, tilde v_s, and m_t are identified.", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract.youngCoefficientBookkeeping records the two sigma_eta^2/8 Young splits and the exact coefficients.", "SALD.generalMovingTargetDiscreteYoungFisherShareScalar formalizes only the real identity that one quarter of (sigma_eta^2/2)*FI is (sigma_eta^2/8)*FI.", "SALD.generalMovingTargetDiscreteTwoYoungFisherBudgetScalar formalizes only the real budget that two sigma_eta^2/8 shares leave -(sigma_eta^2/4)*FI.", "SALD.generalMovingTargetDiscreteResidualYoungCoefficientScalar formalizes only the residual Young coefficient rewrite with epsilon=sigma_eta^2/4 after sigma_eta^(-2) is identified.", "SALD.cycle28GeneralVaSaldDerivativeSideMiddleObligation keeps this middle map synchronized with sald.general_moving_target_discrete.derivative_side_conditions." ] citedResultInterfaces := [ "lem:frozen_delta_cross_lip is still an obligation through SALD.generalMovingTargetDiscreteFrozenDeltaObligation; this packet only records its use at appendix.tex:1503-1511.", "eq:LSI-KL-FI starts after the two Young splits, at appendix.tex:1526-1542, and remains the inherited density-test obligation.", "lem:dv_variation starts at appendix.tex:1544 and remains source-cited through the residual DV finite-log-mgf witness.", "No SLT theorem applies to the frozen/residual algebra or the local Young coefficient bookkeeping." ] obligations := [ "sald.general_moving_target_discrete.cycle28_derivative_side_middle", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "probability.lsi_to_kl_fi", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.forward_kl.schedule_time_change", "sald.general_moving_target_discrete.kl_derivative" ] lowerPacket := [ "Target exactly SALD.generalMovingTargetDiscreteDerivativeSideConditionContract / SALD.generalMovingTargetDiscreteDerivativeSideConditionObligation / sald.general_moving_target_discrete.derivative_side_conditions.", "Preferred lower sub-slice: prove or refine the frozen/residual algebra in appendix.tex:1469-1478, then use the compiled scalar Young bookkeeping helpers only after the analytic Young and FI identities are supplied.", "Keep the frozen-delta lemma, LSI-to-KL/FI, residual DV finite-log-mgf, s-to-t time change, and final Gronwall/display bridge as separate dependencies.", "Do not add regularity, endpoint, positivity, finite-energy, or coefficient assumptions to thm:general-moving-target-SALD-discrete; refine named obligations instead." ] reviewerChecklist := [ "Every source step from appendix.tex:1469-1511 is classified as Lean contract, compiled scalar helper, cited result, or named obligation.", "SALD.generalVaSaldDiscreteContract lists SALD.cycle28GeneralVaSaldDerivativeSideMiddleObligation and still keeps the theorem contractOnly.", "SALD.generalVaSaldDiscreteProofDag contains ASTIS.SALD.general_moving_target_discrete.cycle28_middle_derivative_side_conditions between the upper packet and derivative_side_conditions block.", "SALD.saldDependenciesForLabel \"thm:general-moving-target-SALD-discrete\" includes SALD.cycle28GeneralVaSaldMiddleContract, sald.general_moving_target_discrete.cycle28_derivative_side_middle, and the three compiled scalar Young bookkeeping helpers.", "The theorem display, source coefficients, source file selection, and analytic dependency statuses are unchanged." ] status := ProofStatus.obligation def cycle16UnifiedForwardKlTransportBridgeMiddleContract : UnifiedForwardKlTransportBridgeMiddleContract where residualIdentitySource := saldGuidedResidualMainSource correctionEquationSource := saldCorrectionFieldSource transportBridgeSource := saldUnifiedTransportBridgeSource specializationProofSource := saldUnifiedForwardKlProofSource objective := "Translate the cycle-16 upper packet into a lower-ready line ledger for the unified VA-SALD transport bridge, without changing the theorem statement or proving the general theorem." sourceAlgebraLedger := [ "main_body.tex:359-363 states partial_t pi_t+div(pi_t*u_t)=-pi_t*(g_t-E_pi_t[g_t]) and points back to prop:guided_path_residual.", "main_body.tex:364-367 introduces w_t through div(pi_t*w_t)=pi_t*(g_t-E_pi_t[g_t]).", "main_body.tex:368 concludes by cancellation that u_t+w_t is a transport velocity field for pi_t.", "appendix.tex:949-951 proves thm:unified-forward-KL only by specializing thm:general-moving-target-SALD with c_t<-u_t." ] leanInterfaceMap := [ "SALD.guidedResidualIdentityContract and sald.guided_path_residual.identity provide the centered residual identity.", "SALD.unifiedForwardKlSpecializationContract records the correction-field equation, v_t=u_t+w_t, c_t=u_t, and m_t=w_t.", "SALD.cycle16UnifiedForwardKlTransportBridgeLowerContract isolates the signed cancellation and divergence-linearity backend for the lower slice.", "SALD.unifiedForwardKlTransportBridgeObligation is the lower target for the local divergence cancellation.", "SALD.unifiedForwardKlSpecializationObligation remains the theorem-level specialization after the transport bridge is available." ] citedResultInterfaces := [ "No SLT theorem applies to this bridge; it is local continuity-equation and divergence algebra.", "The later residual DV, Gronwall, and discrete EM uses stay under their existing obligations." ] obligations := [ "sald.guided_path_residual.identity", "sald.unified_forward_kl.transport_bridge_lower", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization" ] lowerPacket := [ "Target sald.unified_forward_kl.transport_velocity_bridge only.", "Prove or refine only the cancellation between the two displayed equations in main_body.tex:359-368.", "Use SALD.cycle16UnifiedForwardKlTransportBridgeLowerContract for the exact residual equation, correction equation, divergence-linearity, and cancellation ledger.", "Keep existence, regularity, boundary conditions, and weak/divergence interpretation for w_t as explicit local obligations.", "After the bridge, record v_t=u_t+w_t, c_t=u_t, and m_t=w_t; do not start DV, Gronwall, or discrete EM proof search." ] reviewerChecklist := [ "The line ledger cites main_body.tex:359-368 and appendix.tex:949-951, not sald_version_2.tex.", "SALD.unifiedForwardKlTransportBridgeObligation remains an obligation and is not replaced by a direct VA-SALD proof.", "SALD.unifiedForwardKlSpecializationObligation still depends on sald.unified_forward_kl.transport_velocity_bridge.", "No correction-field existence theorem or SLT reuse claim is marked formalized." ] status := ProofStatus.obligation def cycle16UnifiedForwardKlTransportBridgeLowerContract : UnifiedForwardKlTransportBridgeLowerContract where sourceBlock := saldUnifiedTransportBridgeSource parentPacket := "SALD.cycle16UnifiedForwardKlTransportBridgeMiddleContract" targetObligation := "sald.unified_forward_kl.transport_velocity_bridge" residualEquation := "main_body.tex:359-363 gives partial_t pi_t + div(pi_t*u_t) = -pi_t*(g_t - E_pi_t[g_t]) from prop:guided_path_residual." correctionEquation := "main_body.tex:364-367 assumes div(pi_t*w_t) = pi_t*(g_t - E_pi_t[g_t]) for the correction field w_t." divergenceLinearity := "A future Lean proof needs a weak/product-divergence backend rewriting div(pi_t*(u_t+w_t)) as div(pi_t*u_t)+div(pi_t*w_t), with the required integrability and boundary conditions." cancellationStep := "Adding the residual and correction displays cancels -pi_t*(g_t-E_pi_t[g_t]) with +pi_t*(g_t-E_pi_t[g_t]), yielding partial_t pi_t + div(pi_t*u_t) + div(pi_t*w_t) = 0." transportVelocityConclusion := "After the divergence-linearity rewrite, the source conclusion is partial_t pi_t + div(pi_t*(u_t+w_t)) = 0, so v_t=u_t+w_t transports pi_t." specializationIdentifications := "For appendix.tex:949-951, set c_t=u_t in thm:general-moving-target-SALD; with v_t=u_t+w_t, the residual m_t=v_t-c_t is w_t." exclusions := [ "Do not prove existence, uniqueness, regularity, or boundary conditions for w_t in this lower slice.", "Do not prove thm:unified-forward-KL directly; the source route remains specialization of thm:general-moving-target-SALD.", "Do not enter residual DV, Gronwall, pure-contraction, or discrete EM proof search." ] dependencies := [ "SALD.cycle16UnifiedForwardKlTransportBridgeMiddleContract", "SALD.unifiedForwardKlSpecializationContract", "sald.guided_path_residual.identity", "eq:poisson-eq", "TransportVelocityContract" ] sourceGaps := [ "The paper states the correction equation but does not construct w_t or prove its regularity.", "The divergence algebra is written in classical notation; Lean still needs a weak or classical divergence interface for pi_t*u_t, pi_t*w_t, and pi_t*(u_t+w_t).", "Boundary/no-flux or decay assumptions needed to interpret the continuity equation are not separated in the source theorem." ] status := ProofStatus.obligation def cycle13FirstAppendixVocabularyPacket : FirstAppendixVocabularyPacket where objective := "Refresh the source-index and first appendix/vocabulary contracts for lem:gronwall, lem:dv_variation, def:PI, and eq:LSI-KL-FI, so lower work can refine one backend without drifting from the original paper." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "def:PI", "eq:LSI-KL-FI" ] modeDiscipline := [ "faithfulPaper: use /home/nitanda_sub/mark/repos/sald/paper/appendix.tex:47-94 and main_body.tex:202-215, with sald_version_2.tex excluded.", "Preserve the source Gronwall signs, DV supremum formula and finite-log-mgf condition, PI constant convention, and LSI-to-KL/FI coefficient 1/(2*C_LSI).", "Treat Gronwall as a local real-analysis obligation, DV as source-cited plus local instantiation obligations, PI as a definition plus downstream velocity-norm obligation, and LSI-to-KL/FI as a density-test obligation." ] nonGoals := [ "Do not restate or prove any forward-KL, discrete forward-KL, guided, or general VA-SALD theorem in this cycle.", "Do not replace the source route by Pinsker, Talagrand, Girsanov, or PI-to-LSI reasoning.", "Do not add endpoint, smoothness, finite-log-mgf, absolute-continuity, or Sobolev assumptions silently to theorem statements.", "Do not mark Gronwall, DV, PI velocity bounds, or LSI-to-KL/FI as formalized until they build locally." ] lowerPacket := [ "Target exactly one first-layer interface: SALD.saldGronwallEndpointCalculusContract, SALD.saldDvFiniteLogMgfContract, SALD.saldPiVelocityNormDependencyContract, or SALD.saldLsiKlFiDensityTestContract.", "Before editing, check research-wiki/source-index/SALD_original.jsonl for the source label and line, then preserve the matching SourceAnchor in Lean.", "If the backend is not ready, refine the named ProofObligation only; do not promote the claim or add theorem-level assumptions.", "Keep SALD.firstFaithfulLabels and SALD.saldFirstProofDag synchronized with the four focus labels." ] reviewerChecklist := [ "python3 tools/astis.py source-index ASTIS-SALD-001 indexes lem:gronwall, lem:dv_variation, def:PI, and eq:LSI-KL-FI and excludes sald_version_2.tex.", "SALD.cycle13FirstAppendixSourceIndexAuditContract maps each focus label to its Lean contract and obligation status.", "SALD.saldFirstProofDag exposes the first-layer dependencies without changing any theorem status to formalized.", "The conversion window, proof-obligation ledger, and SLT audit all state that this cycle is source-index/contract synchronization only.", "The fake-proof scan finds no axiom, sorry, admit, Prop := True, or := trivial proof closure." ] status := ProofStatus.obligation def cycle13FirstAppendixSourceIndexAuditContract : FirstAppendixSourceIndexAuditContract where sourceIndexPath := "research-wiki/source-index/SALD_original.jsonl" sourceRoot := saldPaperRoot excludedFiles := ["sald_version_2.tex"] indexedLabels := [ "appendix.tex:47 lem:gronwall", "appendix.tex:73 lem:dv_variation", "appendix.tex:86 def:PI", "main_body.tex:202 eq:LSI-KL-FI" ] sourceLineMap := [ "lem:gronwall -> appendix.tex:47-71, integrating-factor proof on [0,t1].", "lem:dv_variation -> appendix.tex:73-79, Boucheron Corollary 4.15 with finite log-mgf condition.", "def:PI -> appendix.tex:86-94, variance bound with constant C_PI > 0.", "eq:LSI-KL-FI -> main_body.tex:202-215, LSI definition, phi=sqrt(rho/pi), KL/FI vocabulary, and coefficient 1/(2*C_LSI)." ] leanContractMap := [ "lem:gronwall -> SALD.gronwallContract, SALD.saldGronwallCandidateContract, SALD.saldGronwallEndpointCalculusContract.", "lem:dv_variation -> SALD.dvContract, SALD.saldDvFiniteLogMgfContract, dvVariationalObligation saldDvVariationSource.", "def:PI -> SALD.piDefinitionContract, SALD.saldPIContract, SALD.saldPiVelocityNormDependencyContract.", "eq:LSI-KL-FI -> SALD.lsiKlFiVocabularyContract, SALD.saldKLContract, SALD.saldFIContract, SALD.saldLSIContract, SALD.saldLsiKlFiDensityTestContract." ] obligationMap := [ "lem:gronwall remains sald.gronwall.integrating_factor and sald.gronwall.endpoint_calculus obligations.", "lem:dv_variation remains probability.dv_variational_formula source-cited plus sald.dv_variation.finite_log_mgf_interface.", "def:PI is contract-only; appendix.tex:96-151 velocity-norm use remains sald.pi.velocity_norm_backend.", "eq:LSI-KL-FI remains sald.lsi_kl_fi.density_test_interface and probability.lsi_to_kl_fi obligations." ] lowerPacket := [ "Choose one focus label and refine its existing contract/obligation only.", "Use the source-index line as the first citation in any conversion-window or proof-obligation update.", "Keep sald_version_2.tex excluded and do not add labels outside the original source root.", "Leave theorem contracts for thm:forward-KL and later SALD results unchanged." ] reviewerChecklist := [ "The source-index file contains all four focus labels with the listed line numbers.", "No SourceAnchor path points to sald_version_2.tex.", "Each focus label maps to a Lean-facing contract and an honest non-formalized status.", "The mandatory source-index and check commands pass." ] status := ProofStatus.obligation def cycle13FirstAppendixMiddleAuditContract : FirstAppendixMiddleAuditContract where sourceIndexPath := "research-wiki/source-index/SALD_original.jsonl" focusLabels := [ "lem:gronwall", "lem:dv_variation", "def:PI", "eq:LSI-KL-FI" ] sourceReadWindows := [ "appendix.tex:47-71: Gronwall statement and integrating-factor proof.", "appendix.tex:73-79: DV variational formula cited from Boucheron Cor. 4.15.", "appendix.tex:86-151: PI definition plus velocity-norm Sobolev/Riesz proof route.", "main_body.tex:202-215: LSI definition, phi=sqrt(rho/pi), and KL/FI vocabulary." ] sourceStepMap := [ "lem:gronwall: start from dK/dt <= -a_t*K_t+b_t, differentiate exp(int_0^t a)*K, integrate from 0 to t1, then rewrite exp(-int_0^t1 a)*exp(int_0^t a) as exp(-int_t^t1 a); the source assumes continuous a,b and differentiable K on [0,t1].", "lem:dv_variation: use the cited equality KL(nu||mu)=sup_Z(E_nu Z-log E_mu exp Z) over finite-log-mgf tests; later SALD uses only instantiations with squared velocity or residual fields.", "def:PI: record Var_mu(phi) <= C_PI^{-1}*int ||nabla phi||^2 dmu, then map the appendix Sobolev/Riesz velocity-norm proof to a separate backend obligation instead of folding it into the definition.", "eq:LSI-KL-FI: apply LSI to phi=sqrt(rho/pi), identify entropy with KL and the gradient term with one quarter of FI, preserving the coefficient 1/(2*C_LSI)." ] leanStepMap := [ "lem:gronwall -> SALD.saldGronwallCandidateContract, SALD.saldGronwallEndpointCalculusContract, SALD.saldGronwallExponentRewriteContract, sald.gronwall.integrating_factor, sald.gronwall.endpoint_calculus, and sald.gronwall.exponent_rewrite.", "lem:dv_variation -> SALD.dvContract, SALD.saldDvFiniteLogMgfContract, probability.dv_variational_formula, and sald.dv_variation.finite_log_mgf_interface.", "def:PI -> SALD.saldPIContract, SALD.piDefinitionContract, SALD.saldPiVelocityNormDependencyContract, and sald.pi.velocity_norm_backend.", "eq:LSI-KL-FI -> SALD.saldKLContract, SALD.saldFIContract, SALD.saldLSIContract, SALD.saldLsiKlFiBridgeContract, SALD.saldLsiKlFiDensityTestContract, sald.lsi_kl_fi.density_test_interface, and probability.lsi_to_kl_fi." ] citedResultMap := [ "DV is the only external cited result in the focus window; it remains source-cited from Boucheron Cor. 4.15 and is not imported from SLT or Mathlib.", "Gronwall is a local real-calculus obligation; possible Mathlib interval-integral and derivative lemmas are backend candidates only.", "PI and LSI/KL/FI vocabulary require local measure/Sobolev/density-test backends; no SLT theorem is marked as formalized." ] obligationMap := [ "Gronwall blocked on endpoint-safe differentiability or absolute-continuity formulation, interval-integral FTC, order integration, endpoint evaluation, and the separated exponent rewrite sald.gronwall.exponent_rewrite.", "DV blocked on common probability space, measurable test variables, finite log-mgf witnesses, and alpha-complexity monotonicity for theorem-specific squared-norm tests.", "PI velocity bound blocked on weighted mean-zero Sobolev Hilbert structure, bounded functional T_mu, Riesz representation, weak PDE regularity, and boundary handling.", "LSI/KL/FI blocked on Radon-Nikodym density vocabulary, admissibility or approximation for sqrt(rho/pi), entropy rewrite, FI chain rule, and coefficient audit." ] lowerPacket := [ "Preferred lower target: SALD.saldGronwallEndpointCalculusContract and its lower sub-obligation SALD.saldGronwallExponentRewriteContract / sald.gronwall.exponent_rewrite, because the source proof is self-contained real calculus and does not require the probability backend.", "Alternative lower targets: SALD.saldDvFiniteLogMgfContract, SALD.saldPiVelocityNormDependencyContract, or SALD.saldLsiKlFiDensityTestContract.", "Refine one interface only; if a full backend proof is not ready, add a narrower ProofObligation instead of changing theorem statements.", "Preserve the SourceAnchor from SALD_original.jsonl and leave sald_version_2.tex excluded." ] reviewerChecklist := [ "The middle map cites the same four source-index labels and line windows as the upper packet.", "Each source proof step is classified as Lean contract, cited result, or named obligation.", "No first-layer status is upgraded to formalized and no later SALD theorem target is edited.", "Conversion window, proof-obligation ledger, SLT audit, and saldFirstProofDag mention the middle map." ] status := ProofStatus.obligation /-- Cycle-17 upper packet for rebaselining the first appendix/vocabulary layer. This returns to the source-index focus after the cycle-16 unified transport bridge work. It is workflow data only: the four source labels remain fixed, the existing first-layer contracts remain the Lean targets, and every analytic backend keeps its honest non-formalized status. -/ def cycle17FirstAppendixVocabularyPacket : FirstAppendixVocabularyPacket where objective := "Rebaseline the source-index and first appendix/vocabulary contracts for lem:gronwall, lem:dv_variation, def:PI, and eq:LSI-KL-FI after the cycle-16 unified transport bridge, before assigning any new analytic proof search." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "def:PI", "eq:LSI-KL-FI" ] modeDiscipline := [ "faithfulPaper: use only /home/nitanda_sub/mark/repos/sald/paper/appendix.tex:47-94 and main_body.tex:202-215 for this packet; sald_version_2.tex remains excluded.", "Preserve the source Gronwall sign convention, DV finite-log-mgf supremum formula, PI constant convention C_PI^{-1}, and LSI-to-KL/FI coefficient 1/(2*C_LSI).", "Keep this as Phase 1 transcript and obligation refinement; do not reorganize reusable APIs or generalize beyond the original paper." ] nonGoals := [ "Do not edit thm:forward-KL, thm:forward-KL-discrete, thm:general-moving-target-SALD, thm:unified-forward-KL, or thm:general-moving-target-SALD-discrete.", "Do not replace the paper route by Pinsker, Talagrand, PI-to-LSI, Girsanov, or a direct VA-SALD proof.", "Do not add endpoint, smoothness, finite-log-mgf, absolute-continuity, Sobolev, or boundary assumptions to theorem statements; keep them as named source gaps or obligations.", "Do not mark Gronwall, DV, PI velocity bounds, or LSI-to-KL/FI formalized unless a local Lean proof builds." ] lowerPacket := [ "Middle must keep two-way synchronization across Lean, conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, and the exact TeX line windows.", "Preferred lower target: tighten SALD.cycle13FirstAppendixSourceIndexAuditContract / sald.first_appendix.source_index_audit against SALD_original.jsonl, then hand off exactly one existing analytic sub-obligation.", "If an analytic lower slice is chosen, prefer SALD.saldGronwallExponentRewriteContract / sald.gronwall.exponent_rewrite because it is a local real-algebra substep of appendix.tex:63-69.", "Alternative lower slices remain SALD.saldDvFiniteLogMgfContract, SALD.saldPiVelocityNormDependencyContract, and SALD.saldLsiKlFiDensityTestContract; refine only one and keep statuses unchanged." ] reviewerChecklist := [ "python3 tools/astis.py source-index ASTIS-SALD-001 refreshes SALD_original.jsonl with lem:gronwall, lem:dv_variation, def:PI, and eq:LSI-KL-FI while excluding sald_version_2.tex.", "SALD.saldFirstProofDag still reports Gronwall obligation, DV sourceCited, PI contractOnly, and LSI/KL/FI obligation.", "The conversion window and proof-obligation ledger identify cycle 17 as source-index/contract synchronization, not a proof of the analytic backends.", "No fake proof closure appears and python3 tools/astis.py check passes." ] status := ProofStatus.obligation /-- Cycle-17 middle source-to-Lean rebaseline for the first appendix layer. This translates the cycle-17 upper source-index packet into a lower-ready source map. It deliberately reuses the existing first-layer contracts and selects only the Gronwall exponent rewrite as the next narrow analytic slice. -/ def cycle17FirstAppendixMiddleAuditContract : FirstAppendixMiddleAuditContract where sourceIndexPath := "research-wiki/source-index/SALD_original.jsonl" focusLabels := [ "lem:gronwall", "lem:dv_variation", "def:PI", "eq:LSI-KL-FI" ] sourceReadWindows := [ "appendix.tex:47-71: Gronwall statement, integrating-factor derivative, integration from 0 to t1, and exponent rewrite at lines 63-69.", "appendix.tex:73-79: Donsker--Varadhan variational formula cited from Boucheron Cor. 4.15 with finite log-mgf tests.", "appendix.tex:86-94 plus appendix.tex:96-151: PI definition and the downstream Sobolev/Riesz velocity-norm route.", "main_body.tex:202-215: LSI definition, the phi=sqrt(rho/pi) bridge, KL/FI vocabulary, and the coefficient 1/(2*C_LSI)." ] sourceStepMap := [ "lem:gronwall: the middle target is the final display rewrite, after multiplying by exp(-int_0^t1 a), from exp(-int_0^t1 a)*exp(int_0^t a) to exp(-int_t^t1 a).", "lem:dv_variation: keep the equality as a cited formula and expose only common-space, measurable-test, and finite-log-mgf interfaces for later SALD instantiations.", "def:PI: keep the definition contract-only and route the mean-zero Sobolev, bounded-functional, Riesz, weak-PDE, and velocity-bound steps to sald.pi.velocity_norm_backend.", "eq:LSI-KL-FI: preserve the paper substitution phi=sqrt(rho/pi), density normalization, entropy rewrite, FI chain rule, and final coefficient audit." ] leanStepMap := [ "lem:gronwall -> SALD.saldGronwallExponentRewriteContract, SALD.gronwallExponentRewriteObligation, and the parent SALD.saldGronwallEndpointCalculusContract.", "lem:dv_variation -> SALD.dvContract, SALD.saldDvFiniteLogMgfContract, probability.dv_variational_formula, and sald.dv_variation.finite_log_mgf_interface.", "def:PI -> SALD.saldPIContract, SALD.piDefinitionContract, SALD.saldPiVelocityNormDependencyContract, and sald.pi.velocity_norm_backend.", "eq:LSI-KL-FI -> SALD.saldLsiKlFiBridgeContract, SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiVocabularyContract, sald.lsi_kl_fi.density_test_interface, and probability.lsi_to_kl_fi." ] citedResultMap := [ "DV remains the only externally cited result in this focus window; it is still Boucheron Cor. 4.15 / possible SLT entropy_duality pattern only, not a local import.", "Gronwall exponent rewriting is local real interval-integral and Real.exp algebra; the scalar Real.exp product substep is compiled locally, and no SLT result applies.", "PI and LSI/KL/FI use local measure, Sobolev, and density-test obligations; no SLT theorem is marked formalized." ] obligationMap := [ "sald.gronwall.exponent_rewrite: interval-integral additivity on 0<=t<=t1, compiled scalar exponent algebra, compiled outer-integral congruence, and remaining theorem-specific adjacent interval-integrability.", "sald.dv_variation.finite_log_mgf_interface: common measurable space, measurable Z, finite log E_mu[exp Z], and theorem-specific alpha-complexity witnesses.", "sald.pi.velocity_norm_backend: weighted mean-zero Sobolev Hilbert backend, bounded T_mu, Riesz representation, weak PDE interpretation, and boundary regularity.", "sald.lsi_kl_fi.density_test_interface: Radon-Nikodym density, smooth/admissible sqrt test, entropy identity, FI chain rule, and coefficient audit." ] lowerPacket := [ "Target SALD.saldGronwallExponentRewriteContract / sald.gronwall.exponent_rewrite only.", "Preserve appendix.tex:63-69 exactly: no sign assumption on a or b, no change to the additive b_t integral, and no change to the final Gronwall bound.", "If the interval-integral backend is not ready, keep the scalar exponent lemma as the only formalized substep and refine the remaining obligation rather than proving lem:gronwall or changing theorem statements.", "Leave DV source-cited, PI contract-only, and LSI/KL/FI as obligations." ] reviewerChecklist := [ "SALD_original.jsonl indexes the four focus labels and excludes sald_version_2.tex.", "SALD.saldFirstProofDag dependencies for the four labels include the cycle-17 middle rebaseline.", "Conversion window, proof-obligation ledger, and SLT audit name this middle packet and its Gronwall exponent-rewrite lower target.", "No analytic backend is promoted to formalized and the mandatory check passes." ] status := ProofStatus.obligation /-- Cycle-21 upper packet for the first appendix/vocabulary source-index layer. This returns to the original first-DAG labels after the cycle-20 discrete general VA-SALD scalar coefficient work. It is an upper-role selection packet: the source theorem statements stay fixed, the source-index and conversion window must remain synchronized, and lower work should refine one existing first-layer obligation rather than reopening a theorem proof. -/ def cycle21FirstAppendixVocabularyPacket : FirstAppendixVocabularyPacket where objective := "Rebaseline the source-index and first appendix/vocabulary contracts for lem:gronwall, lem:dv_variation, def:PI, and eq:LSI-KL-FI after the cycle-20 discrete general VA-SALD scalar coefficient work, before assigning any new theorem-level proof search." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "def:PI", "eq:LSI-KL-FI" ] modeDiscipline := [ "faithfulPaper: use only /home/nitanda_sub/mark/repos/sald/paper/appendix.tex:47-94, appendix.tex:96-151 for the PI velocity-norm dependency, and main_body.tex:202-215; sald_version_2.tex remains excluded.", "Preserve the source Gronwall signs and endpoint display, DV finite-log-mgf variational formula, PI constant convention C_PI^{-1}, and LSI-to-KL/FI coefficient 1/(2*C_LSI).", "Keep Phase 1 as transcript and proof-obligation refinement: Gronwall and LSI-to-KL/FI remain obligations, DV remains source-cited plus instantiation obligations, and PI remains contract-only with a separate velocity-norm backend." ] nonGoals := [ "Do not edit thm:forward-KL, thm:forward-KL-discrete, prop:guided_path_residual, thm:general-moving-target-SALD, thm:unified-forward-KL, or thm:general-moving-target-SALD-discrete.", "Do not replace the paper route by Pinsker, Talagrand, PI-to-LSI, Girsanov, or a direct VA-SALD theorem proof.", "Do not add endpoint, smoothness, finite-log-mgf, absolute-continuity, Sobolev, or boundary assumptions to theorem statements; keep them as named obligations or source gaps.", "Do not mark Gronwall, DV, PI velocity bounds, or LSI-to-KL/FI formalized unless a local Lean proof builds." ] lowerPacket := [ "Middle must keep Lean, conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, research-wiki/source-index/SALD_original.jsonl, and the exact TeX windows synchronized in both directions.", "Preferred lower target: SALD.saldGronwallEndpointCalculusContract / SALD.saldGronwallExponentRewriteContract / sald.gronwall.exponent_rewrite, because cycle 17 and cycle 18 already isolated compiled scalar exponent helpers while theorem-specific adjacent interval-integrability remains open.", "Alternative lower targets are SALD.saldDvFiniteLogMgfContract, SALD.saldPiVelocityNormDependencyContract, and SALD.saldLsiKlFiDensityTestContract; choose exactly one and refine only its named obligation.", "If the backend is blocked, record the exact source-contract gap instead of changing theorem statements or promoting a cited result." ] reviewerChecklist := [ "python3 tools/astis.py source-index ASTIS-SALD-001 refreshes SALD_original.jsonl with lem:gronwall, lem:dv_variation, def:PI, and eq:LSI-KL-FI while excluding sald_version_2.tex.", "SALD.saldFirstProofDag still reports Gronwall obligation, DV sourceCited, PI contractOnly, and LSI/KL/FI obligation, with no later theorem proof status upgraded.", "The conversion window, proof-obligation ledger, and SLT audit identify cycle 21 as source-index/first-appendix contract synchronization, not analytic proof closure.", "The mandatory gate python3 tools/astis.py check passes and the fake-proof scan remains clean." ] status := ProofStatus.obligation /-- Cycle-21 middle source-to-Lean audit for the first appendix layer. This is the middle-role transcript for the cycle-21 upper packet. It rereads the exact TeX windows, maps each proof step to an existing Lean-facing contract, cited result, or proof obligation, and selects a single lower target without changing theorem statements or analytic statuses. -/ def cycle21FirstAppendixMiddleAuditContract : FirstAppendixMiddleAuditContract where sourceIndexPath := "research-wiki/source-index/SALD_original.jsonl" focusLabels := [ "lem:gronwall", "lem:dv_variation", "def:PI", "eq:LSI-KL-FI" ] sourceReadWindows := [ "appendix.tex:47-71: Gronwall statement, integrating-factor derivative, integration from 0 to t1, and final exponent rewrite.", "appendix.tex:73-79: Donsker--Varadhan formula cited as Boucheron Cor. 4.15, with same-space probabilities and finite log-mgf tests.", "appendix.tex:86-151: PI definition, weighted mean-zero Sobolev space, bounded functional, Riesz step, and velocity-norm applications.", "main_body.tex:202-215: LSI definition, rho << pi density route with phi=sqrt(rho/pi), KL/FI comparison, and KL/FI vocabulary." ] sourceStepMap := [ "lem:gronwall: lines 47-52 state the differential inequality and endpoint bound; lines 58-61 differentiate exp(int_0^t a)*K; line 63 integrates; lines 65-69 multiply by exp(-int_0^t1 a) and rewrite the integral factor to exp(-int_t^t1 a).", "lem:dv_variation: lines 73-79 give the cited variational equality KL(nu||mu)=sup_Z(E_nu Z-log E_mu exp Z), with the finite log-mgf side condition as the only local instantiation interface.", "def:PI: lines 86-94 define PI by Var_mu(phi) <= C_PI^{-1} int ||nabla phi||^2 dmu; lines 96-151 use that definition for weighted Sobolev norm equivalence, weak PDE/Riesz representation, and A_0 velocity bounds.", "eq:LSI-KL-FI: lines 202-215 define LSI, substitute phi=sqrt(rho/pi), record KL <= FI/(2*C_LSI), and define the KL/FI integrals under rho << pi." ] leanStepMap := [ "lem:gronwall -> SALD.saldGronwallCandidateContract, SALD.saldGronwallEndpointCalculusContract, SALD.saldGronwallExponentRewriteContract, sald.gronwall.integrating_factor, sald.gronwall.endpoint_calculus, and sald.gronwall.exponent_rewrite.", "lem:dv_variation -> SALD.dvContract, SALD.saldDvFiniteLogMgfContract, probability.dv_variational_formula, and sald.dv_variation.finite_log_mgf_interface.", "def:PI -> SALD.saldPIContract, SALD.piDefinitionContract, SALD.saldPiVelocityNormDependencyContract, and sald.pi.velocity_norm_backend.", "eq:LSI-KL-FI -> SALD.saldKLContract, SALD.saldFIContract, SALD.saldLSIContract, SALD.saldLsiKlFiBridgeContract, SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiVocabularyContract, sald.lsi_kl_fi.density_test_interface, and probability.lsi_to_kl_fi." ] citedResultMap := [ "DV is source-cited from Boucheron Cor. 4.15; an SLT entropy_duality pattern is only a future port candidate and is not imported.", "Gronwall is local real/interval-integral calculus; existing scalar exponent helpers are formalized substeps only, not a proof of the lemma.", "PI velocity bounds and LSI-to-KL/FI remain local measure/Sobolev/density-test obligations; no SLT theorem is marked formalized." ] obligationMap := [ "sald.gronwall.exponent_rewrite remains blocked on endpoint-safe calculus/FTC choices, adjacent interval-integrability in theorem contexts, and congruence inside the b_t integral.", "sald.dv_variation.finite_log_mgf_interface remains blocked on common measurable space, measurable Z, finite log E_mu[exp Z], and alpha-complexity witnesses for squared fields.", "sald.pi.velocity_norm_backend remains blocked on weighted mean-zero Sobolev Hilbert structure, bounded T_mu, Riesz representation, weak PDE interpretation, and boundary regularity.", "sald.lsi_kl_fi.density_test_interface remains blocked on Radon-Nikodym density conventions, admissibility/approximation for sqrt(rho/pi), entropy rewrite, FI chain rule, and coefficient audit." ] lowerPacket := [ "Target exactly SALD.saldGronwallExponentRewriteContract / sald.gronwall.exponent_rewrite as the first lower slice.", "Permitted work: refine endpoint-safe derivative/FTC assumptions, adjacent interval-integrability, or the remaining integral-congruence obligation for appendix.tex:63-69.", "Keep the already formalized scalar helpers as substeps only; do not mark lem:gronwall formalized unless the full local theorem builds.", "Leave DV source-cited, PI contract-only, LSI/KL/FI obligation, and all forward-KL or VA-SALD theorem statements unchanged." ] reviewerChecklist := [ "SALD_original.jsonl indexes lem:gronwall, lem:dv_variation, def:PI, and eq:LSI-KL-FI from the original source and excludes sald_version_2.tex.", "SALD.saldFirstProofDag dependencies for the four focus labels include SALD.cycle21FirstAppendixMiddleAuditContract.", "The conversion window, proof-obligation ledger, SLT audit, and dialogue board identify cycle 21 middle as source-to-Lean synchronization, not analytic proof closure.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-25 upper packet for the first appendix/vocabulary layer. This returns to the source-index focus after the cycle-24 continuous general VA-SALD Gronwall coefficient work. It chooses the PI velocity-norm dependency as the next lower slice while keeping the four first-layer source labels and all analytic statuses fixed. -/ def cycle25FirstAppendixVocabularyPacket : FirstAppendixVocabularyPacket where objective := "Rebaseline the source-index and first appendix/vocabulary contracts for lem:gronwall, lem:dv_variation, def:PI, and eq:LSI-KL-FI after the cycle-24 general VA-SALD coefficient work, then select the def:PI velocity-norm dependency as the next lower obligation refinement." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "def:PI", "eq:LSI-KL-FI" ] modeDiscipline := [ "faithfulPaper: use only /home/nitanda_sub/mark/repos/sald/paper/appendix.tex:47-151 and main_body.tex:202-215 for this packet; sald_version_2.tex remains excluded.", "Preserve the source Gronwall signs and endpoint display, DV finite-log-mgf variational formula, PI constant convention C_PI^{-1}, and LSI-to-KL/FI coefficient 1/(2*C_LSI).", "Keep this as Phase 1 transcript and proof-obligation refinement: Gronwall and LSI-to-KL/FI remain obligations, DV remains source-cited plus instantiation obligations, and PI remains contract-only with the velocity-norm proof route tracked separately." ] nonGoals := [ "Do not edit thm:forward-KL, thm:forward-KL-discrete, prop:guided_path_residual, thm:general-moving-target-SALD, thm:unified-forward-KL, or thm:general-moving-target-SALD-discrete.", "Do not replace the paper route by Pinsker, Talagrand, PI-to-LSI, Girsanov, or a direct VA-SALD theorem proof.", "Do not add endpoint, smoothness, finite-log-mgf, absolute-continuity, Sobolev, Hilbert-space, weak-PDE, or boundary assumptions to theorem statements; keep them as named obligations or source gaps.", "Do not mark Gronwall, DV, PI velocity bounds, or LSI-to-KL/FI formalized unless a local Lean proof builds." ] lowerPacket := [ "Middle must keep Lean, conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, research-wiki/source-index/SALD_original.jsonl, and the exact TeX windows synchronized in both directions.", "Target exactly SALD.saldPiVelocityNormDependencyContract / SALD.piVelocityNormBackendObligation / sald.pi.velocity_norm_backend as the first lower slice.", "First sub-slice: appendix.tex:96-129, covering dot H^1(mu), the mean-zero interface, PI norm equivalence, and boundedness of T_mu before the Riesz representation step.", "If the Sobolev or weak-PDE backend is blocked, refine the named obligation with the exact source-contract gap instead of changing theorem statements or promoting the PI velocity bound." ] reviewerChecklist := [ "python3 tools/astis.py source-index ASTIS-SALD-001 refreshes SALD_original.jsonl with lem:gronwall, lem:dv_variation, def:PI, and eq:LSI-KL-FI while excluding sald_version_2.tex.", "SALD.saldFirstProofDag dependencies for the four focus labels include SALD.cycle25FirstAppendixVocabularyPacket and still report Gronwall obligation, DV sourceCited, PI contractOnly, and LSI/KL/FI obligation.", "The conversion window and proof-obligation ledger identify cycle 25 as source-index/first-appendix contract synchronization, with PI velocity-norm backend refinement as the lower target.", "The mandatory gate python3 tools/astis.py check passes and the fake-proof scan remains clean." ] status := ProofStatus.obligation /-- Cycle-25 middle source-to-Lean map for the first appendix layer. This translates the upper-selected PI velocity-norm backend into a lower-ready sub-slice while keeping the Gronwall, DV, PI, and LSI/KL/FI statuses fixed. -/ def cycle25FirstAppendixMiddleAuditContract : FirstAppendixMiddleAuditContract where sourceIndexPath := "research-wiki/source-index/SALD_original.jsonl" focusLabels := [ "lem:gronwall", "lem:dv_variation", "def:PI", "eq:LSI-KL-FI" ] sourceReadWindows := [ "appendix.tex:47-71: Gronwall statement and integrating-factor proof; unchanged status guard for this cycle.", "appendix.tex:73-79: Donsker--Varadhan formula cited as Boucheron Cor. 4.15; unchanged source-cited dependency.", "appendix.tex:86-151: PI definition plus the weighted Sobolev/Riesz velocity-norm route, with the lower sub-slice starting at appendix.tex:96-129.", "main_body.tex:202-215: LSI definition, phi=sqrt(rho/pi), KL/FI vocabulary, and coefficient 1/(2*C_LSI); unchanged obligation." ] sourceStepMap := [ "lem:gronwall: keep dK/dt <= -a_t*K_t+b_t and the endpoint exponent display as existing Gronwall obligations; no Gronwall proof search is selected in cycle 25.", "lem:dv_variation: keep the cited variational equality over finite-log-mgf random variables; local theorem instantiations still owe common-space, measurability, and finite-log-mgf witnesses.", "def:PI: lines 86-94 define PI; lines 96-103 define dot H^1(mu) and its gradient inner product; lines 104-112 use PI to identify the gradient norm as equivalent to the weighted H^1 norm on the mean-zero space; lines 114-129 introduce the weak PDE and the first Cauchy-Schwarz bound for T_mu before the Riesz step.", "def:PI follow-on: lines 130-138 are the immediate PI operator-norm and Riesz continuation after the selected first sub-slice; lower should expose the interface but not prove the Riesz backend unless it builds locally.", "eq:LSI-KL-FI: keep the source route phi=sqrt(rho/pi), entropy rewrite, FI chain rule, and coefficient audit in the existing density-test obligation." ] leanStepMap := [ "lem:gronwall -> SALD.saldGronwallCandidateContract, SALD.saldGronwallEndpointCalculusContract, SALD.saldGronwallExponentRewriteContract, and existing sald.gronwall.* obligations.", "lem:dv_variation -> SALD.dvContract, SALD.saldDvFiniteLogMgfContract, probability.dv_variational_formula, and sald.dv_variation.finite_log_mgf_interface.", "def:PI -> SALD.saldPIContract, SALD.piDefinitionContract, SALD.saldPiVelocityNormDependencyContract, SALD.piVelocityNormBackendObligation, and sald.pi.velocity_norm_backend.", "cycle 25 PI lower target -> SALD.cycle25FirstAppendixMiddleAuditContract and SALD.cycle25FirstAppendixPiVelocityNormMiddleObligation, refining only the appendix.tex:96-129 source-to-Lean interface.", "eq:LSI-KL-FI -> SALD.saldKLContract, SALD.saldFIContract, SALD.saldLSIContract, SALD.saldLsiKlFiDensityTestContract, sald.lsi_kl_fi.density_test_interface, and probability.lsi_to_kl_fi." ] citedResultMap := [ "DV remains the only external cited result in the first-layer window; no SLT theorem is imported or promoted.", "The PI velocity-norm route is local analysis: weighted Sobolev Hilbert structure, PI norm equivalence, bounded functional, Riesz representation, weak PDE, and boundary/regularity interfaces.", "Gronwall and LSI/KL/FI remain local Mathlib/measure obligations outside this lower slice." ] obligationMap := [ "sald.pi.velocity_norm_backend first sub-slice: formalize or specify dot H^1(mu), the mean-zero condition E_mu[psi]=0, the gradient inner product, the PI-to-H1 norm equivalence, and boundedness of T_mu before Riesz.", "sald.pi.velocity_norm_backend follow-on: line 119 has a phi/psi notation mismatch in the source, and lines 130-138 require the operator-norm, Riesz representation, weak-PDE interpretation, and boundary/regularity backend.", "sald.gronwall.exponent_rewrite, sald.dv_variation.finite_log_mgf_interface, and sald.lsi_kl_fi.density_test_interface remain unchanged obligations/source-cited dependencies.", "No theorem-level forward-KL, discrete, guided, general VA-SALD, or unified theorem statement is changed by this map." ] lowerPacket := [ "Target exactly SALD.saldPiVelocityNormDependencyContract / SALD.piVelocityNormBackendObligation / sald.pi.velocity_norm_backend.", "Start with appendix.tex:96-129: dot H^1(mu), mean-zero interface, gradient inner product, PI norm equivalence, weak-form notation, and the Cauchy-Schwarz boundedness of T_mu.", "Record line 119's phi/psi mismatch and the line 130-138 operator-norm/Riesz continuation as explicit source-contract gaps if the lower proof cannot close them.", "Do not reopen Gronwall, DV, LSI-to-KL/FI, forward-KL, guided, general VA-SALD, unified, or discrete proof search in this lower attempt." ] reviewerChecklist := [ "SALD_original.jsonl indexes lem:gronwall, lem:dv_variation, def:PI, lem:velocity-norm-bound, and eq:LSI-KL-FI from the original files while excluding sald_version_2.tex.", "SALD.saldFirstProofDag dependencies for the four first-layer labels include SALD.cycle25FirstAppendixMiddleAuditContract.", "def:PI dependencies include SALD.cycle25FirstAppendixPiVelocityNormMiddleObligation and still report PI as contractOnly, with velocity-norm as an obligation.", "Conversion window, proof-obligation ledger, SLT audit, and dialogue board classify cycle 25 middle as source-to-Lean synchronization, not analytic proof closure." ] status := ProofStatus.obligation /-- Cycle-25 middle obligation for the selected PI velocity-norm sub-slice. -/ def cycle25FirstAppendixPiVelocityNormMiddleObligation : ProofObligation where id := "sald.first_appendix.cycle25_pi_velocity_norm_middle" statement := "Maintain the cycle 25 lower-ready map for appendix.tex lines 96-129: define the weighted mean-zero Sobolev interface dot H^1(mu), expose the mean-zero/variance bridge and PI norm equivalence, and state the bounded-functional T_mu step before the Riesz representation theorem is used." source := saldPiVelocityNormSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle25FirstAppendixVocabularyPacket", "SALD.cycle25FirstAppendixMiddleAuditContract", "SALD.saldPIContract", "SALD.piDefinitionContract", "SALD.saldPiVelocityNormDependencyContract", "SALD.piVelocityNormBackendObligation", "sald.pi.velocity_norm_backend" ] note := "This is a middle-role synchronization obligation. It preserves PI as contract-only, keeps the velocity-norm bound as an obligation, and records line 119 notation plus the line 130-138 operator-norm/Riesz continuation as source-contract gaps rather than theorem assumptions." /-- Cycle-25 lower obligation after compiling the scalar PI/Cauchy--Schwarz core. -/ def cycle25PiVelocityNormLowerObligation : ProofObligation where id := "sald.first_appendix.cycle25_pi_velocity_norm_lower" statement := "Instantiate the compiled scalar cores for appendix.tex lines 104-129 with genuine weighted-Sobolev data: mean-zero variance identity, PI square-root norm bound, L2 pairing/Cauchy-Schwarz estimate for T_mu(psi), nonnegative L2 norms, and admissibility of the source test functions before the Riesz step." source := saldPiVelocityNormSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle25FirstAppendixVocabularyPacket", "SALD.cycle25FirstAppendixMiddleAuditContract", "SALD.cycle25FirstAppendixPiVelocityNormMiddleObligation", "SALD.saldPiVelocityNormDependencyContract", "SALD.piVelocityNormMeanZeroH1UpperScalar", "SALD.piVelocityNormBoundedFunctionalScalar", "sald.pi.velocity_norm_backend" ] note := "Cycle 25 lower closes only two theorem-independent real-order substeps: the weighted H1 upper-bound algebra and the Cauchy-Schwarz/PI scalar propagation for T_mu. It does not construct dot H^1(mu), prove the Riesz representation step, interpret the weak PDE, or mark Lemma lem:velocity-norm-bound formalized." /-- Cycle-29 upper packet for the first appendix/vocabulary layer. This returns to the first-DAG source-index layer after the cycle-28 guided general VA-SALD derivative-side algebra. It selects the LSI/KL/FI density-test bridge as the next lower obligation refinement while preserving all source labels, constants, and non-formalized statuses. -/ def cycle29FirstAppendixVocabularyPacket : FirstAppendixVocabularyPacket where objective := "Rebaseline the source-index and first appendix/vocabulary contracts for lem:gronwall, lem:dv_variation, def:PI, and eq:LSI-KL-FI after the cycle-28 guided/general derivative-side algebra, then select the LSI/KL/FI density-test bridge as the next lower obligation refinement." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "def:PI", "eq:LSI-KL-FI" ] modeDiscipline := [ "faithfulPaper: use only /home/nitanda_sub/mark/repos/sald/paper/appendix.tex:47-151 and main_body.tex:202-215 for this packet; sald_version_2.tex remains excluded.", "Preserve the source Gronwall signs and endpoint display, the DV finite-log-mgf variational formula, the PI constant convention C_PI^{-1}, and the LSI-to-KL/FI coefficient 1/(2*C_LSI).", "Keep this as Phase 1 transcript and proof-obligation refinement: Gronwall and LSI-to-KL/FI remain obligations, DV remains source-cited plus instantiation obligations, and PI remains contract-only with the velocity-norm backend tracked separately." ] nonGoals := [ "Do not edit thm:forward-KL, thm:forward-KL-discrete, prop:guided_path_residual, thm:general-moving-target-SALD, thm:unified-forward-KL, or thm:general-moving-target-SALD-discrete.", "Do not replace the paper route by Pinsker, Talagrand, PI-to-LSI, Girsanov, entropy transport, or a direct VA-SALD theorem proof.", "Do not add smooth-density, absolute-continuity, finite-KL/FI, test-function admissibility, approximation, boundary, or Sobolev assumptions to theorem statements; keep them as named obligations or source gaps.", "Do not mark Gronwall, DV, PI velocity bounds, or LSI-to-KL/FI formalized unless a local Lean proof builds." ] lowerPacket := [ "Middle must keep Lean, conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, research-wiki/source-index/SALD_original.jsonl, and the exact TeX windows synchronized in both directions.", "Target exactly SALD.saldLsiKlFiDensityTestContract / SALD.lsiKlFiDensityTestObligation / sald.lsi_kl_fi.density_test_interface as the first lower slice.", "First sub-slice: main_body.tex:208-215, covering rho << pi, the density ratio r=rho/pi, phi=sqrt(r), normalization int phi^2 dpi=1, entropy rewrite to KL, FI chain rule with the one-quarter factor, and the final coefficient 1/(2*C_LSI).", "If the density, smooth-test, approximation, or FI-chain-rule backend is blocked, refine the named obligation with the exact source-contract gap instead of changing theorem statements or promoting the LSI comparison." ] reviewerChecklist := [ "python3 tools/astis.py source-index ASTIS-SALD-001 refreshes SALD_original.jsonl with lem:gronwall, lem:dv_variation, def:PI, and eq:LSI-KL-FI while excluding sald_version_2.tex.", "SALD.saldFirstProofDag dependencies for the four focus labels include SALD.cycle29FirstAppendixVocabularyPacket and still report Gronwall obligation, DV sourceCited, PI contractOnly, and LSI/KL/FI obligation.", "The conversion window and proof-obligation ledger identify cycle 29 as source-index/first-appendix contract synchronization, with LSI/KL/FI density-test refinement as the lower target.", "The mandatory gate python3 tools/astis.py check passes and the fake-proof scan remains clean." ] status := ProofStatus.obligation /-- Cycle-29 middle source-to-Lean map for the first appendix layer. This translates the upper-selected LSI/KL/FI density-test bridge into a lower-ready source map. It keeps Gronwall, DV, PI, and the later SALD theorem statements at their existing statuses and only refines the density-test obligation behind `eq:LSI-KL-FI`. -/ def cycle29FirstAppendixMiddleAuditContract : FirstAppendixMiddleAuditContract where sourceIndexPath := "research-wiki/source-index/SALD_original.jsonl" focusLabels := [ "lem:gronwall", "lem:dv_variation", "def:PI", "eq:LSI-KL-FI" ] sourceReadWindows := [ "appendix.tex:47-71: Gronwall statement and integrating-factor proof; unchanged status guard for this cycle.", "appendix.tex:73-79: Donsker--Varadhan formula cited as Boucheron Cor. 4.15; unchanged source-cited dependency.", "appendix.tex:86-151: PI definition plus the weighted Sobolev/Riesz velocity-norm route; unchanged contract-only definition and backend obligation.", "main_body.tex:202-215: LSI definition, rho << pi, phi=sqrt(rho/pi), KL/FI vocabulary, and the coefficient 1/(2*C_LSI), with the selected lower sub-slice at lines 208-215." ] sourceStepMap := [ "lem:gronwall: keep dK/dt <= -a_t*K_t+b_t and the endpoint exponential display as existing Gronwall obligations; no Gronwall proof search is selected in cycle 29.", "lem:dv_variation: keep the cited variational equality over finite-log-mgf random variables; local theorem instantiations still owe common-space, measurability, and finite-log-mgf witnesses.", "def:PI: keep lines 86-94 as the PI definition and lines 96-151 as the separate velocity-norm Sobolev/Riesz backend; no PI-to-LSI route is introduced.", "eq:LSI-KL-FI: lines 208-215 require rho << pi, the Radon-Nikodym density r=rho/pi, the LSI test phi=sqrt(r), normalization int phi^2 dpi=1, entropy identity with KL, FI chain rule with factor 1/4, and the final coefficient 1/(2*C_LSI)." ] leanStepMap := [ "lem:gronwall -> SALD.saldGronwallCandidateContract, SALD.saldGronwallEndpointCalculusContract, SALD.saldGronwallExponentRewriteContract, and existing sald.gronwall.* obligations.", "lem:dv_variation -> SALD.dvContract, SALD.saldDvFiniteLogMgfContract, probability.dv_variational_formula, and sald.dv_variation.finite_log_mgf_interface.", "def:PI -> SALD.saldPIContract, SALD.piDefinitionContract, SALD.saldPiVelocityNormDependencyContract, and sald.pi.velocity_norm_backend.", "eq:LSI-KL-FI -> SALD.saldKLContract, SALD.saldFIContract, SALD.saldLSIContract, SALD.saldLsiKlFiBridgeContract, SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, SALD.cycle29LsiKlFiDensityTestMiddleObligation, sald.lsi_kl_fi.density_test_interface, and probability.lsi_to_kl_fi." ] citedResultMap := [ "DV remains the only external cited result in the first-layer source window; no SLT theorem is imported or promoted.", "The LSI/KL/FI bridge is local measure and density-test analysis, not an SLT reuse result and not a theorem-level VA-SALD proof.", "Gronwall and PI remain local real-analysis/Sobolev obligations outside this lower slice." ] obligationMap := [ "sald.lsi_kl_fi.density_test_interface first sub-slice: expose the Radon-Nikodym density ratio r=d rho/d pi and normalization int r dpi=1 for probability densities.", "sald.lsi_kl_fi.density_test_interface admissibility sub-slice: justify phi=sqrt(r) as a smooth/admissible LSI test function or record an approximation/closure lemma.", "sald.lsi_kl_fi.density_test_interface identity sub-slice: rewrite integral phi^2 log(phi^2) dpi as KL(rho||pi) and integral ||nabla phi||^2 dpi as (1/4)*FI(rho||pi), preserving finite KL/FI hypotheses and zero-density conventions.", "sald.lsi_kl_fi.density_test_interface coefficient sub-slice: combine the source LSI factor 2/C_LSI with the FI chain-rule factor 1/4 to obtain exactly FI/(2*C_LSI); SALD.lsiKlFiCoefficientAuditScalar formalizes only this scalar algebra, while the analytic inputs remain obligations." ] lowerPacket := [ "Target exactly SALD.saldLsiKlFiDensityTestContract / SALD.lsiKlFiDensityTestObligation / sald.lsi_kl_fi.density_test_interface.", "Start with main_body.tex:208-215: rho << pi, r=rho/pi, phi=sqrt(r), normalization, entropy rewrite, FI chain rule, finite KL/FI interfaces, and coefficient audit.", "If a backend is missing, refine SALD.cycle29LsiKlFiDensityTestMiddleObligation with the exact density, admissibility, approximation, zero-density, or chain-rule gap.", "Do not reopen Gronwall, DV, PI velocity-norm, forward-KL, guided, general VA-SALD, unified, or discrete theorem proof search in this lower attempt." ] reviewerChecklist := [ "SALD_original.jsonl indexes lem:gronwall, lem:dv_variation, def:PI, lem:velocity-norm-bound, and eq:LSI-KL-FI from the original files while excluding sald_version_2.tex.", "SALD.saldFirstProofDag dependencies for the four first-layer labels include SALD.cycle29FirstAppendixMiddleAuditContract.", "eq:LSI-KL-FI dependencies include SALD.cycle29LsiKlFiDensityTestMiddleObligation and still report LSI/KL/FI as an obligation.", "Conversion window, proof-obligation ledger, SLT audit, and dialogue board classify cycle 29 middle as source-to-Lean synchronization, not analytic proof closure." ] status := ProofStatus.obligation /-- Cycle-29 middle obligation for the selected LSI/KL/FI density-test sub-slice. -/ def cycle29LsiKlFiDensityTestMiddleObligation : ProofObligation where id := "sald.lsi_kl_fi.cycle29_density_test_middle" statement := "Maintain the cycle 29 lower-ready map for main_body.tex lines 208-215: expose rho << pi, r=d rho/d pi, phi=sqrt(r), normalization, entropy-to-KL rewrite, FI chain rule with the one-quarter factor, finite KL/FI interfaces, and the exact coefficient KL(rho||pi) <= FI(rho||pi)/(2*C_LSI)." source := saldKlFiLsiSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle29FirstAppendixVocabularyPacket", "SALD.cycle29FirstAppendixMiddleAuditContract", "SALD.saldKLContract", "SALD.saldFIContract", "SALD.saldLSIContract", "SALD.saldLsiKlFiBridgeContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiCoefficientAuditScalar", "SALD.lsiKlFiDensityTestObligation", "probability.lsi_to_kl_fi" ] note := "This is a middle-role synchronization obligation. It narrows the lower target to the source density-test bridge; cycle 29 lower formalizes only the final scalar coefficient audit, and does not prove LSI-to-KL/FI, change theorem hypotheses, or import an SLT result." /-- Cycle-29 lower obligation for the LSI/KL/FI density-test coefficient slice. -/ def cycle29LsiKlFiDensityTestLowerObligation : ProofObligation where id := "sald.lsi_kl_fi.cycle29_density_test_lower" statement := "Use SALD.lsiKlFiCoefficientAuditScalar only after the source density-test obligations have produced the LSI inequality with Dirichlet term and the FI chain-rule identity dirichlet=(1/4)*FI. The Radon-Nikodym density, normalization, smooth/admissible sqrt test, entropy rewrite, zero-density convention, and FI chain rule remain open analytic obligations." source := saldKlFiLsiSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle29FirstAppendixVocabularyPacket", "SALD.cycle29FirstAppendixMiddleAuditContract", "SALD.cycle29LsiKlFiDensityTestMiddleObligation", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiCoefficientAuditScalar", "SALD.lsiKlFiDensityTestObligation", "probability.lsi_to_kl_fi" ] note := "Lower cycle 29 compiled the real-algebra coefficient sublemma for main_body.tex:205-210, preserving the exact 1/(2*C_LSI) source constant. This does not close the LSI-to-KL/FI theorem." /-- Cycle-33 middle obligation for the proof-producing density-test scalar slice. -/ def cycle33LsiKlFiDensityTestMiddleObligation : ProofObligation where id := "sald.lsi_kl_fi.cycle33_density_test_middle" statement := "Translate main_body.tex lines 208-215 into proof-producing scalar density-test code for phi=sqrt(r): nonnegative r gives phi^2=r, hence the pointwise entropy integrand phi^2*log(phi^2)=r*log(r), and once an integral backend identifies testMass with densityMass, density normalization gives the LSI test normalization. The full Radon-Nikodym backend, smooth/admissible sqrt test, finite KL/FI, zero-density convention, and FI chain rule remain obligations." source := saldKlFiLsiSource status := ProofStatus.obligation dependsOn := [ "AutoSamplingTheory.lsiKlFiSqrtDensitySquareScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityEntropyIntegrandScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityNormalizationScalar", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiDensityTestObligation", "probability.lsi_to_kl_fi" ] note := "Cycle 33 closes only theorem-independent scalar pieces of the source substitution phi=sqrt(r). It does not prove LSI, the KL/FI integral identities, or the Fisher-information chain rule." /-- Cycle-33 lower obligation for the normalized LSI-test scalar bridge. -/ def cycle33LsiKlFiDensityTestLowerObligation : ProofObligation where id := "sald.lsi_kl_fi.cycle33_density_test_lower" statement := "Use SALD.lsiKlFiDensityTestBridgeScalar after the analytic backend has supplied the normalized LSI test inequality for phi=sqrt(r), the entropy-to-KL identity, and the Fisher chain-rule identity dirichlet=(1/4)*FI. The Radon-Nikodym density, integral transport, smooth/admissible sqrt test or approximation, finite KL/FI, zero-density convention, and FI chain rule remain open obligations." source := saldKlFiLsiSource status := ProofStatus.obligation dependsOn := [ "SALD.lsiKlFiDensityTestBridgeScalar", "SALD.lsiKlFiCoefficientAuditScalar", "AutoSamplingTheory.lsiKlFiSqrtDensitySquareScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityEntropyIntegrandScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityNormalizationScalar", "SALD.cycle33LsiKlFiDensityTestMiddleObligation", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiDensityTestObligation", "probability.lsi_to_kl_fi" ] note := "Lower cycle 33 formalizes only the theorem-independent real-order handoff from an already normalized LSI test inequality plus entropy/FI identifications to the displayed KL/FI coefficient. It does not close eq:LSI-KL-FI." /-- Cycle-38 upper packet for the LSI/KL/FI proof-closure target. This packet follows the current proof-closure order after cycle 36 advanced Gronwall assembly and cycle 37 advanced the one-sided Donsker--Varadhan backend. It selects `eq:LSI-KL-FI` as item (3) and narrows lower work to the remaining density-test chain-rule/admissibility bridge in `main_body.tex:202-215`. -/ def cycle38LsiKlFiUpperPacket : FirstAppendixVocabularyPacket where objective := "Proof-closure priority check before assigning lower work: (1) lem:gronwall was advanced in cycle 36 but remains an obligation pending endpoint-safe differentiability/derivative witnesses; (2) lem:dv_variation was advanced in cycle 37 by a Mathlib-backed one-sided tilted-measure backend while the Boucheron supremum equality remains sourceCited; (3) cycle 38 therefore selects eq:LSI-KL-FI; (4) the forward-KL Fokker-Planck/KL derivative identity and (5) the EM interpolation Fokker-Planck backend remain later closure targets." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "thm:forward-KL", "thm:forward-KL-discrete" ] modeDiscipline := [ "faithfulPaper Phase 1: use only the original main_body.tex:202-215 LSI, KL, and FI bridge; sald_version_2.tex remains excluded.", "Keep the paper statement fixed: pi satisfies LSI with constant C_LSI > 0, rho << pi, phi=sqrt(rho/pi), and the final comparison is KL(rho||pi) <= FI(rho||pi)/(2*C_LSI).", "Use the cycle 29 and cycle 33 compiled scalar helpers only after the source density-test inputs are supplied; they do not prove Radon-Nikodym construction, smooth/admissible sqrt tests, entropy integrability, or the Fisher-information chain rule.", "If the Sobolev or measure-theoretic chain rule is too large for the local Mathlib state, create a precise source-cited theorem interface with explicit density, positivity/zero-set, differentiability or approximation, and finite KL/FI hypotheses, and keep its status below formalized." ] nonGoals := [ "Do not run a source-index rebaseline unless a reviewer identifies a blocking source-anchor defect.", "Do not reopen Gronwall, Donsker--Varadhan, forward-KL derivative, EM interpolation, PI velocity-norm, guided, general VA-SALD, unified, or accumulated-error proof search in this upper packet.", "Do not replace the source LSI route with PI, Pinsker, Talagrand, transport, or theorem-level forward-KL reasoning.", "Do not add hidden smoothness, positivity, finite-KL, finite-FI, boundedness, or approximation assumptions to thm:forward-KL or any downstream theorem statement.", "Do not mark probability.lsi_to_kl_fi, SALD.lsiKlFiDensityTestObligation, or SALD.saldStatusForLabel \"eq:LSI-KL-FI\" formalized." ] lowerPacket := [ "Middle must keep two-way Lean/Markdown/LaTeX synchronization for main_body.tex:202-215, AutoSamplingTheory/Probability.lean, AutoSamplingTheory/SALD.lean, conversion-windows/ASTIS-SALD-001.md, and proof-obligations/ASTIS-SALD-001.md.", "Target exactly SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, and sald.lsi_kl_fi.density_test_interface.", "First lower attempt should be proof-producing: after the cycle 33 sqrt-density square, entropy-integrand, normalization, and normalized-test scalar bridge, prove one narrow pointwise or vector-norm handoff for the Fisher chain rule, or a theorem-independent scalar integral bridge that turns a supplied Dirichlet=(1/4)*FI identity into the existing SALD.lsiKlFiDensityTestBridgeScalar input.", "If the full chain rule for sqrt(rho/pi) is blocked, introduce a precise source-cited or obligation interface naming the Radon-Nikodym density, positivity or zero-density convention, smooth/admissible sqrt test or approximation theorem, finite KL/FI requirements, and the identity integral ||nabla sqrt(r)||^2 d pi = (1/4)*FI(rho||pi).", "Do not consume SALD.lsiKlFiCoefficientAuditScalar or SALD.lsiKlFiDensityTestBridgeScalar until the entropy-to-KL and Fisher-chain inputs are explicit assumptions, compiled lemmas, or source-cited interfaces." ] reviewerChecklist := [ "The cycle explicitly chooses proof-closure item (3), eq:LSI-KL-FI, after recording current Gronwall and DV statuses.", "Any new Lean proof is tied to main_body.tex:202-215 and is only a density/test-function, entropy, Fisher-chain, or scalar coefficient sublemma; no downstream theorem receives hidden assumptions.", "SALD.lsiKlFiVocabularyContract, SALD.lsiKlFiDensityTestObligation, probability.lsi_to_kl_fi, and SALD.saldStatusForLabel \"eq:LSI-KL-FI\" remain ProofStatus.obligation.", "No axiom, sorry, admit, Prop := True, := trivial, source-file drift, SLT import, or alternate non-LSI proof route is introduced.", "The mandatory gate python3 tools/astis.py check passes." ] status := ProofStatus.obligation /-- Cycle-38 upper workflow obligation for the LSI/KL/FI density-test bridge. -/ def cycle38LsiKlFiUpperObligation : ProofObligation where id := "sald.lsi_kl_fi.cycle38_upper_packet" statement := "Maintain the cycle 38 upper packet for main_body.tex lines 202-215: after cycle 36 Gronwall and cycle 37 Donsker-Varadhan progress, select proof-closure item (3), eq:LSI-KL-FI, and direct middle/lower work toward a proof-producing Fisher chain-rule/admissibility bridge for phi=sqrt(rho/pi), while keeping the full LSI-to-KL/FI theorem as an obligation until the analytic density-test backend builds." source := saldKlFiLsiSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle38LsiKlFiUpperPacket", "SALD.lsiKlFiVocabularyContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiDensityTestObligation", "SALD.cycle29LsiKlFiDensityTestMiddleObligation", "SALD.cycle29LsiKlFiDensityTestLowerObligation", "SALD.cycle33LsiKlFiDensityTestMiddleObligation", "SALD.cycle33LsiKlFiDensityTestLowerObligation", "AutoSamplingTheory.lsiKlFiSqrtDensitySquareScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityEntropyIntegrandScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityNormalizationScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainOfDerivativesScalar", "SALD.lsiKlFiCoefficientAuditScalar", "SALD.lsiKlFiDensityTestBridgeScalar", "SALD.lsiKlFiHalfFisherScalar", "SALD.lsiKlFiDensityTestHalfFisherScalar", "probability.lsi_to_kl_fi" ] note := "This is an upper-role assignment packet, not a proof of LSI-to-KL/FI. The source theorem remains an obligation; the next lower work should first try a compiled Fisher-chain or admissibility sublemma and otherwise record a precise source-cited interface." /-- Cycle-38 middle obligation for the LSI/KL/FI Fisher-chain scalar slice. -/ def cycle38LsiKlFiMiddleObligation : ProofObligation where id := "sald.lsi_kl_fi.cycle38_middle_fisher_chain" statement := "Translate main_body.tex lines 208-215 beyond the cycle-33 sqrt-density scalar helpers: for positive density ratio r, AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar proves the local 1/4 Fisher-integrand coefficient from d sqrt(r) and d log(r), and SALD.lsiKlFiDensityTestHalfFisherScalar turns a normalized LSI density-test bridge into the half-Fisher form C_LSI*KL <= (1/2)*FI used by the forward-KL derivative. The vector-gradient, integral, admissibility/approximation, zero-density, finite KL/FI, and Radon-Nikodym backends remain obligations." source := saldKlFiLsiSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle38LsiKlFiUpperPacket", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainOfDerivativesScalar", "SALD.lsiKlFiHalfFisherScalar", "SALD.lsiKlFiDensityTestHalfFisherScalar", "SALD.lsiKlFiDensityTestBridgeScalar", "SALD.lsiKlFiCoefficientAuditScalar", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiDensityTestObligation", "probability.lsi_to_kl_fi" ] note := "Middle cycle 38 adds theorem-independent scalar Fisher-chain and half-Fisher coefficient lemmas tied to the source substitution phi=sqrt(rho/pi). It does not prove the full LSI-to-KL/FI comparison, the vector Sobolev chain rule, or any downstream SALD theorem." /-- Cycle-38 lower obligation after compiling a finite-coordinate Fisher-chain handoff. -/ def cycle38LsiKlFiLowerObligation : ProofObligation where id := "sald.lsi_kl_fi.cycle38_lower_finite_sum_fisher_chain" statement := "Use AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumScalar and AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumHandoffScalar as the theorem-independent finite-coordinate bridge for main_body.tex lines 208-215: once the analytic backend supplies coordinate derivative identities for sqrt(rho/pi) and log(rho/pi), plus the Dirichlet and Fisher finite-sum identifications, the source input dirichlet=(1/4)*FI follows. Radon-Nikodym density, vector Sobolev gradients, integral transport, admissibility/approximation, zero-density conventions, and finite KL/FI remain obligations." source := saldKlFiLsiSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle38LsiKlFiUpperPacket", "SALD.cycle38LsiKlFiMiddleObligation", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainOfDerivativesScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumHandoffScalar", "SALD.lsiKlFiDensityTestBridgeScalar", "SALD.lsiKlFiDensityTestHalfFisherScalar", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiDensityTestObligation", "probability.lsi_to_kl_fi" ] note := "Lower cycle 38 compiles only the finite-coordinate sum and Dirichlet/Fisher handoff after explicit derivative and finite-sum hypotheses are supplied. It does not mark eq:LSI-KL-FI, probability.lsi_to_kl_fi, or the Sobolev/integral density-test backend formalized." /-- Cycle-43 upper packet for the remaining LSI/KL/FI density-test backend. This packet follows the current proof-closure sprint after cycle 41 narrowed Gronwall endpoint calculus and cycle 42 narrowed the selected-test Donsker--Varadhan backend. It keeps the active target at item (3), `eq:LSI-KL-FI`, and assigns lower work to the measure-level bridge that is still missing after the scalar and finite-coordinate cycle-38 helpers. -/ def cycle43LsiKlFiUpperPacket : FirstAppendixVocabularyPacket where objective := "Proof-closure priority check before assigning lower work: (1) lem:gronwall has endpoint-safe local calculus progress from cycle 41 but remains an obligation until the paper's concise differentiability hypothesis is connected to the closed-interval or absolute-continuity backend; (2) lem:dv_variation has cycle 42 selected scaled-test finite-mgf and one-sided energy sublemmas while the Boucheron supremum equality remains sourceCited; (3) this cycle therefore selects eq:LSI-KL-FI, specifically the density/test-function and coefficient sublemmas in main_body.tex:202-215; (4) the forward-KL Fokker-Planck/KL derivative identity and (5) the EM interpolation Fokker-Planck backend remain later closure targets." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "thm:forward-KL", "thm:forward-KL-discrete" ] modeDiscipline := [ "faithfulPaper Phase 1: use only the original main_body.tex:202-215 LSI, KL, and FI bridge; sald_version_2.tex remains excluded.", "Keep the source theorem fixed: pi satisfies LSI with constant C_LSI > 0, rho << pi, phi=sqrt(rho/pi), KL and FI are the displayed integrals, and the comparison is exactly KL(rho||pi) <= FI(rho||pi)/(2*C_LSI).", "Build proof-producing code only for source-shaped density/test-function, normalization, entropy-identity, Fisher-chain, or coefficient sublemmas; leave any large analytic theorem as a precise source-cited or obligation interface.", "Keep probability.lsi_to_kl_fi, SALD.lsiKlFiDensityTestObligation, and SALD.saldStatusForLabel \"eq:LSI-KL-FI\" below formalized status until the full Radon-Nikodym, admissibility, integral, and Sobolev-chain backend compiles." ] nonGoals := [ "Do not run a source-index rebaseline unless a reviewer identifies a blocking source-anchor defect.", "Do not reopen Gronwall, Donsker-Varadhan, forward-KL derivative, EM interpolation, PI velocity-norm, guided, general VA-SALD, unified, or accumulated-error proof search in this upper packet.", "Do not replace the source LSI route with PI, Pinsker, Talagrand, transport, or theorem-level forward-KL reasoning.", "Do not add hidden smoothness, positivity, finite-KL, finite-FI, boundedness, approximation, or zero-density hypotheses to thm:forward-KL or any downstream theorem statement.", "Do not consume the cycle-29 coefficient audit or cycle-33/38 scalar bridges unless the analytic premises are explicit assumptions, compiled lemmas, or source-cited interfaces." ] lowerPacket := [ "Middle must keep two-way Lean/Markdown/LaTeX synchronization for main_body.tex:202-215, AutoSamplingTheory/Probability.lean, AutoSamplingTheory/SALD.lean, conversion-windows/ASTIS-SALD-001.md, and proof-obligations/ASTIS-SALD-001.md.", "Target exactly SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, and sald.lsi_kl_fi.density_test_interface; do not change theorem statements.", "First proof-producing lower attempt: add a narrow lemma or interface for one of the remaining measure-level inputs after cycle 38: Radon-Nikodym density normalization int r d pi=1, entropy integral transport integral r log r d pi = KL(rho||pi), admissibility or approximation of phi=sqrt(r), zero-density convention, or the vector/integral Fisher chain rule int ||nabla sqrt(r)||^2 d pi = (1/4)*FI(rho||pi).", "If the full Sobolev or measure-theoretic backend is too large for local Mathlib, record a precise source-cited theorem interface naming absolute continuity, nonnegative density ratio, admissible sqrt test or approximation, finite KL/FI, zero-density handling, and the exact integral identities; keep the interface status below formalized.", "After any lower proof, translate the accepted Lean declaration back into the conversion window and proof-obligation ledger, and keep SLT audit entries at no-slt-import unless an actual built dependency is introduced." ] reviewerChecklist := [ "The packet explicitly checks the closure order (Gronwall, DV, LSI/KL/FI, forward-KL derivative, EM interpolation) and selects item (3), eq:LSI-KL-FI.", "Any new Lean proof is tied to main_body.tex:202-215 and is only a local density/test-function, normalization, entropy, Fisher-chain, or coefficient sublemma; no downstream theorem receives hidden assumptions.", "SALD.lsiKlFiVocabularyContract, SALD.lsiKlFiDensityTestObligation, probability.lsi_to_kl_fi, and SALD.saldStatusForLabel \"eq:LSI-KL-FI\" remain ProofStatus.obligation.", "No axiom, sorry, admit, Prop := True, := trivial, source-file drift, SLT import, or alternate non-LSI proof route is introduced.", "The mandatory gate python3 tools/astis.py check passes." ] status := ProofStatus.obligation /-- Cycle-43 upper workflow obligation for the LSI/KL/FI density-test backend. -/ def cycle43LsiKlFiUpperObligation : ProofObligation where id := "sald.lsi_kl_fi.cycle43_upper_packet" statement := "Maintain the cycle 43 upper packet for main_body.tex lines 202-215: after cycle 41 Gronwall endpoint-safe progress and cycle 42 Donsker-Varadhan selected-test progress, select proof-closure item (3), eq:LSI-KL-FI, and direct middle/lower work toward the remaining Radon-Nikodym normalization, entropy integral transport, admissibility/approximation, zero-density, and vector/integral Fisher-chain interfaces for phi=sqrt(rho/pi), while keeping the full LSI-to-KL/FI theorem as an obligation until the analytic density-test backend builds." source := saldKlFiLsiSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle43LsiKlFiUpperPacket", "SALD.lsiKlFiVocabularyContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiDensityTestObligation", "SALD.cycle29LsiKlFiDensityTestMiddleObligation", "SALD.cycle29LsiKlFiDensityTestLowerObligation", "SALD.cycle33LsiKlFiDensityTestMiddleObligation", "SALD.cycle33LsiKlFiDensityTestLowerObligation", "SALD.cycle38LsiKlFiUpperObligation", "SALD.cycle38LsiKlFiMiddleObligation", "SALD.cycle38LsiKlFiLowerObligation", "AutoSamplingTheory.lsiKlFiSqrtDensitySquareScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityEntropyIntegrandScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityNormalizationScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainOfDerivativesScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumHandoffScalar", "AutoSamplingTheory.lsiKlFiRnDerivLIntegralMassOne", "AutoSamplingTheory.lsiKlFiRnDerivDensityMassOne", "AutoSamplingTheory.lsiKlFiSqrtRnDerivTestMassOne", "AutoSamplingTheory.lsiKlFiRnDerivEntropyIntegral", "AutoSamplingTheory.lsiKlFiSqrtRnDerivEntropyIntegral", "SALD.lsiKlFiCoefficientAuditScalar", "SALD.lsiKlFiDensityTestBridgeScalar", "SALD.lsiKlFiHalfFisherScalar", "SALD.lsiKlFiDensityTestHalfFisherScalar", "probability.lsi_to_kl_fi" ] note := "This is an upper-role assignment packet, not a proof of LSI-to-KL/FI. Lower work should try one proof-producing density/test-function backend lemma first and otherwise record a precise source-cited interface; eq:LSI-KL-FI remains an obligation." /-- Cycle-43 middle density-normalization and entropy-transport obligation. The accompanying declarations in `AutoSamplingTheory/Probability.lean` formalize the Mathlib-backed Radon-Nikodym mass and entropy transport pieces for the source substitution `phi=sqrt(rho/pi)`. The remaining Sobolev admissibility and Fisher-chain integral theorem are still obligations. -/ def cycle43LsiKlFiMiddleObligation : ProofObligation where id := "sald.lsi_kl_fi.cycle43_middle_density_entropy" statement := "Cycle 43 middle compiles the Radon-Nikodym density normalization and entropy-transport sublemmas for main_body.tex lines 208-215: rho << pi gives unit mass for d rho/d pi, the sqrt density test has unit LSI mass, and the sqrt-test entropy integral is transported to the KL log-likelihood integral. This does not prove smooth/admissible sqrt-test approximation, the vector/integral Fisher chain rule, or the full probability.lsi_to_kl_fi theorem." source := saldKlFiLsiSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle43LsiKlFiUpperPacket", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiDensityTestObligation", "AutoSamplingTheory.lsiKlFiRnDerivLIntegralMassOne", "AutoSamplingTheory.lsiKlFiRnDerivDensityMassOne", "AutoSamplingTheory.lsiKlFiSqrtRnDerivTestMassOne", "AutoSamplingTheory.lsiKlFiRnDerivEntropyIntegral", "AutoSamplingTheory.lsiKlFiSqrtRnDerivEntropyIntegral", "AutoSamplingTheory.lsiKlFiSqrtDensitySquareScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityEntropyIntegrandScalar", "probability.lsi_to_kl_fi" ] note := "This is formalized measure-level density and entropy plumbing only. eq:LSI-KL-FI remains an obligation because the LSI admissibility/approximation step and the vector/integral Fisher-information chain rule have not been closed." /-- Cycle-43 lower finite-coordinate integral Fisher-chain obligation. The accompanying declarations in `AutoSamplingTheory/Probability.lean` push the cycle-38 finite-coordinate Fisher-chain identity through an arbitrary measure under a.e. positivity and supplied coordinate derivative identities. This narrows the source Fisher-chain backend, but it still does not identify the finite-coordinate integrals with the paper's full vector-gradient Fisher information. -/ def cycle43LsiKlFiLowerObligation : ProofObligation where id := "sald.lsi_kl_fi.cycle43_lower_integral_fisher_chain" statement := "Cycle 43 lower compiles a finite-coordinate integral Fisher-chain handoff for main_body.tex lines 208-215: if r>0 almost everywhere and the coordinate derivative identities for sqrt(r) and log(r) hold almost everywhere, then the integral of the squared sqrt-density coordinate derivatives equals the integral of (1/4)*r times the squared log-density coordinate derivatives; after supplied finite-coordinate Dirichlet and Fisher identifications, this yields the exact dirichlet=(1/4)*FI scalar input consumed by the existing LSI/KL/FI bridge. This still leaves vector-gradient equivalence, zero-density Sobolev handling, smooth/admissible sqrt-test approximation, finite KL/FI theorem interfaces, and full probability.lsi_to_kl_fi open." source := saldKlFiLsiSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle43LsiKlFiUpperPacket", "SALD.cycle43LsiKlFiMiddleObligation", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiDensityTestObligation", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainOfDerivativesScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumHandoffScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralFiniteSum", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralHandoffScalar", "SALD.lsiKlFiDensityTestBridgeScalar", "SALD.lsiKlFiDensityTestHalfFisherScalar", "probability.lsi_to_kl_fi" ] note := "This is a proof-producing lower slice for the Fisher side of the source density-test bridge only. eq:LSI-KL-FI remains an obligation because the full Sobolev/vector-gradient backend and sqrt-test admissibility/approximation are not closed." /-- Cycle-44 upper ledger for the main SALD theorem-skeleton sprint. The cycle focus is not another isolated scalar lemma. This packet checks the five slow analytic interfaces, keeps their unproved backends below formalized status, and wires them into the theorem route requested by the task contract. -/ def cycle44MainSkeletonAnalyticInterfaceLedger : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldForwardKlProofSource objective := "Main skeleton sprint 1: make the five source-cited or obligation-level analytic interfaces explicit, then wire them into the faithful SALD theorem route without changing paper statements, constants, or source labels." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "proof:thm:forward-KL:derivative", "proof:thm:forward-KL-discrete:conditional-fp", "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Gronwall: SALD.saldGronwallCandidateContract, SALD.saldGronwallEndpointCalculusContract, SALD.saldGronwallExponentRewriteContract, and the cycle 36/41 local wrappers expose endpoint-safe differentiability, FTC, interval-integrability, endpoint evaluation, and exponent algebra; full source lemma remains ProofStatus.obligation.", "Donsker--Varadhan: dvVariationalFormulaInterface saldDvVariationSource and SALD.saldDvFiniteLogMgfContract expose common probability space, absolute-continuity, finite-KL/log-likelihood, selected-test measurability, finite-log-mgf, and one-sided selected-test consequences; the Boucheron supremum equality remains ProofStatus.sourceCited.", "LSI/KL/FI: SALD.saldLsiKlFiDensityTestContract plus cycle 43 density/entropy and integral Fisher-chain obligations expose Radon-Nikodym density, zero-density convention, admissible sqrt test or approximation, entropy identity, and Fisher chain-rule assumptions; probability.lsi_to_kl_fi remains ProofStatus.obligation.", "Continuous forward-KL Fokker--Planck/KL derivative: SALD.forwardKlDerivativeCandidateContract and SALD.forwardKlDerivativeSideConditionContract expose mass conservation, KL differentiation under the integral, SALD Fokker--Planck, integration by parts, target transport, LSI handoff, and inverse-schedule calculus; sald.forward_kl.kl_derivative remains ProofStatus.obligation.", "EM interpolation Fokker--Planck: SALD.discreteForwardKlEmInterpolationSideConditionContract and SALD.generalMovingTargetDiscreteDerivativeSideConditionContract expose endpoint laws, conditional drift, interpolation Fokker--Planck, Laplacian split, stitched-interval regularity, and EM common-space/absolute-continuity; the conditional-FP theorem remains ProofStatus.obligation." ] theoremRoute := [ "1. thm:forward-KL consumes the continuous KL derivative interface, eq:LSI-KL-FI, lem:dv_variation, def:alpha-complexity, lem:gronwall, endpoint schedule identities, and coefficient-chain obligations.", "2. thm:forward-KL-discrete reuses thm:forward-KL dependencies and additionally consumes the EM endpoint/conditional-FP backend, frozen-delta obligation, discrete DV witness, stitched Gronwall, and accumulated-error bridge.", "3. prop:guided_path_residual stays as the guided residual identity route with normalizer, differentiation-under-integral, and divergence-cancellation obligations; it is not promoted by the analytic backend ledger.", "4. thm:general-moving-target-SALD consumes the continuous KL derivative pattern with sigma-weighted coefficients, residual DV witness for m_t, and the same Gronwall/LSI/DV interfaces.", "5. thm:unified-forward-KL specializes thm:general-moving-target-SALD using prop:guided_path_residual, the correction-field transport bridge, and m_t=w_t; no direct alternate proof is introduced.", "6. thm:general-moving-target-SALD-discrete consumes the general EM interpolation/conditional-FP backend, frozen-delta lemma, residual DV witness, LSI, Gronwall, and the discrete stitching interfaces." ] modeDiscipline := [ "faithfulPaper Phase 1: use only the original /home/nitanda_sub/mark/repos/sald/paper sources and keep sald_version_2.tex excluded.", "Do not weaken, strengthen, or restate the theorem displays; source gaps are recorded as obligations or source-cited interfaces.", "A backend can be sourceCited or obligation-level, but it cannot be marked formalized unless every analytic dependency builds locally.", "The theorem skeleton route is prioritized over new isolated scalar sublemmas until these six theorem nodes have stable dependency wiring." ] nonGoals := [ "Do not run a source-index rebaseline except for the task acceptance gate or a reviewer-reported source-anchor defect.", "Do not start systematic SLT or measure-theory backfill before the theorem route is stable.", "Do not import or mark an SLT theorem as formalized for DV, LSI, concentration, or EM one-step analysis in this cycle.", "Do not add endpoint, density, smoothness, absolute-continuity, finite-KL/FI, finite-log-mgf, boundary, or stitched-interval hypotheses to paper theorem statements as hidden assumptions.", "Do not replace the source proof route with PI, Pinsker, Talagrand, Girsanov, path-space comparison, or a direct theorem proof." ] lowerPacket := [ "Middle should keep conversion-windows/ASTIS-SALD-001.md and proof-obligations/ASTIS-SALD-001.md synchronized with the five-interface DAG table and the six theorem-route nodes.", "Lower should target exactly one named backend interface, preferring the continuous forward-KL density/boundary KL derivative or the EM conditional-FP common-space/conditional-law interface after this ledger is accepted.", "If a backend is too large, sharpen the source-cited or obligation interface with common space, absolute continuity, finite quantities, admissibility, endpoint, and regularity hypotheses instead of changing theorem statements.", "Any proof-producing work must be theorem-route useful and tied to one of the five slow interfaces; scalar-only lemmas are deferred unless they directly unblock a listed backend." ] reviewerChecklist := [ "The five analytic interfaces are named with source anchors and statuses below formalized except for already compiled local sublemmas.", "The theorem contracts and saldDependenciesForLabel entries for the six theorem-route labels point to the cycle 44 ledger.", "No source statement, source constant, source file selection, theorem status, or external reuse status is changed.", "No axiom, sorry, admit, Prop := True, := trivial, hidden assumption, or alternate proof route is introduced.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.continuousSaldContract", "SALD.discreteSaldContract", "SALD.guidedResidualContract", "SALD.generalVaSaldContract", "SALD.unifiedForwardKlContract", "SALD.generalVaSaldDiscreteContract" ] status := ProofStatus.obligation /-- Cycle-44 upper obligation for the main skeleton analytic interface ledger. -/ def cycle44MainSkeletonAnalyticInterfaceObligation : ProofObligation where id := "sald.main_skeleton.cycle44_analytic_interface_ledger" statement := "Cycle 44 upper records the main-skeleton analytic interface ledger: Gronwall endpoint-safe differentiability/FTC, Donsker-Varadhan common-space and finite-log-mgf, LSI/KL/FI density-test and Fisher-chain, continuous forward-KL Fokker-Planck/KL derivative, and EM interpolation endpoint/conditional-law Fokker-Planck are explicit source-cited or obligation interfaces before the theorem route is wired through thm:forward-KL, thm:forward-KL-discrete, prop:guided_path_residual, thm:general-moving-target-SALD, thm:unified-forward-KL, and thm:general-moving-target-SALD-discrete." source := saldForwardKlProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle44MainSkeletonAnalyticInterfaceLedger", "SALD.saldGronwallCandidateContract", "SALD.saldGronwallEndpointCalculusContract", "SALD.saldGronwallExponentRewriteContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.forwardKlProofDag", "SALD.discreteForwardKlProofDag", "SALD.generalVaSaldProofDag", "SALD.generalVaSaldDiscreteProofDag" ] note := "This is an upper-role route ledger, not a new analytic proof. It deliberately leaves unproved backends as obligation or sourceCited and does not promote any theorem skeleton to formalized status." /-- Cycle-44 proof-DAG pane for the five analytic interfaces and theorem route. -/ def cycle44MainSkeletonAnalyticInterfaceDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle44.gronwall_interface" interface := "Endpoint-safe Gronwall interface: continuous coefficients, differentiable or absolutely continuous K, FTC/order integration, endpoint evaluation, and exponent rewrite with source signs unchanged." source := saldGronwallSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.saldGronwallCandidateContract", "SALD.saldGronwallEndpointCalculusContract", "SALD.saldGronwallExponentRewriteContract", "SALD.cycle41GronwallMiddleObligation", "SALD.cycle41GronwallLowerObligation" ] reusedBy := ["thm:forward-KL", "thm:forward-KL-discrete", "thm:general-moving-target-SALD", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle44.dv_interface" interface := "Donsker-Varadhan interface: same probability space, absolute continuity, finite KL/log-likelihood, measurable selected tests, finite log-mgf, and selected-test one-sided consequences; full Boucheron supremum equality stays cited." source := saldDvVariationSource targetLean := "AutoSamplingTheory/Probability.lean" dependsOn := [ "dvVariationalFormulaInterface saldDvVariationSource", "probability.dv_variational_formula", "SALD.saldDvFiniteLogMgfContract", "SALD.cycle42DvVariationMiddleObligation", "SALD.cycle42DvVariationLowerObligation" ] reusedBy := ["thm:forward-KL", "thm:forward-KL-discrete", "thm:general-moving-target-SALD", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.sourceCited }, { id := "ASTIS.SALD.cycle44.lsi_kl_fi_interface" interface := "LSI-to-KL/FI interface: Radon-Nikodym density, zero-set convention, admissible sqrt-density test or approximation, entropy identity, Fisher chain rule, and finite KL/FI assumptions." source := saldKlFiLsiSource targetLean := "AutoSamplingTheory/Probability.lean" dependsOn := [ "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiDensityTestObligation", "SALD.cycle43LsiKlFiMiddleObligation", "SALD.cycle43LsiKlFiLowerObligation", "probability.lsi_to_kl_fi" ] reusedBy := ["thm:forward-KL", "thm:forward-KL-discrete", "thm:general-moving-target-SALD", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle44.continuous_forward_kl_derivative_interface" interface := "Continuous forward-KL Fokker--Planck/KL derivative backend: law/density regularity, SALD Fokker--Planck, integration by parts, target transport, LSI handoff, and inverse-schedule time change." source := saldForwardKlDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "SALD.cycle39ForwardKlDerivativeUpperObligation", "SALD.cycle39ForwardKlDerivativeMiddleObligation" ] reusedBy := ["thm:forward-KL", "thm:general-moving-target-SALD"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle44.em_interpolation_fp_interface" interface := "Euler--Maruyama interpolation Fokker--Planck backend: endpoint laws, conditional drift, conditional-law density, interpolation Fokker--Planck equation, Laplacian split, and stitched interval regularity." source := saldForwardKlDiscreteConditionalFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.discreteForwardKlEmEndpointObligation", "SALD.discreteForwardKlEmConditionalFpObligation", "SALD.discreteForwardKlEmInterpolationObligation", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteEmInterpolationObligation" ] reusedBy := ["thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle44.theorem_skeleton_route" interface := "Wire the five analytic interfaces into the faithful theorem order: forward-KL, discrete forward-KL, guided residual, continuous general moving target, unified forward-KL, and discrete general moving target." source := saldForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle44MainSkeletonAnalyticInterfaceLedger", "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.continuousSaldContract", "SALD.discreteSaldContract", "SALD.guidedResidualContract", "SALD.generalVaSaldContract", "SALD.unifiedForwardKlContract", "SALD.generalVaSaldDiscreteContract" ] reusedBy := ["ASTIS-SALD-001 main skeleton closure"] status := ProofStatus.obligation } ] /-- Cycle-49 upper readiness check for the five slow analytic interfaces. This sharpens the cycle-44 ledger after the theorem-level route wrappers from cycles 45--48 are in place. It is intentionally route data: the unresolved analytic backends stay as obligations or source-cited facts. -/ def cycle49MainSkeletonAnalyticReadinessLedger : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteDerivativeSource objective := "Cycle 49 upper: re-check the five slow analytic backends after the theorem skeleton route is wired, record the exact source-cited interface expected from each backend, and assign the next lower packet to the discrete general EM endpoint/conditional-law Fokker--Planck slice." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "proof:thm:forward-KL:derivative", "proof:thm:general-moving-target-SALD-discrete:em-fp", "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Gronwall backend: SALD.saldGronwallCandidateContract and SALD.saldGronwallEndpointCalculusContract now name the closed-interval derivative semantics, FTC/order-integration requirement, endpoint evaluation, interval-integrability, and exponent-rewrite handoffs; the full source lemma remains ProofStatus.obligation.", "DV backend: dvVariationalFormulaInterface saldDvVariationSource and SALD.saldDvFiniteLogMgfContract name common measurable space, nu << mu, finite KL/log-likelihood, selected-test measurability, finite log-mgf, and the one-sided selected-test consequence; the Boucheron equality remains ProofStatus.sourceCited.", "LSI/KL/FI backend: SALD.saldLsiKlFiDensityTestContract names rho << pi, Radon-Nikodym density r, zero-density convention, sqrt(r) test admissibility or approximation, entropy identity, finite KL/FI, and Fisher chain rule; probability.lsi_to_kl_fi remains ProofStatus.obligation.", "Continuous Fokker--Planck/KL derivative backend: SALD.forwardKlDerivativeCandidateContract, SALD.forwardKlDerivativeSideConditionContract, and SALD.generalMovingTargetDerivativeCandidateContract name law/density regularity, mass conservation, KL differentiation under the integral, SALD/general Fokker--Planck equations, integration by parts, target transport, LSI handoff, and inverse-schedule calculus; the KL derivative blocks remain ProofStatus.obligation.", "EM interpolation Fokker--Planck backend: SALD.discreteForwardKlEmInterpolationSideConditionContract, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff, SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation, and SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation name endpoint laws, regular conditional drift, conditional-law density/absolute-continuity, weak conditional Fokker--Planck, Laplacian split, stitched intervals, and common-space assumptions; the analytic backend remains ProofStatus.obligation." ] theoremRoute := [ "1. thm:forward-KL is already routed through the continuous KL derivative, LSI/KL/FI, DV velocity witness, Gronwall, and endpoint schedule obligations.", "2. thm:forward-KL-discrete is already routed through EM endpoint/conditional-FP, discrete KL derivative, frozen-delta/LSI, DV velocity, Gronwall accumulation, and accumulated-error obligations.", "3. prop:guided_path_residual is already routed through the normalizer derivative and residual identity obligations; it remains contractOnly and feeds only the unified specialization.", "4. thm:general-moving-target-SALD is already routed through the continuous general KL derivative, residual DV, sigma-weighted Gronwall, and pure-contraction obligations.", "5. thm:unified-forward-KL is already routed as a specialization of thm:general-moving-target-SALD via prop:guided_path_residual, eq:poisson-eq, and the transport bridge.", "6. thm:general-moving-target-SALD-discrete is the next lower target: appendix.tex:1354-1387 now has a compiled named-interpolation endpoint-law handoff, while conditional-law density and weak conditional-Fokker--Planck remain obligations before frozen/residual algebra or Gronwall display work continues." ] modeDiscipline := [ "faithfulPaper Phase 1 only: preserve the original source statements and constants from main_body.tex and appendix.tex, with sald_version_2.tex excluded.", "Do not add endpoint, density, absolute-continuity, finite-log-mgf, smoothness, boundary, or stitched-interval assumptions to theorem statements; record them in obligations.", "Do not promote Gronwall, DV, LSI/KL/FI, continuous KL derivative, or EM interpolation Fokker--Planck beyond obligation/sourceCited until a local proof or imported theorem builds.", "Keep theorem-level skeleton closure ahead of broad SLT/measure-theory backfill." ] nonGoals := [ "No source-index rebaseline beyond the required acceptance command.", "No new scalar-only lemma unless it directly discharges the selected EM endpoint/conditional-law backend.", "No SLT import or formalized reuse claim for entropy duality, LSI, concentration, or one-step EM analysis in this packet.", "No alternate proof route replacing derivative -> LSI -> DV -> Gronwall or the paper EM/frozen-delta path." ] lowerPacket := [ "Lower target: SALD.generalMovingTargetDiscreteDerivativeCandidateContract / SALD.generalMovingTargetDiscreteDerivativeObligation / sald.general_moving_target_discrete.kl_derivative.", "First sub-slice: appendix.tex:1354-1387, especially endpoint laws for hat rho_s, named-law representations for rho_k^eta and rho_{k+1}^eta, regular conditional drift bar b_{k,s}, density/absolute-continuity, and weak conditional Fokker--Planck.", "Use SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation for the named-process endpoint-law bookkeeping once pointwise interpolation identities and law representations are supplied; do not treat it as the Brownian/EM construction or conditional-FP proof.", "Preserve the cycle-28 frozen/residual algebra and the two sigma_eta^2/8 Young shares for the next sub-slice after the EM interface is stable." ] reviewerChecklist := [ "All five analytic backends have named Lean-facing interfaces and statuses below formalized, except already compiled local algebra helpers.", "The six theorem contracts include this cycle-49 readiness obligation while remaining contractOnly where appropriate.", "The selected lower packet points to appendix.tex:1354-1387 and not to a source-index rebaseline or broad SLT backfill.", "No source theorem statement, coefficient, source label, or theorem proofStatus is changed.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle44MainSkeletonAnalyticInterfaceLedger", "SALD.cycle45ForwardKlSkeletonObligation", "SALD.cycle46DiscreteForwardKlSkeletonObligation", "SALD.cycle47GuidedGeneralSkeletonObligation", "SALD.cycle48UnifiedDiscreteSkeletonObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation" ] status := ProofStatus.obligation /-- Cycle-49 upper obligation selecting the next theorem-level backend. -/ def cycle49MainSkeletonAnalyticReadinessObligation : ProofObligation where id := "sald.main_skeleton.cycle49_analytic_readiness" statement := "Cycle 49 upper re-checks the five source-cited or obligation-level analytic interfaces after the theorem wrappers are present, confirms that the six theorem skeletons are routed through them without statement changes, and selects appendix.tex:1354-1387 for the next lower backfill of the discrete general EM endpoint/conditional-law/Fokker--Planck backend." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle49MainSkeletonAnalyticReadinessLedger", "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle45ForwardKlSkeletonObligation", "SALD.cycle46DiscreteForwardKlSkeletonObligation", "SALD.cycle47GuidedGeneralSkeletonObligation", "SALD.cycle48UnifiedDiscreteSkeletonObligation", "SALD.generalMovingTargetDiscreteDerivativeObligation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "This is an upper-role readiness and handoff packet. It does not prove or promote any analytic backend; the only proof-producing content remains the previously compiled local algebra and endpoint-law handoff lemmas." /-- Cycle-49 middle audit for the analytic-readiness ledger. This is the source-to-Lean synchronization layer after the upper readiness packet. It checks the five analytic interfaces against the current theorem contracts, records the theorem consumers, and keeps the next lower target on the discrete general EM endpoint/conditional-law Fokker--Planck slice. -/ def cycle49MainSkeletonAnalyticMiddleContract : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteDerivativeSource objective := "Cycle 49 middle: synchronize the post-route analytic readiness ledger with the conversion window, proof obligations, and theorem DAGs; verify that the five slow analytic interfaces are consumed by the six theorem skeletons in source order; and hand off appendix.tex:1354-1387 as the lower-ready EM endpoint/conditional-law/Fokker--Planck slice." sourceLabels := [ "appendix.tex:47-79", "main_body.tex:202-215", "appendix.tex:168-252", "appendix.tex:724-951", "main_body.tex:359-395", "appendix.tex:1313-1603", "appendix.tex:1354-1387" ] analyticInterfaces := [ "Gronwall is source label lem:gronwall with endpoint-safe differentiability/FTC, interval-integrability, endpoint evaluation, and exponent-rewrite obligations in SALD.saldGronwallEndpointCalculusContract; theorem consumers are the continuous, discrete, continuous-general, and discrete-general Gronwall blocks.", "DV is source label lem:dv_variation with same-space probability measures, nu << mu, finite KL/log-likelihood, selected-test measurability, finite log-mgf, and alpha-scaling witnesses exposed by dvVariationalFormulaInterface saldDvVariationSource and SALD.saldDvFiniteLogMgfContract; the Boucheron equality remains sourceCited.", "LSI/KL/FI is source label eq:LSI-KL-FI with density rho << pi, Radon-Nikodym density, zero-set convention, admissible sqrt-density test or approximation, entropy identity, finite KL/FI, and Fisher chain rule exposed by SALD.saldLsiKlFiDensityTestContract.", "Continuous Fokker--Planck/KL derivative is consumed by thm:forward-KL and thm:general-moving-target-SALD through SALD.forwardKlDerivativeSideConditionContract and SALD.generalMovingTargetDerivativeCandidateContract; density, boundary, integration-by-parts, and inverse-schedule backends stay obligations.", "EM interpolation Fokker--Planck is consumed by thm:forward-KL-discrete and thm:general-moving-target-SALD-discrete through SALD.discreteForwardKlEmInterpolationSideConditionContract, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, the endpoint-law handoff, the named-interpolation endpoint helper, and the cycle-48 EM endpoint/conditional-law audit." ] theoremRoute := [ "forward-KL: appendix.tex:168-252 routes derivative -> LSI -> DV velocity -> Gronwall; unchanged main_body.tex:240-247 statement remains contractOnly.", "forward-KL-discrete: appendix.tex:260-592 routes EM conditional-FP -> frozen defect/LSI -> DV velocity -> Gronwall -> accumulated constants; unchanged main_body.tex:301-323 statement remains contractOnly.", "guided residual: appendix.tex:619-704 routes normalizer derivative and centered residual identity; it remains contractOnly and feeds the unified transport bridge.", "general moving-target: appendix.tex:724-951 routes continuous general Fokker--Planck/KL derivative -> LSI -> residual DV -> sigma-weighted Gronwall -> pure contraction.", "unified forward-KL: main_body.tex:359-395 and appendix.tex:949-951 route only through the guided residual/correction-field transport bridge and the continuous general theorem specialization c_t <- u_t.", "general moving-target discrete: appendix.tex:1313-1603 routes the general EM endpoint/conditional-FP backend -> frozen delta -> KL derivative/LSI -> residual DV -> Gronwall/stitching; appendix.tex:1354-1387 is the immediate lower slice." ] modeDiscipline := [ "faithfulPaper Phase 1: use only original main_body.tex, appendix.tex, and iteration_complexity.tex; sald_version_2.tex remains excluded.", "Keep every unproved backend below formalized status; compiled scalar or endpoint-law helpers are dependencies only.", "Record endpoint, density, absolute-continuity, finite-KL/FI, finite-log-mgf, boundary, conditional-law, and stitched-interval facts as obligations rather than hidden theorem assumptions.", "Do not start broad SLT or SDE library backfill until the theorem skeleton route and this middle audit remain green." ] nonGoals := [ "Do not restate theorem displays, source constants, alpha ranges, slowdown assumptions, or source labels.", "Do not replace the derivative -> LSI -> DV -> Gronwall route or the paper EM/frozen-delta route.", "Do not add scalar-only sublemmas unless they directly discharge appendix.tex:1354-1387 endpoint or conditional-law obligations.", "Do not mark the EM endpoint-law handoff as a Brownian construction, regular conditional drift, density, or weak Fokker--Planck proof." ] lowerPacket := [ "Target exactly SALD.generalMovingTargetDiscreteDerivativeCandidateContract / SALD.generalMovingTargetDiscreteDerivativeObligation / sald.general_moving_target_discrete.kl_derivative.", "First lower sub-slice: appendix.tex:1354-1387, with named-law representations for hat rho_s, rho_k^eta, and rho_{k+1}^eta; endpoint-law equalities; common-space and absolute-continuity assumptions for hat rho_s and tilde pi_s; regular conditional drift bar b_{k,s}; and the weak conditional Fokker--Planck equation.", "Use SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation only after the concrete named-process law representations and pointwise endpoint identities are supplied; it is not the stochastic construction or conditional-FP proof.", "Keep appendix.tex:1469-1511 frozen/residual algebra and the two sigma_eta^2/8 Young shares as the next sub-slice after the endpoint/conditional-law interface is stable.", "Local SLT material may guide disintegration or one-step patterns as reference only; do not import or promote SLT results." ] reviewerChecklist := [ "SALD.cycle49MainSkeletonAnalyticMiddleObligation is listed by all six theorem contracts and the cycle-49 DAG pane.", "The conversion window and proof-obligation ledger contain the cycle-49 middle route audit and the appendix.tex:1354-1387 lower packet.", "The five analytic backends remain sourceCited or obligation-level, with no theorem proofStatus promoted.", "No source theorem statement, coefficient, source label, theorem route, or external reuse status is changed.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle49MainSkeletonAnalyticReadinessLedger", "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle45ForwardKlSkeletonMiddleObligation", "SALD.cycle46DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle47GuidedGeneralSkeletonMiddleObligation", "SALD.cycle48UnifiedDiscreteSkeletonMiddleObligation", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] status := ProofStatus.obligation /-- Cycle-49 middle obligation tying the analytic-readiness audit to lower work and the Markdown conversion window. -/ def cycle49MainSkeletonAnalyticMiddleObligation : ProofObligation where id := "sald.main_skeleton.cycle49_middle_route_audit" statement := "Cycle 49 middle synchronizes the post-route analytic readiness ledger with the theorem DAGs, proof-obligation ledger, and conversion window: Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL derivative, and EM interpolation Fokker-Planck remain explicit source-cited or obligation interfaces, all six theorem skeletons consume them without statement changes, and appendix.tex:1354-1387 remains the lower-ready discrete general EM endpoint/conditional-law/Fokker-Planck slice." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle49MainSkeletonAnalyticMiddleContract", "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle45ForwardKlSkeletonMiddleObligation", "SALD.cycle46DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle47GuidedGeneralSkeletonMiddleObligation", "SALD.cycle48UnifiedDiscreteSkeletonMiddleObligation", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "This is middle-role source-to-Lean synchronization, not an analytic proof. It adds no hidden assumptions to theorem statements and does not promote any Gronwall, DV, LSI, continuous derivative, conditional-law, or EM Fokker--Planck backend." /-- Cycle-49 proof-DAG pane for the post-route analytic readiness check. -/ def cycle49MainSkeletonAnalyticReadinessDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle49.analytic_readiness" interface := "Post-route check that Gronwall, DV, LSI/KL/FI, continuous KL derivative, and EM interpolation Fokker--Planck have explicit source-cited or obligation interfaces before lower work resumes." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle49MainSkeletonAnalyticReadinessLedger", "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle44MainSkeletonAnalyticInterfaceDag" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle49.middle_route_audit" interface := "Middle source-to-Lean audit for the post-route analytic readiness ledger: verify the five backend interfaces against the six theorem consumers and keep appendix.tex:1354-1387 as the next lower EM endpoint/conditional-law/Fokker--Planck slice." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle49MainSkeletonAnalyticMiddleContract", "SALD.cycle49MainSkeletonAnalyticMiddleObligation", "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "cycle49 lower packet" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle49.lower_packet.general_discrete_em_fp" interface := "Lower packet for appendix.tex:1354-1387: use the named-interpolation endpoint-law handoff for concrete named laws and pointwise endpoint identities, then expose regular conditional drift, density/AC, common space, and weak conditional Fokker--Planck before the KL derivative proof." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeObligation", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] reusedBy := ["thm:general-moving-target-SALD-discrete", "cycle49 lower"] status := ProofStatus.obligation } ] /-- Cycle-45 upper packet for the continuous forward-KL theorem skeleton. This keeps the cycle-44 global interface ledger in place and wires those interfaces into the specific `thm:forward-KL` route requested for main skeleton sprint 2. It is workflow data only; the analytic backends remain obligations or source-cited interfaces. -/ def cycle45ForwardKlSkeletonUpperPacket : ForwardKlUpperPacket where objective := "Main skeleton sprint 2: wire the five source-cited analytic interfaces into the faithful proof skeleton for thm:forward-KL, matching main_body.tex:238-247 and appendix.tex:164-252 without changing constants, theorem statements, or source labels." sourceLabels := [ "thm:forward-KL", "eq:SALD", "eq:FP-eq", "eq:LSI-KL-FI", "def:alpha-complexity", "lem:dv_variation", "lem:gronwall", "proof:thm:forward-KL:derivative", "proof:thm:forward-KL:dv-energy", "proof:thm:forward-KL:gronwall" ] modeDiscipline := [ "faithfulPaper Phase 1: use only the original main_body.tex:238-247 theorem display and appendix.tex:164-252 proof; sald_version_2.tex remains out of scope.", "Before lower work, the five slow backends are checked as explicit interfaces: Gronwall endpoint calculus, DV common-space/finite-log-mgf, LSI/KL/FI density-test, continuous KL derivative/Fokker-Planck, and EM interpolation Fokker-Planck for downstream discrete reuse.", "The continuous theorem route is derivative -> LSI -> inverse time change -> DV -> Gronwall, with K(t), a(t), b(t), the factor 1/2, and both final exponent displays unchanged.", "All missing density, boundary, finite-KL/FI, finite-log-mgf, endpoint, positivity, and coefficient-regularity facts remain named obligations rather than hidden theorem assumptions." ] nonGoals := [ "Do not restate thm:forward-KL or promote SALD.continuousSaldContract above contractOnly.", "Do not prove or replace the Gronwall, DV, LSI/KL/FI, Fokker-Planck/KL derivative, or EM interpolation analytic backends in this upper packet.", "Do not introduce a different entropy method, a different slowdown normalization, or a merged one-exponential theorem display.", "Do not start systematic SLT or measure-theory backfill before the theorem skeleton route is accepted." ] lowerPacket := [ "Middle should synchronize the conversion window and proof-obligation row for the new cycle-45 forward-KL route wrapper.", "Lower should target exactly one remaining named backend already on the route: sald.forward_kl.density_boundary_regular, sald.forward_kl.schedule_time_change, sald.forward_kl.dv_finite_log_mgf_witness, or sald.forward_kl.gronwall_side_conditions.", "If the backend is too large, sharpen that interface with source-cited common-space, absolute-continuity, finite quantity, endpoint, positivity, and regularity witnesses instead of changing the theorem statement.", "Keep the discrete EM interpolation backend visible as a sibling slow interface for theorem-route stability, but do not spend this cycle on discrete coefficient or accumulated-error work." ] reviewerChecklist := [ "SALD.continuousSaldContract lists SALD.cycle45ForwardKlSkeletonObligation while remaining contractOnly.", "SALD.forwardKlProofDag contains ASTIS.SALD.forward_KL.cycle45_theorem_skeleton_route before the lower derivative, DV, and Gronwall nodes.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes the cycle-45 packet, obligation, and DAG block.", "No analytic dependency is marked formalized, no source theorem statement or constant changes, and no fake proof closure is introduced.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-45 obligation tying the continuous forward-KL theorem skeleton to the five source-cited analytic interfaces. -/ def cycle45ForwardKlSkeletonObligation : ProofObligation where id := "sald.forward_kl.cycle45_theorem_skeleton_route" statement := "Cycle 45 upper wires the source-cited analytic interfaces into thm:forward-KL: the continuous KL derivative/Fokker-Planck backend yields the pre-DV K'(t) inequality; eq:LSI-KL-FI supplies the LSI contraction coefficient; lem:dv_variation with the alpha-complexity finite-log-mgf witness supplies the velocity-energy bound; lem:gronwall with endpoint schedule identities and coefficient side conditions yields exactly the main_body.tex:243-246 display. The EM interpolation Fokker-Planck interface remains checked as the downstream discrete slow backend but is not promoted or used to alter the continuous theorem." source := saldForwardKlProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle45ForwardKlSkeletonUpperPacket", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlMiddleSourceToLeanMapObligation", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "SALD.forwardKlGronwallInstantiationContract", "sald.forward_kl.gronwall_application", "sald.forward_kl.endpoint_schedule_identities", "sald.forward_kl.gronwall_side_conditions", "SALD.discreteForwardKlEmInterpolationSideConditionContract" ] note := "This is a theorem-route wrapper for main skeleton sprint 2. It keeps Gronwall, DV, LSI/KL/FI, continuous KL derivative, and EM interpolation backends below formalized status and does not change the paper theorem display." /-- Cycle-45 proof-DAG pane for the continuous forward-KL theorem route. -/ def cycle45ForwardKlSkeletonDag : List ProofDagBlock := [ { id := "ASTIS.SALD.forward_KL.cycle45_theorem_skeleton_route" interface := "Continuous forward-KL theorem-level route: source statement, derivative/Fokker-Planck interface, LSI handoff, DV finite-log-mgf witness, and Gronwall endpoint/exponent side conditions compose to the exact main_body.tex:243-246 display." source := saldForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle45ForwardKlSkeletonUpperPacket", "SALD.cycle45ForwardKlSkeletonObligation", "SALD.cycle45ForwardKlSkeletonMiddleContract", "SALD.cycle45ForwardKlSkeletonMiddleObligation", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "SALD.forwardKlDvEnergyCandidateContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract" ] reusedBy := ["thm:forward-KL", "thm:forward-KL-discrete theorem-route reuse"] status := ProofStatus.obligation } ] /-- Cycle-45 middle audit for the continuous forward-KL theorem skeleton. This is the middle-role source-to-Lean synchronization layer for the upper route wrapper. It checks that the theorem statement and appendix proof are consumed through the already named analytic interfaces in the paper order, and selects one lower-ready backend without changing the theorem display. -/ def cycle45ForwardKlSkeletonMiddleContract : ForwardKlMiddleSourceToLeanContract where sourceStatement := saldForwardKlSource sourceProof := saldForwardKlProofSource derivativeSource := saldForwardKlDerivativeSource dvSource := saldForwardKlDvEnergySource gronwallSource := saldForwardKlGronwallSource objective := "Middle source-to-Lean audit for main skeleton sprint 2: verify that cycle45ForwardKlSkeletonObligation consumes the continuous KL derivative, LSI/KL/FI, DV finite-log-mgf, and Gronwall endpoint interfaces in the exact appendix.tex:168-252 order, while main_body.tex:238-247 remains the unchanged theorem statement." sourceStepMap := [ "main_body.tex:238-247 fixes the statement: pi_t has C_LSI(t)>=0, finite E_alpha0(pi_t,v_t), alpha in (0,alpha0], rho_s is the SALD law, and the terminal bound has the source two-exponential initial term plus residual integral.", "appendix.tex:168-185 differentiates KL(rho_s||tilde pi_s), uses mass conservation, substitutes the SALD Fokker-Planck equation, and identifies the first term as -FI through integration by parts.", "appendix.tex:187-208 transports tilde pi_s by tilde v_s=dot{t}(s)v_{t(s)}, evaluates the second term, then applies Cauchy--Schwarz and Young with the exact 1/2 and 1/2 split.", "appendix.tex:210-228 applies eq:LSI-KL-FI and the inverse schedule chain rule to get the t-time pre-DV inequality with coefficient (1/2)*dot{s}(t)^(-1).", "appendix.tex:230-241 applies lem:dv_variation to Z=alpha*||v_t||^2 and rewrites the log-mgf as E_alpha(pi_t,v_t), preserving the alpha^(-1) coefficient.", "appendix.tex:244-252 applies lem:gronwall with a(t)=dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1), then splits the initial exponent and drops the LSI term from the residual exponent exactly as in the source display." ] leanStepMap := [ "Keep SALD.continuousForwardKlStatementContract and SALD.continuousSaldContract unchanged; the middle contract only audits the route wrapper.", "Use SALD.cycle44MainSkeletonAnalyticInterfaceLedger and SALD.cycle45ForwardKlSkeletonObligation as the parent theorem-skeleton interfaces.", "Route appendix.tex:168-228 through SALD.forwardKlDerivativeCandidateContract, SALD.forwardKlDerivativeSideConditionContract, sald.forward_kl.density_boundary_regular, sald.forward_kl.schedule_time_change, and sald.forward_kl.kl_derivative.", "Route the LSI step through SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi; the density-test, zero-set, admissibility, entropy, and Fisher-chain backends remain obligations.", "Route appendix.tex:230-241 through SALD.forwardKlDvFiniteLogMgfWitnessContract, sald.forward_kl.dv_finite_log_mgf_witness, and sald.forward_kl.dv_energy_bound; the DV formula remains source-cited and the alpha0-to-alpha witness remains an obligation.", "Route appendix.tex:244-252 through SALD.forwardKlGronwallInstantiationContract, SALD.forwardKlGronwallSideConditionContract, and sald.forward_kl.gronwall_application.", "Keep SALD.discreteForwardKlEmInterpolationSideConditionContract visible only as the downstream slow backend checked by the cycle-44 ledger; it is not used to change the continuous theorem.", "Select the next lower target as sald.forward_kl.gronwall_side_conditions, because it is the theorem-level display-matching backend after derivative, LSI, and DV interfaces are already named." ] citedResultInterfaces := [ "lem:dv_variation remains source-cited through the DV interface; no SLT entropy-duality theorem is imported or marked formalized.", "lem:gronwall remains the endpoint-safe real-analysis obligation with FTC, coefficient regularity, endpoint, and exponent-splitting side conditions.", "eq:LSI-KL-FI remains the density-test and Fisher-chain obligation; existing compiled scalar and RN-density helpers are dependencies only.", "The continuous Fokker-Planck/KL derivative remains a local SDE/measure-analysis obligation; no Fokker-Planck backend is promoted by this middle audit.", "The EM interpolation Fokker-Planck backend remains a downstream discrete obligation, not a continuous forward-KL assumption." ] obligations := [ "sald.forward_kl.cycle45_theorem_skeleton_route", "sald.forward_kl.cycle45_middle_route_audit", "sald.forward_kl.kl_derivative", "probability.lsi_to_kl_fi", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "sald.forward_kl.gronwall_side_conditions", "sald.forward_kl.gronwall_application", "sald.discrete_forward_kl.em_interpolation_fp" ] lowerPacket := [ "Target exactly SALD.forwardKlGronwallSideConditionContract / SALD.forwardKlGronwallSideConditionObligation / sald.forward_kl.gronwall_side_conditions.", "First lower sub-slice: expose coefficient regularity and adjacent interval-integrability for dot{s}(t)*C_LSI(t), (1/2)*dot{s}(t)^(-1)*alpha^(-1), and (1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t).", "Second lower sub-slice: connect the theorem-specific exponent split to SALD.forwardKlGronwallExpProductRewriteIntegralCongrOfPieces without changing signs, endpoint limits, or constants.", "Leave endpoint K(0)/K(T) rewrites, residual-exponent monotonicity, DV finite-log-mgf, LSI density-test, continuous KL derivative, and full Gronwall as obligations unless an exact compiled proof is added.", "Do not work on discrete EM interpolation in this lower packet; it remains only the downstream sibling slow interface for theorem-route stability." ] reviewerChecklist := [ "SALD.continuousSaldContract lists SALD.cycle45ForwardKlSkeletonMiddleObligation while remaining contractOnly.", "SALD.forwardKlProofDag contains ASTIS.SALD.forward_KL.cycle45_middle_route_audit after the cycle-45 theorem skeleton route and before lower derivative/DV/Gronwall nodes.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes SALD.cycle45ForwardKlSkeletonMiddleContract, SALD.cycle45ForwardKlSkeletonMiddleObligation, and sald.forward_kl.cycle45_middle_route_audit.", "No analytic backend status is promoted, no theorem statement or source constant changes, and no alternate proof route is introduced.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-45 middle obligation tying the forward-KL route audit to lower work. -/ def cycle45ForwardKlSkeletonMiddleObligation : ProofObligation where id := "sald.forward_kl.cycle45_middle_route_audit" statement := "Cycle 45 middle audits the continuous thm:forward-KL theorem skeleton after the upper wrapper: main_body.tex:238-247 is kept unchanged, appendix.tex:168-252 is routed through the named continuous KL derivative, LSI/KL/FI, DV finite-log-mgf, and Gronwall endpoint/exponent interfaces in paper order, and the next lower target is the theorem-level Gronwall side-condition backend sald.forward_kl.gronwall_side_conditions. All slow analytic interfaces remain obligation or source-cited." source := saldForwardKlProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle45ForwardKlSkeletonUpperPacket", "SALD.cycle45ForwardKlSkeletonObligation", "SALD.cycle45ForwardKlSkeletonMiddleContract", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "sald.forward_kl.gronwall_side_conditions", "sald.forward_kl.gronwall_application", "SALD.discreteForwardKlEmInterpolationSideConditionContract" ] note := "This is a middle workflow obligation and source-to-Lean route audit, not an analytic proof. It does not add hidden smoothness, density, endpoint, absolute-continuity, finite-log-mgf, or interval-integrability assumptions to thm:forward-KL." /-- Cycle-50 upper packet for the continuous forward-KL theorem skeleton. After the cycle-49 post-route readiness audit, this packet re-enters the specific continuous `thm:forward-KL` route. It records the upper-role check that the five slow analytic interfaces are precise enough to be consumed by the theorem skeleton, while keeping the theorem contract-only. -/ def cycle50ForwardKlSkeletonUpperPacket : ForwardKlUpperPacket where objective := "Cycle 50 upper: consume the cycle-49 analytic-readiness audit and re-wire the faithful thm:forward-KL skeleton through the five source-cited analytic interfaces, matching main_body.tex:238-247 and appendix.tex:164-252 without changing constants, theorem statements, or source labels." sourceLabels := [ "thm:forward-KL", "proof:thm:forward-KL", "proof:thm:forward-KL:derivative", "proof:thm:forward-KL:dv-energy", "proof:thm:forward-KL:gronwall", "eq:LSI-KL-FI", "lem:dv_variation", "lem:gronwall", "def:alpha-complexity", "proof:thm:forward-KL-discrete:conditional-fp" ] modeDiscipline := [ "faithfulPaper Phase 1: use only the original main_body.tex:238-247 statement and appendix.tex:164-252 proof; sald_version_2.tex remains excluded.", "The five slow backends are checked before lower work: endpoint-safe Gronwall, common-space DV with finite log-mgf, LSI/KL/FI density-test and Fisher chain rule, continuous KL derivative/Fokker--Planck, and downstream EM interpolation Fokker--Planck.", "The continuous theorem route remains derivative -> LSI -> inverse schedule time change -> DV velocity energy -> Gronwall, with K(t), a(t), b(t), the 1/2 factors, alpha range, and both final exponent displays unchanged.", "Endpoint, density, absolute-continuity, finite-KL/FI, finite-log-mgf, boundary, coefficient-regularity, and interval-integrability facts are named obligations, not hidden hypotheses of thm:forward-KL." ] nonGoals := [ "Do not restate thm:forward-KL, change SALD.continuousForwardKlStatementContract, or promote SALD.continuousSaldContract above contractOnly.", "Do not prove or replace Gronwall, DV, LSI/KL/FI, the continuous Fokker--Planck/KL derivative, or the EM interpolation backend in this upper packet.", "Do not merge the two source exponent factors, change the coefficient (1/2)*dot{s}(t)^(-1)*alpha^(-1), or replace the paper route with another entropy method.", "Do not start broad SLT, concentration, or SDE backfill before this theorem-route wiring remains synchronized in Lean and Markdown." ] lowerPacket := [ "Middle should synchronize this cycle-50 upper route with conversion-windows/ASTIS-SALD-001.md and proof-obligations/ASTIS-SALD-001.md.", "If lower proof-producing work resumes on continuous forward-KL, target exactly SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative over appendix.tex:168-228.", "The first lower sub-slice is the source KL derivative interface: law/density regularity, mass conservation, SALD Fokker--Planck, integration by parts, target transport, and inverse-schedule calculus.", "Keep DV, LSI/KL/FI, Gronwall side conditions, endpoint schedule identities, and EM interpolation as separate named obligations unless an exact compiled proof is added." ] reviewerChecklist := [ "SALD.continuousSaldContract lists SALD.cycle50ForwardKlSkeletonObligation while remaining contractOnly.", "SALD.forwardKlProofDag contains ASTIS.SALD.forward_KL.cycle50_theorem_skeleton_route and still routes through the cycle-45 theorem wrapper and cycle-49 readiness audit.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes the cycle-50 upper packet, obligation, and DAG block.", "All five slow analytic backends remain ProofStatus.obligation or ProofStatus.sourceCited unless their full analytic dependencies build locally.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-50 obligation tying the post-readiness audit back to `thm:forward-KL`. -/ def cycle50ForwardKlSkeletonObligation : ProofObligation where id := "sald.forward_kl.cycle50_theorem_skeleton_route" statement := "Cycle 50 upper re-checks and wires the faithful continuous thm:forward-KL skeleton after the cycle-49 readiness audit: main_body.tex:238-247 stays fixed, appendix.tex:168-228 routes through the continuous KL derivative/Fokker-Planck interface and eq:LSI-KL-FI, appendix.tex:230-241 routes through the DV finite-log-mgf witness and alpha-complexity interface, and appendix.tex:244-252 routes through lem:gronwall with endpoint and coefficient side conditions. The EM interpolation Fokker-Planck interface remains visible only as the downstream discrete slow backend." source := saldForwardKlProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle50ForwardKlSkeletonUpperPacket", "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle49MainSkeletonAnalyticMiddleObligation", "SALD.cycle45ForwardKlSkeletonObligation", "SALD.cycle45ForwardKlSkeletonMiddleObligation", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "sald.forward_kl.endpoint_schedule_identities", "sald.forward_kl.gronwall_application", "sald.forward_kl.gronwall_side_conditions", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp" ] note := "This is an upper-role route closure record, not an analytic proof. It preserves the source theorem statement and keeps all unresolved Gronwall, DV, LSI/KL/FI, KL-derivative, and EM interpolation backends below formalized status." /-- Cycle-50 middle audit for the continuous forward-KL theorem skeleton. This is the middle-role synchronization layer after the cycle-49 readiness audit and the cycle-50 upper route wrapper. It keeps the theorem statement fixed, checks the paper-order route, and selects the continuous KL derivative/Fokker--Planck backend as the lower-ready theorem-level target. -/ def cycle50ForwardKlSkeletonMiddleContract : ForwardKlMiddleSourceToLeanContract where sourceStatement := saldForwardKlSource sourceProof := saldForwardKlProofSource derivativeSource := saldForwardKlDerivativeSource dvSource := saldForwardKlDvEnergySource gronwallSource := saldForwardKlGronwallSource objective := "Cycle 50 middle: synchronize the post-readiness thm:forward-KL route after SALD.cycle50ForwardKlSkeletonObligation, verify the exact main_body.tex:238-247 statement and appendix.tex:168-252 derivative -> LSI -> DV -> Gronwall proof order, and select sald.forward_kl.kl_derivative as the next lower backend without changing constants or theorem status." sourceStepMap := [ "main_body.tex:238-247 fixes the theorem statement: C_LSI(t)>=0, finite E_alpha0(pi_t,v_t), alpha in (0,alpha0], SALD law rho_s, and the two-exponential initial term plus residual alpha-complexity integral.", "appendix.tex:168-185 differentiates KL(rho_s||tilde pi_s), uses int partial_s rho_s dx=0, substitutes the SALD Fokker-Planck equation, and identifies the first term as -FI by integration by parts.", "appendix.tex:187-208 builds the slowed target transport velocity tilde v_s=dot t(s)*v_{t(s)}, evaluates the second term, and applies Cauchy--Schwarz/Young with the exact 1/2 share.", "appendix.tex:210-228 applies eq:LSI-KL-FI and inverse-schedule calculus to obtain the t-time pre-DV inequality with coefficient (1/2)*dot{s}(t)^(-1).", "appendix.tex:230-241 invokes lem:dv_variation with Z=alpha*||v_t||^2 and rewrites the log-mgf as E_alpha(pi_t,v_t), preserving alpha^(-1).", "appendix.tex:244-252 applies lem:gronwall with a(t)=dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1) and b(t)=(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t), then performs only the source exponent split/drop." ] leanStepMap := [ "Use SALD.cycle49MainSkeletonAnalyticReadinessObligation, SALD.cycle49MainSkeletonAnalyticMiddleObligation, SALD.cycle45ForwardKlSkeletonMiddleObligation, and SALD.cycle50ForwardKlSkeletonObligation as parent route checks.", "Keep SALD.continuousForwardKlStatementContract and SALD.continuousSaldContract contractOnly; this middle contract adds no theorem hypotheses.", "Route appendix.tex:168-228 through SALD.forwardKlDerivativeCandidateContract, SALD.forwardKlDerivativeSideConditionContract, sald.forward_kl.density_boundary_regular, sald.forward_kl.schedule_time_change, and sald.forward_kl.kl_derivative.", "Route appendix.tex:210-217 through SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi; density, zero-set, admissibility, entropy, and Fisher-chain assumptions remain obligations.", "Route appendix.tex:230-241 through SALD.forwardKlDvFiniteLogMgfWitnessContract and sald.forward_kl.dv_finite_log_mgf_witness before using sald.forward_kl.dv_energy_bound.", "Route appendix.tex:244-252 through SALD.forwardKlGronwallInstantiationContract, SALD.forwardKlGronwallSideConditionContract, sald.forward_kl.gronwall_side_conditions, and sald.forward_kl.gronwall_application.", "Keep SALD.discreteForwardKlEmInterpolationSideConditionContract and sald.discrete_forward_kl.em_interpolation_fp visible only as downstream discrete slow interfaces.", "Select the next lower target as SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative over appendix.tex:168-228." ] citedResultInterfaces := [ "lem:gronwall remains the endpoint-safe real-analysis obligation with coefficient regularity, FTC/order integration, endpoint rewrites, and exponent-display side conditions.", "lem:dv_variation remains source-cited; no SLT entropy-duality theorem is imported or marked formalized.", "eq:LSI-KL-FI remains the density-test, entropy-identity, zero-set, admissibility, and Fisher-chain obligation.", "The continuous Fokker-Planck/KL derivative is the selected local SDE/measure-analysis backend; this middle audit does not promote it.", "The EM interpolation Fokker-Planck interface remains a downstream discrete obligation and is not a continuous theorem assumption." ] obligations := [ "sald.forward_kl.cycle50_theorem_skeleton_route", "sald.forward_kl.cycle50_middle_route_audit", "sald.forward_kl.kl_derivative", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "probability.lsi_to_kl_fi", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "sald.forward_kl.gronwall_side_conditions", "sald.forward_kl.gronwall_application", "sald.discrete_forward_kl.em_interpolation_fp" ] lowerPacket := [ "Target exactly SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative.", "First lower sub-slice: appendix.tex:168-185 mass conservation, KL differentiation under the integral, SALD Fokker-Planck substitution, boundary/no-flux integration by parts, and -FI identification.", "Second lower sub-slice: appendix.tex:187-208 target transport velocity, integration by parts for the target term, Cauchy--Schwarz/Young, and the L2 velocity term.", "Third lower sub-slice: appendix.tex:218-228 inverse-schedule chain rule and dot{s}(t)*dot t(s(t))^2=dot{s}(t)^(-1).", "Do not add density, AC, endpoint, finite-KL/FI, finite-log-mgf, coefficient-regularity, or schedule assumptions to thm:forward-KL; keep blocked facts as named obligations." ] reviewerChecklist := [ "SALD.continuousSaldContract lists SALD.cycle50ForwardKlSkeletonMiddleObligation while remaining contractOnly.", "SALD.forwardKlProofDag contains ASTIS.SALD.forward_KL.cycle50_middle_route_audit after the cycle-50 route node.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes the cycle-50 middle contract, obligation, and named audit obligation.", "No theorem statement, source constant, source label, source file selection, or analytic backend status is changed.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-50 middle obligation selecting the continuous KL derivative backend. -/ def cycle50ForwardKlSkeletonMiddleObligation : ProofObligation where id := "sald.forward_kl.cycle50_middle_route_audit" statement := "Cycle 50 middle audits the post-readiness continuous thm:forward-KL skeleton: main_body.tex:238-247 remains unchanged, appendix.tex:168-252 is routed through the named continuous KL derivative/Fokker-Planck, LSI/KL/FI, DV finite-log-mgf, and Gronwall endpoint/exponent interfaces in source order, and the next lower target is sald.forward_kl.kl_derivative over appendix.tex:168-228. All slow analytic interfaces remain obligations or source-cited." source := saldForwardKlProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle50ForwardKlSkeletonUpperPacket", "SALD.cycle50ForwardKlSkeletonObligation", "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle49MainSkeletonAnalyticMiddleObligation", "SALD.cycle45ForwardKlSkeletonObligation", "SALD.cycle45ForwardKlSkeletonMiddleObligation", "SALD.cycle50ForwardKlSkeletonMiddleContract", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "sald.forward_kl.gronwall_side_conditions", "sald.forward_kl.gronwall_application", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp" ] note := "This is a middle workflow obligation and lower packet selector, not an analytic proof. It preserves the source theorem statement and does not add hidden density, boundary, absolute-continuity, finite-log-mgf, schedule, endpoint, or coefficient-regularity hypotheses." /-- Cycle-50 lower obligation for the continuous derivative/DV scalar handoff. -/ def cycle50ForwardKlDerivativeLowerObligation : ProofObligation where id := "sald.forward_kl.cycle50_derivative_dv_lower" statement := "Cycle 50 lower compiles the source-shaped scalar handoff from appendix.tex:168-241: once the KL derivative/Fokker-Planck backend, the source KL/FI comparison, the inverse-schedule velocity scaling, and the selected-test DV estimate are supplied explicitly, SALD.forwardKlDerivativeDvGronwallCoefficientOfKlFiVelocityScalingScalar derives the pre-Gronwall differential inequality with coefficient dot{s}(t)*C_LSI(t) - (1/2)*dot{s}(t)^(-1)*alpha^(-1) and residual (1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t)." source := saldForwardKlProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle50ForwardKlSkeletonMiddleContract", "SALD.cycle50ForwardKlSkeletonMiddleObligation", "SALD.forwardKlDerivativeDvGronwallCoefficientOfKlFiVelocityScalingScalar", "SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar", "SALD.forwardKlPostDvGronwallCoefficientOfScheduleScalar", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "sald.forward_kl.kl_derivative", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "probability.lsi_to_kl_fi", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound" ] note := "This lower item is formalized only as Real/order coefficient bookkeeping. It does not prove mass conservation, differentiation under the KL integral, Fokker-Planck, integration by parts, LSI density-test, finite log-mgf, source-cited DV, inverse-function calculus, Gronwall, or thm:forward-KL." /-- Cycle-50 proof-DAG pane for the continuous forward-KL theorem route. -/ def cycle50ForwardKlSkeletonDag : List ProofDagBlock := [ { id := "ASTIS.SALD.forward_KL.cycle50_theorem_skeleton_route" interface := "Post-readiness continuous forward-KL route: compose the cycle-49 five-backend check, the cycle-45 theorem wrapper, derivative/Fokker-Planck, LSI/KL/FI, DV finite-log-mgf, and Gronwall endpoint/exponent interfaces into the exact source theorem display." source := saldForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle50ForwardKlSkeletonUpperPacket", "SALD.cycle50ForwardKlSkeletonObligation", "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle49MainSkeletonAnalyticMiddleObligation", "SALD.cycle45ForwardKlSkeletonObligation", "SALD.cycle45ForwardKlSkeletonMiddleObligation", "SALD.cycle50ForwardKlSkeletonMiddleContract", "SALD.cycle50ForwardKlSkeletonMiddleObligation", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract" ] reusedBy := ["thm:forward-KL", "ASTIS-SALD-001 cycle 50"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle50_middle_route_audit" interface := "Middle audit after the cycle-50 post-readiness route: verify the exact source-order derivative -> LSI -> DV -> Gronwall map and select the continuous KL derivative/Fokker-Planck backend as the next lower target." source := saldForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle50ForwardKlSkeletonMiddleContract", "SALD.cycle50ForwardKlSkeletonMiddleObligation", "SALD.cycle50ForwardKlSkeletonObligation", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "sald.forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract" ] reusedBy := ["thm:forward-KL", "cycle 50 lower derivative packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle50_derivative_dv_lower" interface := "Lower proof-producing scalar handoff for appendix.tex:168-241: compose the selected KL derivative route, source KL/FI comparison, inverse-schedule velocity scaling, and selected-test DV input into the pre-Gronwall differential inequality with the paper's exact coefficient." source := saldForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle50ForwardKlDerivativeLowerObligation", "SALD.forwardKlDerivativeDvGronwallCoefficientOfKlFiVelocityScalingScalar", "SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar", "SALD.forwardKlPostDvGronwallCoefficientOfScheduleScalar", "SALD.forwardKlDerivativeObligation", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "probability.lsi_to_kl_fi", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound" ] reusedBy := ["thm:forward-KL", "sald.forward_kl.kl_derivative", "sald.forward_kl.dv_energy_bound"] status := ProofStatus.obligation } ] /-- Cycle-51 upper packet for the discrete forward-KL theorem skeleton. This is the post-cycle-50 return to `thm:forward-KL-discrete`. It checks the five slow analytic interfaces again, then records the discrete theorem route as a consumer of the already named EM/Fokker--Planck interfaces rather than a request to prove those interfaces in this upper cycle. -/ def cycle51DiscreteForwardKlSkeletonUpperPacket : DiscreteForwardKlUpperPacket where objective := "Cycle 51 upper: consume the cycle-49 analytic readiness audit and the cycle-50 continuous derivative/DV handoff, then re-wire thm:forward-KL-discrete through the explicit EM endpoint/conditional-Fokker-Planck, frozen-defect/LSI, DV velocity, Gronwall, and accumulated-error interfaces matching main_body.tex:299-323 and appendix.tex:260-592." sourceLabels := [ "thm:forward-KL-discrete", "proof:thm:forward-KL-discrete", "eq:frozen_interp_terminal_disc_prop_additive_final", "proof:thm:forward-KL-discrete:conditional-fp", "proof:thm:forward-KL-discrete:derivative", "proof:thm:forward-KL-discrete:dv-velocity", "proof:thm:forward-KL-discrete:gronwall", "proof:thm:forward-KL-discrete:accumulated-error", "thm:forward-KL", "proof:thm:forward-KL:derivative", "eq:LSI-KL-FI", "lem:dv_variation", "lem:gronwall" ] modeDiscipline := [ "faithfulPaper Phase 1: use only original main_body.tex:299-323 and appendix.tex:260-592 for the discrete theorem; sald_version_2.tex remains excluded.", "Before lower work, the five slow backends are explicitly checked: SALD.saldGronwallEndpointCalculusContract for endpoint-safe Gronwall, dvVariationalFormulaInterface saldDvVariationSource plus SALD.saldDvFiniteLogMgfContract for DV, SALD.saldLsiKlFiDensityTestContract for LSI/KL/FI, SALD.forwardKlDerivativeCandidateContract plus cycle-50 scalar handoff for continuous derivative reuse, and SALD.discreteForwardKlEmInterpolationSideConditionContract for EM endpoint/conditional-law Fokker-Planck.", "The discrete route remains EM interpolation -> conditional-FP/KL derivative -> frozen score defect and LSI -> DV velocity bound -> s-to-t time change -> Gronwall -> linear-slowdown accumulated-error collection.", "The theorem constants T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'} are preserved exactly." ] nonGoals := [ "Do not restate thm:forward-KL-discrete or promote SALD.discreteSaldContract above contractOnly.", "Do not prove or replace the EM conditional-Fokker-Planck backend, frozen-defect lemma, LSI/KL/FI density-test, source-cited DV formula, Gronwall, endpoint stitching, or accumulated-error bridge in this upper packet.", "Do not add density, absolute-continuity, endpoint, finite-log-mgf, coefficient-regularity, or stitched-interval assumptions to the theorem statement.", "Do not start broad SLT or measure-theory backfill before the discrete theorem-level route and lower packet are synchronized." ] lowerPacket := [ "Middle should synchronize this cycle-51 discrete route with conversion-windows/ASTIS-SALD-001.md and proof-obligations/ASTIS-SALD-001.md, preserving two-way Lean/Markdown/source mapping.", "Lower should target exactly SALD.discreteForwardKlDerivativeCandidateContract / SALD.discreteForwardKlDerivativeObligation / sald.discrete_forward_kl.kl_derivative over appendix.tex:334-491.", "First lower sub-slice: use the existing source-cited EM endpoint and conditional-Fokker-Planck interfaces from appendix.tex:334-385 as inputs to the KL derivative identity; do not attempt to prove the conditional-FP theorem from scratch in this cycle.", "Second lower sub-slice: preserve the frozen-cross coefficient, the moving Young share, eq:LSI-KL-FI, and the DV velocity witness through appendix.tex:454-523 before any Gronwall display work.", "If a measure-theory fact is missing, refine the named EM, density, absolute-continuity, or stitched-interval obligation instead of changing the theorem statement." ] reviewerChecklist := [ "SALD.discreteSaldContract lists SALD.cycle51DiscreteForwardKlSkeletonObligation while remaining contractOnly.", "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle51_theorem_interface_route after the cycle-46 route audit and before lower EM/defect/DV/Gronwall nodes.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes the cycle-51 upper packet, obligation, DAG node, and the cycle-50 derivative/DV lower handoff as a reusable coefficient dependency.", "All five slow analytic backends remain ProofStatus.obligation or ProofStatus.sourceCited unless their full analytic dependencies build locally.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-51 obligation tying the post-cycle-50 discrete route back to the source-cited EM/Fokker--Planck interfaces. -/ def cycle51DiscreteForwardKlSkeletonObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle51_theorem_interface_route" statement := "Cycle 51 upper re-checks and wires thm:forward-KL-discrete after the cycle-50 continuous derivative/DV handoff: main_body.tex:299-323 stays fixed, appendix.tex:260-385 is consumed through the EM endpoint/conditional-Fokker-Planck interfaces, appendix.tex:388-491 through the discrete KL derivative with frozen defect and eq:LSI-KL-FI, appendix.tex:493-523 through the EM DV velocity witness, and appendix.tex:526-592 through Gronwall and the accumulated-error bridge. The EM/Fokker-Planck backend is used as a source-cited obligation interface, not reproved in this cycle." source := saldForwardKlDiscreteProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle51DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle49MainSkeletonAnalyticMiddleObligation", "SALD.cycle46DiscreteForwardKlSkeletonObligation", "SALD.cycle46DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle50ForwardKlDerivativeLowerObligation", "SALD.forwardKlDerivativeDvGronwallCoefficientOfKlFiVelocityScalingScalar", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "sald.discrete_forward_kl.kl_derivative", "SALD.frozenDeltaCrossLipSaldContract", "sald.discrete_forward_kl.frozen_delta_cross_lip", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.discreteForwardKlGronwallInstantiationContract", "sald.discrete_forward_kl.gronwall_accumulation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.coefficient_chain_audit", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "thm:forward-KL" ] note := "This is an upper-role interface-route record, not an analytic proof. It adds no hidden EM, density, absolute-continuity, finite-log-mgf, endpoint, boundary, interval-integrability, or coefficient assumptions to thm:forward-KL-discrete." /-- Cycle-51 middle audit for the discrete forward-KL theorem route. This is the middle-role synchronization layer for the current sprint. It keeps the upper theorem route fixed, records the appendix line map around the discrete KL derivative backend, and selects lower work on `sald.discrete_forward_kl.kl_derivative` without attempting to prove the EM conditional Fokker--Planck theorem here. -/ def cycle51DiscreteForwardKlSkeletonMiddleContract : DiscreteForwardKlEmDefectAccumulationMiddleContract where sourceStatement := saldForwardKlDiscreteSource interpolationSource := saldForwardKlDiscreteInterpolationSource derivativeSource := saldForwardKlDiscreteDerivativeSource gronwallSource := saldForwardKlDiscreteGronwallSource accumulatedSource := saldForwardKlDiscreteAccumulatedErrorSource objective := "Cycle 51 middle: synchronize the post-cycle-50 thm:forward-KL-discrete route after SALD.cycle51DiscreteForwardKlSkeletonObligation, verify main_body.tex:299-323 and appendix.tex:260-592 in source order, and select sald.discrete_forward_kl.kl_derivative over appendix.tex:334-491 as the next lower backend while consuming EM endpoint/conditional-FP interfaces as source-cited obligations." sourceStepMap := [ "main_body.tex:299-323 fixes linear slowdown t(s)=s/r, reuses thm:forward-KL assumptions, adds score Lipschitz and alpha0' complexity hypotheses, assumes 4*eta^2*L_space^2<1/2, and states the exact barGamma/barDelta terminal KL bound.", "appendix.tex:260-330 defines the frozen EM interpolation and the SALD frozen score-defect lemma; the lemma is still an obligation via the later general frozen-delta result specialized with c=0 and sigma_eta(t)=sqrt(2).", "appendix.tex:334-385 supplies the inputs to the derivative backend: endpoint laws for hat rho, the conditional frozen drift bar b_{k,s}, the conditional-drift Fokker-Planck equation, and the Laplacian split relative to tilde pi_s.", "appendix.tex:388-452 differentiates KL(hat rho_s||tilde pi_s), evaluates the EM Fokker-Planck contribution by integration by parts, and imports the target transport term from the continuous route.", "appendix.tex:454-491 inserts the frozen-cross estimate, the moving Young bound, and eq:LSI-KL-FI to obtain the s-time pre-DV inequality.", "appendix.tex:493-523 remains the separate DV velocity witness for nu=hat rho_s, mu=tilde pi_s, and Z=alpha*||v_{t(s)}||^2.", "appendix.tex:526-592 remains the separate time-change, Gronwall, and accumulated-error bridge to the main-body display." ] leanStepMap := [ "Keep SALD.discreteForwardKlStatementContract and SALD.discreteSaldContract unchanged; this middle record only audits the route.", "Use SALD.cycle49MainSkeletonAnalyticReadinessObligation, SALD.cycle46DiscreteForwardKlSkeletonMiddleObligation, and SALD.cycle51DiscreteForwardKlSkeletonObligation as parent route checks.", "Consume appendix.tex:334-385 through SALD.discreteForwardKlEmInterpolationSideConditionContract, sald.discrete_forward_kl.em_endpoint_laws, sald.discrete_forward_kl.conditional_drift_density, sald.discrete_forward_kl.em_conditional_fokker_planck, and sald.discrete_forward_kl.em_interpolation_fp.", "Route appendix.tex:388-491 through SALD.discreteForwardKlDerivativeCandidateContract, SALD.discreteForwardKlDerivativeObligation, SALD.frozenDeltaCrossLipSaldContract, and SALD.saldLsiKlFiDensityTestContract.", "Keep appendix.tex:493-523 on SALD.discreteForwardKlDvFiniteLogMgfWitnessContract and sald.discrete_forward_kl.dv_velocity_bound rather than folding DV into the derivative backend.", "Keep appendix.tex:526-592 on SALD.discreteForwardKlGronwallInstantiationContract and SALD.discreteForwardKlAccumulatedErrorBridgeContract.", "Select lower work on sald.discrete_forward_kl.kl_derivative, first using the existing EM endpoint/conditional-FP interfaces as hypotheses instead of reproving them." ] citedResultInterfaces := [ "EM endpoint and conditional-Fokker-Planck backends are consumed as source-cited obligation interfaces; no Brownian-law, disintegration, or weak-FP theorem is promoted.", "eq:LSI-KL-FI remains the density-test, entropy, zero-set, and Fisher-chain obligation used to replace one half-FI by C_LSI*K.", "lem:dv_variation remains source-cited and is used only through the separate discrete DV finite-log-mgf witness after the derivative backend.", "lem:gronwall remains the endpoint-safe real-analysis obligation with stitched-interval regularity and coefficient integrability.", "The cycle-50 continuous scalar handoff is reusable coefficient data only and does not close the discrete KL derivative." ] obligations := [ "sald.discrete_forward_kl.cycle51_theorem_interface_route", "sald.discrete_forward_kl.cycle51_middle_route_audit", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.frozen_delta_cross_lip", "sald.discrete_forward_kl.kl_derivative", "probability.lsi_to_kl_fi", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.accumulated_error_bridge" ] lowerPacket := [ "Target exactly SALD.discreteForwardKlDerivativeCandidateContract / SALD.discreteForwardKlDerivativeObligation / sald.discrete_forward_kl.kl_derivative.", "First lower sub-slice: appendix.tex:334-385 as inputs, namely endpoint laws, conditional drift density, conditional-Fokker-Planck equation, and Laplacian split; do not prove the EM Fokker-Planck theorem from scratch.", "Second lower sub-slice: appendix.tex:388-491, using the supplied EM-FP interface to derive the KL derivative identity, frozen-cross contribution, moving Young share, and LSI handoff.", "Leave appendix.tex:493-523 DV, appendix.tex:526-592 Gronwall, endpoint stitching, residual exponent, and barGamma/barDelta collection in their existing obligations.", "If a backend is missing, sharpen the named EM, density, absolute-continuity, boundary, or stitched-interval obligation instead of changing thm:forward-KL-discrete." ] reviewerChecklist := [ "SALD.discreteSaldContract lists SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation while remaining contractOnly.", "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle51_middle_route_audit after the cycle51 theorem-interface route.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle51DiscreteForwardKlSkeletonMiddleContract, SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation, and sald.discrete_forward_kl.cycle51_middle_route_audit.", "No EM/Fokker-Planck, KL derivative, LSI, DV, Gronwall, or accumulated-error backend is promoted to formalized status.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-51 middle obligation tying the discrete route audit to the derivative lower packet. -/ def cycle51DiscreteForwardKlSkeletonMiddleObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle51_middle_route_audit" statement := "Cycle 51 middle audits the discrete thm:forward-KL-discrete route after the upper interface wrapper: main_body.tex:299-323 remains unchanged, appendix.tex:334-491 is selected as the next derivative backend, and appendix.tex:334-385 EM endpoint/conditional-Fokker-Planck interfaces are consumed explicitly as source-cited obligations rather than reproved. The follow-on DV, Gronwall, stitched-interval, residual-exponent, and accumulated-error interfaces remain separate obligations." source := saldForwardKlDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle51DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle51DiscreteForwardKlSkeletonObligation", "SALD.cycle51DiscreteForwardKlSkeletonMiddleContract", "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle49MainSkeletonAnalyticMiddleObligation", "SALD.cycle46DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle50ForwardKlDerivativeLowerObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "SALD.discreteForwardKlDerivativeObligation", "SALD.frozenDeltaCrossLipSaldContract", "sald.discrete_forward_kl.frozen_delta_cross_lip", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.discreteForwardKlGronwallInstantiationContract", "sald.discrete_forward_kl.gronwall_accumulation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "sald.discrete_forward_kl.accumulated_error_bridge", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI" ] note := "This is a middle workflow obligation and source-to-Lean route audit, not an analytic proof. It does not add hidden endpoint, density, absolute-continuity, finite-log-mgf, conditional-law, boundary, stitched-interval, or coefficient assumptions to thm:forward-KL-discrete." /-- Cycle-32 upper packet for the DV proof-closure sprint. This packet explicitly checks the proof-closure order and selects `lem:dv_variation` only after the cycle-31 reviewer left `lem:gronwall` as a partial local proof with remaining endpoint/integrability obligations. The DV result is cited by the paper, so this cycle narrows the source-cited interface and lower handoff without upgrading it to a local Lean theorem. -/ def cycle32DvVariationUpperPacket : FirstAppendixVocabularyPacket where objective := "Proof-closure priority check: (1) lem:gronwall remains an obligation after cycle 31 partial sublemmas; this cycle follows the requested item (2) lem:dv_variation by sharpening appendix.tex:73-79 into a precise source-cited DV interface before returning to (3) eq:LSI-KL-FI, (4) forward-KL Fokker-Planck/KL derivative, and (5) EM interpolation Fokker-Planck." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "thm:forward-KL", "thm:forward-KL-discrete" ] modeDiscipline := [ "faithfulPaper Phase 1: reproduce only appendix.tex:73-79 and the paper-cited Boucheron Corollary 4.15 interface; do not change any SALD theorem statement.", "Keep the exact DV display KL(nu||mu)=sup_Z {E_nu[Z]-log E_mu[exp Z]} and the finite-log-mgf predicate log E_mu[exp Z] < +infty.", "Treat DV as sourceCited until a local Mathlib/SLT port builds; downstream theorem blocks may use it only as an explicit dependency with their own common-space, measurability, absolute-continuity, and finite-log-mgf witnesses.", "Do not spend this cycle on source-index rebaseline unless a reviewer identifies a blocking source-anchor defect." ] nonGoals := [ "Do not mark probability.dv_variational_formula, SALD.dvContract, or lem:dv_variation formalized.", "Do not replace DV with Pinsker, Talagrand, LSI, Girsanov, or any other entropy inequality.", "Do not reopen broad transcript expansion, polished article export, PI velocity-norm work, LSI density-test proof search, forward-KL derivative proof search, or EM interpolation proof search in this upper packet.", "Do not add hidden boundedness, smoothness, or exponential-integrability assumptions to the source theorem statements." ] lowerPacket := [ "Middle must keep two-way Lean/Markdown/LaTeX synchronization for appendix.tex:73-79, AutoSamplingTheory/Probability.lean, AutoSamplingTheory/SALD.lean, conversion-windows/ASTIS-SALD-001.md, and proof-obligations/ASTIS-SALD-001.md.", "Lower should first check whether the current local Mathlib state already exposes a usable entropy-duality theorem. If not, do not axiomatize it; refine the source-cited interface around dvVariationalFormulaInterface saldDvVariationSource and SALD.saldDvFiniteLogMgfContract.", "Target exactly the DV interface slice: same-space probability measures mu and nu, real measurable tests Z, finite predicate log E_mu[exp Z] < +infty, equality as a supremum over such tests, and the one-sided consequence E_nu[Z] <= KL(nu||mu)+log E_mu[exp Z].", "Keep theorem-specific SALD applications as obligations: forward-KL, discrete forward-KL, and general moving-target blocks must still supply common-space, absolute-continuity, measurability, finite-log-mgf, alpha0-to-alpha monotonicity, and positive-alpha scaling." ] reviewerChecklist := [ "SALD.dvContract and SALD.saldStatusForLabel \"lem:dv_variation\" still report ProofStatus.sourceCited.", "AutoSamplingTheory/Probability.lean contains only contract/interface data for DV, not an axiom or fake proof closure.", "SALD.saldDependenciesForLabel \"lem:dv_variation\" lists dvVariationalFormulaInterface and the cycle-32 DV interface obligation as explicit dependencies.", "The conversion window, proof-obligation ledger, cited-results audit, and dialogue board all classify cycle 32 as source-cited DV interface refinement, not proof formalization.", "The mandatory gate python3 tools/astis.py check passes and the forbidden-proof scan remains clean." ] status := ProofStatus.obligation /-- Cycle-32 source-cited interface obligation for the cited DV formula. -/ def cycle32DvVariationInterfaceObligation : ProofObligation where id := "sald.dv_variation.cycle32_source_cited_interface" statement := "Maintain the precise source-cited interface for appendix.tex lines 73-79: same-space probability measures mu and nu, real measurable tests Z, finite predicate log E_mu[exp Z] < +infty, equality KL(nu||mu)=sup_Z(E_nu[Z]-log E_mu[exp Z]), and the one-sided consequence used by SALD theorem blocks." source := saldDvVariationSource status := ProofStatus.sourceCited dependsOn := [ "SALD.cycle32DvVariationUpperPacket", "SALD.dvContract", "SALD.saldDvFiniteLogMgfContract", "dvVariationalFormulaInterface saldDvVariationSource", "probability.dv_variational_formula", "KLContract" ] note := "This is not a local proof of Donsker--Varadhan. It exposes the paper-cited Boucheron Corollary 4.15 dependency as a precise interface, while theorem-specific finite-log-mgf and common-space witnesses remain obligations." /-- Cycle-32 middle source-to-Lean map for the cited DV formula. The local Mathlib audit found KL and tilted-measure infrastructure, but no ready theorem matching the Boucheron/SALD entropy-duality display. This contract therefore keeps the equality source-cited and adds only the compiled scalar rearrangement needed after the cited variational upper bound is available. -/ def cycle32DvVariationMiddleAuditContract : FirstAppendixMiddleAuditContract where sourceIndexPath := "research-wiki/source-index/SALD_original.jsonl" focusLabels := [ "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "thm:forward-KL", "thm:forward-KL-discrete" ] sourceReadWindows := [ "appendix.tex:73-79", "Mathlib MeasureTheory/Measure/Tilted.lean: tilted-measure infrastructure only", "Mathlib InformationTheory/KullbackLeibler/Basic.lean: KL infrastructure only" ] sourceStepMap := [ "appendix.tex:73 states mu and nu are probability distributions on the same space; map to dvVariationalFormulaInterface.probabilityMeasures and SALD.saldDvFiniteLogMgfContract.commonSpaceInterface.", "appendix.tex:75-77 states the Boucheron Corollary 4.15 equality over finite-log-mgf tests; map to probability.dv_variational_formula and keep sourceCited.", "The one-sided SALD use is the algebraic consequence of an admissible-test upper bound E_nu[Z]-logMgf <= KL; map the algebra only to AutoSamplingTheory.dvVariationalOneSidedConsequenceScalar." ] leanStepMap := [ "dvVariationalFormulaInterface saldDvVariationSource records the cited equality and finite-log-mgf predicate.", "AutoSamplingTheory.dvVariationalOneSidedConsequenceScalar compiles the rearrangement expectation - logMgf <= kl -> expectation <= kl + logMgf.", "No local Mathlib theorem was found that closes the full DV variational equality in the current v4.29.1 checkout; Measure.tilted and InformationTheory.klDiv are future backend candidates only." ] citedResultMap := [ "Boucheron, Lugosi, and Massart, Concentration Inequalities, Corollary 4.15, cited at appendix.tex:73.", "SLT entropy_duality remains reference-only and is not imported or marked formalized." ] obligationMap := [ "probability.dv_variational_formula remains source-cited.", "sald.dv_variation.finite_log_mgf_interface still owes common-space, measurability, finite-log-mgf, and alpha-complexity witnesses for theorem-specific SALD tests.", "forward-KL, discrete forward-KL, general moving-target, and general discrete theorem blocks still owe their own DV instantiation side conditions." ] lowerPacket := [ "Do not search broad SDE/Sampling libraries in this cycle.", "If continuing DV, target only a precise theorem statement around Mathlib klDiv/tilted measures or a finite-log-mgf witness sublemma; do not introduce axioms.", "Use AutoSamplingTheory.dvVariationalOneSidedConsequenceScalar only after a cited or formal DV bound supplies expectation - logMgf <= KL." ] reviewerChecklist := [ "SALD.dvContract and saldStatusForLabel \"lem:dv_variation\" stay sourceCited.", "The new theorem is a scalar Real rearrangement only and does not claim the DV equality.", "No SLT theorem is imported or marked formalized.", "python3 tools/astis.py check passes." ] status := ProofStatus.obligation def cycle32DvVariationMiddleObligation : ProofObligation where id := "sald.dv_variation.cycle32_middle_source_to_lean" statement := "Maintain the middle source-to-Lean map for appendix.tex lines 73-79 after the local Mathlib audit: DV equality remains Boucheron-source-cited, while the one-sided consequence uses only the compiled scalar rearrangement from E_nu[Z]-logMgf <= KL to E_nu[Z] <= KL+logMgf." source := saldDvVariationSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle32DvVariationUpperPacket", "SALD.cycle32DvVariationInterfaceObligation", "SALD.cycle32DvVariationMiddleAuditContract", "AutoSamplingTheory.dvVariationalOneSidedConsequenceScalar", "probability.dv_variational_formula", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract" ] note := "The only new compiled proof is real-order algebra. The analytic entropy-duality theorem, common-space witness, measurable-test witness, and finite-log-mgf witness are not closed." /-- Cycle-32 lower scalar bridge for the cited DV formula. This records the proof-producing lower slice: from a bounded set of admissible variational values, membership of the selected test, and the source-cited supremum identity, Lean derives the one-sided real inequality used downstream. The cited entropy-duality theorem and theorem-specific admissibility witnesses remain obligations. -/ def cycle32DvVariationLowerObligation : ProofObligation where id := "sald.dv_variation.cycle32_lower_supremum_bridge" statement := "Maintain the lower scalar bridge for appendix.tex lines 73-79: once the cited DV formula identifies KL(nu||mu) with the supremum over admissible finite-log-mgf test values, and the selected test value E_nu[Z]-logMgf is in that bounded admissible set, AutoSamplingTheory.dvVariationalOneSidedFromSupremumScalar derives E_nu[Z] <= KL(nu||mu)+logMgf." source := saldDvVariationSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle32DvVariationUpperPacket", "SALD.cycle32DvVariationInterfaceObligation", "SALD.cycle32DvVariationMiddleAuditContract", "SALD.cycle32DvVariationMiddleObligation", "AutoSamplingTheory.dvVariationalOneSidedConsequenceScalar", "AutoSamplingTheory.dvVariationalOneSidedFromSupremumScalar", "probability.dv_variational_formula", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "sald.dv_variation.finite_log_mgf_interface" ] note := "The compiled theorem proves only the order-theoretic supremum-to-one-sided consequence over Real values. The bounded admissible-value set, common probability space, measurability, finite log-mgf, and the Boucheron DV equality itself are still explicit cited/obligation inputs." /-- Cycle-37 upper packet for the cited Donsker--Varadhan proof target. This packet follows the current proof-closure order after cycle 36 advanced the Gronwall assembly under explicit Mathlib side conditions. It selects `lem:dv_variation` as item (2), keeps Boucheron Corollary 4.15 source-cited, and asks middle/lower work to sharpen the Lean theorem interface or prove only backend sublemmas that genuinely build locally. -/ def cycle37DvVariationUpperPacket : FirstAppendixVocabularyPacket where objective := "Proof-closure priority check before assigning lower work: (1) lem:gronwall was advanced in cycle 36 but remains an obligation pending the endpoint-safe differentiability/derivative-witness bridge; (2) cycle 37 therefore selects lem:dv_variation; (3) eq:LSI-KL-FI, (4) the forward-KL Fokker-Planck/KL derivative identity, and (5) the EM interpolation Fokker-Planck backend remain later proof-closure targets." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "thm:forward-KL", "thm:forward-KL-discrete" ] modeDiscipline := [ "faithfulPaper Phase 1: reproduce exactly appendix.tex:73-79 and its cited Boucheron Corollary 4.15 dependency; sald_version_2.tex remains excluded.", "Keep the paper statement fixed: mu and nu are probability distributions on the same space, the supremum ranges over real random variables Z with log E_mu[exp Z] < +infty, and the display is KL(nu||mu)=sup_Z(E_nu[Z]-log E_mu[exp Z]).", "If a local Mathlib proof is too large, keep probability.dv_variational_formula source-cited and expose a Mathlib-shaped theorem interface with explicit common-space, absolute-continuity, measurability, finite-KL, and finite-log-mgf hypotheses.", "Do not spend this cycle on source-index rebaseline unless a reviewer finds a blocking source-anchor defect; the current anchor is appendix.tex:73-79 from the original paper." ] nonGoals := [ "Do not mark SALD.dvContract, probability.dv_variational_formula, or saldStatusForLabel \"lem:dv_variation\" formalized unless a compiled Lean theorem replaces the source-cited result.", "Do not replace Donsker--Varadhan with Pinsker, Talagrand, LSI, Girsanov, path-space comparison, or a theorem-specific Cauchy-only bound.", "Do not add new finite-mgf, absolute-continuity, smoothness, boundedness, or state-space assumptions to thm:forward-KL or any downstream theorem statement.", "Do not reopen LSI/KL/FI, forward-KL derivative, EM interpolation, PI velocity-norm, guided, general VA-SALD, unified, or accumulated-error proof search in this upper packet." ] lowerPacket := [ "Middle must keep two-way Lean/Markdown/LaTeX synchronization for appendix.tex:73-79, AutoSamplingTheory/Probability.lean, AutoSamplingTheory/SALD.lean, conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, and research-wiki/cited-results/SLT_reuse_audit.md.", "First lower target: one declaration/interface only, centered on dvVariationalFormulaInterface saldDvVariationSource and probability.dv_variational_formula; either prove a genuinely local Mathlib-backed sublemma or refine the source-cited theorem interface without changing its status.", "Use the local Mathlib audit as the starting point: InformationTheory.KullbackLeibler.Basic supplies klDiv infrastructure and MeasureTheory.Measure.Tilted supplies tilted-measure/log-likelihood infrastructure, but no ready entropy-duality theorem matching appendix.tex:73-79 has been found.", "Keep the compiled scalar bridges AutoSamplingTheory.dvVariationalOneSidedConsequenceScalar and AutoSamplingTheory.dvVariationalOneSidedFromSupremumScalar as post-DV order lemmas only; do not present them as a proof of the variational equality.", "Downstream SALD applications must continue to depend explicitly on theorem-specific common-space, absolute-continuity, measurability, finite-log-mgf, alpha0-to-alpha monotonicity, and positive-alpha scaling obligations." ] reviewerChecklist := [ "SALD.dvContract remains ProofStatus.sourceCited and includes the cycle 37 workflow obligation only as assignment data.", "SALD.saldDependenciesForLabel \"lem:dv_variation\" names the cycle 37 packet and keeps the cycle 32 scalar bridges as post-DV consequences, not theorem closures.", "The conversion window, proof-obligation ledger, and SLT audit classify cycle 37 as source-cited DV equality/proof-target sharpening, with no SLT import and no claim that the Boucheron supremum formula is formalized.", "No axiom, sorry, admit, Prop := True, := trivial, hidden theorem assumption, source-file drift, or alternate entropy route is introduced.", "The mandatory gate python3 tools/astis.py check passes." ] status := ProofStatus.obligation /-- Cycle-37 upper workflow obligation for the cited DV interface target. -/ def cycle37DvVariationUpperObligation : ProofObligation where id := "sald.dv_variation.cycle37_upper_packet" statement := "Maintain the cycle 37 upper packet for appendix.tex lines 73-79: after the cycle 36 Gronwall assembly, return to proof-closure priority item (2), lem:dv_variation, and direct middle/lower work toward either a genuinely compiling Mathlib-backed entropy-duality interface or a sharper source-cited theorem interface for the paper's Boucheron Corollary 4.15 display." source := saldDvVariationSource status := ProofStatus.sourceCited dependsOn := [ "SALD.cycle37DvVariationUpperPacket", "SALD.dvContract", "SALD.cycle32DvVariationUpperPacket", "SALD.cycle32DvVariationInterfaceObligation", "SALD.cycle32DvVariationMiddleObligation", "SALD.cycle32DvVariationLowerObligation", "AutoSamplingTheory.dvVariationalOneSidedConsequenceScalar", "AutoSamplingTheory.dvVariationalOneSidedFromSupremumScalar", "dvVariationalFormulaInterface saldDvVariationSource", "probability.dv_variational_formula", "SALD.saldDvFiniteLogMgfContract", "sald.dv_variation.finite_log_mgf_interface", "Mathlib InformationTheory.KullbackLeibler.Basic", "Mathlib MeasureTheory.Measure.Tilted" ] note := "This is an upper-role assignment packet, not a proof of Donsker--Varadhan. The cited equality remains source-cited unless a local Lean theorem builds; the existing scalar lemmas are only post-DV order consequences." /-- Cycle-37 middle source-to-Lean map for the cited DV theorem. This records the proof-producing Mathlib-backed sublemma now available for the one-sided admissible-test inequality. It does not promote the paper-cited supremum equality to a local theorem. -/ def cycle37DvVariationMiddleAuditContract : FirstAppendixMiddleAuditContract where sourceIndexPath := "research-wiki/source-index/SALD_original.jsonl" focusLabels := [ "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "thm:forward-KL", "thm:forward-KL-discrete" ] sourceReadWindows := [ "appendix.tex:73-79", "Mathlib InformationTheory/KullbackLeibler/Basic.lean: klDiv and Gibbs nonnegativity", "Mathlib MeasureTheory/Measure/LogLikelihoodRatio.lean: tilted-right log-likelihood integral identity", "Mathlib MeasureTheory/Measure/Tilted.lean: exponential tilting and absolute continuity" ] sourceStepMap := [ "appendix.tex:73-74 fixes the common probability-space orientation KL(nu||mu); map to probability measures nu and mu plus hnu_mu : nu << mu for the one-sided backend.", "appendix.tex:75-78 fixes the finite-log-mgf test class; map a selected test Z to hZ_nu, hexp_mu, and hllr hypotheses before applying the local one-sided theorem.", "The full equality KL(nu||mu)=sup_Z(E_nu[Z]-log E_mu[exp Z]) remains Boucheron-source-cited through probability.dv_variational_formula.", "The admissible-test upper bound E_nu[Z]-log E_mu[exp Z] <= KL(nu||mu) is now proved locally under explicit Mathlib hypotheses by exponential tilting." ] leanStepMap := [ "AutoSamplingTheory.dvVariationalOneSidedOfTiltedRight compiles the one-sided DV inequality using klDiv, mu.tilted Z, absolute continuity into the tilted law, Gibbs nonnegativity, and the tilted-right llr integral identity.", "AutoSamplingTheory.dvVariationalTiltedRightOneSidedConsequence composes that tilted inequality with the scalar rearrangement to produce the paper-consumed form E_nu[Z] <= KL(nu||mu)+log E_mu[exp Z] under the same selected-test hypotheses.", "AutoSamplingTheory.dvVariationalOneSidedConsequenceScalar and AutoSamplingTheory.dvVariationalOneSidedFromSupremumScalar remain scalar/order bridges, now alongside the stronger Mathlib-backed one-sided theorem.", "No local theorem proves the supremum identity or existence/optimality of maximizing tests; dvVariationalFormulaInterface saldDvVariationSource and probability.dv_variational_formula remain sourceCited." ] citedResultMap := [ "Boucheron, Lugosi, and Massart, Concentration Inequalities, Corollary 4.15, cited at appendix.tex:73, still supplies the full variational equality.", "Mathlib tilted-measure and KL lemmas supply only the admissible-test inequality under explicit integrability/absolute-continuity hypotheses.", "SLT entropy_duality remains reference-only and is not imported or marked formalized." ] obligationMap := [ "probability.dv_variational_formula remains source-cited for the equality and supremum over all finite-log-mgf tests.", "sald.dv_variation.finite_log_mgf_interface still owes theorem-specific common-space, absolute-continuity, measurability, finite-log-mgf, and finite-KL/log-likelihood integrability witnesses.", "Downstream forward-KL and general-moving-target theorem blocks may use the one-sided theorem only after supplying their own selected-test hypotheses; no theorem statement receives new hidden assumptions." ] lowerPacket := [ "If lower continues this DV slice, target exactly the theorem-instance witness layer: show how a SALD squared-velocity or residual test supplies hnu_mu, hZ_nu, hexp_mu, and hllr for AutoSamplingTheory.dvVariationalOneSidedOfTiltedRight.", "Do not claim the Boucheron supremum equality is formalized; keep dvVariationalFormulaInterface and probability.dv_variational_formula sourceCited.", "Do not move to LSI/KL/FI, forward-KL derivative, or EM interpolation until the DV interface review accepts this one-sided backend classification." ] reviewerChecklist := [ "AutoSamplingTheory.dvVariationalOneSidedOfTiltedRight is a real theorem with no axiom/sorry/admit/fake closure.", "SALD.dvContract and saldStatusForLabel \"lem:dv_variation\" remain sourceCited because the supremum equality is not proved.", "Cycle 37 notes classify the new theorem as one-sided admissible-test inequality only, not Donsker--Varadhan equality.", "No SLT theorem is imported or marked formalized, and no downstream SALD theorem assumptions are changed." ] status := ProofStatus.obligation /-- Cycle-37 middle obligation tracking the new one-sided tilted backend. -/ def cycle37DvVariationMiddleObligation : ProofObligation where id := "sald.dv_variation.cycle37_middle_one_sided_tilted" statement := "Maintain the cycle 37 middle DV map for appendix.tex lines 73-79: AutoSamplingTheory.dvVariationalOneSidedOfTiltedRight proves the one-sided admissible-test inequality E_nu[Z]-log E_mu[exp Z] <= KL(nu||mu) under explicit Mathlib absolute-continuity, integrability, finite-log-mgf, and log-likelihood hypotheses, while the Boucheron supremum equality remains source-cited." source := saldDvVariationSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle37DvVariationUpperPacket", "SALD.cycle37DvVariationUpperObligation", "SALD.cycle37DvVariationMiddleAuditContract", "AutoSamplingTheory.dvVariationalOneSidedOfTiltedRight", "AutoSamplingTheory.dvVariationalTiltedRightOneSidedConsequence", "AutoSamplingTheory.dvVariationalOneSidedConsequenceScalar", "AutoSamplingTheory.dvVariationalOneSidedFromSupremumScalar", "dvVariationalFormulaInterface saldDvVariationSource", "probability.dv_variational_formula", "SALD.saldDvFiniteLogMgfContract", "sald.dv_variation.finite_log_mgf_interface", "Mathlib InformationTheory.KullbackLeibler.Basic", "Mathlib MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib MeasureTheory.Measure.Tilted" ] note := "This closes a genuine local one-sided backend but not lem:dv_variation as stated in the paper. The equality with the supremum over all finite-log-mgf tests remains a Boucheron-source-cited dependency." /-- Cycle-37 lower obligation tracking the composed one-sided DV consequence. -/ def cycle37DvVariationLowerObligation : ProofObligation where id := "sald.dv_variation.cycle37_lower_one_sided_consequence" statement := "Maintain the cycle 37 lower DV consequence for appendix.tex lines 73-79: AutoSamplingTheory.dvVariationalTiltedRightOneSidedConsequence derives E_nu[Z] <= KL(nu||mu)+log E_mu[exp Z] from the Mathlib tilted one-sided backend plus the scalar rearrangement, under explicit selected-test hypotheses; the Boucheron supremum equality remains source-cited." source := saldDvVariationSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle37DvVariationUpperPacket", "SALD.cycle37DvVariationUpperObligation", "SALD.cycle37DvVariationMiddleAuditContract", "SALD.cycle37DvVariationMiddleObligation", "AutoSamplingTheory.dvVariationalOneSidedOfTiltedRight", "AutoSamplingTheory.dvVariationalTiltedRightOneSidedConsequence", "AutoSamplingTheory.dvVariationalOneSidedConsequenceScalar", "dvVariationalFormulaInterface saldDvVariationSource", "probability.dv_variational_formula", "SALD.saldDvFiniteLogMgfContract", "sald.dv_variation.finite_log_mgf_interface" ] note := "This is a proof-producing lower consequence of the one-sided tilted backend. It does not prove the paper's supremum equality, does not discharge SALD squared-velocity or residual test hypotheses, and does not add assumptions to downstream theorem statements." /-- Cycle-42 middle source-to-Lean map for selected DV tests. This records the proof-producing middle slice for this cycle: `alpha0` finite exponential integrability implies the finite-log-mgf hypothesis for the selected scaled test `Z=alpha*q`, and the existing tilted backend then gives the one-sided admissible-test inequality. The source-cited Boucheron supremum equality is still not promoted to a local theorem. -/ def cycle42DvVariationMiddleAuditContract : FirstAppendixMiddleAuditContract where sourceIndexPath := "research-wiki/source-index/SALD_original.jsonl" focusLabels := [ "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "thm:forward-KL", "thm:forward-KL-discrete" ] sourceReadWindows := [ "appendix.tex:73-79", "appendix.tex:230-241 for the first SALD scaled test Z=alpha*||v_t||^2", "Mathlib Probability/Moments/IntegrableExpMul.lean: alpha0-to-alpha exponential-moment interval lemma", "Mathlib MeasureTheory/Measure/Tilted.lean and LogLikelihoodRatio.lean: one-sided tilted backend" ] sourceStepMap := [ "appendix.tex:73-74 fixes same-space probability measures mu and nu; selected SALD uses keep nu and mu as explicit Measure parameters.", "appendix.tex:75-78 fixes the finite-log-mgf class for real random variables Z; the selected scaled-test interface sets Z=alpha*q.", "The SALD applications supply q as a squared velocity/residual/frozen-defect norm and use alpha0-complexity as the intended finite-mgf witness.", "The Boucheron equality KL(nu||mu)=sup_Z(E_nu[Z]-log E_mu[exp Z]) remains source-cited through probability.dv_variational_formula." ] leanStepMap := [ "AutoSamplingTheory.dvFiniteLogMgfOfLeAlpha proves the local finite-log-mgf monotonicity: Integrable exp(alpha0*q) -> Integrable exp(alpha*q) for 0 <= alpha <= alpha0 under a finite measure.", "AutoSamplingTheory.dvVariationalOneSidedOfScaledTest composes that finite-log-mgf witness with AutoSamplingTheory.dvVariationalOneSidedOfTiltedRight for Z=alpha*q.", "The selected-test theorem keeps hnu_mu, Integrable (alpha*q) under nu, Integrable (llr nu mu) under nu, probability, and sigma-finiteness hypotheses explicit.", "No theorem proves the Boucheron supremum equality or theorem-specific SALD common-space/measurability/log-likelihood witnesses." ] citedResultMap := [ "Boucheron, Lugosi, and Massart, Concentration Inequalities, Corollary 4.15, cited at appendix.tex:73, still supplies only the full DV equality as a source-cited result.", "Mathlib ProbabilityTheory.integrable_exp_mul_of_nonneg_of_le supplies only the finite-mgf monotonicity used by the selected SALD test.", "No SLT theorem is imported or marked formalized." ] obligationMap := [ "probability.dv_variational_formula and SALD.dvContract remain sourceCited.", "sald.dv_variation.finite_log_mgf_interface still owes theorem-specific q, common-space, absolute-continuity, measurability, finite-KL/log-likelihood, and positive-alpha witnesses.", "forward-KL, discrete forward-KL, general moving-target, and general discrete theorem blocks may use AutoSamplingTheory.dvVariationalOneSidedOfScaledTest only after supplying their own selected-test hypotheses." ] lowerPacket := [ "Next lower attempt should instantiate q for one concrete source use, preferably appendix.tex:230-241 with q=||v_t||^2, without changing thm:forward-KL.", "If the theorem-specific hnu_mu, hZ_nu, or hllr hypotheses are unavailable locally, record the exact source gap and keep the interface below formalized.", "Do not mark lem:dv_variation formalized; this cycle closes only a finite-mgf/selected-test one-sided backend." ] reviewerChecklist := [ "AutoSamplingTheory.dvFiniteLogMgfOfLeAlpha and AutoSamplingTheory.dvVariationalOneSidedOfScaledTest compile without fake proof closures.", "SALD.dvContract and saldStatusForLabel \"lem:dv_variation\" remain sourceCited.", "The conversion window and cited-results page classify the new declarations as selected-test consequences, not as the Boucheron supremum formula.", "python3 tools/astis.py check passes." ] status := ProofStatus.obligation /-- Cycle-42 obligation for the selected scaled-test DV interface. -/ def cycle42DvVariationMiddleObligation : ProofObligation where id := "sald.dv_variation.cycle42_selected_scaled_test" statement := "Maintain the cycle 42 selected-test DV interface for appendix.tex lines 73-79: AutoSamplingTheory.dvFiniteLogMgfOfLeAlpha proves the alpha0-to-alpha finite-log-mgf handoff for Z=alpha*q, and AutoSamplingTheory.dvVariationalOneSidedOfScaledTest proves the one-sided admissible-test inequality under explicit selected-test hypotheses; the Boucheron supremum equality remains source-cited." source := saldDvVariationSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle42DvVariationMiddleAuditContract", "AutoSamplingTheory.dvFiniteLogMgfOfLeAlpha", "AutoSamplingTheory.dvVariationalOneSidedOfScaledTest", "AutoSamplingTheory.dvVariationalOneSidedOfTiltedRight", "AutoSamplingTheory.dvVariationalTiltedRightOneSidedConsequence", "probability.dv_variational_formula", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "sald.dv_variation.finite_log_mgf_interface", "Mathlib ProbabilityTheory.integrable_exp_mul_of_nonneg_of_le" ] note := "This is a proof-producing selected-test backend for the finite-log-mgf and one-sided inequality layer. It does not prove the paper's supremum equality, and it does not discharge theorem-specific SALD common-space, absolute-continuity, selected-test integrability, or log-likelihood integrability witnesses." /-- Cycle-42 lower obligation tracking the post-DV scaled energy bound. -/ def cycle42DvVariationLowerObligation : ProofObligation where id := "sald.dv_variation.cycle42_lower_scaled_energy" statement := "Maintain the cycle 42 lower scaled-test DV energy bridge for appendix.tex lines 73-79 and the first SALD use at appendix.tex lines 230-241: AutoSamplingTheory.dvVariationalScaledTestEnergyBound divides the selected-test one-sided inequality for Z=alpha*q by alpha>0 and rewrites the log-mgf quotient as eAlpha; AutoSamplingTheory.dvVariationalScaledTestEnergyBoundWithCoeff preserves the downstream nonnegative coefficient." source := saldForwardKlDvEnergySource status := ProofStatus.obligation dependsOn := [ "SALD.cycle42DvVariationMiddleAuditContract", "SALD.cycle42DvVariationMiddleObligation", "AutoSamplingTheory.dvFiniteLogMgfOfLeAlpha", "AutoSamplingTheory.dvVariationalOneSidedOfScaledTest", "AutoSamplingTheory.dvVariationalScaledTestEnergyBound", "AutoSamplingTheory.dvVariationalScaledTestEnergyBoundWithCoeff", "AutoSamplingTheory.dvVariationalOneSidedOfTiltedRight", "probability.dv_variational_formula", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "SALD.forwardKlDvPositiveAlphaScalingScalar", "SALD.forwardKlDvPositiveAlphaCoefficientScalar" ] note := "This is a proof-producing lower consequence for selected scaled tests only. It does not prove the Boucheron supremum equality, does not mark lem:dv_variation formalized, and does not discharge the theorem-specific common-space, absolute-continuity, q-integrability, alpha0 finite-mgf, or log-likelihood hypotheses." /-- Cycle-18 upper packet returning to the continuous forward-KL chain. This packet uses the accepted cycle-17 scalar Gronwall algebra only as a dependency marker for the continuous theorem's final Gronwall bookkeeping. It does not claim the full Gronwall lemma, DV, LSI-to-KL/FI, or the KL derivative backend has been formalized. -/ def cycle18ForwardKlUpperPacket : ForwardKlUpperPacket where objective := "Return to continuous thm:forward-KL and select the final Gronwall/DV/LSI dependency chain as the next faithful lower target, reusing the cycle-17 scalar exponent algebra only as local bookkeeping inside sald.forward_kl.gronwall_side_conditions." sourceLabels := [ "thm:forward-KL", "eq:SALD", "eq:FP-eq", "eq:LSI-KL-FI", "def:alpha-complexity", "lem:dv_variation", "lem:gronwall" ] modeDiscipline := [ "faithfulPaper: use main_body.tex:238-247 and appendix.tex:164-252 from the original source root, with sald_version_2.tex excluded.", "Keep the theorem statement fixed: LSI constants C_LSI(t)>=0, finite alpha0-complexity for the transport velocity v_t, alpha in (0,alpha0], and the two-term terminal KL display in main_body.tex:243-246.", "Preserve the source route derivative -> LSI -> DV -> Gronwall and the coefficients a(t)=dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1) and b(t)=(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t).", "Treat the cycle-17 scalar Gronwall lemmas and cycle-18 adjacent-interval bridge as partial local algebra only; theorem-specific interval-integrability, coefficient regularity, endpoint rewrites, DV finite-log-mgf, LSI-to-KL/FI, and KL derivative remain obligations or source-cited facts." ] nonGoals := [ "Do not restate or prove thm:forward-KL in this packet.", "Do not replace the paper route with Pinsker, Talagrand, Girsanov, path-space comparison, PI-to-LSI, or a generalized moving-target theorem.", "Do not add endpoint, positivity, density, boundary, finite-log-mgf, absolute-continuity, differentiability, or interval-integrability assumptions silently to the theorem statement.", "Do not mark lem:gronwall, lem:dv_variation, eq:LSI-KL-FI, the KL derivative, or the moving-target dependency chain formalized." ] lowerPacket := [ "Target exactly SALD.forwardKlGronwallSideConditionContract / SALD.forwardKlGronwallSideConditionObligation / sald.forward_kl.gronwall_side_conditions.", "Refine only the final Gronwall side-condition ledger: endpoint K(0)/K(T) rewrites, coefficient regularity for a(t), b(t), exponent split, residual LSI exponent drop, and how the cycle-17 scalar Real.exp helpers fit after adjacent-interval additivity.", "Keep the DV slice as SALD.forwardKlDvFiniteLogMgfWitnessContract and the LSI slice as SALD.saldLsiKlFiDensityTestContract; do not prove or rewrite those backends in the same lower attempt.", "If interval-integral additivity or residual exponent monotonicity is not ready, refine the named obligation instead of changing the theorem display." ] reviewerChecklist := [ "SALD.continuousSaldContract remains contractOnly and still lists the forward-KL moving-target, derivative, DV, coefficient-chain, Gronwall-side-condition, and Gronwall application obligations.", "SALD.forwardKlProofDag routes thm:forward-KL through moving_target_dependencies, coefficient_chain_audit, dv_energy, and gronwall_side_conditions, with cycle-17 scalar Gronwall helpers and the cycle-18 adjacent-interval bridge recorded only as local dependencies.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes SALD.cycle18ForwardKlUpperPacket without removing SALD.cycle14ForwardKlUpperPacket or SALD.cycle14ForwardKlMiddleContract.", "The source index contains thm:forward-KL, eq:LSI-KL-FI, lem:dv_variation, and lem:gronwall, and sald_version_2.tex remains excluded.", "No fake proof closure appears and no analytic dependency is promoted beyond its current obligation/source-cited status." ] status := ProofStatus.obligation /-- Cycle-18 middle packet for the continuous forward-KL Gronwall side conditions. This refines the upper packet into a lower-ready source-to-Lean map for the last Gronwall display of `thm:forward-KL`. It records how the accepted cycle-17 scalar Gronwall algebra can be reused only after the interval-integral side conditions have produced the required scalar equalities. -/ def cycle18ForwardKlMiddleContract : ForwardKlMiddleSourceToLeanContract where sourceStatement := saldForwardKlSource sourceProof := saldForwardKlProofSource derivativeSource := saldForwardKlDerivativeSource dvSource := saldForwardKlDvEnergySource gronwallSource := saldForwardKlGronwallSource objective := "Translate the cycle-18 upper target into a lower-ready map for sald.forward_kl.gronwall_side_conditions, with the final Gronwall endpoint, exponent-splitting, residual-exponent, and scalar-helper bridge kept separate from DV, LSI, KL-derivative, and schedule backends." sourceStepMap := [ "main_body.tex:238-247 fixes the theorem statement, inverse slowdown notation, C_LSI(t)>=0, alpha in (0,alpha0], and the two-exponential terminal display; no new theorem hypotheses are introduced.", "appendix.tex:210-228 supplies the LSI/time-change side of the coefficient a(t), but the LSI-to-KL/FI density-test proof and inverse-schedule calculus remain separate obligations.", "appendix.tex:230-241 supplies the DV side of the coefficient a(t) and b(t) through Z=alpha*||v_t||^2; finite-log-mgf and common-space facts remain in the DV witness obligation.", "appendix.tex:244-248 applies lem:gronwall with a(t)=dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1) and b(t)=(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t).", "appendix.tex:249-252 splits the initial exponent into the LSI contraction factor and alpha factor, then drops only the nonpositive LSI contribution from the residual exponent.", "appendix.tex:63-69 supplies the reusable scalar exponent pattern; cycle-17 formalized the Real additive/exponential algebra, cycle 18 added the adjacent-interval integral bridge, and cycle 21 adds the outer-integral congruence wrapper once adjacent interval-integrability is available." ] leanStepMap := [ "The theorem statement remains SALD.continuousForwardKlStatementContract and SALD.continuousSaldContract; this packet changes neither.", "Endpoint rewrites K(T)=KL(rho_S||pi_T) and K(0)=KL(rho_0||pi_0) remain routed through SALD.forwardKlEndpointScheduleContract and sald.forward_kl.endpoint_schedule_identities.", "The final Gronwall side conditions are routed through SALD.forwardKlGronwallSideConditionContract and SALD.forwardKlGronwallSideConditionObligation.", "The initial exponent split should first produce interval-integral equalities for the two coefficient pieces; the reusable Gronwall interval steps are SALD.gronwallExpProductRewriteIntervalIntegral and SALD.gronwallExpProductRewriteIntegralCongr, which call the cycle-17 scalar helpers after adjacent interval-integrability is available.", "The residual exponent drop needs sign facts C_LSI(u)>=0 and dot{s}(u)>0 plus interval-integral monotonicity; it is not supplied by the cycle-17 scalar helper lemmas.", "Coefficient regularity for a(t), b(t) remains the local interface required before lem:gronwall can be instantiated." ] citedResultInterfaces := [ "lem:dv_variation remains source-cited through Boucheron Cor. 4.15 or a future local entropy-duality port; this packet does not import or formalize it.", "eq:LSI-KL-FI remains the local density-test obligation using phi=sqrt(rho/pi), with coefficient 1/(2*C_LSI) unchanged.", "lem:gronwall remains the local real-analysis obligation; the cycle-17 scalar helper lemmas and cycle-18 interval bridge are only substeps of the exponent rewrite.", "No SLT theorem is used for endpoint rewrites, coefficient regularity, exponent splitting, or residual exponent monotonicity." ] obligations := [ "sald.forward_kl.gronwall_side_conditions", "sald.forward_kl.endpoint_schedule_identities", "sald.forward_kl.coefficient_chain_audit", "sald.forward_kl.gronwall_application", "sald.gronwall.exponent_rewrite", "sald.gronwall.integrating_factor", "SALD.gronwallExponentRewriteObligation", "SALD.forwardKlGronwallSideConditionObligation", "SALD.forwardKlDvFiniteLogMgfWitnessObligation", "SALD.lsiKlFiDensityTestObligation" ] lowerPacket := [ "Target exactly SALD.forwardKlGronwallSideConditionContract / SALD.forwardKlGronwallSideConditionObligation / sald.forward_kl.gronwall_side_conditions.", "Preferred sub-slice: bridge interval-integral additivity/orientation to the cycle-17 scalar helpers for the reusable exponent split, then record theorem-specific interval-integrability and residual-exponent monotonicity obligations separately.", "Do not change SALD.forwardKlDvFiniteLogMgfWitnessContract, SALD.saldLsiKlFiDensityTestContract, SALD.continuousForwardKlStatementContract, or the source theorem display.", "Do not mark endpoint rewrites, coefficient regularity, residual exponent drop, DV, LSI-to-KL/FI, KL derivative, schedule, or full Gronwall formalized unless a compiled local proof is added." ] reviewerChecklist := [ "SALD.forwardKlProofDag lists SALD.cycle18ForwardKlMiddleContract inside the Gronwall side-condition block and keeps the cycle-17 scalar helpers plus SALD.gronwallExpProductRewriteIntervalIntegral and SALD.gronwallExpProductRewriteIntegralCongr as local dependencies only.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes SALD.cycle18ForwardKlMiddleContract while retaining cycle-14 upper/middle and cycle-18 upper dependencies.", "The conversion window and proof-obligation ledger record this packet as source-to-Lean synchronization, not a proof of thm:forward-KL.", "SALD_original.jsonl still indexes thm:forward-KL, eq:LSI-KL-FI, lem:dv_variation, and lem:gronwall while excluding sald_version_2.tex.", "No fake proof closure appears and no analytic dependency is promoted." ] status := ProofStatus.obligation /-- Cycle-22 upper packet for the continuous forward-KL Gronwall side conditions. This packet follows the cycle-21 Gronwall outer-integral congruence refinement. It selects only the theorem-specific coefficient regularity and interval- integrability bridge needed before the compiled Gronwall exponent sublemmas can be used inside the continuous forward-KL display. It does not change `thm:forward-KL` or promote DV, LSI, the KL derivative, endpoint rewrites, or the full Gronwall lemma. -/ def cycle22ForwardKlUpperPacket : ForwardKlUpperPacket where objective := "Keep continuous thm:forward-KL fixed and assign the next lower work to the theorem-specific Gronwall side-condition bridge: coefficient regularity and adjacent interval-integrability for the source a(t), b(t), and exponent pieces, reusing SALD.gronwallExpProductRewriteIntegralCongr only as a local algebra substep." sourceLabels := [ "thm:forward-KL", "eq:SALD", "eq:FP-eq", "eq:LSI-KL-FI", "def:alpha-complexity", "lem:dv_variation", "lem:gronwall" ] modeDiscipline := [ "faithfulPaper: use main_body.tex:238-247, appendix.tex:210-252, and the Gronwall exponent-rewrite source window appendix.tex:63-69 from the original source root; sald_version_2.tex remains excluded.", "Preserve the source theorem statement, the differential inequality after DV, the Gronwall coefficients a(t)=dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1) and b(t)=(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t), and the two-term display in main_body.tex:243-246.", "Treat SALD.forwardKlGronwallExpProductRewriteIntegralCongrOfPieces and SALD.gronwallExpProductRewriteIntegralCongr as compiled local interval-integral algebra only. The theorem still owes piecewise coefficient regularity, endpoint rewrites, residual-exponent monotonicity, KL derivative, DV finite-log-mgf, LSI-to-KL/FI, and full Gronwall backends." ] nonGoals := [ "Do not restate or prove thm:forward-KL in this packet.", "Do not replace the source route derivative -> LSI -> DV -> Gronwall with Pinsker, Talagrand, Girsanov, PI-to-LSI, the general moving-target theorem, or any alternate comparison method.", "Do not add endpoint, positivity, density, boundary, finite-log-mgf, absolute-continuity, differentiability, continuity, or interval-integrability assumptions silently to the theorem statement.", "Do not mark lem:gronwall, lem:dv_variation, eq:LSI-KL-FI, the KL derivative, endpoint schedule identities, or residual-exponent drop formalized." ] lowerPacket := [ "Target exactly SALD.forwardKlGronwallSideConditionContract / SALD.forwardKlGronwallSideConditionObligation / sald.forward_kl.gronwall_side_conditions.", "First lower sub-slice: expose the theorem-specific regularity and adjacent interval-integrability hypotheses for the LSI part dot{s}(t)*C_LSI(t), alpha part (1/2)*dot{s}(t)^(-1)*alpha^(-1), and b(t); SALD.forwardKlGronwallCoeffAdjacentIntervalIntegrable now assembles the a(t) interval-integrability from the two coefficient pieces.", "Classify blockers explicitly: local-lemma for interval-integral additivity/congruence, source-contract-gap for coefficient regularity and endpoint schedule identities, and internal-paper-step for the residual LSI exponent drop.", "Keep DV finite-log-mgf, LSI density-test, KL derivative, endpoint K(0)/K(T) rewrites, and the full Gronwall lemma in their existing obligations unless a separate compiled proof is added." ] reviewerChecklist := [ "SALD.forwardKlProofDag routes ASTIS.SALD.forward_KL.gronwall_side_conditions through SALD.cycle22ForwardKlUpperPacket while retaining cycle-14 and cycle-18 packets.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes SALD.cycle22ForwardKlUpperPacket and still lists the compiled Gronwall scalar/interval helpers plus the forward-KL coefficient assembly lemmas only as dependencies.", "The conversion window and proof-obligation ledger identify cycle 22 as a faithful upper packet for coefficient regularity and interval-integrability, not as a proof of thm:forward-KL.", "SALD_original.jsonl indexes thm:forward-KL, eq:LSI-KL-FI, lem:dv_variation, and lem:gronwall while excluding sald_version_2.tex.", "No fake proof closure appears and no analytic dependency is promoted beyond obligation or source-cited status." ] status := ProofStatus.obligation /-- Cycle-22 middle packet for the continuous forward-KL coefficient bridge. This converts the upper coefficient-regularity objective into a lower-ready source-to-Lean map. The packet is deliberately narrower than the full `thm:forward-KL` proof: it only audits the Gronwall coefficients and the adjacent interval-integrability needed before the compiled exponent congruence can be used in the theorem display. -/ def cycle22ForwardKlMiddleContract : ForwardKlMiddleSourceToLeanContract where sourceStatement := saldForwardKlSource sourceProof := saldForwardKlProofSource derivativeSource := saldForwardKlDerivativeSource dvSource := saldForwardKlDvEnergySource gronwallSource := saldForwardKlGronwallSource objective := "Translate the cycle-22 upper target into a lower-ready map for sald.forward_kl.gronwall_side_conditions: prove or isolate theorem-specific regularity and adjacent interval-integrability for the LSI and alpha pieces of a(t), and for b(t), before applying SALD.forwardKlGronwallExpProductRewriteIntegralCongrOfPieces." sourceStepMap := [ "appendix.tex:210-217 supplies the nonnegative LSI part of a(t), namely dot{s}(t)*C_LSI(t), after the LSI-to-KL/FI bridge.", "appendix.tex:218-228 supplies the inverse-schedule rewrite that turns the velocity coefficient into dot{s}(t)^(-1).", "appendix.tex:230-241 supplies the DV alpha contribution (1/2)*dot{s}(t)^(-1)*alpha^(-1) to a(t) and the source b(t)=(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t).", "appendix.tex:244-248 applies lem:gronwall to K(t)=KL(rho_{s(t)}||pi_t) with exactly those a(t) and b(t); continuity or interval-integrability is implicit in the paper and must be exposed locally.", "appendix.tex:249-250 splits the initial exponent into the LSI contraction factor and alpha factor; the compiled outer-integral congruence is applicable only after adjacent interval-integrability and orientation hypotheses are available.", "appendix.tex:248-252 drops the LSI part from the residual exponential using C_LSI(u)>=0 and dot{s}(u)>0; this remains an internal paper step plus interval-integral monotonicity obligation." ] leanStepMap := [ "Keep SALD.continuousForwardKlStatementContract and SALD.continuousSaldContract unchanged.", "Route the lower target through SALD.forwardKlGronwallSideConditionContract and SALD.forwardKlGronwallSideConditionObligation, not a new theorem statement.", "Treat a(t) as the difference of the LSI coefficient and alpha coefficient; lower work may expose integrability for the pieces before assembling the Gronwall coefficient.", "Treat b(t) as a product of dot{s}(t)^(-1) and E_alpha(pi_t,v_t); finite-log-mgf and measurability still come from the DV witness and alpha-complexity obligations.", "Use SALD.forwardKlGronwallExpProductRewriteIntegralCongrOfPieces only for the exponent-congruence substep after the LSI and alpha coefficient pieces have adjacent interval-integrability; it delegates to SALD.gronwallExpProductRewriteIntegralCongr.", "Keep endpoint rewrites K(0)/K(T), residual-exponent monotonicity, DV, LSI-to-KL/FI, the KL derivative, and full Gronwall as separate obligations unless a compiled proof is added." ] citedResultInterfaces := [ "lem:dv_variation remains source-cited through Boucheron Cor. 4.15 or a future local entropy-duality port; cycle 22 only records its coefficient output.", "eq:LSI-KL-FI remains the local density-test obligation that supplies the LSI contraction coefficient.", "lem:gronwall remains the local real-analysis obligation; SALD.forwardKlGronwallExpProductRewriteIntegralCongrOfPieces and SALD.gronwallExpProductRewriteIntegralCongr are only compiled sublemmas for the final exponent algebra.", "No SLT theorem is imported or marked formalized for coefficient regularity, adjacent interval-integrability, endpoint rewrites, or residual-exponent monotonicity." ] obligations := [ "sald.forward_kl.gronwall_side_conditions", "sald.forward_kl.coefficient_chain_audit", "sald.forward_kl.endpoint_schedule_identities", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.kl_derivative", "probability.lsi_to_kl_fi", "sald.gronwall.exponent_rewrite", "sald.gronwall.integrating_factor", "SALD.forwardKlGronwallSideConditionObligation", "SALD.forwardKlGronwallApplicationObligation", "SALD.forwardKlDvFiniteLogMgfWitnessObligation", "SALD.lsiKlFiDensityTestObligation" ] lowerPacket := [ "Target exactly SALD.forwardKlGronwallSideConditionContract / SALD.forwardKlGronwallSideConditionObligation / sald.forward_kl.gronwall_side_conditions.", "First prove or refine only the coefficient regularity and adjacent interval-integrability needed for dot{s}(t)*C_LSI(t), (1/2)*dot{s}(t)^(-1)*alpha^(-1), and b(t); the a(t) interval-integrability is assembled by SALD.forwardKlGronwallCoeffAdjacentIntervalIntegrable.", "After those hypotheses are exposed, connect the theorem-specific exponent split to SALD.forwardKlGronwallExpProductRewriteIntegralCongrOfPieces without changing the signs, endpoint limits, or source coefficients.", "Leave endpoint K(0)/K(T) rewrites, residual exponent drop, DV finite-log-mgf, LSI density-test, KL derivative, and full Gronwall in their existing obligations unless a separate compiled proof is added." ] reviewerChecklist := [ "SALD.forwardKlProofDag lists SALD.cycle22ForwardKlMiddleContract in the Gronwall side-condition block.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes SALD.cycle22ForwardKlMiddleContract while retaining cycle-14 and cycle-18 packets.", "The conversion window, proof-obligation ledger, and SLT audit identify cycle 22 middle as source-to-Lean synchronization for coefficient regularity, not a proof of thm:forward-KL.", "SALD_original.jsonl still indexes thm:forward-KL, eq:LSI-KL-FI, lem:dv_variation, and lem:gronwall while excluding sald_version_2.tex.", "No fake proof closure appears and no analytic dependency is promoted beyond obligation or source-cited status." ] status := ProofStatus.obligation /-- Cycle-26 upper packet for the continuous forward-KL DV witness. This returns to `thm:forward-KL` after the first-appendix cycle-25 PI work and selects only the theorem-specific Donsker--Varadhan finite-log-mgf/common-space witness as the next lower target. It preserves the existing Gronwall coefficient work from cycle 22 as a dependency and does not promote DV, LSI-to-KL/FI, the KL derivative, or Gronwall to formalized status. -/ def cycle26ForwardKlUpperPacket : ForwardKlUpperPacket where objective := "Return to continuous thm:forward-KL and select the theorem-specific DV finite-log-mgf/common-space witness for Z=alpha*||v_t||^2 as the next lower target, while preserving the existing derivative -> LSI -> DV -> Gronwall chain and the cycle-22 Gronwall coefficient side-condition work." sourceLabels := [ "thm:forward-KL", "def:alpha-complexity", "lem:dv_variation", "eq:LSI-KL-FI", "lem:gronwall", "eq:SALD", "eq:FP-eq" ] modeDiscipline := [ "faithfulPaper: use main_body.tex:218-248 and appendix.tex:73-79, 164-252 from the original source root; sald_version_2.tex remains excluded.", "Keep the theorem statement fixed: finite E_alpha0(pi_t,v_t), alpha in (0,alpha0], the transport velocity v_t of pi_t, and the terminal KL display in main_body.tex:243-246 are not changed.", "Preserve the DV instantiation exactly: nu=rho_{s(t)}, mu=pi_t, Z=alpha*||v_t||^2, with the source coefficient alpha^(-1) and the downstream Gronwall coefficient (1/2)*dot{s}(t)^(-1)*alpha^(-1).", "Treat lem:dv_variation as source-cited, the alpha0-to-alpha finite-log-mgf bridge as a local obligation, and LSI-to-KL/FI, KL derivative, schedule calculus, coefficient regularity, and full Gronwall as separate obligations." ] nonGoals := [ "Do not restate or prove thm:forward-KL in this packet.", "Do not replace the paper route derivative -> LSI -> DV -> Gronwall with Pinsker, Talagrand, Girsanov, PI-to-LSI, path-space comparison, or the general moving-target theorem.", "Do not add finite-log-mgf, absolute-continuity, common-space, endpoint, positivity, differentiability, or integrability assumptions silently to the theorem statement.", "Do not import or mark an SLT entropy-duality theorem formalized; SLT remains a reference pattern until a local port builds.", "Do not reopen the cycle-22 Gronwall coefficient assembly except as a dependency of the final theorem display." ] lowerPacket := [ "Target exactly SALD.forwardKlDvFiniteLogMgfWitnessContract / SALD.forwardKlDvFiniteLogMgfWitnessObligation / sald.forward_kl.dv_finite_log_mgf_witness.", "First lower sub-slice: expose the common state space, absolute-continuity relationship, measurability of ||v_t||^2, and finite log-mgf needed to apply lem:dv_variation with Z=alpha*||v_t||^2.", "Use SALD.forwardKlDvAlphaMonotonicityContract only for the alpha0-to-alpha finite-log-mgf bridge; if the monotonicity/order backend is blocked, record that backend as the lower obligation rather than adding a theorem assumption.", "Preserve positive-alpha scaling when dividing the DV inequality by alpha and keep the downstream coefficient (1/2)*dot{s}(t)^(-1)*alpha^(-1) unchanged.", "Leave LSI density-test, KL derivative, endpoint schedule identities, Gronwall side conditions, residual exponent drop, and full Gronwall in their existing obligations." ] reviewerChecklist := [ "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes SALD.cycle26ForwardKlUpperPacket while retaining cycle-14, cycle-18, and cycle-22 packets.", "SALD.forwardKlProofDag routes ASTIS.SALD.forward_KL.dv_finite_log_mgf_witness through SALD.cycle26ForwardKlUpperPacket before the DV-energy block.", "SALD.forwardKlDvFiniteLogMgfWitnessObligation remains an obligation and depends on SALD.forwardKlDvAlphaMonotonicityContract plus the source-cited DV interface; no theorem assumption is added.", "The conversion window, proof-obligation ledger, and SLT audit classify cycle 26 as upper workflow/source-dependency refinement, not as a proof of DV or thm:forward-KL.", "SALD_original.jsonl indexes thm:forward-KL, def:alpha-complexity, lem:dv_variation, eq:LSI-KL-FI, and lem:gronwall while excluding sald_version_2.tex.", "No fake proof closure appears and no analytic dependency is promoted beyond obligation or source-cited status." ] status := ProofStatus.obligation /-- Cycle-26 middle packet for the continuous forward-KL DV witness. This converts the upper-selected finite-log-mgf/common-space target into a lower-ready source-to-Lean map. It does not prove the Donsker--Varadhan formula, the exponential-moment monotonicity bridge, or `thm:forward-KL`. -/ def cycle26ForwardKlMiddleContract : ForwardKlMiddleSourceToLeanContract where sourceStatement := saldForwardKlSource sourceProof := saldForwardKlProofSource derivativeSource := saldForwardKlDerivativeSource dvSource := saldForwardKlDvEnergySource gronwallSource := saldForwardKlGronwallSource objective := "Translate the cycle-26 upper target into a lower-ready map for sald.forward_kl.dv_finite_log_mgf_witness: isolate common-space/absolute-continuity, measurability of ||v_t||^2, alpha0-to-alpha finite-log-mgf, and positive-alpha scaling before the DV-energy inequality is used." sourceStepMap := [ "main_body.tex:218-228 defines E_alpha(pi_t,v_t)=alpha^(-1)*log E_{pi_t}[exp(alpha*||v_t||^2)] and the integrated alpha-complexity; this supplies the log-mgf expression, not a Lean proof of finiteness.", "main_body.tex:240-241 assumes there is alpha0>0 with E_{alpha0}(pi_t,v_t)<+infty for every t in [0,T]; appendix.tex:230-241 immediately applies the DV formula for any alpha in (0,alpha0].", "appendix.tex:73-79 states lem:dv_variation on probability distributions on the same space with finite log-mgf; in this theorem the intended measures are nu=rho_{s(t)} and mu=pi_t.", "appendix.tex:230-235 chooses the exact DV test function Z_t(x)=alpha*||v_t(x)||^2; lower work must expose measurability and real-valuedness of this squared-velocity test.", "appendix.tex:234-238 gives E_{rho_{s(t)}}[alpha*||v_t||^2] <= KL(rho_{s(t)}||pi_t)+log E_{pi_t}[exp(alpha*||v_t||^2)]; the absolute-continuity/common-space facts are implicit and stay obligations.", "appendix.tex:239-241 divides by alpha>0 and rewrites alpha^(-1)*log E_{pi_t}[exp(alpha*||v_t||^2)] as E_alpha(pi_t,v_t), preserving the downstream coefficient (1/2)*dot{s}(t)^(-1)*alpha^(-1)." ] leanStepMap := [ "Keep SALD.continuousForwardKlStatementContract and SALD.continuousSaldContract unchanged; no new finite-mgf or absolute-continuity theorem hypothesis is added.", "Route the selected lower target through SALD.forwardKlDvFiniteLogMgfWitnessContract, SALD.forwardKlDvFiniteLogMgfWitnessObligation, and sald.forward_kl.dv_finite_log_mgf_witness.", "Use SALD.forwardKlDvAlphaMonotonicityContract and sald.forward_kl.dv_alpha_mgf_monotonicity only for the alpha0-to-alpha finite-log-mgf bridge.", "Use SALD.saldDvFiniteLogMgfContract and probability.dv_variational_formula only as the source-cited DV interface; do not mark entropy duality formalized.", "Use SALD.forwardKlDvPositiveAlphaScalingScalar and SALD.forwardKlDvPositiveAlphaCoefficientScalar only for the scalar division and coefficient-preservation substep after the DV inequality has been supplied.", "Keep the common state space and rho_{s(t)} << pi_t requirements tied to SALD.forwardKlMovingTargetDependencyContract and the KL vocabulary rather than adding a theorem-level assumption.", "Leave the subsequent coefficient audit and Gronwall assembly in SALD.forwardKlDependencyChainAuditContract, SALD.forwardKlGronwallSideConditionContract, and the cycle-22 Gronwall helpers." ] citedResultInterfaces := [ "lem:dv_variation is external-cited-result via Boucheron Cor. 4.15; the SLT entropy_duality theorem remains a future port pattern, not an imported ASTIS dependency.", "The alpha0-to-alpha finite-log-mgf bridge is a local measure/order lemma, not an external cited result.", "The common-space, absolute-continuity, and measurability interfaces are source-contract gaps implicit in the paper's KL and vector-field vocabulary.", "LSI-to-KL/FI, the KL derivative, endpoint schedule identities, coefficient regularity, and full Gronwall remain separate obligations outside this lower slice." ] obligations := [ "sald.forward_kl.cycle26_dv_witness_middle", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_alpha_mgf_monotonicity", "SALD.forwardKlDvPositiveAlphaScalingScalar", "SALD.forwardKlDvPositiveAlphaCoefficientScalar", "sald.dv_variation.finite_log_mgf_interface", "probability.dv_variational_formula", "sald.forward_kl.moving_target_dependency_chain", "sald.forward_kl.dv_energy_bound" ] lowerPacket := [ "Target exactly SALD.forwardKlDvFiniteLogMgfWitnessContract / SALD.forwardKlDvFiniteLogMgfWitnessObligation / sald.forward_kl.dv_finite_log_mgf_witness.", "First sub-slice: provide a Lean-facing interface for the same measurable space of rho_{s(t)} and pi_t, rho_{s(t)} << pi_t, and measurability/real-valuedness of Z_t=alpha*||v_t||^2.", "Second sub-slice if the first is blocked: refine sald.forward_kl.dv_alpha_mgf_monotonicity from finite E_alpha0(pi_t,v_t) to finite log E_{pi_t}[exp(alpha*||v_t||^2)] for 0 LSI -> DV -> Gronwall with Girsanov, Pinsker, Talagrand, path-space comparison, the general moving-target theorem, or a PI-based route.", "Do not reopen the cycle-26 DV finite-log-mgf witness or cycle-29 LSI density-test lower work except as dependencies of the derivative handoff.", "Do not change the Young coefficient 1/2, the LSI coefficient C_LSI(t), the time-change coefficient (1/2)*dot{s}(t)^(-1), or the terminal display in main_body.tex:243-246.", "Do not mark the KL derivative, Fokker--Planck backend, integration by parts, LSI-to-KL/FI, DV, Gronwall, or endpoint schedule identities formalized." ] lowerPacket := [ "Target exactly SALD.forwardKlDerivativeSideConditionContract / SALD.forwardKlDensityBoundaryObligation / sald.forward_kl.density_boundary_regular as the first lower slice.", "First sub-slice: appendix.tex:168-185, expose mass conservation, differentiation under the KL integral, the SALD Fokker--Planck equation, and the integration-by-parts conditions that identify the first derivative term as -FI(rho_s||tilde_pi_s).", "Second sub-slice if the first is blocked: appendix.tex:187-208, expose common state-space/density assumptions, the slowed-target transport identity for tilde_v_s, target-side integration by parts, and Cauchy--Schwarz/Young with the exact 1/2 coefficients.", "Leave appendix.tex:218-228 time-change identities in SALD.forwardKlScheduleTimeChangeObligation unless the lower attempt finishes the density/boundary slice and explicitly records the handoff.", "Keep DV, final coefficient-chain audit, Gronwall side conditions, and theorem-level endpoint rewrites in their existing obligations." ] reviewerChecklist := [ "SALD.forwardKlProofDag contains a cycle-30 derivative-side block before ASTIS.SALD.forward_KL.derivative and before DV/Gronwall blocks.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes SALD.cycle30ForwardKlUpperPacket and sald.forward_kl.cycle30_derivative_side_upper while retaining cycle-14, cycle-18, cycle-22, and cycle-26 packets.", "SALD.forwardKlDerivativeSideConditionContract still classifies mass conservation, differentiation-under-integral, SALD/target integration by parts, Young, and inverse-schedule time change as obligations, not theorem assumptions.", "Conversion window, proof-obligation ledger, and SLT audit identify cycle 30 as derivative-side source-to-Lean routing, not a proof of thm:forward-KL.", "SALD_original.jsonl indexes thm:forward-KL, eq:SALD, eq:FP-eq, eq:LSI-KL-FI, lem:dv_variation, and lem:gronwall from the original files while excluding sald_version_2.tex.", "No fake proof closure appears and no analytic dependency is promoted beyond obligation or source-cited status." ] status := ProofStatus.obligation /-- Cycle-30 middle packet for the continuous forward-KL derivative side. This translates the upper-selected derivative-side target into a lower-ready source-to-Lean map. The first lower slice is only `appendix.tex:168-185`: mass conservation, KL differentiation, the SALD Fokker-Planck equation, and the integration-by-parts identification of the first derivative term with `-FI`. Target-side transport, LSI, inverse-schedule calculus, DV, and Gronwall remain separate obligations. -/ def cycle30ForwardKlMiddleContract : ForwardKlMiddleSourceToLeanContract where sourceStatement := saldForwardKlSource sourceProof := saldForwardKlProofSource derivativeSource := saldForwardKlDerivativeSource dvSource := saldForwardKlDvEnergySource gronwallSource := saldForwardKlGronwallSource objective := "Translate the cycle-30 upper derivative-side target into a lower-ready map for sald.forward_kl.density_boundary_regular, with appendix.tex:168-185 as the first sub-slice and appendix.tex:187-208 as a follow-on density/boundary slice before the separate LSI, time-change, DV, and Gronwall obligations." sourceStepMap := [ "main_body.tex:238-247 fixes the theorem statement, inverse slowdown notation, LSI constants, alpha-complexity assumption, alpha range, and terminal KL display; this middle packet does not change any of them.", "appendix.tex:168-174 differentiates KL(rho_s||tilde_pi_s), passes the derivative through the integral, and uses int partial_s rho_s dx=0; the Lean target must expose mass conservation and differentiation-under-integral hypotheses.", "appendix.tex:176-185 substitutes the SALD Fokker-Planck equation partial_s rho_s=div(rho_s*nabla log(rho_s/tilde_pi_s)) and integrates by parts to identify the first term as -FI(rho_s||tilde_pi_s).", "appendix.tex:187-197 states that v_t transports pi_t and that tilde_v_s=dot{t}(s)*v_{t(s)} transports tilde_pi_s; this is a follow-on transport side condition, not part of the first density/FI slice.", "appendix.tex:199-208 evaluates the target derivative term by integration by parts and Cauchy--Schwarz/Young, preserving the exact split (1/2)*FI+(1/2)*||tilde_v_s||^2.", "appendix.tex:210-228 applies the LSI bridge and inverse-schedule chain rule only after the density/boundary and transport slices have produced the s-derivative inequality." ] leanStepMap := [ "Route the selected lower target through SALD.forwardKlDerivativeSideConditionContract, SALD.forwardKlDensityBoundaryObligation, and sald.forward_kl.density_boundary_regular.", "Use SALD.forwardKlDerivativeCandidateContract only as the parent derivative route; do not promote sald.forward_kl.kl_derivative before the density/boundary side conditions are available.", "Keep FokkerPlanckContract, KLContract, and FIContract as vocabulary/backends for appendix.tex:168-185; the paper's smooth-density, positivity, boundary, and domination assumptions remain explicit source gaps.", "Keep TransportVelocityContract and SALD.forwardKlScheduleTimeChangeObligation as sibling obligations for appendix.tex:187-228.", "Keep SALD.saldLsiKlFiDensityTestContract, SALD.forwardKlDvFiniteLogMgfWitnessContract, and SALD.forwardKlGronwallSideConditionContract unchanged; they are downstream of this derivative-side middle packet.", "This middle packet adds a workflow obligation sald.forward_kl.cycle30_derivative_side_middle and no theorem-level assumption." ] citedResultInterfaces := [ "No SLT theorem applies to the derivative-side density, boundary, mass-conservation, or integration-by-parts interfaces.", "eq:FP-eq is a local Fokker-Planck/SDE backend obligation for SALD, not an external cited theorem.", "eq:LSI-KL-FI remains the cycle-29 density-test obligation and starts after the first derivative slice.", "lem:dv_variation and lem:gronwall are downstream source-cited or local-analysis dependencies and are not reopened in this middle packet." ] obligations := [ "sald.forward_kl.cycle30_derivative_side_middle", "sald.forward_kl.cycle30_derivative_side_upper", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "probability.lsi_to_kl_fi", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.gronwall_side_conditions" ] lowerPacket := [ "Target exactly SALD.forwardKlDerivativeSideConditionContract / SALD.forwardKlDensityBoundaryObligation / sald.forward_kl.density_boundary_regular.", "First lower sub-slice: appendix.tex:168-185 only. Expose mass conservation, differentiation under the KL integral, the SALD Fokker-Planck equation, boundary/no-flux or decay assumptions, and the FI identification.", "If appendix.tex:168-185 is blocked, refine the missing density, domination, positivity, or integration-by-parts interface as a proof obligation rather than adding a theorem assumption.", "Second sub-slice after the first is stable: appendix.tex:187-208 target-side transport, integration by parts, and Young's inequality with the exact 1/2 coefficients.", "Leave appendix.tex:210-228 time change, eq:LSI-KL-FI, the DV finite-log-mgf witness, coefficient-chain audit, and Gronwall side conditions as named downstream obligations." ] reviewerChecklist := [ "SALD.forwardKlProofDag contains ASTIS.SALD.forward_KL.cycle30_derivative_side_middle between the upper packet and the derivative block.", "SALD.continuousSaldContract lists SALD.cycle30ForwardKlDerivativeSideMiddleObligation while keeping thm:forward-KL contract-only.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes SALD.cycle30ForwardKlMiddleContract and sald.forward_kl.cycle30_derivative_side_middle.", "The conversion window, proof-obligation ledger, and SLT audit classify cycle 30 middle as derivative-side source-to-Lean synchronization, not as a proof of the KL derivative.", "No analytic dependency is promoted beyond obligation or source-cited status." ] status := ProofStatus.obligation /-- Cycle-34 upper packet for the continuous forward-KL derivative closure sprint. This packet explicitly checks the proof-closure order and assigns only the next proof-producing derivative slice inside `appendix.tex:168-228`. The new compiled scalar lemmas combine already-supplied analytic inputs; they do not close the Fokker--Planck, integration-by-parts, LSI density-test, or time-change analytic backends. -/ def cycle34ForwardKlDerivativeUpperPacket : ForwardKlUpperPacket where objective := "Proof-closure priority check: (1) lem:gronwall remains a local real-analysis obligation, (2) lem:dv_variation remains source-cited with scalar consequences, (3) eq:LSI-KL-FI remains an obligation after cycle 33 scalar density-test lemmas, so this cycle follows item (4) by narrowing the continuous forward-KL Fokker-Planck/KL derivative block to proof-producing scalar lemmas for appendix.tex:168-217 before assigning the EM interpolation backend." sourceLabels := [ "thm:forward-KL", "eq:SALD", "eq:FP-eq", "eq:LSI-KL-FI", "lem:dv_variation", "lem:gronwall" ] modeDiscipline := [ "faithfulPaper: use main_body.tex:238-247 and appendix.tex:168-228 from the original source root; sald_version_2.tex remains excluded.", "Keep the theorem statement and paper route fixed: KL differentiation, SALD Fokker-Planck, slowed-target transport, Cauchy--Schwarz/Young, LSI, inverse-schedule time change, DV, then Gronwall.", "Use SALD.forwardKlPostYoungDerivativeBoundScalar and SALD.forwardKlLsiDerivativeBoundScalar only after the analytic identities they require are supplied by the named derivative and LSI obligations.", "Keep mass conservation, differentiation under the integral, Fokker--Planck, boundary/no-flux, target transport, LSI-to-KL/FI, inverse-schedule calculus, DV, and Gronwall below formalized status unless a compiled local proof replaces the obligation." ] nonGoals := [ "Do not rebaseline the source index or expand transcript ledgers unless a reviewer finds a blocking source-anchor defect.", "Do not restate or prove thm:forward-KL in this packet.", "Do not add smoothness, density, absolute-continuity, endpoint, positivity, boundary, LSI, inverse-function, finite-log-mgf, or interval-integrability assumptions to the theorem statement.", "Do not replace the source route with Girsanov, Pinsker, Talagrand, path-space comparison, PI-to-LSI, or the general moving-target theorem.", "Do not promote the KL derivative, Fokker--Planck backend, integration by parts, LSI-to-KL/FI, schedule time change, DV, or Gronwall." ] lowerPacket := [ "Target exactly SALD.forwardKlPostYoungDerivativeBoundScalar and SALD.forwardKlLsiDerivativeBoundScalar as the first proof-producing lower slice, registered by SALD.cycle34ForwardKlDerivativeScalarObligation / sald.forward_kl.cycle34_derivative_scalar.", "Middle must keep the source-to-Lean map synchronized for appendix.tex:168-217: first-term identity from cycle 30, target-side Young bound, and LSI half-Fisher comparison.", "If lower continues beyond the compiled scalar lemmas, the next sub-slice is appendix.tex:187-208 target-side transport/integration-by-parts and the exact Young 1/2 split, still as inputs to the scalar lemma.", "For appendix.tex:218-228, use SALD.forwardKlTimeChangedDerivativeBoundScalar only after sald.forward_kl.schedule_time_change supplies the analytic chain-rule, velocity-square scaling, inverse-derivative, and positivity inputs.", "Keep DV, coefficient-chain audit, Gronwall side conditions, endpoint rewrites, and EM interpolation work out of this lower attempt." ] reviewerChecklist := [ "SALD.forwardKlPostYoungDerivativeBoundScalar proves only the Real arithmetic from firstTerm=-FI and targetTerm <= (1/2)*FI+(1/2)*velocitySq to the post-Young derivative bound.", "SALD.forwardKlLsiDerivativeBoundScalar proves only the Real-order LSI substitution from C_LSI*K <= (1/2)*FI to the post-LSI derivative bound.", "SALD.forwardKlTimeChangedDerivativeBoundScalar proves only the Real-order inverse-schedule handoff after the analytic chain rule, velocity scaling, and inverse-derivative identities are supplied.", "SALD.forwardKlProofDag routes the cycle-34 derivative scalar block before ASTIS.SALD.forward_KL.derivative and keeps sald.forward_kl.kl_derivative as an obligation.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes the cycle-34 packet, obligation, and scalar lemmas without removing cycle-30 derivative-side obligations.", "No source-cited analytic theorem is marked formalized and the mandatory ASTIS check passes." ] status := ProofStatus.obligation /-- Cycle-34 middle map for the continuous forward-KL derivative scalar closure. This translates the upper packet into the specific Lean handoff for `appendix.tex:218-228`. The compiled theorem is pure real arithmetic; the chain rule, inverse-function identity, velocity scaling, and positivity facts remain in `sald.forward_kl.schedule_time_change`. -/ def cycle34ForwardKlDerivativeMiddleContract : ForwardKlMiddleSourceToLeanContract where sourceStatement := saldForwardKlSource sourceProof := saldForwardKlProofSource derivativeSource := saldForwardKlDerivativeSource dvSource := saldForwardKlDvEnergySource gronwallSource := saldForwardKlGronwallSource objective := "Translate appendix.tex:218-228 into a compiled scalar time-change handoff for the continuous forward-KL derivative, after the post-Young and LSI scalar inequalities from appendix.tex:168-217 have been supplied." sourceStepMap := [ "appendix.tex:210-217 supplies the s-time inequality dK/ds <= -C_LSI(t(s))*K_s+(1/2)*||tilde v_s||^2 through the existing post-Young and LSI scalar lemmas.", "appendix.tex:218-222 applies the analytic chain rule d/dt K(s(t))=dot{s}(t)*dK/ds at s=s(t); Lean uses this as the input hdKdt.", "appendix.tex:191-197 gives tilde v_s=dot{t}(s)*v_{t(s)}, so the L2 square scales as ||tilde v_s||^2=dot{t}(s)^2*||v_t||^2; Lean uses this as the input hvelocity.", "appendix.tex:223-228 rewrites dot{s}(t)*dot{t}(s(t))^2 to dot{s}(t)^(-1), requiring dot{t}(s(t))=dot{s}(t)^(-1), dot{s}(t) != 0, and nonnegative dot{s}(t).", "The resulting t-time pre-DV inequality is dK/dt <= -dot{s}(t)*C_LSI(t)*K(t)+(1/2)*dot{s}(t)^(-1)*||v_t||^2." ] leanStepMap := [ "SALD.forwardKlTimeChangedDerivativeBoundScalar compiles the real-order handoff from the supplied s-time bound to the t-time pre-DV inequality.", "SALD.forwardKlPostYoungDerivativeBoundScalar and SALD.forwardKlLsiDerivativeBoundScalar remain the compiled inputs for appendix.tex:168-217.", "SALD.forwardKlScheduleTimeChangeObligation still owns the analytic chain rule, inverse-schedule derivative identity, velocity-square scaling, and dot{s}(t) positivity/nonzero facts.", "SALD.forwardKlDerivativeObligation may use the scalar lemma only after the density/boundary backend, LSI comparison, and schedule-time-change inputs are available." ] citedResultInterfaces := [ "No SLT theorem applies to this scalar time-change handoff.", "eq:LSI-KL-FI, lem:dv_variation, and lem:gronwall remain separate obligations/source-cited dependencies and are not promoted by this middle packet." ] obligations := [ "sald.forward_kl.cycle34_derivative_middle", "sald.forward_kl.cycle34_derivative_scalar", "sald.forward_kl.schedule_time_change", "sald.forward_kl.density_boundary_regular", "probability.lsi_to_kl_fi", "sald.forward_kl.kl_derivative" ] lowerPacket := [ "If continuing the derivative sprint, target exactly SALD.forwardKlScheduleTimeChangeObligation / sald.forward_kl.schedule_time_change.", "First lower sub-slice: prove or precisely interface the analytic chain-rule input d/dt K(s(t))=dot{s}(t)*dK/ds and the inverse derivative identity dot{t}(s(t))=dot{s}(t)^(-1).", "Second sub-slice: expose the L2 velocity-square scaling ||tilde v_s||^2=dot{t}(s)^2*||v_t||^2 under the slowed target transport identity.", "Use SALD.forwardKlTimeChangedDerivativeBoundScalar only after those inputs are supplied; do not mark the schedule backend, KL derivative theorem, DV, Gronwall, or thm:forward-KL formalized." ] reviewerChecklist := [ "SALD.forwardKlTimeChangedDerivativeBoundScalar is a scalar Real theorem and does not prove the analytic chain rule or inverse-function theorem.", "SALD.forwardKlScheduleTimeChangeObligation remains obligation status and lists the scalar theorem as a downstream bookkeeping helper.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes the cycle-34 middle contract and the new scalar lemma.", "No source theorem statement or coefficient is changed, and the mandatory ASTIS check passes." ] status := ProofStatus.obligation /-- Cycle-39 upper packet for the continuous forward-KL derivative sprint. This packet follows the current proof-closure focus: keep the source theorem fixed and translate `appendix.tex:168-228` into the forward-KL Fokker--Planck/KL derivative identity before returning to ledger expansion or the EM interpolation backend. -/ def cycle39ForwardKlDerivativeUpperPacket : ForwardKlUpperPacket where objective := "Proof-closure priority check before lower assignment: (1) lem:gronwall has cycle 36 local assembly progress but remains an obligation, (2) lem:dv_variation has one-sided scalar consequences while the Boucheron equality remains source-cited, (3) eq:LSI-KL-FI has cycle 38 finite-coordinate Fisher-chain progress but the vector/integral density-test backend remains an obligation, so this cycle follows the requested item (4): close theorem-specific continuous forward-KL Fokker-Planck/KL derivative lemmas for appendix.tex:168-228 before any more ledger work or the item (5) EM interpolation Fokker-Planck backend." sourceLabels := [ "thm:forward-KL", "eq:SALD", "eq:FP-eq", "eq:LSI-KL-FI", "lem:dv_variation", "lem:gronwall" ] modeDiscipline := [ "faithfulPaper: use main_body.tex:238-247 and appendix.tex:168-228 from the original source root; sald_version_2.tex remains excluded.", "Keep the proof route fixed: KL differentiation, SALD Fokker-Planck first term, slowed-target transport, Cauchy--Schwarz/Young, LSI, inverse-schedule time change, DV, then Gronwall.", "Use SALD.forwardKlPreDvDerivativeBoundScalar only as a composition of supplied scalar/analytic inputs; it does not prove the Fokker-Planck, integration-by-parts, LSI density-test, or inverse-schedule backends.", "Keep Gronwall, DV, LSI/KL/FI, the full KL derivative theorem, and EM interpolation below formalized status unless a compiled local proof for exactly that interface is added." ] nonGoals := [ "Do not rebaseline the source index or expand transcript ledgers unless a reviewer finds a blocking source-anchor defect.", "Do not restate, weaken, or prove thm:forward-KL in this packet.", "Do not add smoothness, density, positivity, absolute-continuity, endpoint, boundary, LSI, inverse-function, finite-log-mgf, or interval-integrability assumptions to the theorem statement.", "Do not replace the paper route with Girsanov, path-space comparison, the general moving-target theorem, or a PI-based route.", "Do not start the Euler--Maruyama interpolation Fokker-Planck backend in this lower packet." ] lowerPacket := [ "First lower target: prove or verify a theorem-specific scalar composition around SALD.forwardKlPreDvDerivativeBoundScalar, with all analytic premises explicit and no theorem-status promotion.", "Second lower target only after the scalar composition is stable: refine SALD.forwardKlDensityBoundaryObligation / sald.forward_kl.density_boundary_regular for appendix.tex:168-185, exposing mass conservation, KL differentiation under the integral, the SALD Fokker-Planck equation, boundary/no-flux or decay, and FI identification.", "Third lower target if the first two are stable: refine SALD.forwardKlScheduleTimeChangeObligation / sald.forward_kl.schedule_time_change for appendix.tex:191-228, including the chain rule, velocity-square scaling, inverse derivative, and dot{s}(t) positivity/nonzero inputs used by SALD.forwardKlTimeChangedDerivativeBoundScalar.", "Middle must keep AutoSamplingTheory/SALD.lean, conversion-windows/ASTIS-SALD-001.md, and proof-obligations/ASTIS-SALD-001.md synchronized against the same appendix.tex:168-228 source window.", "Leave DV finite-log-mgf, coefficient-chain audit, Gronwall side conditions, endpoint rewrites, and EM interpolation work as sibling obligations." ] reviewerChecklist := [ "SALD.forwardKlPreDvDerivativeBoundScalar compiles and is only scalar Real/order composition from explicit KL derivative, first-term, Cauchy, LSI, chain-rule, velocity-scaling, and inverse-schedule premises.", "SALD.cycle39ForwardKlDerivativeUpperPacket and sald.forward_kl.cycle39_derivative_upper are listed in the forward-KL contract, proof DAG, and saldDependenciesForLabel \"thm:forward-KL\".", "sald.forward_kl.kl_derivative, sald.forward_kl.density_boundary_regular, sald.forward_kl.schedule_time_change, probability.lsi_to_kl_fi, lem:dv_variation, and lem:gronwall remain obligations or source-cited dependencies.", "No source-index rebaseline, theorem-constant change, hidden analytic assumption, alternate proof route, or fake proof closure appears.", "The mandatory ASTIS check passes." ] status := ProofStatus.obligation /-- Cycle-39 middle map for the source-shaped derivative schedule handoff. This translates the upper packet into proof-producing scalar targets for `appendix.tex:191-228`: the inverse-schedule product identity, the slowed-velocity square scaling, and the composed pre-DV derivative inequality. It does not prove the analytic inverse-function theorem, L2 velocity scaling, KL chain rule, Fokker--Planck equation, or LSI density-test backend. -/ def cycle39ForwardKlDerivativeMiddleContract : ForwardKlMiddleSourceToLeanContract where sourceStatement := saldForwardKlSource sourceProof := saldForwardKlProofSource derivativeSource := saldForwardKlDerivativeSource dvSource := saldForwardKlDvEnergySource gronwallSource := saldForwardKlGronwallSource objective := "Translate appendix.tex:168-228 into source-shaped scalar derivative lemmas: keep the existing KL derivative, first-term, target Cauchy, and LSI premises explicit, and reduce the schedule-side premises to dot{s}(t)*dot{t}(s(t))=1 plus norm-square scaling for tilde v_s=dot{t}(s)v_{t(s)}." sourceStepMap := [ "appendix.tex:168-185 still supplies the KL derivative display and first-term Fisher identity only through the density/boundary obligation.", "appendix.tex:199-208 still supplies the target-side Cauchy input before the existing Young scalar lemma can be used.", "appendix.tex:210-217 still supplies the source KL/FI comparison through probability.lsi_to_kl_fi and the cycle 38 density-test scalar handoff; SALD.forwardKlLsiDerivativeBoundOfKlFiScalar converts it to the half-Fisher derivative form.", "appendix.tex:191-197 defines tilde v_s=dot{t}(s)v_{t(s)}; the new scalar lemma SALD.forwardKlVelocitySquareScalingScalar proves the square scaling after the analytic norm identities are supplied.", "appendix.tex:218-228 uses the inverse schedule; SALD.forwardKlInverseScheduleDerivativeScalar and SALD.forwardKlTimeChangeSquareCoefficientRewriteOfProductScalar prove the Real algebra from the supplied product identity dotS*dotT=1.", "The composed source-shaped pipeline is SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar, which still assumes all analytic premises explicitly." ] leanStepMap := [ "Use SALD.forwardKlInverseScheduleDerivativeScalar only after sald.forward_kl.schedule_time_change supplies dotS*dotT=1 from inverse-function calculus.", "Use SALD.forwardKlVelocitySquareScalingScalar only after the slowed-target transport/L2 backend supplies scalar norm-square identities for tildeVelocitySq and velocitySq.", "Use SALD.forwardKlTimeChangedDerivativeBoundOfProductScalar and SALD.forwardKlPreDvDerivativeBoundOfProductScalar to avoid requiring lower work to pre-rewrite dotT as dotS^-1.", "Use SALD.forwardKlPreDvDerivativeBoundOfVelocityScalingScalar when the LSI input has already been converted to the half-Fisher premise.", "Use SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar when the LSI backend supplies the paper's source-shaped KL/FI comparison rather than the already-multiplied half-Fisher premise.", "Keep SALD.forwardKlDensityBoundaryObligation, SALD.forwardKlScheduleTimeChangeObligation, probability.lsi_to_kl_fi, and sald.forward_kl.kl_derivative at obligation status." ] citedResultInterfaces := [ "No SLT theorem applies to the inverse-schedule or velocity-square scalar algebra.", "eq:LSI-KL-FI remains an open density-test backend; lem:dv_variation and lem:gronwall are downstream and not used by the pre-DV scalar pipeline.", "The source-cited analytic facts are not promoted: inverse-function calculus, Fokker--Planck, integration by parts, and L2 transport remain named obligations." ] obligations := [ "sald.forward_kl.cycle39_derivative_middle", "sald.forward_kl.cycle39_derivative_upper", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "probability.lsi_to_kl_fi", "sald.forward_kl.kl_derivative" ] lowerPacket := [ "First lower slice: verify or reuse SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar as the source-shaped scalar target for appendix.tex:168-228.", "Second lower slice: refine sald.forward_kl.schedule_time_change to supply dotS*dotT=1, nonnegativity of dotS, the KL chain rule hdKdt, and scalar norm-square inputs from tilde v_s=dotT*v.", "Third lower slice: refine sald.forward_kl.density_boundary_regular for appendix.tex:168-185 if the KL derivative display or first-term FI identity is still missing.", "Do not touch DV, Gronwall, endpoint rewrites, coefficient-chain audit, or EM interpolation in this lower packet." ] reviewerChecklist := [ "SALD.forwardKlInverseScheduleDerivativeScalar, SALD.forwardKlVelocitySquareScalingScalar, SALD.forwardKlLsiDerivativeBoundOfKlFiScalar, and SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar compile and are scalar Real lemmas only.", "The middle packet does not mark sald.forward_kl.schedule_time_change, sald.forward_kl.density_boundary_regular, probability.lsi_to_kl_fi, or sald.forward_kl.kl_derivative formalized.", "The theorem statement and all source coefficients in appendix.tex:168-228 remain unchanged.", "No source-index rebaseline or external SLT/SDE import is introduced." ] status := ProofStatus.obligation /-- Cycle-35 upper packet for the discrete EM interpolation Fokker--Planck sprint. The earlier proof-closure items have current scalar or source-cited slices, so this packet returns to item (5): the Euler--Maruyama interpolation endpoint and conditional-drift Fokker--Planck backend in `appendix.tex:260-385`. It reuses the cycle-15 EM spine instead of broadening the transcript. -/ def cycle35DiscreteForwardKlEmFpUpperPacket : DiscreteForwardKlUpperPacket where objective := "Proof-closure priority check: (1) lem:gronwall remains open after cycle 31 local real-analysis sublemmas, (2) lem:dv_variation remains source-cited with cycle 32 scalar consequences, (3) eq:LSI-KL-FI remains open after cycle 33 density-test scalar lemmas, and (4) the continuous forward-KL derivative has cycle 34 scalar handoffs but still depends on analytic Fokker-Planck inputs; therefore this cycle follows item (5), the Euler-Maruyama interpolation Fokker-Planck backend for appendix.tex:260-385." sourceLabels := [ "thm:forward-KL-discrete", "eq:frozen_interp_terminal_disc_prop_additive_final", "proof:thm:forward-KL-discrete:em-interpolation-fp", "lem:frozen_delta_cross_lip_sald", "eq:LSI-KL-FI", "lem:dv_variation", "lem:gronwall" ] modeDiscipline := [ "faithfulPaper: use main_body.tex:299-323 and appendix.tex:260-385 from the original source root; sald_version_2.tex remains excluded.", "Preserve the source order: define the frozen interpolation hat X_s, use endpoint laws on [s_k,s_{k+1}], define bar b_{k,s} by conditional expectation, invoke partial_s hat rho_s = -div(hat rho_s*bar b_{k,s}) + Delta hat rho_s, split the Laplacian relative to tilde pi_s, then hand off to the KL derivative block.", "Keep thm:forward-KL-discrete, t(s)=s/r, Gamma, Delta, barGamma, barDelta, alpha, alpha', eta, r, and the step-size condition unchanged.", "Treat endpoint laws, conditional expectation/disintegration, density/positivity, Fokker-Planck, Laplacian split, boundary integration by parts, stitched regularity, LSI, DV, and Gronwall as obligations unless a compiled local proof replaces exactly one named interface." ] nonGoals := [ "Do not rebaseline the source index or expand the whole discrete theorem transcript unless a reviewer finds a blocking source-anchor defect.", "Do not prove or restate thm:forward-KL-discrete in this packet.", "Do not jump to frozen Gamma/Delta, DV velocity, Gronwall accumulation, or accumulated-error collection before the EM conditional-FP backend is narrowed.", "Do not import or mark an SLT one_step_discretization theorem formalized; it remains at most a route reference.", "Do not close the conditional Fokker-Planck backend by adding theorem-level smoothness, density, disintegration, or boundary hypotheses." ] lowerPacket := [ "Target exactly SALD.discreteForwardKlEmInterpolationSideConditionContract and SALD.discreteForwardKlEmConditionalFpObligation, registered by SALD.cycle35DiscreteForwardKlEmFpUpperObligation / sald.discrete_forward_kl.cycle35_em_fp_upper.", "Middle must keep the two-way source map synchronized for appendix.tex:260-385: endpoint interpolation laws, bar b_{k,s} conditional drift, conditional-drift Fokker-Planck equation, Laplacian split relative to tilde pi_s, and KL-derivative handoff.", "Lower should attempt one proof-producing Lean interface first: either an endpoint-law algebra lemma for the frozen interpolation at s=s_k and s=s_{k+1}, or a narrow conditional-drift measurability/density interface for bar b_{k,s}.", "If the analytic Fokker-Planck theorem is too large for local Mathlib, create a precise source-cited theorem interface depending on the existing conditional-drift density obligation; keep its status below formalized.", "Keep endpoint stitching, frozen one-step Gamma/Delta estimates, LSI, DV, Gronwall, and accumulated-error constants as sibling obligations outside the lower attempt." ] reviewerChecklist := [ "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle35_em_fp_upper before derivative, DV, and Gronwall blocks.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle35DiscreteForwardKlEmFpUpperPacket, sald.discrete_forward_kl.cycle35_em_fp_upper, and the existing EM endpoint/conditional-FP obligations.", "SALD.discreteSaldContract lists SALD.cycle35DiscreteForwardKlEmFpUpperObligation without removing cycle-15 EM endpoint, conditional-drift density, conditional-FP, or stitched-interval obligations.", "No theorem constants, source files, source route, or analytic dependency statuses are changed.", "The mandatory source-index and ASTIS checks pass, and no fake proof closure pattern appears." ] status := ProofStatus.obligation /-- Cycle-35 middle packet for the EM interpolation Fokker--Planck sprint. This translates `appendix.tex:260-385` into lower-ready Lean targets while keeping the analytic endpoint-law and conditional-drift Fokker--Planck backends open. The compiled endpoint and divergence regrouping lemmas are local algebra only; they do not prove stochastic laws, disintegration, density regularity, or integration by parts. -/ def cycle35DiscreteForwardKlEmFpMiddleContract : DiscreteForwardKlEmDefectAccumulationMiddleContract where sourceStatement := saldForwardKlDiscreteSource interpolationSource := saldForwardKlDiscreteInterpolationSource derivativeSource := saldForwardKlDiscreteConditionalFpSource gronwallSource := saldForwardKlDiscreteGronwallSource accumulatedSource := saldForwardKlDiscreteAccumulatedErrorSource objective := "Translate the cycle-35 EM interpolation Fokker-Planck target into proof-producing local algebra plus precise analytic obligations: endpoint vector identities for appendix.tex:260-266, conditional drift and density interfaces for appendix.tex:347-364, and divergence-regrouping algebra for appendix.tex:377-385." sourceStepMap := [ "appendix.tex:260-266 defines the frozen interpolation hat X_s = X_k^eta+(s-s_k)*nabla log pi_{t_k}(X_k^eta)+sqrt(2)*(W_s-W_{s_k}).", "appendix.tex:334-335 uses the endpoint laws hat rho_{s_k}=rho_k^eta and hat rho_{s_{k+1}}=rho_{k+1}^eta before differentiating KL on one EM interval.", "appendix.tex:347-354 defines the frozen conditional drift bar b_{k,s}(x)=E[nabla log pi_{t_k}(X_k^eta) | hat X_s=x].", "appendix.tex:357-364 invokes the conditional-drift Fokker-Planck equation partial_s hat rho_s=-div(hat rho_s*bar b_{k,s})+Delta hat rho_s.", "appendix.tex:365-385 splits Delta hat rho_s relative to tilde pi_s and regroups the drift as grad log tilde pi_s - bar b_{k,s} before the KL derivative handoff." ] leanStepMap := [ "SALD.discreteForwardKlEmInterpolationLeftEndpointVector compiles only the left endpoint algebra after the Brownian increment and time increment vanish.", "SALD.discreteForwardKlEmInterpolationRightEndpointVector compiles only the right endpoint algebra after the mesh identity and EM update definition are supplied.", "SALD.discreteForwardKlConditionalFpDivergenceDriftSplit compiles the additive regrouping of the divergence terms after linearity and the analytic Laplacian split are supplied.", "SALD.discreteForwardKlEmInterpolationSideConditionContract still owns endpoint law matching, conditional frozen drift, interpolation Fokker-Planck, density regularity, and stitched intervals.", "SALD.cycle15DiscreteForwardKlConditionalDriftDensityContract and SALD.cycle15DiscreteForwardKlEmConditionalFpLowerContract remain the analytic interfaces for bar b_{k,s} and the conditional Fokker-Planck theorem." ] citedResultInterfaces := [ "No SLT theorem is imported for the endpoint algebra or divergence regrouping.", "SLT one_step_discretization remains only a reference pattern for future EM analytic work; it is not used as a dependency.", "The conditional-drift Fokker-Planck theorem remains a local analytic obligation below formalized status." ] obligations := [ "sald.discrete_forward_kl.cycle35_em_fp_middle", "sald.discrete_forward_kl.cycle35_em_fp_upper", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.stitched_interval_regularity" ] lowerPacket := [ "Use the endpoint-vector lemmas only as pointwise algebra inside sald.discrete_forward_kl.em_endpoint_laws; the law equality still needs the stochastic endpoint interface.", "Use the divergence regrouping lemma only after the conditional-drift FP equation, Laplacian split, and linearity of divergence have been supplied.", "Next lower proof-producing target: connect the endpoint vector lemmas to a source-cited endpoint-law interface, or refine the regular conditional law/measurability contract for bar b_{k,s}.", "Do not move to frozen Gamma/Delta, DV velocity, Gronwall, or accumulated-error collection until the EM conditional-FP backend has a precise theorem interface." ] reviewerChecklist := [ "The three new Lean theorems are theorem-independent algebra and make no claim about stochastic laws, densities, Fokker-Planck validity, or integration by parts.", "SALD.discreteForwardKlProofDag and SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" expose the cycle-35 middle packet and the three compiled helper lemmas.", "The endpoint, conditional-drift density, conditional-FP, EM interpolation, and stitched-interval obligations remain obligation status.", "No theorem constants, source route, or source file selection changes." ] status := ProofStatus.obligation /-- Cycle-40 middle packet for the EM endpoint and conditional-FP backend. This keeps the proof-closure priority on item (5) and refines the cycle-35 EM spine with law-level endpoint handoffs. The conditional-drift Fokker--Planck theorem, density/disintegration, Laplacian chain rule, and integration-by-parts backend remain explicit obligations. -/ def cycle40DiscreteForwardKlEmFpMiddleContract : DiscreteForwardKlEmDefectAccumulationMiddleContract where sourceStatement := saldForwardKlDiscreteSource interpolationSource := saldForwardKlDiscreteInterpolationSource derivativeSource := saldForwardKlDiscreteConditionalFpSource gronwallSource := saldForwardKlDiscreteGronwallSource accumulatedSource := saldForwardKlDiscreteAccumulatedErrorSource objective := "Cycle 40 middle priority check: Gronwall, DV, LSI/KL/FI, and the continuous forward-KL derivative have current source-cited or scalar progress but still leave analytic backends open, so this packet follows item (5) and translates appendix.tex:260-385 into endpoint-law handoffs plus the existing conditional-drift Fokker-Planck obligation." sourceStepMap := [ "appendix.tex:260-266 defines hat X_s = X_k^eta+(s-s_k)*nabla log pi_{t_k}(X_k^eta)+sqrt(2)*(W_s-W_{s_k}).", "appendix.tex:334-335 uses the endpoint laws hat rho_{s_k}=rho_k^eta and hat rho_{s_{k+1}}=rho_{k+1}^eta before differentiating KL.", "appendix.tex:347-354 defines bar b_{k,s}(x)=E[nabla log pi_{t_k}(X_k^eta) | hat X_s=x].", "appendix.tex:357-364 invokes the conditional-drift Fokker-Planck equation for hat rho_s.", "appendix.tex:365-385 splits Delta hat rho_s relative to tilde pi_s and regroups the drift as grad log tilde pi_s - bar b_{k,s}." ] leanStepMap := [ "SALD.discreteForwardKlEmInterpolationLeftEndpointVector and SALD.discreteForwardKlEmInterpolationRightEndpointVector remain the pointwise endpoint algebra.", "SALD.discreteForwardKlLawEqOfPointwise turns any supplied pointwise equality of random variables into equality under an abstract law operator.", "SALD.discreteForwardKlEmInterpolationLeftEndpointLawHandoff and SALD.discreteForwardKlEmInterpolationRightEndpointLawHandoff are the new endpoint-law handoffs for sald.discrete_forward_kl.em_endpoint_laws.", "SALD.discreteForwardKlEmEndpointLawPairHandoff is the lower endpoint-law pair: after named representations for hat rho_s, rho_k^eta, and rho_{k+1}^eta are supplied, it proves both endpoint laws used at appendix.tex:334-335.", "SALD.discreteForwardKlConditionalFpLaplacianSplitHandoff remains the source regrouping after hfp and hlap are supplied.", "SALD.discreteForwardKlEmConditionalFpObligation remains the precise source-cited interface for conditional drift, density, Fokker-Planck, and Laplacian split." ] citedResultInterfaces := [ "No SLT theorem is imported or marked formalized for EM endpoint laws.", "The one_step_discretization pattern remains only a possible future reference route for analytic EM facts.", "The conditional-drift Fokker-Planck theorem is local SDE/measure analysis and stays below formalized status." ] obligations := [ "sald.discrete_forward_kl.cycle40_em_fp_middle", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.stitched_interval_regularity" ] lowerPacket := [ "First lower target: instantiate SALD.discreteForwardKlEmInterpolationLeftEndpointLawHandoff and SALD.discreteForwardKlEmInterpolationRightEndpointLawHandoff with the repository's eventual law notation for hat rho_s, rho_k^eta, and rho_{k+1}^eta; SALD.discreteForwardKlEmEndpointLawPairHandoff now proves the pair from explicit named-law representation hypotheses.", "Second lower target: refine sald.discrete_forward_kl.conditional_drift_density for the regular conditional law and measurability/integrability of bar b_{k,s}.", "Third lower target only if the interface is stable: state the source-cited conditional-drift Fokker-Planck theorem that supplies hfp and the Laplacian split consumed by SALD.discreteForwardKlConditionalFpLaplacianSplitHandoff.", "Keep frozen Gamma/Delta, LSI, DV, Gronwall, coefficient-chain, and accumulated-error work outside this lower packet." ] reviewerChecklist := [ "The endpoint-law handoff lemmas compile and are only abstract funext/congruence plus pointwise endpoint algebra.", "SALD.discreteForwardKlEmEndpointLawPairHandoff compiles only under explicit named-law representation hypotheses for hat rho_s, rho_k^eta, and rho_{k+1}^eta.", "The packet does not construct Brownian motion, regular conditional laws, densities, or the Fokker-Planck theorem.", "sald.discrete_forward_kl.em_endpoint_laws, conditional_drift_density, em_conditional_fokker_planck, and em_interpolation_fp remain obligations.", "The source theorem, constants, step-size condition, alpha ranges, and source files are unchanged.", "The mandatory source-index and ASTIS checks pass." ] status := ProofStatus.obligation def generalVaSaldEulerMaruyamaContract : EulerMaruyamaContract where id := "ASTIS.SALD.general_moving_target_discrete.euler_maruyama" updateFormula := "X_{k+1}^eta = X_k^eta + eta*(dot{t}_k*c_{t_k}(X_k^eta)+(sigma_{t_k}^2/2)*nabla log pi_{t_k}(X_k^eta)) + sigma_{t_k}*sqrt(eta)*xi_k." interpolationFormula := "For s in [s_k,s_{k+1}], hat X_s = X_k^eta + (s-s_k)*(dot{t}_k*c_{t_k}(X_k^eta)+(sigma_eta^2/2)*nabla log pi_{t_k}(X_k^eta)) + sigma_eta*(W_s-W_{s_k})." source := saldGeneralMovingTargetDiscreteEmSource status := ProofStatus.contractOnly def generalFrozenDeltaCrossLipContract : GeneralFrozenDeltaCrossLipContract where sourceBlock := saldFrozenDeltaCrossLipGeneralSource frozenError := "delta_pi^VA(x)=dot{t}(s)*c_t(x)+(sigma_eta^2/2)*nabla log pi_t(x)-E[dot{t}_k*c_{t_k}(X_k^eta)+(sigma_eta^2/2)*nabla log pi_{t_k}(X_k^eta) | hat X_s=x]." assumptions := [ "space Lipschitzness of c_t with L_{c,space} and score with L_{pi,space}", "time Lipschitzness along s for c_t and score with growth function M", "finite alpha0' exponential complexities for c_t, nabla log pi_t, and 1+M", "linear slowdown or constant dot{t}(s)=dot{s}(t)^(-1) on [0,S]" ] stepSizeCondition := "4*eta^2*(dot{t}(s)*L_{c,space}+(sigma_eta(t(s))^2/2)*L_{pi,space})^2 < 1/2." bound := "-int hat rho_s <= (sigma_eta(t(s))^2/8)*FI(hat rho_s||tilde pi_s) + 2*eta^2*alpha'^(-1)*Gamma(t(s))*KL(hat rho_s||tilde pi_s) + 2*eta*Delta(t(s))." gammaDefinition := "Gamma(t)=sigma_eta(t)^(-2)*{4*dot{s}(t)^(-2)*L_c_time^2+sigma_eta(t)^4*L_pi_time^2+8*(4*dot{s}(t)^(-2)*L_c_space^2+sigma_eta(t)^4*L_pi_space^2)*(4*dot{s}(t)^(-2)+sigma_eta(t)^4+eta^2*(4*dot{s}(t)^(-2)*L_c_time^2+sigma_eta(t)^4*L_pi_time^2))}." deltaDefinition := "Delta(t)=eta*sigma_eta(t)^(-2)*{8*(4*dot{s}(t)^(-2)*L_c_space^2+sigma_eta(t)^4*L_pi_space^2)*(4*dot{s}(t)^(-2)*E_alpha'(pi_t,c_t)+sigma_eta(t)^4*E_alpha'(pi_t,nabla log pi_t))+(4*dot{s}(t)^(-2)*L_c_time^2+sigma_eta(t)^4*L_pi_time^2)*(1+8*eta^2*(4*dot{s}(t)^(-2)*L_c_space^2+sigma_eta(t)^4*L_pi_space^2))*E_alpha'(pi_t,1+M)}+4*d*(4*dot{s}(t)^(-2)*L_c_space^2+sigma_eta(t)^4*L_pi_space^2)." proofRoute := [ "appendix.tex:1112-1123 rewrites the cross term using the joint law of X_k^eta and hat X_s, then applies Young with coefficient sigma_eta^2/8.", "appendix.tex:1125-1158 splits the frozen-field difference into time and space increments using the Lipschitz hypotheses.", "appendix.tex:1161-1210 bounds the EM interpolation increment and resolves the self-referential term using the step-size condition.", "appendix.tex:1213-1275 controls phi_{t_k} in L2(hat rho_s) and applies DV to c_t, the score, and 1+M.", "appendix.tex:1277-1306 collects the resulting Gamma and Delta terms." ] dependencies := [ "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "eq:general_discrete_delta_def", "lem:dv_variation", "def:alpha-complexity" ] sourceGaps := [ "the proof uses conditional distributions of (X_k^eta,hat X_s) and L2 estimates under hat rho_s without a local probability backend", "the statement assumes dot{t}(s) is constant but Gamma and Delta are written in dot{s}(t) notation; the Lean route must expose the constant inverse-schedule interface", "finite log-mgf monotonicity for alpha' <= alpha0' is needed for the three DV uses" ] status := ProofStatus.obligation def generalMovingTargetDiscreteStatementContract : GeneralMovingTargetDiscreteStatementContract where theoremLabel := "thm:general-moving-target-SALD-discrete" sourceStatement := saldGeneralMovingTargetDiscreteSource sourceProof := saldGeneralMovingTargetDiscreteSource emUpdate := "Discrete-time general VA-SALD eq:SALD_general_EM with drift dot{t}_k*c_{t_k}+(sigma_{t_k}^2/2)*nabla log pi_{t_k} and noise sigma_{t_k}*sqrt(eta)*xi_k." interpolation := "Continuous frozen interpolation eq:general_moving_target_SALD_frozen_interp with hat rho_{s_k}=rho_k^eta and hat rho_{s_{k+1}}=rho_{k+1}^eta." frozenError := "General VA frozen-field error delta_pi^VA from eq:general_discrete_delta_def, bounded by lem:frozen_delta_cross_lip." constantSchedule := "The theorem assumes dot{t}(s)=dot{s}(t)^(-1) is constant, matching the discrete-time linear-slowdown discussion." assumptions := "Same conditions as thm:general-moving-target-SALD plus the space/time Lipschitz, exponential-complexity, and step-size assumptions of lem:frozen_delta_cross_lip." alphaRange := "For any alpha in (0,alpha0] and alpha' in (0,alpha0']." terminalBound := "KL(rho_K^eta||pi_T) is bounded by the Gronwall expression with a(t)=(sigma_eta(t)^2/2)*dot{s}(t)*C_LSI(t)-2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t) and b(t)=2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t)+2*dot{s}(t)*eta*Delta(t)." proofQuantity := "K(t)=KL(hat rho_{s(t)}||pi_t), with K(T)=KL(rho_K^eta||pi_T) by endpoint law matching." differentialInequality := "dK/dt <= -((sigma_eta(t)^2/2)*dot{s}(t)*C_LSI(t)-2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t))*K(t)+2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t)+2*dot{s}(t)*eta*Delta(t)." proofSteps := [ "appendix.tex:953-1024 defines the EM update, frozen interpolation, sigma_eta, phi_t, and delta_pi^VA.", "appendix.tex:1026-1307 proves lem:frozen_delta_cross_lip for the general VA frozen-field error.", "appendix.tex:1354-1488 differentiates KL(hat rho_s||tilde pi_s), inserts the interpolation Fokker--Planck equation, and decomposes the cross term into delta_pi^VA plus dot{t}*m_t.", "appendix.tex:1493-1542 applies Young to the m_t term, applies the frozen-delta lemma, and uses LSI.", "appendix.tex:1544-1598 applies DV to m_t and changes from s to t.", "appendix.tex:1600 applies lem:gronwall to obtain eq:general_moving_target_KL_bound_discrete." ] dependencies := [ "thm:general-moving-target-SALD", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "eq:general_discrete_delta_def", "lem:frozen_delta_cross_lip", "lem:dv_variation", "lem:gronwall", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions" ] status := ProofStatus.contractOnly def generalMovingTargetDiscreteDerivativeCandidateContract : GeneralMovingTargetDiscreteDerivativeCandidateContract where sourceBlock := saldGeneralMovingTargetDiscreteDerivativeSource interpolationLaw := "For s in [s_k,s_{k+1}], hat rho_s=Law(hat X_s), with endpoint laws rho_k^eta and rho_{k+1}^eta." frozenConditionalDrift := "bar b_{k,s}(x)=E[dot{t}_k*c_{t_k}(X_k^eta)+(sigma_eta^2/2)*nabla log pi_{t_k}(X_k^eta) | hat X_s=x]." fokkerPlanck := "partial_s hat rho_s = -div(hat rho_s*bar b_{k,s})+(sigma_eta^2/2)*Delta hat rho_s, then Delta hat rho_s is split relative to A_s=nabla log(hat rho_s/tilde pi_s); SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract records the weak-test source statement and SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff plus SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff compile only its test-indexed sign/coefficient packaging, while SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff and SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff consume the supplied analytic FP/Laplacian identities for algebraic regrouping." klDerivativeIdentity := "d/ds KL(hat rho_s||tilde pi_s)=int partial_s hat rho_s*log(hat rho_s/tilde pi_s) - int (hat rho_s/tilde pi_s)*partial_s tilde pi_s; SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract records the cycle-73 handoff that applies the weak FP identity to the log-ratio test before integration by parts, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns isolates the normalized source-sign-to-KL substitution, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns composes that handoff through the cycle-72 admissible source-sign wrapper, and SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces composes it through the cycle-77 generator-piece source-sign wrapper." frozenResidualDecomposition := "(sigma_eta^2/2)*nabla log tilde pi_s - bar b_{k,s} + tilde v_s = delta_pi^VA + dot{t}(s)*m_{t(s)}." mYoungStep := "-dot{t}(s)*int hat rho_s <= (sigma_eta^2/8)*FI + 2*sigma_eta^(-2)*dot{t}(s)^2*||m_{t(s)}||_{L2(hat rho_s)}^2." frozenDeltaStep := "lem:frozen_delta_cross_lip bounds -int hat rho_s by (sigma_eta^2/8)*FI + 2*Gamma*eta^2*alpha'^(-1)*KL + 2*Delta*eta." lsiStep := "After the two Young splits, LSI converts -(sigma_eta^2/4)*FI into -(sigma_eta^2/2)*C_LSI*K." dvStep := "Apply DV with Z=alpha*||m_t||^2, using the discrete residual finite-log-mgf witness, to replace ||m_t||_{L2(hat rho_s)}^2 by alpha^(-1)*K+E_alpha(pi_t,m_t)." outputSInequality := "d/ds K_s <= -((sigma_eta^2/2)*C_LSI(t(s))-2*sigma_eta^(-2)*dot{t}(s)^2*alpha^(-1)-2*Gamma(t(s))*eta^2*alpha'^(-1))*K_s + 2*sigma_eta^(-2)*dot{t}(s)^2*E_alpha(pi_{t(s)},m_{t(s)}) + 2*Delta(t(s))*eta." timeChangedInequality := "d/dt K(t) <= -((sigma_eta(t)^2/2)*dot{s}(t)*C_LSI(t)-2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t))*K(t) + 2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t)+2*dot{s}(t)*Delta(t)*eta." requiredRegularity := [ "endpoint law matching for the general VA-SALD EM interpolation", "conditional-drift weak Fokker--Planck equation with drift sign -div(hat rho_s*bar b_{k,s}) and diffusion sign +(sigma_eta^2/2)*Delta hat rho_s", "density positivity, finite KL/FI, differentiation-under-integral, and integration-by-parts on every EM subinterval", "constant inverse schedule and dot{s}(t)>0", "positive sigma_eta(t) and finite residual L2/log-mgf terms" ] sourceGaps := [ "the interpolation Fokker--Planck equation and conditional drift are invoked as standard facts rather than stated as a lemma", "the cycle-72 and cycle-69 lower theorems prove only weak-test/source-sign coefficient packaging, including the explicit admissible-test predicate variant, after the analytic weak FP identity is supplied; cycle 73 then substitutes that supplied weak FP identity into the differentiated KL display at the log-ratio test, with a lower wrapper composing through the cycle-72 admissible source-sign theorem; cycle 78 isolates the normalized source-sign-to-KL substitution and composes the same KL handoff through the cycle-77 generator-piece source-sign theorem, but none of these prove the weak FP theorem, conditional law, density, log-ratio admissibility, or integration-by-parts theorem", "the proof applies a global Gronwall step after interval-wise estimates without isolating stitched-interval regularity", "the source says to substitute Gamma(t) before the final inequality; the collection of all constants must be kept synchronized with lem:frozen_delta_cross_lip" ] dependencies := [ "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "eq:general_discrete_delta_def", "lem:frozen_delta_cross_lip", "eq:LSI-KL-FI", "lem:dv_variation", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.derivative_side_conditions", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces", "SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff", "sald.general_moving_target_discrete.cycle69_em_fp_source_signs_lower", "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "sald.general_moving_target_discrete.cycle54_em_fp_lower", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change" ] status := ProofStatus.obligation def generalMovingTargetDiscreteConditionalDriftContract : GeneralMovingTargetDiscreteConditionalDriftContract where sourceBlock := saldGeneralMovingTargetDiscreteDerivativeSource parentInterface := "sald.general_moving_target_discrete.em_interpolation_fp / appendix.tex:1358-1387" randomVariables := "On a fixed interval [s_k,s_{k+1}], use the common probability space carrying X_k^eta, the frozen EM interpolation hat X_s, c_{t_k}(X_k^eta), and nabla log pi_{t_k}(X_k^eta)." conditioningMap := "Condition on hat X_s=x, with x in the Euclidean state space supporting hat rho_s=Law(hat X_s)." conditionalLawKernel := "Provide a regular conditional law of X_k^eta given hat X_s=x, or an equivalent kernel supporting conditional expectations of both drift summands." driftSummands := "dot t_k*c_{t_k}(X_k^eta) and (sigma_eta^2/2)*nabla log pi_{t_k}(X_k^eta), matching appendix.tex:1368-1377." conditionalExpectationLinearity := "The selected conditional expectation must be linear for integrable vector-valued summands; SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination compiles only this algebra after linearity is supplied." selectedDriftField := "bar b_{k,s}(x)=E[dot t_k*c_{t_k}(X_k^eta)+(sigma_eta^2/2)*nabla log pi_{t_k}(X_k^eta) | hat X_s=x], defined pointwise or hat-rho_s-a.e. and usable as the drift field in divergence form." measurabilityInterface := "x |-> bar b_{k,s}(x), and the two selected conditional-expectation summands, must be measurable with respect to the state-space sigma algebra; SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents derives bar b_{k,s} regularity from supplied component regularity and closure rules, while SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff transfers any already-supplied combo predicate by equality." integrabilityInterface := "Both summands must be conditionally integrable; hat rho_s*bar b_{k,s} must have enough local integrability for the weak divergence term; cycle 70 records this as a conditional-law/measurability obligation, not as a proved disintegration theorem." fokkerPlanckInput := "This interface supplies only the regular conditional drift input to partial_s hat rho_s = -div(hat rho_s*bar b_{k,s}) + (sigma_eta^2/2)*Delta hat rho_s; it does not prove the weak Fokker--Planck equation." exclusions := [ "Do not prove or mark the regular conditional law itself as formalized in this contract.", "Do not prove density/absolute-continuity for hat rho_s or tilde pi_s here.", "Do not prove the weak Fokker-Planck equation, KL differentiation, boundary integration by parts, Laplacian split, LSI, DV, or Gronwall here.", "Do not add these requirements as hidden hypotheses to thm:general-moving-target-SALD-discrete." ] dependencies := [ "eq:general_moving_target_SALD_frozen_interp", "SALD.generalVaSaldEulerMaruyamaContract", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination", "SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "sald.general_moving_target_discrete.em_interpolation_fp" ] sourceGaps := [ "The source defines bar b_{k,s} by conditional expectation but does not spell out the regular conditional probability/kernel.", "The source does not separately state measurability and integrability of the selected drift field.", "The source immediately invokes the associated Fokker-Planck equation; Lean needs this drift interface before the weak FP identity can be stated." ] status := ProofStatus.obligation def generalMovingTargetDiscreteConditionalLawMeasurabilityContract : GeneralMovingTargetDiscreteConditionalLawMeasurabilityContract where sourceBlock := saldGeneralMovingTargetDiscreteDerivativeSource parentInterface := "sald.general_moving_target_discrete.em_interpolation_fp / appendix.tex:1358-1387" commonSpace := "A fixed EM interval [s_k,s_{k+1}] uses one probability space carrying X_k^eta, hat X_s, c_{t_k}(X_k^eta), nabla log pi_{t_k}(X_k^eta), and the Brownian increment in eq:general_moving_target_SALD_frozen_interp." interpolationLaw := "hat rho_s is Law(hat X_s), with endpoint bookkeeping supplied separately by SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation and the paired endpoint helpers." conditionalKernel := "A regular conditional kernel for X_k^eta given hat X_s=x, or an equivalent kernel for the joint law of (X_k^eta,hat X_s), must be supplied before forming the conditional expectations." kernelCompatibility := "The kernel must disintegrate the joint law and be compatible with the hat rho_s marginal, so conditional expectations are defined hat-rho_s-a.e. on the same state space as the weak FP equation." componentConditionalFields := "Name condC_{k,s}(x)=E[c_{t_k}(X_k^eta)|hat X_s=x] and condScore_{k,s}(x)=E[nabla log pi_{t_k}(X_k^eta)|hat X_s=x] with the same conditional-expectation version." selectedDriftField := "Name bar b_{k,s}(x)=E[dot t_k*c_{t_k}(X_k^eta)+(sigma_eta^2/2)*nabla log pi_{t_k}(X_k^eta)|hat X_s=x]; SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents compiles only the algebraic rewrite to dot t_k*condC_{k,s}+(sigma_eta^2/2)*condScore_{k,s} after linearity is supplied." measurabilitySideConditions := "The kernel must be measurable in x and the named fields condC_{k,s}, condScore_{k,s}, and bar b_{k,s} must be measurable; SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents derives regularity of bar b_{k,s} from supplied component regularity and add/smul closure, while SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff transfers any supplied congruence-stable measurability predicate from the named component combination to bar b_{k,s}." integrabilitySideConditions := "Both source summands must be conditionally integrable under the kernel, and bar b_{k,s} must be integrable enough against hat rho_s to define div(hat rho_s*bar b_{k,s}) in the chosen weak-test class." weakFpHandoff := "This contract stops before the weak conditional Fokker-Planck theorem; SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract records the required weak-test statement, while cycle 72, cycle 69, cycle 54, and cycle 73 wrappers consume supplied weak FP and Laplacian/KL-derivative identities without constructing them. The cycle-72 admissible-test wrapper keeps the chosen test predicate explicit, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract keeps the log-ratio test admissibility separate, and SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns composes the two supplied handoffs." exclusions := [ "Do not mark the regular conditional kernel or disintegration theorem as formalized here.", "Do not infer density or absolute-continuity of hat rho_s or tilde pi_s from this conditional-law interface.", "Do not prove the weak Fokker-Planck equation, KL differentiation, integration by parts, LSI, DV, Gronwall, or theorem closure in this packet.", "Do not add the conditional-law requirements as hidden assumptions to thm:forward-KL-discrete or thm:general-moving-target-SALD-discrete." ] dependencies := [ "eq:general_moving_target_SALD_frozen_interp", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination", "SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns", "SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff", "SALD.cycle69GeneralMovingTargetDiscreteEmFpSourceSignsLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] sourceGaps := [ "appendix.tex defines bar b_{k,s} by conditional expectation but does not state a regular conditional probability theorem.", "appendix.tex immediately passes from the conditional drift definition to the associated Fokker-Planck equation, so Lean needs separate measurability and integrability hypotheses.", "No local Mathlib or SLT theorem has yet been ported for this disintegration/conditional-expectation backend." ] status := ProofStatus.obligation def generalMovingTargetDiscreteEndpointConditionalCompatibilityContract : GeneralMovingTargetDiscreteEndpointConditionalCompatibilityContract where sourceBlock := saldGeneralMovingTargetDiscreteDerivativeSource parentInterface := "sald.general_moving_target_discrete.em_interpolation_fp / appendix.tex:1358-1387" commonSpace := "For fixed k and s in [s_k,s_{k+1}], use the common probability space carrying X_k^eta and hat X_s. The joint law is represented as Measure.map (fun omega => (X_k^eta omega, hat X_s omega)) P." jointLawRepresentation := "The conditional kernel for X_k^eta given hat X_s=x must be compatible with this joint law, not with an unrelated copy of the state space." hatRhoMarginal := "The second marginal of the joint law is the named interpolation law hat rho_s=Law(hat X_s). SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap compiles this Measure.map projection from explicit measurability and the named-law equality." conditionalKernelCompatibility := "A future analytic backend must supply a kernel/disintegration predicate for the joint law and its second marginal before conditional expectations of the two frozen drift summands are formed." compatibilityTransport := "SALD.generalMovingTargetDiscreteConditionalKernelCompatibilityOfJointMapMarginal compiles equality transport from the joint law's Measure.map Prod.snd marginal to the named hat rho_s marginal under a supplied congruence rule, while SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityOfJointMap packages the common lower use case directly as marginal equality plus transported kernel compatibility." weakFpUse := "The weak Fokker-Planck statement recorded by SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract may consume the kernel only after this marginal compatibility is aligned with the same hat rho_s that appears in partial_s hat rho_s." exclusions := [ "Do not treat this as a proof of existence of a regular conditional probability.", "Do not prove conditional expectation linearity, measurability, integrability, density, absolute-continuity, or weak Fokker-Planck here.", "Do not promote sald.general_moving_target_discrete.em_interpolation_fp or any theorem contract to formalized status.", "Do not import or claim an SLT disintegration theorem." ] dependencies := [ "SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap", "SALD.generalMovingTargetDiscreteConditionalKernelCompatibilityOfJointMapMarginal", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityOfJointMap", "AutoSamplingTheory.lawMapProdSnd", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "sald.general_moving_target_discrete.em_interpolation_fp" ] sourceGaps := [ "appendix.tex:1368-1377 defines the conditional drift but does not spell out the disintegration theorem or marginal compatibility proof.", "appendix.tex:1379-1387 invokes the associated Fokker-Planck equation immediately after the conditional drift definition.", "The local Lean work proves only Measure.map marginal bookkeeping and predicate transport, not the analytic conditional-law backend." ] status := ProofStatus.obligation def generalMovingTargetDiscreteWeakConditionalFpSourceSignContract : GeneralMovingTargetDiscreteWeakConditionalFpSourceSignContract where sourceBlock := saldGeneralMovingTargetDiscreteWeakFpSource parentInterface := "sald.general_moving_target_discrete.em_interpolation_fp / appendix.tex:1358-1387" fixedInterval := "Fix k and s in [s_k,s_{k+1}] after hat rho_s=Law(hat X_s) and the endpoint-to-conditional marginal compatibility have been aligned." namedLaw := "The weak equation is for the same named interpolation law hat rho_s that appears in partial_s hat rho_s and in the conditional-kernel marginal from cycle 71." conditionalDrift := "The drift field is the cycle-70/71 selected bar b_{k,s}(x)=E[dot t_k*c_{t_k}(X_k^eta)+(sigma_eta^2/2)*nabla log pi_{t_k}(X_k^eta)|hat X_s=x], with measurability and integrability supplied explicitly." weakTestClass := "Use a smooth compactly supported or otherwise admissible weak-test class for which partial_s hat rho_s, div(hat rho_s*bar b_{k,s}), and Delta hat rho_s are all meaningful distributions." weakEquation := "For every admissible test phi, the supplied weak-FP identity must represent partial_s hat rho_s as -div(hat rho_s*bar b_{k,s}) plus (sigma_eta^2/2)*Delta hat rho_s; SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff splits that input into a supplied generator/time-derivative identity and a supplied generator source expansion, while SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff further splits the source expansion into drift and diffusion actions under explicit conditional-law, density, test, and boundary hypotheses. SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff and SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff compile the test-indexed source-sign/coefficient packaging once the analytic identity is supplied, and SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns consumes the normalized source signs at the log-ratio test." driftSignConvention := "The divergence term is negative exactly as in appendix.tex:1384: -nabla dot (hat rho_s bar b_{k,s})." diffusionSignConvention := "The Laplacian term is positive exactly as in appendix.tex:1385-1386: +(sigma_eta^2/2) Delta hat rho_s." explicitHypotheses := [ "common probability space and joint law of (X_k^eta,hat X_s)", "regular conditional kernel compatible with the named hat rho_s marginal", "bar b_{k,s} measurability and local integrability against hat rho_s", "density/absolute-continuity and time regularity for hat rho_s on the fixed EM interval", "admissible weak-test class with integration-by-parts and Laplacian dual actions", "generator identity for the frozen EM interpolation on admissible tests", "source expansion of that generator as -div(hat rho_s*bar b_{k,s}) plus sigmaCoeff*Delta hat rho_s", "optional component split of the source expansion into separate drift and diffusion actions", "diffusion coefficient identified with sigma_eta^2/2" ] downstreamHandoffs := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces", "SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "sald.general_moving_target_discrete.kl_derivative" ] exclusions := [ "Do not construct the Brownian/EM law or regular conditional kernel in this contract.", "Do not prove density, absolute-continuity, or integration-by-parts side conditions here.", "Do not prove the KL derivative, LSI/KL/FI, DV, Gronwall, or theorem closure.", "Do not promote sald.general_moving_target_discrete.em_interpolation_fp above ProofStatus.obligation." ] dependencies := [ "eq:general_moving_target_SALD_frozen_interp", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces", "SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "sald.general_moving_target_discrete.em_interpolation_fp" ] sourceGaps := [ "appendix.tex:1379-1387 invokes the Fokker--Planck equation associated with the frozen interpolation but does not state the weak-test theorem or its regularity hypotheses.", "The generator-level theorem that turns the frozen EM interpolation into the weak time derivative of hat rho_s is still supplied as an analytic hypothesis.", "The source does not spell out the density, local integrability, or boundary/integration-by-parts assumptions required to interpret the divergence and Laplacian terms distributionally.", "The source passes immediately from this FP equation to the Laplacian split and KL derivative handoff." ] status := ProofStatus.obligation def generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract : GeneralMovingTargetDiscreteKlDerivativeWeakFpHandoffContract where sourceBlock := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource parentInterface := "sald.general_moving_target_discrete.em_interpolation_fp / appendix.tex:1358-1387" differentiatedKlFormula := "eq:general_KL_derivative_0_discrete: d/ds KL(hat rho_s||tilde pi_s)=int partial_s hat rho_s*log(hat rho_s/tilde pi_s) - int (hat rho_s/tilde pi_s)*partial_s tilde pi_s, using int partial_s hat rho_s dx=0." selectedWeakTest := "Use the log-density-ratio test phi_s=log(hat rho_s/tilde pi_s), or a justified admissible approximation, in the weak conditional Fokker--Planck identity for hat rho_s." admissibilityInterface := "The log-ratio test must belong to the admissible weak-test class, with density positivity, finite KL/FI, local Sobolev/smooth approximation, and boundary behavior sufficient for the divergence and Laplacian dual actions." weakFpSubstitution := "Substitute partial_s hat rho_s paired with phi_s by the supplied weak FP action -div(hat rho_s*bar b_{k,s}) paired with phi_s plus (sigma_eta^2/2)*Delta hat rho_s paired with phi_s; SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar compiles only this scalar substitution under explicit hypotheses, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns compiles the same substitution from already-normalized admissible weak-FP source signs, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction records both the log-ratio weak-FP action and the resulting dK display, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns composes through the cycle-72 admissible source-sign theorem, and SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces composes it through the cycle-77 generator-piece source-sign handoff." sourceSignedDerivative := "The resulting derivative display preserves the source signs: negative drift-divergence action, positive sigma_eta^2/2 Laplacian action, and the unchanged target-time term -int (hat rho_s/tilde pi_s)*partial_s tilde pi_s." integrationByPartsHandoff := "After this handoff, a separate analytic backend must justify integration by parts, the Laplacian split Delta hat rho_s=div(hat rho_s*A_s)+div(hat rho_s*nabla log tilde pi_s), and the Fisher-information identification before the frozen/residual Young and LSI steps." downstreamDerivativeInterface := "Feeds SALD.generalMovingTargetDiscreteDerivativeCandidateContract and sald.general_moving_target_discrete.kl_derivative; it is also reusable by the discrete forward-KL EM interpolation backend with its simpler frozen drift." explicitHypotheses := [ "hat rho_s and tilde pi_s live on the same measurable/smooth state space and hat rho_s is absolutely continuous enough for a log-density ratio", "finite KL and enough time regularity to differentiate KL and drop int partial_s hat rho_s dx by mass conservation", "the log-density-ratio test is admissible or approximated in the weak-test class", "the weak FP identity from cycle 72 applies to that test with sigmaCoeff=sigma_eta^2/2", "boundary/no-flux or decay hypotheses support the later integration-by-parts handoff" ] downstreamHandoffs := [ "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces", "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "sald.general_moving_target_discrete.kl_derivative", "sald.discrete_forward_kl.kl_derivative" ] exclusions := [ "Do not prove the weak conditional Fokker--Planck theorem in this handoff.", "Do not prove admissibility of the log-density-ratio test, density positivity, absolute continuity, finite KL/FI, or integration by parts here.", "Do not promote LSI/KL/FI, DV, Gronwall, EM interpolation, or either discrete theorem contract above its current status.", "Do not add the log-ratio admissibility or boundary requirements as hidden assumptions to a theorem statement." ] dependencies := [ "eq:general_KL_derivative_0_discrete", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] sourceGaps := [ "appendix.tex:1358-1387 differentiates KL and invokes the associated Fokker--Planck equation without stating the log-ratio weak-test admissibility theorem.", "The source does not isolate the density/absolute-continuity and boundary hypotheses needed to substitute the weak FP identity into the KL derivative.", "The following integration-by-parts and Fisher-information identifications remain separate analytic obligations." ] status := ProofStatus.obligation def generalMovingTargetDiscreteDerivativeSideConditionContract : GeneralMovingTargetDiscreteDerivativeSideConditionContract where sourceBlock := saldGeneralMovingTargetDiscreteDerivativeSource endpointLawInterface := "For each EM interval s in [s_k,s_{k+1}], hat rho_s=Law(hat X_s), with hat rho_{s_k}=rho_k^eta and hat rho_{s_{k+1}}=rho_{k+1}^eta before stitching K(t) globally. SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation compiles this endpoint bookkeeping once named law representations and pointwise interpolation identities are supplied." conditionalDriftInterface := "bar b_{k,s}(x)=E[dot t_k*c_{t_k}(X_k^eta)+(sigma_eta^2/2)*nabla log pi_{t_k}(X_k^eta) | hat X_s=x], defined against the joint law of (X_k^eta,hat X_s) through SALD.generalMovingTargetDiscreteConditionalDriftContract, SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract, and SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract; only the Measure.map marginal transport plus abstract named-component and regularity handoffs are compiled locally." fokkerPlanckSplitInterface := "partial_s hat rho_s=-div(hat rho_s*bar b_{k,s})+(sigma_eta^2/2)*Delta hat rho_s, then Delta hat rho_s=div(hat rho_s*A_s)+div(hat rho_s*nabla log tilde pi_s); SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract states the weak-test FP source-sign interface, SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff, SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff, and SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff preserve the source signs and sigma_eta^2/2 coefficient under explicit weak-FP hypotheses, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract records the log-ratio-test substitution into eq:general_KL_derivative_0_discrete, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns isolates the normalized weak-FP-to-KL handoff, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction keeps the log-ratio weak-FP action adjacent to the dK display, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns composes that substitution through the cycle-72 admissible source-sign wrapper, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces composes it through the cycle-77 generator-piece source-sign wrapper, and SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff proves the sigma-weighted regrouping under abstract divergence linearity." transportVelocityInterface := "If v_t generates pi_t, then tilde v_s=dot t(s)*v_{t(s)} generates tilde pi_s=pi_{t(s)} on the same interval." frozenResidualAlgebra := "(sigma_eta^2/2)*nabla log tilde pi_s - bar b_{k,s} + tilde v_s = delta_pi_{t(s)}^VA + dot t(s)*m_{t(s)}, using m_t=v_t-c_t and eq:general_discrete_delta_def." youngCoefficientBookkeeping := "The source splits the two cross terms with sigma_eta^2/8 each: the m_t term gives 2*sigma_eta^(-2)*dot t(s)^2*||m||^2 and lem:frozen_delta_cross_lip gives 2*Gamma*eta^2*alpha'^(-1)*K+2*Delta*eta." lsiBookkeeping := "After the two sigma_eta^2/8 Young contributions, the remaining -(sigma_eta^2/4)*FI is converted by LSI into -(sigma_eta^2/2)*C_LSI*K." dvFiniteMgfInterface := "DV is applied to Z=alpha*||m_{t(s)}||^2 under hat rho_s versus tilde pi_s, requiring SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract: alpha0 finite log-mgf, monotonicity for alpha <= alpha0, EM common-space/absolute-continuity, and positive-alpha scaling." timeChangeAndStitchingInterface := "K(t)=KL(hat rho_{s(t)}||pi_t), dK/dt=dot s(t)*d/ds KL at s=s(t), dot t(s(t))=dot s(t)^(-1), and endpoint laws stitch the interval estimates for Gronwall." requiredRegularity := [ "conditional law/disintegration for (X_k^eta,hat X_s) and bar b_{k,s}", "weak-test Fokker-Planck identity with negative drift-divergence and positive sigma_eta^2/2 Laplacian signs", "smooth positive densities for hat rho_s and tilde pi_s on each EM interval", "mass conservation, differentiation under the integral, and integration by parts for the KL derivative", "transport-velocity regularity for v_t and the slowed path tilde pi_s", "positive sigma_eta(t), dot s(t), and finite KL/FI/residual-energy quantities" ] sourceGaps := [ "appendix.tex lines 1354-1387 invoke endpoint laws and the conditional-drift Fokker--Planck equation without a standalone lemma", "appendix.tex lines 1469-1478 use the frozen/residual algebra after defining delta_pi^VA, but Lean needs an explicit rewrite interface", "appendix.tex lines 1493-1542 rely on the exact sigma_eta^2/8 + sigma_eta^2/8 Young split to obtain the later doubled residual coefficients", "appendix.tex lines 1573-1600 pass from interval-wise s-derivatives to a global t-Gronwall inequality without spelling out stitched regularity" ] dependencies := [ "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "eq:general_discrete_delta_def", "SALD.generalMovingTargetDiscreteFrozenResidualAlgebraVector", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination", "SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap", "SALD.generalMovingTargetDiscreteConditionalKernelCompatibilityOfJointMapMarginal", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityOfJointMap", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces", "SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff", "sald.general_moving_target_discrete.cycle69_em_fp_source_signs_lower", "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "sald.general_moving_target_discrete.cycle54_em_fp_lower", "lem:frozen_delta_cross_lip", "TransportVelocityContract", "FokkerPlanckContract", "KLContract", "FIContract", "probability.dv_variational_formula", "sald.forward_kl.schedule_time_change" ] status := ProofStatus.obligation def generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract : GeneralMovingTargetDiscreteDvFiniteLogMgfWitnessContract where sourceBlock := saldGeneralMovingTargetDiscreteResidualDvSource theoremStatement := saldGeneralMovingTargetDiscreteSource dvMeasures := "appendix.tex:1544-1552 applies lem:dv_variation with nu=hat rho_s and mu=tilde pi_s=pi_{t(s)} under the general VA-SALD EM interpolation." testFunction := "Z_s(x)=alpha*||m_{t(s)}(x)||^2, where m_t=v_t-c_t is the residual field from the continuous general theorem." finiteAlpha0Assumption := "The discrete theorem imports the continuous general assumptions, including finite E_{alpha0}(pi_t,m_t), and allows alpha in (0,alpha0]." alphaMonotonicityBridge := "Reuse the continuous residual witness for finite log E_{pi_{t(s)}}[exp(alpha*||m_{t(s)}||^2)] before invoking DV on the EM interpolation law." interpolationLawInterface := "hat rho_s is the law of the frozen EM interpolation and tilde pi_s=pi_{t(s)} on each stitched interval, as tracked by SALD.generalMovingTargetDiscreteDerivativeSideConditionContract." commonSpaceAndAbsoluteContinuity := "The DV step requires hat rho_s and tilde pi_s on the same measurable space with the absolute-continuity/density interface needed for KL(hat rho_s||tilde pi_s)." measurabilityInterface := "m_{t(s)} and ||m_{t(s)}||^2 must be measurable under both hat rho_s and tilde pi_s; this remains part of the local analytic side-condition backend." scalingStep := "After DV bounds E_{hat rho_s}[alpha*||m_{t(s)}||^2], divide by alpha>0 and rewrite the log-mgf as E_alpha(pi_{t(s)},m_{t(s)})." outputBound := "||m_{t(s)}||_{L2(hat rho_s)}^2 <= alpha^(-1)*KL(hat rho_s||tilde pi_s)+E_alpha(pi_{t(s)},m_{t(s)})." coefficientUse := "Multiplication by 2*sigma_eta^(-2)*dot t(s)^2 yields the doubled residual coefficients that become 2*sigma_eta(t)^(-2)*dot{s}(t)^(-1) after time change." dependencies := [ "lem:dv_variation", "def:alpha-complexity", "SALD.saldDvFiniteLogMgfContract", "sald.dv_variation.finite_log_mgf_interface", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.derivative_side_conditions" ] sourceGaps := [ "the source applies DV in one display and does not isolate EM absolute-continuity or common-space facts", "the continuous finite alpha0-complexity assumption for m_t is reused along t(s), but the alpha0-to-alpha monotonicity bridge is not stated separately", "the proof must preserve the factor 2*sigma_eta^(-2)*dot t(s)^2 before the time-change rewrite" ] lowerPacket := [ "Target this contract or sald.general_moving_target_discrete.dv_finite_log_mgf_witness only.", "Refine one backend: EM common-space/absolute-continuity, measurability of ||m_{t(s)}||^2, reuse of the continuous residual finite-log-mgf witness, positive-alpha scaling, or preservation of the doubled residual coefficient.", "Do not add new hypotheses to thm:general-moving-target-SALD-discrete and do not modify the theorem display at appendix.tex:1316-1347." ] status := ProofStatus.obligation def generalMovingTargetDiscreteGronwallInstantiationContract : GeneralMovingTargetDiscreteGronwallInstantiationContract where sourceBlock := saldGeneralMovingTargetDiscreteGronwallSource quantityK := "K(t)=KL(hat rho_{s(t)}||pi_t), with K(T)=KL(rho_K^eta||pi_T)." gronwallA := "a(t)=(sigma_eta(t)^2/2)*dot{s}(t)*C_LSI(t)-2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t)." gronwallB := "b(t)=2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t)+2*dot{s}(t)*eta*Delta(t)." theoremBound := "The source bound eq:general_moving_target_KL_bound_discrete is exactly the Gronwall bound with this a(t) and b(t), without the later exponent simplifications used in main-body corollaries." constantScheduleInterface := "The theorem assumes dot{t}(s)=dot{s}(t)^(-1) is constant; Lean must represent the endpoint laws and K(t) over the stitched EM intervals under this schedule." requiredRegularity := [ "interval-integrability of C_LSI, sigma_eta, Gamma, Delta, and E_alpha(pi_t,m_t)", "piecewise differentiability or absolute continuity of K(t) after stitching EM intervals", "positivity of sigma_eta(t) and dot{s}(t)" ] sourceGaps := [ "the theorem assumes a constant inverse schedule but does not spell out the schedule API needed by the Gronwall backend", "the final proof says only that applying lem:gronwall finishes the proof; all regularity and endpoint matching remain obligations" ] dependencies := [ "lem:gronwall", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation def generalMovingTargetDiscreteGronwallSideConditionContract : GeneralMovingTargetDiscreteGronwallSideConditionContract where sourceBlock := saldGeneralMovingTargetDiscreteGronwallSideConditionSource theoremStatement := saldGeneralMovingTargetDiscreteSource endpointLawIdentities := "The global K(t) used at appendix.tex:1573-1583 must be stitched from EM interval laws with s(0)=0, s(T)=S, hat rho_{s_k}=rho_k^eta, hat rho_{s_{k+1}}=rho_{k+1}^eta, and pi_{t(s(t))}=pi_t." constantScheduleIdentities := "The theorem assumes dot{t}(s)=dot{s}(t)^(-1) is constant; the time-change from appendix.tex:1579-1583 must rewrite dot{s}(t)*dot{t}(s(t))^2 as dot{s}(t)^(-1) and dot{s}(t)*Gamma/Delta terms exactly as in lines 1588-1597." terminalKlIdentification := "After Gronwall, K(T) is identified with KL(rho_K^eta||pi_T), using endpoint matching of the last frozen interpolation interval and the target endpoint t=T." initialKlIdentification := "The initial Gronwall term K(0) is identified with KL(rho_0||pi_0), using X_0^eta~rho_0 and the first interpolation endpoint." stitchedRegularity := "The interval-wise derivative inequalities for KL(hat rho_s||tilde pi_s) must assemble into a piecewise differentiable or absolutely continuous K(t) admissible for lem:gronwall." coefficientRegularity := "a(t)=(sigma_eta(t)^2/2)*dot{s}(t)*C_LSI(t)-2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t) and b(t)=2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t)+2*dot{s}(t)*eta*Delta(t) need the continuity or interval-integrability required by lem:gronwall." gronwallDisplayMatch := "The Gronwall output with this a(t), b(t), K(0), and K(T) is the theorem display eq:general_moving_target_KL_bound_discrete at appendix.tex:1316-1347; no extra exponent split or residual LSI drop is used here." residualCoefficientAudit := "The DV residual term 2*sigma_eta^(-2)*dot{t}(s)^2*(alpha^(-1)*K+E_alpha) becomes the two theorem coefficients 2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*alpha^(-1) in a(t) and 2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t) in b(t)." frozenDeltaCoefficientAudit := "The frozen-delta terms 2*Gamma(t(s))*eta^2*alpha'^(-1)*K_s and 2*Delta(t(s))*eta become 2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t) in a(t) and 2*dot{s}(t)*eta*Delta(t) in b(t)." dependencies := [ "lem:gronwall", "sald.general_moving_target_discrete.constant_schedule_stitching", "SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar", "SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput", "SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "sald.forward_kl.schedule_time_change" ] sourceGaps := [ "the source proof does not isolate endpoint identifications K(T)=KL(rho_K^eta||pi_T) and K(0)=KL(rho_0||pi_0) as a separate lemma", "the constant inverse-schedule assumption is stated in the theorem, while the proof uses it inside the coefficient rewrite after the s-to-t time change", "the final Gronwall call requires stitched-interval regularity and coefficient integrability that are not stated explicitly", "the exact doubled residual and frozen-delta coefficients must be preserved when passing from appendix.tex:1558-1570 to appendix.tex:1586-1597" ] lowerPacket := [ "Target this contract or the named discrete gronwall-side-condition obligation only.", "Preserve the theorem display at appendix.tex:1316-1347 and the differential inequality at appendix.tex:1586-1597.", "Reuse SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput and SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar for the local display-matching pieces, and refine endpoint stitching, constant-schedule coefficient rewrites, coefficient regularity, or exact Gronwall-display matching as obligations unless compiled Lean lemmas prove them.", "Do not add new hypotheses to thm:general-moving-target-SALD-discrete or simplify the doubled residual coefficients." ] status := ProofStatus.obligation def saldPIContract : PIContract where measureName := "mu" constantName := "C_PI" statement := "For all smooth phi, Var_mu[phi] <= (1/C_PI) integral ||nabla phi||^2 dmu." source := saldPiSource status := ProofStatus.contractOnly def lsiKlFiDensityTestObligation : ProofObligation where id := "sald.lsi_kl_fi.density_test_interface" statement := "Formalize the source step main_body.tex lines 208-215: for rho << pi, set phi=sqrt(rho/pi), verify the LSI normalization and admissibility conditions, identify the entropy term with KL(rho||pi), and use the Fisher-information chain rule to obtain KL(rho||pi) <= FI(rho||pi)/(2*C_LSI)." source := saldKlFiLsiSource status := ProofStatus.obligation dependsOn := [ "KLContract", "FIContract", "LSIContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.cycle29FirstAppendixMiddleAuditContract", "SALD.cycle29LsiKlFiDensityTestMiddleObligation", "SALD.cycle29LsiKlFiDensityTestLowerObligation", "SALD.cycle33LsiKlFiDensityTestMiddleObligation", "SALD.cycle33LsiKlFiDensityTestLowerObligation", "SALD.cycle38LsiKlFiUpperPacket", "SALD.cycle38LsiKlFiUpperObligation", "SALD.cycle38LsiKlFiMiddleObligation", "SALD.cycle38LsiKlFiLowerObligation", "SALD.cycle43LsiKlFiUpperPacket", "SALD.cycle43LsiKlFiUpperObligation", "SALD.cycle43LsiKlFiMiddleObligation", "SALD.cycle43LsiKlFiLowerObligation", "AutoSamplingTheory.lsiKlFiSqrtDensitySquareScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityEntropyIntegrandScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityNormalizationScalar", "AutoSamplingTheory.lsiKlFiRnDerivLIntegralMassOne", "AutoSamplingTheory.lsiKlFiRnDerivDensityMassOne", "AutoSamplingTheory.lsiKlFiSqrtRnDerivTestMassOne", "AutoSamplingTheory.lsiKlFiRnDerivEntropyIntegral", "AutoSamplingTheory.lsiKlFiSqrtRnDerivEntropyIntegral", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainOfDerivativesScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumHandoffScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralFiniteSum", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralHandoffScalar", "SALD.lsiKlFiCoefficientAuditScalar", "SALD.lsiKlFiDensityTestBridgeScalar", "probability.lsi_to_kl_fi" ] note := "saldLsiKlFiDensityTestContract records the smooth-density, absolute-continuity, finite-KL/FI, test-function, coefficient, and approximation interfaces; cycle 29 formalized only the final real-algebra coefficient audit, cycle 33 formalizes scalar sqrt-density square, entropy-integrand, normalization handoff, and normalized-test scalar bridge lemmas, cycle 38 adds pointwise plus finite-coordinate Fisher-chain handoffs, cycle 43 middle formalizes the Mathlib Radon-Nikodym mass and entropy-transport sublemmas, and cycle 43 lower formalizes a finite-coordinate a.e. integral Fisher-chain handoff while leaving sqrt-test admissibility/approximation, zero-density Sobolev handling, vector-gradient equivalence, finite theorem-level KL/FI interfaces, and the full LSI-to-KL/FI theorem open. Do not replace the source LSI route with PI, Pinsker, or Talagrand." def dvFiniteLogMgfInterfaceObligation : ProofObligation where id := "sald.dv_variation.finite_log_mgf_interface" statement := "Formalize the first-layer interface needed to instantiate lem:dv_variation: common measurable probability space for mu and nu, measurable random variable Z, finite log E_mu[exp Z], and the alpha-complexity witness used by SALD squared-velocity tests." source := saldDvVariationSource status := ProofStatus.obligation dependsOn := ["probability.dv_variational_formula", "SALD.saldDvFiniteLogMgfContract", "def:alpha-complexity"] note := "The DV theorem itself remains source-cited. This obligation only exposes the finite-log-mgf and common-space side conditions needed by later theorem-specific DV-energy obligations." def piVelocityNormBackendObligation : ProofObligation where id := "sald.pi.velocity_norm_backend" statement := "Formalize appendix lines 96-151: weighted mean-zero Sobolev space under PI, norm equivalence, boundedness of T_mu, Riesz representation for the weak PDE div(mu*v)=-g*mu, and the velocity bound ||v||_{L2(mu)} <= C_PI^{-1/2}||g||_{L2(mu)}." source := saldPiVelocityNormSource status := ProofStatus.obligation dependsOn := [ "PIContract", "SALD.saldPiVelocityNormDependencyContract", "SALD.cycle25FirstAppendixMiddleAuditContract", "SALD.cycle25FirstAppendixPiVelocityNormMiddleObligation", "SALD.cycle25PiVelocityNormLowerObligation", "SALD.piVelocityNormMeanZeroH1UpperScalar", "SALD.piVelocityNormBoundedFunctionalScalar", "weighted Sobolev backend", "Riesz representation theorem" ] note := "The PI definition remains contract-only; this obligation tracks the downstream analytic backend used to estimate moving-target and guided-path energy. Cycle 25 lower formalizes only the scalar real-order propagation inside appendix.tex:104-129, with the weighted Sobolev, L2 pairing, line 130-138 operator-norm/Riesz continuation, weak-PDE, and velocity-bound interfaces still obligation data." /-- Cycle-36 upper packet for returning to the Gronwall proof-closure target. This packet deliberately selects proof-closure priority item (1), `lem:gronwall`, after cycle 35 finished a local EM interpolation algebra pass. It is workflow data only: it fixes the next middle/lower objective without claiming that the global closed-interval calculus backend has been formalized. -/ def cycle36GronwallUpperPacket : FirstAppendixVocabularyPacket where objective := "Proof-closure priority check before assigning lower work: (1) lem:gronwall remains open after cycle 31 local derivative, order-integration, endpoint, and exponent sublemmas; (2) lem:dv_variation remains source-cited with cycle 32 scalar consequences; (3) eq:LSI-KL-FI remains open after cycle 33 density-test scalar lemmas; (4) the forward-KL Fokker-Planck/KL derivative identity still depends on analytic backends after cycle 34 scalar handoffs; and (5) the EM interpolation Fokker-Planck backend has cycle 35 endpoint/divergence algebra only. Cycle 36 therefore returns to item (1): translate appendix.tex:47-71 into a proof-producing Gronwall assembly target." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "thm:forward-KL", "thm:forward-KL-discrete" ] modeDiscipline := [ "faithfulPaper Phase 1: use only the original appendix.tex:47-71 Gronwall lemma and proof; sald_version_2.tex remains excluded.", "Keep the source theorem fixed: continuous a_t and b_t, differentiable K_t on [0,t1], differential inequality dK/dt <= -a_t*K_t+b_t, and the exact final bound with exp(-int_0^t1 a) and exp(-int_t^t1 a).", "Use the existing compiled local helpers as dependencies, not as permission to change the lemma statement or import an alternate Gronwall theorem.", "If the closed-interval derivative/FTC backend is too large for the current local Mathlib state, expose a precise source-cited or Mathlib-backed theorem interface and keep lem:gronwall below formalized status." ] nonGoals := [ "Do not rebaseline the source index unless a reviewer identifies a blocking source-anchor defect.", "Do not work on DV, LSI/KL/FI, forward-KL derivative, EM interpolation, PI velocity-norm, guided, general VA-SALD, unified, or accumulated-error proof search in this upper packet.", "Do not add sign assumptions on a or b, monotonicity assumptions, extra endpoint hypotheses, or theorem-level smoothness assumptions beyond an explicit calculus interface.", "Do not mark SALD.gronwallContract, sald.gronwall.integrating_factor, sald.gronwall.endpoint_calculus, or sald.gronwall.exponent_rewrite formalized until a compiled Lean theorem closes the full source display." ] lowerPacket := [ "Middle must keep two-way Lean/Markdown/LaTeX synchronization for appendix.tex:47-71, AutoSamplingTheory/SALD.lean, conversion-windows/ASTIS-SALD-001.md, and proof-obligations/ASTIS-SALD-001.md.", "Target exactly SALD.saldGronwallCandidateContract, SALD.saldGronwallEndpointCalculusContract, SALD.saldGronwallExponentRewriteContract, and the obligations sald.gronwall.integrating_factor, sald.gronwall.endpoint_calculus, sald.gronwall.exponent_rewrite.", "First proof-producing lower attempt: assemble a theorem-level Gronwall display under explicit global interval hypotheses by using SALD.gronwallIntegratingFactorDerivativeLeOfIntegral, SALD.gronwallOrderIntegrationOfHasDerivAt, SALD.gronwallEndpointEvaluationScalar, SALD.gronwallEndpointMultiplyByExpNegScalar, and SALD.gronwallExpProductRewriteIntegralCongr.", "If endpoint-safe differentiability on [0,t1] is blocked, introduce one narrow interface for the closed-interval or absolute-continuity FTC step, with hypotheses naming the required interval-integrability of f', g, a, and b; leave that interface as obligation/source-cited rather than changing the source lemma.", "Keep downstream theorem-specific Gronwall side conditions for forward-KL, discrete forward-KL, and general moving-target SALD as separate sibling obligations." ] reviewerChecklist := [ "SALD.gronwallContract still has ProofStatus.obligation and lists the cycle 36 upper obligation only as workflow data.", "SALD.saldDependenciesForLabel \"lem:gronwall\" includes SALD.cycle36GronwallUpperPacket and sald.gronwall.cycle36_upper_packet while retaining the cycle 31 compiled helper declarations.", "The conversion window and proof-obligation ledger identify cycle 36 as Gronwall proof-closure item (1), not source-index rebaseline or theorem-status promotion.", "No alternate Gronwall theorem, hidden sign assumption, source-file drift, or fake proof closure is introduced.", "The mandatory gate python3 tools/astis.py check passes." ] status := ProofStatus.obligation /-- Cycle-36 upper workflow obligation for the Gronwall proof-closure packet. -/ def cycle36GronwallUpperObligation : ProofObligation where id := "sald.gronwall.cycle36_upper_packet" statement := "Maintain the cycle 36 upper packet for appendix.tex lines 47-71: return to proof-closure priority item (1), lem:gronwall, and direct middle/lower work toward a proof-producing assembly of the original integrating-factor proof using the existing compiled local real-calculus helpers while keeping the full Gronwall lemma at obligation status until the closed-interval calculus backend builds." source := saldGronwallSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle36GronwallUpperPacket", "SALD.gronwallContract", "SALD.saldGronwallCandidateContract", "SALD.saldGronwallEndpointCalculusContract", "SALD.saldGronwallExponentRewriteContract", "SALD.gronwallIntegratingFactorDerivativeLeOfIntegral", "SALD.gronwallOrderIntegrationOfHasDerivAt", "SALD.gronwallEndpointEvaluationScalar", "SALD.gronwallEndpointMultiplyByExpNegScalar", "SALD.gronwallExpProductRewriteIntegralCongr", "sald.gronwall.integrating_factor", "sald.gronwall.endpoint_calculus", "sald.gronwall.exponent_rewrite" ] note := "This is an upper-role assignment packet. It does not close lem:gronwall; the remaining proof work is the global closed-interval derivative/FTC/integrability assembly plus final exponent-congruence inputs under the exact source signs and constants." /-- Cycle-36 middle proof-producing Gronwall assembly record. -/ def cycle36GronwallMiddleObligation : ProofObligation where id := "sald.gronwall.cycle36_middle_assembly" statement := "Record the cycle 36 proof-producing assembly for appendix.tex lines 58-69: SALD.gronwallEndpointIntegralRewrite moves the endpoint inverse integrating factor through the b_t integral, SALD.gronwallIntegratingFactorBoundOfDerivatives assembles the full displayed Gronwall bound from explicit global derivative/integrability hypotheses, SALD.gronwallIntegratingFactorBoundOfIntegral discharges the derivative of int_0^t a using Mathlib's interval-integral FTC at each point, and SALD.gronwallIntegratingFactorBoundOfContinuousData proves the displayed bound from global continuity of a, b, K, and K' plus the derivative inequality." source := saldGronwallSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle36GronwallUpperPacket", "SALD.gronwallEndpointIntegralRewrite", "SALD.gronwallIntegratingFactorBoundOfDerivatives", "SALD.gronwallIntegratingFactorBoundOfIntegral", "SALD.gronwallIntegratingFactorProductDerivative", "SALD.gronwallIntegratingFactorDerivativeInequalityScalar", "SALD.gronwallOrderIntegrationOfHasDerivAt", "SALD.gronwallEndpointEvaluationScalar", "SALD.gronwallEndpointMultiplyByExpNegScalar", "SALD.gronwallExpProductRewriteIntegralCongr", "SALD.gronwallCoefficientSideConditionsOfContinuous", "SALD.gronwallIntegratingFactorBoundOfContinuousData", "sald.gronwall.endpoint_calculus", "sald.gronwall.exponent_rewrite" ] note := "The Lean theorems are formalized local Gronwall assembly under explicit Mathlib side conditions. The lower continuous-data wrapper removes the separate integrability and coefficient-FTC assumptions when a, b, K, and K' are globally continuous, but it still does not prove that the paper's concise differentiability-on-[0,t1] hypothesis supplies a globally continuous derivative witness or the endpoint-safe derivative semantics. SALD.gronwallContract remains ProofStatus.obligation." /-- Cycle-41 middle proof-producing Gronwall derivative-source wrapper. -/ def cycle41GronwallMiddleObligation : ProofObligation where id := "sald.gronwall.cycle41_deriv_wrapper" statement := "Record the cycle 41 proof-producing Gronwall wrapper for appendix.tex lines 47-71: SALD.gronwallIntegratingFactorBoundOfDifferentiable rewrites the paper's dK/dt as deriv K and proves the displayed bound from continuous a,b, differentiable K, the source inequality with deriv K, and one explicit interval-integrability hypothesis for the product-derivative integrand; SALD.gronwallIntegratingFactorBoundOfC1 discharges that remaining integrability condition when deriv K is continuous." source := saldGronwallSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle36GronwallMiddleObligation", "SALD.gronwallIntegratingFactorBoundOfDifferentiable", "SALD.gronwallIntegratingFactorBoundOfC1", "SALD.gronwallIntegratingFactorBoundOfIntegral", "SALD.gronwallCoefficientSideConditionsOfContinuous", "SALD.gronwallEndpointIntegralRewrite", "sald.gronwall.endpoint_calculus", "sald.gronwall.integrating_factor" ] note := "This narrows the remaining source bridge from an arbitrary K' witness to the actual Lean derivative deriv K. It does not mark lem:gronwall formalized, because the paper's bare phrase differentiable on [0,t1] still needs an endpoint-safe or absolute-continuity interpretation strong enough to justify the FTC/integrability side condition without silently assuming C1 regularity." /-- Cycle-41 lower endpoint-safe Gronwall interior-derivative assembly record. -/ def cycle41GronwallLowerObligation : ProofObligation where id := "sald.gronwall.cycle41_interior_endpoint_bridge" statement := "Record the cycle 41 lower proof-producing endpoint-safe Gronwall bridge for appendix.tex lines 62-69: SALD.gronwallOrderIntegrationOfHasDerivRight uses Mathlib's right-derivative FTC with continuity on [0,t1] and derivatives only on (0,t1); SALD.gronwallIntegratingFactorBoundOfInteriorDerivatives assembles the displayed Gronwall bound with no endpoint derivative hypotheses on the integrating-factor product; SALD.gronwallIntegratingFactorBoundOfInteriorContinuousData and SALD.gronwallIntegratingFactorBoundOfInteriorC1 discharge the continuity/integrability side conditions under a C1-compatible source interpretation." source := saldGronwallSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle41GronwallMiddleObligation", "SALD.gronwallOrderIntegrationOfHasDerivRight", "SALD.gronwallIntegratingFactorBoundOfInteriorDerivatives", "SALD.gronwallIntegratingFactorBoundOfInteriorContinuousData", "SALD.gronwallIntegratingFactorBoundOfInteriorC1", "SALD.gronwallEndpointEvaluationScalar", "SALD.gronwallEndpointMultiplyByExpNegScalar", "SALD.gronwallEndpointIntegralRewrite", "sald.gronwall.endpoint_calculus", "sald.gronwall.integrating_factor" ] note := "This closes one endpoint-safe calculus layer by eliminating endpoint derivative hypotheses from the FTC/order-integration assembly. SALD.gronwallContract remains ProofStatus.obligation because the paper's phrase differentiable on [0,t1] still has to be identified with this C1-compatible or absolute-continuity backend before the source lemma can be marked formalized." def gronwallAnalyticObligation : ProofObligation where id := "sald.gronwall.integrating_factor" statement := "Formalize appendix Lemma lem:gronwall using the displayed integrating-factor proof for differentiable real K and continuous real a_t, b_t on [0,t1]." source := saldGronwallSource status := ProofStatus.obligation dependsOn := [ "SALD.gronwallIntegratingFactorProductDerivative", "SALD.gronwallIntegratingFactorDerivativeInequalityScalar", "SALD.gronwallIntegratingFactorDerivativeLe", "SALD.gronwallIntegratingFactorDerivativeLeOfIntegral", "SALD.gronwallOrderIntegrationOfHasDerivAt", "SALD.gronwallOrderIntegrationOfHasDerivRight", "SALD.gronwallEndpointEvaluationScalar", "SALD.gronwallEndpointMultiplyByExpNegScalar", "SALD.gronwallEndpointIntegralRewrite", "SALD.gronwallIntegratingFactorBoundOfDerivatives", "SALD.gronwallIntegratingFactorBoundOfIntegral", "SALD.gronwallIntegratingFactorBoundOfInteriorDerivatives", "SALD.gronwallCoefficientSideConditionsOfContinuous", "SALD.gronwallIntegratingFactorBoundOfContinuousData", "SALD.gronwallIntegratingFactorBoundOfDifferentiable", "SALD.gronwallIntegratingFactorBoundOfC1", "SALD.gronwallIntegratingFactorBoundOfInteriorContinuousData", "SALD.gronwallIntegratingFactorBoundOfInteriorC1", "SALD.cycle41GronwallMiddleObligation", "SALD.cycle41GronwallLowerObligation" ] note := "Do not change the inequality shape or constants. Cycle 31 closes the pointwise appendix.tex:58-61 derivative/product/scalar inequality slice plus the appendix.tex:62-65 order-integration and endpoint scalar algebra. Cycle 36 middle adds a compiled global assembly under explicit Mathlib side conditions, and cycle 36 lower adds a continuous-data wrapper discharging the coefficient and integrand integrability side conditions under global continuity of a, b, K, and K'. Cycle 41 middle rewrites the source derivative as deriv K and provides a C1-style wrapper; cycle 41 lower adds an endpoint-safe right-derivative assembly that no longer differentiates at the closed endpoints. The full source lemma still needs endpoint-safe production of the FTC/integrability backend from the paper's concise differentiability assumption." def gronwallEndpointCalculusObligation : ProofObligation where id := "sald.gronwall.endpoint_calculus" statement := "Formalize the endpoint-safe calculus side conditions in appendix lines 55-69: choose a closed-interval derivative or absolute-continuity backend, differentiate the integrating factor exp(int_0^t a), integrate the derivative inequality from 0 to t1, evaluate endpoints, and rewrite exp(-int_0^t1 a)*exp(int_0^t a) as exp(-int_t^t1 a)." source := saldGronwallSource status := ProofStatus.obligation dependsOn := [ "SALD.saldGronwallCandidateContract", "SALD.saldGronwallEndpointCalculusContract", "SALD.gronwallIntegratingFactorProductDerivative", "SALD.gronwallIntegratingFactorDerivativeInequalityScalar", "SALD.gronwallIntegratingFactorDerivativeLe", "SALD.gronwallIntegratingFactorDerivativeLeOfIntegral", "SALD.gronwallOrderIntegrationOfHasDerivAt", "SALD.gronwallOrderIntegrationOfHasDerivRight", "SALD.gronwallEndpointEvaluationScalar", "SALD.gronwallEndpointMultiplyByExpNegScalar", "SALD.gronwallEndpointIntegralRewrite", "SALD.gronwallIntegratingFactorBoundOfDerivatives", "SALD.gronwallIntegratingFactorBoundOfIntegral", "SALD.gronwallIntegratingFactorBoundOfInteriorDerivatives", "SALD.gronwallCoefficientSideConditionsOfContinuous", "SALD.gronwallIntegratingFactorBoundOfContinuousData", "SALD.gronwallIntegratingFactorBoundOfDifferentiable", "SALD.gronwallIntegratingFactorBoundOfC1", "SALD.gronwallIntegratingFactorBoundOfInteriorContinuousData", "SALD.gronwallIntegratingFactorBoundOfInteriorC1", "SALD.cycle41GronwallMiddleObligation", "SALD.cycle41GronwallLowerObligation", "Real intervalIntegral backend", "Real.exp algebra" ] note := "This obligation refines the local Gronwall proof backend only. Cycle 31 proves the pointwise derivative inequality after local FTC and HasDerivAt inputs, and adds a compiled interval-order integration lemma plus endpoint scalar algebra. Cycle 36 middle proves the displayed bound under explicit global derivative/integrability hypotheses; cycle 36 lower proves the same display from continuous a, b, K, and K' plus the derivative inequality. Cycle 41 middle proves the same display with the paper derivative written as deriv K; cycle 41 lower adds the right-derivative FTC/order-integration assembly with no endpoint derivative hypotheses, while keeping the C1 or absolute-continuity interpretation explicit. It does not add sign assumptions on a or b, and it does not mark lem:gronwall formalized because the source-to-Mathlib closed-interval differentiability bridge remains open." def gronwallExponentRewriteObligation : ProofObligation where id := "sald.gronwall.exponent_rewrite" statement := "Formalize the final exponent algebra in appendix lines 63-69: after multiplying by exp(-int_0^t1 a), rewrite exp(-int_0^t1 a)*exp(int_0^t a) as exp(-int_t^t1 a) inside the b_t integral using interval-integral additivity on 0<=t<=t1, the compiled scalar Real.exp product lemma, and interval-integral congruence." source := saldGronwallExponentRewriteSource status := ProofStatus.obligation dependsOn := [ "SALD.saldGronwallEndpointCalculusContract", "SALD.saldGronwallExponentRewriteContract", "SALD.gronwallNegIntegralRewriteScalar", "SALD.gronwallExpProductRewriteScalar", "SALD.gronwallIntervalIntegralAdditivityScalar", "SALD.gronwallExpProductRewriteIntervalIntegral", "SALD.gronwallExpProductRewriteIntegralCongr", "SALD.gronwallEndpointIntegralRewrite" ] note := "This is a lower sub-obligation of sald.gronwall.endpoint_calculus. Cycle 17 lower formalized scalar negation/Real.exp algebra; cycle 18 lower added the adjacent-interval bridge from intervalIntegral.integral_add_adjacent_intervals to that scalar helper; cycle 21 lower added the interval congruence wrapper SALD.gronwallExpProductRewriteIntegralCongr; cycle 36 middle added SALD.gronwallEndpointIntegralRewrite to package the final integral term. Proving the adjacent interval-integrability from source regularity remains an obligation. No sign assumptions on a or b were added." def firstAppendixSourceIndexAuditObligation : ProofObligation where id := "sald.first_appendix.source_index_audit" statement := "Maintain the first appendix/vocabulary source-index audit for lem:gronwall, lem:dv_variation, def:PI, and eq:LSI-KL-FI: each source label must be indexed from the original SALD paper, mapped to an existing Lean-facing contract, and left at its honest non-formalized status unless a local proof builds." source := saldAppendixSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle13FirstAppendixVocabularyPacket", "SALD.cycle13FirstAppendixSourceIndexAuditContract", "SALD.cycle17FirstAppendixVocabularyPacket", "SALD.cycle17FirstAppendixMiddleAuditContract", "SALD.cycle21FirstAppendixVocabularyPacket", "SALD.cycle21FirstAppendixMiddleAuditContract", "SALD.cycle25FirstAppendixVocabularyPacket", "SALD.cycle25FirstAppendixMiddleAuditContract", "SALD.cycle29FirstAppendixVocabularyPacket", "SALD.saldFirstProofDag", "SALD.gronwallContract", "SALD.dvContract", "SALD.piDefinitionContract", "SALD.lsiKlFiVocabularyContract", "research-wiki/source-index/SALD_original.jsonl" ] note := "This is a source-index synchronization obligation, not a mathematical theorem. Cycles 17, 21, 25, and 29 reuse it for the same first-appendix focus while keeping sald_version_2.tex excluded and not promoting Gronwall, DV, PI velocity bounds, or LSI-to-KL/FI beyond their current statuses." def firstAppendixMiddleAuditObligation : ProofObligation where id := "sald.first_appendix.middle_source_to_lean_map" statement := "Maintain the first appendix middle source-to-Lean map for lem:gronwall, lem:dv_variation, def:PI, and eq:LSI-KL-FI: every source proof step in the focus windows must be classified as an existing Lean contract, a cited result, or a named proof obligation, with a lower-ready target that does not change later SALD theorem statements." source := saldAppendixSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle13FirstAppendixSourceIndexAuditContract", "SALD.cycle13FirstAppendixMiddleAuditContract", "SALD.cycle17FirstAppendixVocabularyPacket", "SALD.cycle17FirstAppendixMiddleAuditContract", "SALD.cycle21FirstAppendixVocabularyPacket", "SALD.cycle21FirstAppendixMiddleAuditContract", "SALD.cycle25FirstAppendixVocabularyPacket", "SALD.cycle25FirstAppendixMiddleAuditContract", "SALD.cycle25FirstAppendixPiVelocityNormMiddleObligation", "SALD.cycle25PiVelocityNormLowerObligation", "SALD.cycle29FirstAppendixVocabularyPacket", "SALD.cycle29FirstAppendixMiddleAuditContract", "SALD.cycle29LsiKlFiDensityTestMiddleObligation", "SALD.cycle29LsiKlFiDensityTestLowerObligation", "SALD.cycle36GronwallUpperPacket", "SALD.cycle36GronwallUpperObligation", "SALD.cycle36GronwallMiddleObligation", "SALD.cycle41GronwallMiddleObligation", "SALD.cycle37DvVariationUpperPacket", "SALD.cycle37DvVariationUpperObligation", "SALD.cycle37DvVariationMiddleAuditContract", "SALD.cycle37DvVariationMiddleObligation", "SALD.cycle38LsiKlFiUpperPacket", "SALD.cycle38LsiKlFiUpperObligation", "AutoSamplingTheory.dvVariationalOneSidedOfTiltedRight", "AutoSamplingTheory.dvVariationalTiltedRightOneSidedConsequence", "SALD.lsiKlFiCoefficientAuditScalar", "SALD.piVelocityNormMeanZeroH1UpperScalar", "SALD.piVelocityNormBoundedFunctionalScalar", "sald.first_appendix.source_index_audit", "SALD.saldGronwallEndpointCalculusContract", "SALD.saldGronwallExponentRewriteContract", "SALD.gronwallIntegratingFactorProductDerivative", "SALD.gronwallIntegratingFactorDerivativeInequalityScalar", "SALD.gronwallIntegratingFactorDerivativeLe", "SALD.gronwallIntegratingFactorDerivativeLeOfIntegral", "SALD.gronwallEndpointIntegralRewrite", "SALD.gronwallIntegratingFactorBoundOfDerivatives", "SALD.gronwallIntegratingFactorBoundOfIntegral", "SALD.gronwallCoefficientSideConditionsOfContinuous", "SALD.gronwallIntegratingFactorBoundOfContinuousData", "SALD.gronwallIntegratingFactorBoundOfDifferentiable", "SALD.gronwallIntegratingFactorBoundOfC1", "SALD.saldPiVelocityNormDependencyContract", "SALD.saldDvFiniteLogMgfContract", "SALD.saldLsiKlFiDensityTestContract", "research-wiki/source-index/SALD_original.jsonl" ] note := "This is a middle-role workflow obligation. Cycles 17 and 21 rebaseline the map for the same first-layer packet; cycle 25 adds a lower-ready PI velocity-norm source map; cycle 29 adds a lower-ready LSI/KL/FI density-test source map and a scalar coefficient lower lemma; cycle 31 adds proof-producing Gronwall line 58-65 helpers; cycle 36 adds proof-producing Gronwall assembly theorems under explicit Mathlib and continuous-data side conditions; cycle 41 rewrites the Gronwall derivative as deriv K and adds a C1-style proof-producing wrapper while leaving the endpoint-safe source bridge open; cycle 37 returns to the DV target and adds proof-producing one-sided tilted-measure backend and lower one-sided consequence while preserving the source-cited status of the Boucheron supremum equality; cycle 38 returns to eq:LSI-KL-FI and selects the Fisher chain-rule/admissibility bridge after the cycle 33 sqrt-density scalar helpers." def forwardKlMiddleSourceToLeanMapObligation : ProofObligation where id := "sald.forward_kl.middle_source_to_lean_map" statement := "Maintain the cycle-14/18/22 source-to-Lean maps, the cycle-26 upper/middle DV-witness packets, the cycle-30 derivative-side upper/middle packets, the cycle-34 derivative scalar/time-change packet, and the cycle-39 source-shaped derivative middle packet for continuous thm:forward-KL: appendix lines 168-252 and main_body lines 218-248 must map to the existing statement contract, derivative side conditions, LSI bridge, DV witness, coefficient-chain audit, Gronwall side conditions, and lower packet without changing the theorem statement or promoting analytic backends." source := saldForwardKlProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle14ForwardKlUpperPacket", "SALD.cycle14ForwardKlMiddleContract", "SALD.cycle18ForwardKlUpperPacket", "SALD.cycle18ForwardKlMiddleContract", "SALD.cycle22ForwardKlUpperPacket", "SALD.cycle22ForwardKlMiddleContract", "SALD.cycle26ForwardKlUpperPacket", "SALD.cycle26ForwardKlMiddleContract", "SALD.cycle30ForwardKlUpperPacket", "SALD.cycle30ForwardKlMiddleContract", "SALD.cycle34ForwardKlDerivativeUpperPacket", "SALD.cycle34ForwardKlDerivativeMiddleContract", "SALD.cycle39ForwardKlDerivativeUpperPacket", "SALD.cycle39ForwardKlDerivativeMiddleContract", "sald.forward_kl.cycle30_derivative_side_upper", "sald.forward_kl.cycle30_derivative_side_middle", "sald.forward_kl.cycle30_density_boundary_lower", "sald.forward_kl.cycle34_derivative_scalar", "sald.forward_kl.cycle34_derivative_middle", "sald.forward_kl.cycle39_derivative_upper", "sald.forward_kl.cycle39_derivative_middle", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlFirstTermFisherSubstitutionScalar", "SALD.forwardKlTargetTransportYoungBoundScalar", "SALD.forwardKlPostYoungDerivativeBoundScalar", "SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar", "SALD.forwardKlLsiDerivativeBoundScalar", "SALD.forwardKlTimeChangedDerivativeBoundScalar", "SALD.forwardKlInverseScheduleDerivativeScalar", "SALD.forwardKlTimeChangeSquareCoefficientRewriteOfProductScalar", "SALD.forwardKlVelocitySquareScalingScalar", "SALD.forwardKlTimeChangedDerivativeBoundOfProductScalar", "SALD.forwardKlPreDvDerivativeBoundScalar", "SALD.forwardKlPreDvDerivativeBoundOfProductScalar", "SALD.forwardKlPreDvDerivativeBoundOfVelocityScalingScalar", "sald.forward_kl.endpoint_schedule_identities", "sald.forward_kl.moving_target_dependency_chain", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "probability.lsi_to_kl_fi", "sald.forward_kl.cycle26_dv_witness_middle", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "sald.forward_kl.coefficient_chain_audit", "sald.forward_kl.gronwall_application", "sald.forward_kl.gronwall_side_conditions" ] note := "This is a workflow obligation for middle-role synchronization. Cycle 14 selected endpoint schedule identities, cycles 18 and 22 selected Gronwall side-condition refinements, cycle 26 selects and maps the DV finite-log-mgf witness, cycle 30 returns to the derivative-side density/boundary and slowed-target transport interface, cycle 34 adds scalar post-Young/LSI/time-change handoffs, and cycle 39 adds source-shaped scalar schedule/velocity lemmas for appendix.tex:191-228; DV, LSI, Gronwall, schedule calculus, and KL derivative facts remain unformalized." def cycle30ForwardKlDerivativeSideUpperObligation : ProofObligation where id := "sald.forward_kl.cycle30_derivative_side_upper" statement := "Maintain the cycle 30 upper-role target for appendix.tex lines 168-228: lower work should first refine SALD.forwardKlDerivativeSideConditionContract, SALD.forwardKlDensityBoundaryObligation, and sald.forward_kl.density_boundary_regular for mass conservation, differentiation under the KL integral, SALD/target integration by parts, slowed-target transport, and the exact Young 1/2 coefficients, while leaving the inverse-schedule time-change, LSI, DV, and Gronwall backends as separate obligations." source := saldForwardKlDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle30ForwardKlUpperPacket", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlMovingTargetDependencyContract", "SALD.forwardKlEndpointScheduleContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiDensityTestObligation", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "sald.forward_kl.moving_target_dependency_chain", "eq:SALD", "eq:FP-eq", "eq:LSI-KL-FI", "TransportVelocityContract" ] note := "This is an upper workflow obligation, not a proof. It keeps the source theorem and constants fixed, directs lower work to the derivative side-condition interface, and prevents density, boundary, inverse-schedule, LSI, DV, or Gronwall assumptions from being silently added to thm:forward-KL." def cycle30ForwardKlDerivativeSideMiddleObligation : ProofObligation where id := "sald.forward_kl.cycle30_derivative_side_middle" statement := "Maintain the cycle 30 middle source-to-Lean map for appendix.tex lines 168-208, with appendix.tex:168-185 as the first lower sub-slice: mass conservation, differentiation under the KL integral, the SALD Fokker-Planck equation, boundary/integration-by-parts conditions, and the identification of the first derivative term with -FI(rho_s||tilde_pi_s)." source := saldForwardKlDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle30ForwardKlUpperPacket", "SALD.cycle30ForwardKlMiddleContract", "sald.forward_kl.cycle30_derivative_side_upper", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDensityBoundaryObligation", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.cycle30_density_boundary_lower", "SALD.forwardKlFirstTermFisherSubstitutionScalar", "sald.forward_kl.kl_derivative", "eq:SALD", "eq:FP-eq", "KLContract", "FIContract", "FokkerPlanckContract", "TransportVelocityContract" ] note := "This is a middle workflow obligation, not a proof. It narrows lower work to the density/boundary backend before the target-side transport, LSI-to-KL/FI, inverse-schedule time change, DV witness, coefficient-chain audit, and Gronwall side conditions are attempted." def forwardKlDensityBoundaryObligation : ProofObligation where id := "sald.forward_kl.density_boundary_regular" statement := "Formalize the density regularity, mass conservation, differentiation-under-the-integral, and boundary/integration-by-parts conditions used in appendix lines 168-208 for the KL derivative computation." source := saldForwardKlDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle30ForwardKlUpperPacket", "SALD.cycle30ForwardKlMiddleContract", "sald.forward_kl.cycle30_derivative_side_upper", "sald.forward_kl.cycle30_derivative_side_middle", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlFirstTermFisherSubstitutionScalar", "KLContract", "FIContract", "eq:FP-eq", "TransportVelocityContract" ] note := "The paper uses these analytic side conditions implicitly; forwardKlDerivativeSideConditionContract and the cycle 30 middle packet record them before any attempt to prove the derivative inequality." def cycle30ForwardKlDensityBoundaryLowerObligation : ProofObligation where id := "sald.forward_kl.cycle30_density_boundary_lower" statement := "Record the cycle 30 lower scalar substitution for appendix.tex lines 168-185: once mass conservation and differentiation under the KL integral give dK/ds as the first SALD term plus the target term, and the SALD Fokker-Planck/integration-by-parts backend identifies the first term as -FI(rho_s||tilde_pi_s), SALD.forwardKlFirstTermFisherSubstitutionScalar rewrites the derivative display as -FI plus the target term. Smooth densities, positivity, domination, mass conservation, Fokker-Planck, boundary/no-flux, and the FI identification remain obligations." source := saldForwardKlDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle30ForwardKlUpperPacket", "SALD.cycle30ForwardKlMiddleContract", "sald.forward_kl.cycle30_derivative_side_upper", "sald.forward_kl.cycle30_derivative_side_middle", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlFirstTermFisherSubstitutionScalar", "sald.forward_kl.density_boundary_regular", "eq:FP-eq", "KLContract", "FIContract", "FokkerPlanckContract" ] note := "Lower cycle 30 closes only theorem-independent Real equality substitution after the analytic derivative and first-term identities have been supplied. It does not prove the KL derivative, SALD Fokker-Planck equation, integration by parts, Fisher-information identity, target-side transport, LSI, DV, time change, or thm:forward-KL." /-- Cycle-34 upper/lower scalar obligation for the derivative closure sprint. -/ def cycle34ForwardKlDerivativeScalarObligation : ProofObligation where id := "sald.forward_kl.cycle34_derivative_scalar" statement := "Use the compiled scalar lemmas for appendix.tex lines 168-228: SALD.forwardKlTargetTransportYoungBoundScalar closes the Real Young step from the supplied Cauchy target bound; SALD.forwardKlPostYoungDerivativeBoundScalar and SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar combine the KL derivative display, first-term Fisher identity, and target-side bound to get dK/ds <= -(1/2)*FI+(1/2)*||tilde v_s||^2; SALD.forwardKlLsiDerivativeBoundScalar then uses the supplied LSI half-Fisher comparison C_LSI*K <= (1/2)*FI to obtain dK/ds <= -C_LSI*K+(1/2)*||tilde v_s||^2; SALD.forwardKlTimeChangedDerivativeBoundScalar performs the real-order inverse-schedule handoff once the analytic chain rule, velocity-square scaling, and inverse-derivative identities are supplied. The analytic Fokker-Planck, integration-by-parts, target transport, Cauchy input, LSI density-test, and time-change backends remain obligations." source := saldForwardKlDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle34ForwardKlDerivativeUpperPacket", "SALD.cycle34ForwardKlDerivativeMiddleContract", "SALD.forwardKlTargetTransportYoungBoundScalar", "SALD.forwardKlPostYoungDerivativeBoundScalar", "SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar", "SALD.forwardKlLsiDerivativeBoundScalar", "SALD.forwardKlLsiDerivativeBoundOfKlFiScalar", "SALD.forwardKlTimeChangedDerivativeBoundScalar", "SALD.forwardKlFirstTermFisherSubstitutionScalar", "SALD.cycle30ForwardKlDensityBoundaryLowerObligation", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDensityBoundaryObligation", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiDensityTestObligation", "sald.forward_kl.density_boundary_regular", "probability.lsi_to_kl_fi", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative" ] note := "Cycle 34 closes only Real arithmetic/order handoffs inside the derivative block, including the target-side Young coefficient and the time-change coefficient after their analytic inputs are supplied. It does not close the KL derivative theorem, the Fokker-Planck equation, target transport/Cauchy backend, LSI-to-KL/FI, inverse-schedule calculus, DV, Gronwall, or thm:forward-KL." def cycle34ForwardKlTargetYoungLowerObligation : ProofObligation where id := "sald.forward_kl.cycle34_target_young_lower" statement := "Maintain the lower scalar target-side Young slice for appendix.tex lines 199-208: once target transport, integration by parts, Cauchy--Schwarz, and L2/FI identifications supply targetTerm <= sqrt(FI)*sqrt(||tilde v_s||^2) with nonnegative FI and velocity square, SALD.forwardKlTargetTransportYoungBoundScalar proves targetTerm <= (1/2)*FI+(1/2)*||tilde v_s||^2, and SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar feeds that bound into the derivative bookkeeping." source := saldForwardKlDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle34ForwardKlDerivativeUpperPacket", "SALD.cycle34ForwardKlDerivativeMiddleContract", "SALD.forwardKlTargetTransportYoungBoundScalar", "SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar", "SALD.forwardKlPostYoungDerivativeBoundScalar", "SALD.forwardKlFirstTermFisherSubstitutionScalar", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDensityBoundaryObligation", "sald.forward_kl.cycle34_derivative_scalar", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.kl_derivative" ] note := "Lower cycle 34 formalizes only the Real Young inequality and its composition with existing scalar derivative bookkeeping. The target-side transport identity, integration by parts, Cauchy--Schwarz estimate, Fisher/L2 square identifications, density regularity, LSI, schedule time change, DV, and Gronwall remain explicit obligations." def cycle34ForwardKlDerivativeMiddleObligation : ProofObligation where id := "sald.forward_kl.cycle34_derivative_middle" statement := "Maintain the cycle 34 middle source-to-Lean map for appendix.tex lines 218-228: after the s-time post-LSI derivative inequality is available, the analytic schedule backend must supply d/dt K(s(t))=dot{s}(t)*dK/ds, ||tilde v_s||^2=dot{t}(s)^2*||v_t||^2, dot{t}(s(t))=dot{s}(t)^(-1), dot{s}(t) nonnegative and nonzero; then SALD.forwardKlTimeChangedDerivativeBoundScalar gives the t-time pre-DV inequality with coefficient (1/2)*dot{s}(t)^(-1)." source := saldForwardKlDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle34ForwardKlDerivativeUpperPacket", "SALD.cycle34ForwardKlDerivativeMiddleContract", "SALD.forwardKlPostYoungDerivativeBoundScalar", "SALD.forwardKlLsiDerivativeBoundScalar", "SALD.forwardKlTimeChangedDerivativeBoundScalar", "SALD.cycle34ForwardKlTargetYoungLowerObligation", "SALD.forwardKlScheduleTimeChangeObligation", "SALD.forwardKlDerivativeSideConditionContract", "sald.forward_kl.cycle34_target_young_lower", "sald.forward_kl.cycle34_derivative_scalar", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative" ] note := "This middle obligation is a synchronization and lower-packet record. The new compiled lemma is scalar real/order algebra only; the chain rule, inverse-function calculus, target velocity scaling, density/boundary backend, LSI, DV, and Gronwall remain explicit obligations." def cycle39ForwardKlDerivativeUpperObligation : ProofObligation where id := "sald.forward_kl.cycle39_derivative_upper" statement := "Maintain the cycle 39 upper proof-closure packet for appendix.tex lines 168-228: prioritize the continuous forward-KL Fokker-Planck/KL derivative identity, use SALD.forwardKlPreDvDerivativeBoundScalar as the theorem-specific scalar composition from explicit analytic inputs, and then refine density/boundary and schedule-time-change interfaces before any new ledger expansion or EM interpolation work." source := saldForwardKlDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle39ForwardKlDerivativeUpperPacket", "SALD.cycle39ForwardKlDerivativeMiddleContract", "SALD.forwardKlPreDvDerivativeBoundScalar", "SALD.forwardKlPreDvDerivativeBoundOfProductScalar", "SALD.forwardKlPreDvDerivativeBoundOfVelocityScalingScalar", "SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar", "SALD.forwardKlInverseScheduleDerivativeScalar", "SALD.forwardKlTimeChangeSquareCoefficientRewriteOfProductScalar", "SALD.forwardKlVelocitySquareScalingScalar", "SALD.forwardKlTimeChangedDerivativeBoundOfProductScalar", "SALD.forwardKlFirstTermFisherSubstitutionScalar", "SALD.forwardKlTargetTransportYoungBoundScalar", "SALD.forwardKlPostYoungDerivativeBoundScalar", "SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar", "SALD.forwardKlLsiDerivativeBoundScalar", "SALD.forwardKlTimeChangedDerivativeBoundScalar", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "SALD.saldLsiKlFiDensityTestContract", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "probability.lsi_to_kl_fi", "sald.forward_kl.kl_derivative", "eq:FP-eq", "TransportVelocityContract" ] note := "Cycle 39 upper work records the current proof-closure priority and adds a compiled scalar pipeline. Middle synchronization adds source-shaped inverse-schedule and velocity-square scalar lemmas; lower synchronization adds the source-shaped KL/FI-to-half-Fisher scalar handoff. This does not prove mass conservation, differentiation under the integral, the SALD Fokker-Planck equation, integration by parts, target transport, LSI-to-KL/FI, inverse-schedule calculus, DV, Gronwall, EM interpolation, or thm:forward-KL." def cycle39ForwardKlDerivativeMiddleObligation : ProofObligation where id := "sald.forward_kl.cycle39_derivative_middle" statement := "Maintain the cycle 39 middle/lower source-to-Lean map for appendix.tex lines 191-228 and the surrounding pre-DV derivative pipeline: SALD.forwardKlInverseScheduleDerivativeScalar, SALD.forwardKlTimeChangeSquareCoefficientRewriteOfProductScalar, SALD.forwardKlVelocitySquareScalingScalar, SALD.forwardKlTimeChangedDerivativeBoundOfProductScalar, SALD.forwardKlPreDvDerivativeBoundOfProductScalar, SALD.forwardKlPreDvDerivativeBoundOfVelocityScalingScalar, SALD.forwardKlLsiDerivativeBoundOfKlFiScalar, and SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar close only the Real algebra after explicit analytic chain-rule, inverse-schedule, velocity-square, KL derivative, target Cauchy, and source KL/FI comparison premises are supplied." source := saldForwardKlDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle39ForwardKlDerivativeUpperPacket", "SALD.cycle39ForwardKlDerivativeMiddleContract", "SALD.cycle39ForwardKlDerivativeUpperObligation", "SALD.forwardKlPreDvDerivativeBoundScalar", "SALD.forwardKlInverseScheduleDerivativeScalar", "SALD.forwardKlTimeChangeSquareCoefficientRewriteOfProductScalar", "SALD.forwardKlVelocitySquareScalingScalar", "SALD.forwardKlTimeChangedDerivativeBoundOfProductScalar", "SALD.forwardKlPreDvDerivativeBoundOfProductScalar", "SALD.forwardKlPreDvDerivativeBoundOfVelocityScalingScalar", "SALD.forwardKlLsiDerivativeBoundOfKlFiScalar", "SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar", "SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar", "SALD.forwardKlLsiDerivativeBoundScalar", "SALD.forwardKlTimeChangedDerivativeBoundScalar", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "sald.forward_kl.cycle39_derivative_upper", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "probability.lsi_to_kl_fi", "sald.forward_kl.kl_derivative" ] note := "Cycle 39 middle translates the source time-change lines into source-shaped scalar lemmas: product inverse identity instead of a pre-rewritten derivative inverse, norm-square scaling instead of an opaque velocity-square equality, and source KL/FI comparison instead of a pre-multiplied LSI half-Fisher premise. It does not prove the analytic inverse-function theorem, L2 transport scaling, KL differentiation, Fokker-Planck identity, integration by parts, LSI-to-KL/FI, DV, Gronwall, EM interpolation, or thm:forward-KL." def forwardKlScheduleTimeChangeObligation : ProofObligation where id := "sald.forward_kl.schedule_time_change" statement := "Formalize the inverse-schedule calculus in appendix lines 191-228: tilde_v_s=dot{t}(s)*v_{t(s)}, d/dt K(s(t))=dot{s}(t)*dK/ds, dot{s}(t)*dot{t}(s(t))=1, dot{t}(s(t))=dot{s}(t)^(-1), and dot{s}(t)*dot{t}(s(t))^2=dot{s}(t)^(-1). Once these analytic inputs are supplied, SALD.forwardKlTimeChangedDerivativeBoundScalar and the cycle 39 source-shaped scalar lemmas perform the handoff to the t-time pre-DV inequality." source := saldForwardKlDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle30ForwardKlUpperPacket", "SALD.cycle30ForwardKlMiddleContract", "SALD.cycle34ForwardKlDerivativeMiddleContract", "SALD.cycle34ForwardKlDerivativeMiddleObligation", "sald.forward_kl.cycle30_derivative_side_upper", "sald.forward_kl.cycle30_derivative_side_middle", "sald.forward_kl.cycle34_derivative_middle", "sald.forward_kl.cycle39_derivative_middle", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlTimeChangedDerivativeBoundScalar", "SALD.forwardKlInverseScheduleDerivativeScalar", "SALD.forwardKlTimeChangeSquareCoefficientRewriteOfProductScalar", "SALD.forwardKlVelocitySquareScalingScalar", "SALD.forwardKlTimeChangedDerivativeBoundOfProductScalar", "SALD.forwardKlPreDvDerivativeBoundOfProductScalar", "SALD.forwardKlPreDvDerivativeBoundOfVelocityScalingScalar", "SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar", "TransportVelocityContract", "sald.forward_kl.density_boundary_regular" ] note := "This preserves the source time-change from the s-inequality to the t-inequality without adding an unstated theorem assumption. Cycle 34 formalizes the downstream real-order coefficient rewrite from a pre-rewritten inverse derivative, and cycle 39 adds source-shaped scalar algebra from dotS*dotT=1 and velocity norm-square scaling; the analytic schedule facts remain here." def forwardKlDerivativeObligation : ProofObligation where id := "sald.forward_kl.kl_derivative" statement := "Formalize appendix lines 166-225: differentiate K(t)=KL(rho_{s(t)}||pi_t), use the SALD Fokker-Planck equation, transport-velocity identity for pi_t, integration by parts, and LSI to derive dK/dt <= -dot{s}(t)*C_LSI(t)*K(t) + (1/2)*dot{s}(t)^(-1)*||v_t||_{L2(rho_{s(t)})}^2." source := saldForwardKlDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle30ForwardKlUpperPacket", "SALD.cycle30ForwardKlMiddleContract", "sald.forward_kl.cycle30_derivative_side_upper", "sald.forward_kl.cycle30_derivative_side_middle", "eq:SALD", "eq:FP-eq", "eq:LSI-KL-FI", "TransportVelocityContract", "FokkerPlanckContract", "SALD.forwardKlMassConservationDropScalar", "SALD.forwardKlMassConservationFirstTermFisherScalar", "SALD.forwardKlFirstTermFisherSubstitutionScalar", "SALD.cycle55ForwardKlDerivativeMassLowerObligation", "sald.forward_kl.cycle30_density_boundary_lower", "SALD.forwardKlTargetTransportYoungBoundScalar", "SALD.forwardKlPostYoungDerivativeBoundScalar", "SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar", "SALD.forwardKlLsiDerivativeBoundScalar", "SALD.forwardKlTimeChangedDerivativeBoundScalar", "SALD.forwardKlInverseScheduleDerivativeScalar", "SALD.forwardKlTimeChangeSquareCoefficientRewriteOfProductScalar", "SALD.forwardKlVelocitySquareScalingScalar", "SALD.forwardKlTimeChangedDerivativeBoundOfProductScalar", "SALD.forwardKlPreDvDerivativeBoundScalar", "SALD.forwardKlPreDvDerivativeBoundOfProductScalar", "SALD.forwardKlPreDvDerivativeBoundOfVelocityScalingScalar", "SALD.forwardKlLsiDerivativeBoundOfKlFiScalar", "SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar", "SALD.forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar", "SALD.forwardKlPointwisePreDvDerivativeBoundOfRawKlFiVelocityScaling", "SALD.cycle65ForwardKlDerivativePointwiseLowerObligation", "SALD.forwardKlDerivativeDvGronwallCoefficientOfKlFiVelocityScalingScalar", "SALD.cycle50ForwardKlDerivativeLowerObligation", "SALD.cycle39ForwardKlDerivativeMiddleObligation", "sald.forward_kl.cycle34_target_young_lower", "sald.forward_kl.cycle34_derivative_scalar", "sald.forward_kl.cycle34_derivative_middle", "sald.forward_kl.cycle39_derivative_middle", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change" ] note := "This is the main measure-theoretic KL derivative backend for thm:forward-KL; forwardKlDerivativeCandidateContract and forwardKlDerivativeSideConditionContract record the required schedule, density, boundary, and time-change interfaces before proof search. Cycle 30 selects the density/boundary slice first; cycle 34 closes only the scalar target-side Young, post-Young, LSI, and pre-rewritten time-change coefficient handoffs after their analytic inputs are supplied. Cycle 39 adds source-shaped scalar schedule and velocity-square handoffs, while the analytic KL derivative, Fokker-Planck, integration by parts, LSI, and schedule backends remain obligations." def forwardKlDvEnergyObligation : ProofObligation where id := "sald.forward_kl.dv_energy_bound" statement := "Formalize appendix lines 229-241: apply the Donsker--Varadhan variational formula with Z=alpha*||v_t||^2 to obtain ||v_t||_{L2(rho_{s(t)})}^2 <= alpha^{-1}*K(t)+E_alpha(pi_t,v_t)." source := saldForwardKlDvEnergySource status := ProofStatus.obligation dependsOn := [ "probability.dv_variational_formula", "alpha-complexity definition", "KL/FI density vocabulary", "sald.forward_kl.dv_alpha_mgf_monotonicity", "sald.forward_kl.dv_finite_log_mgf_witness" ] note := "The DV formula itself remains source-cited; forwardKlDvEnergyCandidateContract records the faithful theorem-specific instantiation and finite log-mgf sub-obligation." def forwardKlDvAlphaMonotonicityObligation : ProofObligation where id := "sald.forward_kl.dv_alpha_mgf_monotonicity" statement := "Formalize the alpha0-to-alpha exponential-moment monotonicity used before appendix lines 230-241: from finite E_{alpha0}(pi_t,v_t) and 0 conditional Fokker-Planck/KL derivative -> frozen score defect and LSI -> DV velocity bound -> time change -> Gronwall -> linear-slowdown accumulated-error collection.", "The source constants T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'} are not changed." ] nonGoals := [ "Do not restate thm:forward-KL-discrete or promote SALD.discreteSaldContract above contractOnly.", "Do not prove or replace the Gronwall, DV, LSI/KL/FI, continuous KL derivative, EM conditional-FP, frozen-defect, or stitched-interval analytic backends in this upper packet.", "Do not replace the marginal KL proof with path-space or Girsanov analysis.", "Do not start systematic measure-theory or SLT backfill before this theorem-level discrete route is stable." ] lowerPacket := [ "Middle should synchronize the conversion window and proof-obligation row for this cycle-46 discrete theorem wrapper.", "Lower should target exactly one named backend on the route, preferably SALD.discreteForwardKlEmInterpolationSideConditionContract / sald.discrete_forward_kl.em_conditional_fokker_planck or SALD.discreteForwardKlAccumulatedErrorBridgeContract / sald.discrete_forward_kl.accumulated_error_bridge.", "If the selected backend is too large, sharpen the source-cited or obligation interface with common-space, absolute-continuity, endpoint, density, finite-quantity, coefficient-regularity, and stitched-interval hypotheses instead of changing the theorem statement.", "Keep lem:frozen_delta_cross_lip_sald below formalized until the later general frozen-defect lemma is specialized with c=0 and sigma_eta(t)=sqrt(2)." ] reviewerChecklist := [ "SALD.discreteSaldContract lists SALD.cycle46DiscreteForwardKlSkeletonObligation while remaining contractOnly.", "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle46_theorem_skeleton_route before the lower EM, defect, DV, and Gronwall nodes.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes the cycle-46 packet, obligation, DAG, and route node.", "No analytic backend status is promoted, no theorem statement or source constant changes, and no alternate proof route is introduced.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-46 obligation tying the discrete forward-KL theorem skeleton to the five source-cited analytic interfaces. -/ def cycle46DiscreteForwardKlSkeletonObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle46_theorem_skeleton_route" statement := "Cycle 46 upper wires the source-cited analytic interfaces into thm:forward-KL-discrete: the EM endpoint and conditional-Fokker-Planck interface supplies the interpolated law and derivative setup; the discrete KL derivative block uses the frozen score-defect lemma and eq:LSI-KL-FI; lem:dv_variation with the EM finite-log-mgf witness supplies the velocity-energy bound; lem:gronwall with stitched-interval regularity gives the appendix general-schedule bound; the linear-slowdown bridge recovers exactly the main_body.tex:309-323 display with barGamma and barDelta. All slow analytic interfaces remain obligation or source-cited." source := saldForwardKlDiscreteProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle46DiscreteForwardKlSkeletonUpperPacket", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "sald.discrete_forward_kl.kl_derivative", "SALD.frozenDeltaCrossLipSaldContract", "sald.discrete_forward_kl.frozen_delta_cross_lip", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.discreteForwardKlGronwallInstantiationContract", "sald.discrete_forward_kl.gronwall_accumulation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.coefficient_chain_audit", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "thm:forward-KL" ] note := "This is a theorem-route wrapper for main skeleton sprint 3. It does not add hidden smoothness, endpoint, density, finite-log-mgf, interval-integrability, or coefficient assumptions to thm:forward-KL-discrete." /-- Cycle-46 proof-DAG pane for the discrete forward-KL theorem route. -/ def cycle46DiscreteForwardKlSkeletonDag : List ProofDagBlock := [ { id := "ASTIS.SALD.forward_KL_discrete.cycle46_theorem_skeleton_route" interface := "Discrete forward-KL theorem-level route: statement, EM endpoint/conditional-FP backend, KL derivative with frozen defect and LSI, DV velocity witness, Gronwall accumulation, and linear-slowdown accumulated-error bridge compose to the exact main_body.tex:309-323 display." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle46DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle46DiscreteForwardKlSkeletonObligation", "SALD.cycle46DiscreteForwardKlSkeletonMiddleContract", "SALD.cycle46DiscreteForwardKlSkeletonMiddleObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.discreteForwardKlDerivativeCandidateContract", "SALD.frozenDeltaCrossLipSaldContract", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.saldGronwallCandidateContract", "dvVariationalFormulaInterface saldDvVariationSource" ] reusedBy := ["thm:forward-KL-discrete", "main skeleton sprint 3"] status := ProofStatus.obligation } ] /-- Cycle-46 middle audit for the discrete forward-KL theorem skeleton. This is the middle-role source-to-Lean synchronization layer for the discrete route wrapper. It checks the exact statement and appendix proof order, keeps all slow analysis in named interfaces, and selects a lower-ready backend without changing the source theorem display. -/ def cycle46DiscreteForwardKlSkeletonMiddleContract : DiscreteForwardKlEmDefectAccumulationMiddleContract where sourceStatement := saldForwardKlDiscreteSource interpolationSource := saldForwardKlDiscreteInterpolationSource derivativeSource := saldForwardKlDiscreteDerivativeSource gronwallSource := saldForwardKlDiscreteGronwallSource accumulatedSource := saldForwardKlDiscreteAccumulatedErrorSource objective := "Middle source-to-Lean audit for main skeleton sprint 3: verify that cycle46DiscreteForwardKlSkeletonObligation consumes the EM endpoint/conditional-FP, frozen-defect plus LSI, DV velocity, Gronwall, and accumulated-error interfaces in the exact appendix.tex:260-592 order, while main_body.tex:299-323 remains the unchanged theorem statement." sourceStepMap := [ "main_body.tex:299-323 fixes the linear slowdown theorem: t(s)=s/r with r>=1, the Theorem thm:forward-KL assumptions, Lipschitz conditions, alpha0' finite score and 1+M complexities, 4*eta^2*L_space^2<1/2, alpha in (0,alpha0), alpha' in (0,alpha0'], and the displayed barGamma/barDelta bound.", "appendix.tex:260-330 introduces the frozen EM interpolation, frozen field delta_pi_t, and the SALD frozen-defect lemma obtained from the later general lemma with c=0 and sigma_eta(t)=sqrt(2).", "appendix.tex:334-385 fixes k and s in [s_k,s_{k+1}], identifies the endpoint laws, defines bar b_{k,s}, and uses the EM conditional Fokker-Planck equation plus the Laplacian split.", "appendix.tex:388-491 differentiates KL(hat rho_s||tilde pi_s), evaluates the first and target-velocity terms, inserts the frozen cross bound and the moving Young bound, then applies eq:LSI-KL-FI to get the pre-DV s-time inequality.", "appendix.tex:493-523 applies lem:dv_variation under the EM interpolation law with nu=hat rho_s, mu=tilde pi_s, and Z=alpha*||v_{t(s)}||^2, preserving the dot t(s)^2 and alpha^(-1) coefficients.", "appendix.tex:526-553 changes from s to t and obtains the source coefficient dot s(t)*C_LSI(t)-dot s(t)^(-1)*alpha^(-1)-2*dot s(t)*eta^2*alpha'^(-1)*Gamma(t).", "appendix.tex:557-592 applies lem:gronwall to the general-schedule bound; main_body.tex:309-323 then specializes t(s)=s/r and collects A_alpha(pi,v), barGamma, and barDelta_alpha'." ] leanStepMap := [ "Keep SALD.discreteForwardKlStatementContract and SALD.discreteSaldContract unchanged; the middle contract only audits the route wrapper.", "Use SALD.cycle44MainSkeletonAnalyticInterfaceLedger and SALD.cycle46DiscreteForwardKlSkeletonObligation as the parent theorem-skeleton interfaces.", "Route appendix.tex:260-385 through SALD.discreteForwardKlEmInterpolationSideConditionContract, sald.discrete_forward_kl.em_endpoint_laws, sald.discrete_forward_kl.conditional_drift_density, sald.discrete_forward_kl.em_conditional_fokker_planck, and sald.discrete_forward_kl.em_interpolation_fp.", "Route appendix.tex:268-330 and 454-491 through SALD.frozenDeltaCrossLipSaldContract, sald.discrete_forward_kl.frozen_delta_cross_lip, SALD.discreteForwardKlDerivativeCandidateContract, and sald.discrete_forward_kl.kl_derivative.", "Route the LSI step through SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi; density-test, zero-set, admissibility, entropy, and Fisher-chain backends remain obligations.", "Route appendix.tex:493-523 through SALD.discreteForwardKlDvFiniteLogMgfWitnessContract, sald.discrete_forward_kl.dv_finite_log_mgf_witness, and sald.discrete_forward_kl.dv_velocity_bound; DV remains source-cited.", "Route appendix.tex:526-592 through SALD.discreteForwardKlGronwallInstantiationContract, sald.discrete_forward_kl.gronwall_accumulation, SALD.discreteForwardKlAccumulatedErrorBridgeContract, sald.discrete_forward_kl.linear_slowdown_specialization, sald.discrete_forward_kl.residual_exponent_bound, and sald.discrete_forward_kl.accumulated_error_bridge.", "Select the next lower target as sald.discrete_forward_kl.accumulated_error_bridge, because the theorem-level route is wired and the remaining statement-display risk is the linear-slowdown collection of A_alpha, barGamma, and barDelta." ] citedResultInterfaces := [ "lem:dv_variation remains source-cited through the common-space and finite-log-mgf interface; no entropy-duality theorem is imported or marked formalized.", "lem:gronwall remains the endpoint-safe real-analysis obligation with stitched-interval regularity, coefficient regularity, endpoint rewrites, and exponent side conditions.", "eq:LSI-KL-FI remains the density-test and Fisher-chain obligation; compiled scalar and RN-density helpers are dependencies only.", "The continuous forward-KL derivative/Fokker-Planck route remains a reusable obligation for the target transport term; it is not promoted by this discrete middle audit.", "The EM interpolation endpoint and conditional-law Fokker-Planck backend remains the discrete slow backend; endpoint vector/law algebra is only a local handoff." ] obligations := [ "sald.discrete_forward_kl.cycle46_theorem_skeleton_route", "sald.discrete_forward_kl.cycle46_middle_route_audit", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.frozen_delta_cross_lip", "sald.discrete_forward_kl.kl_derivative", "probability.lsi_to_kl_fi", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.coefficient_chain_audit" ] lowerPacket := [ "Target exactly SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.", "First lower sub-slice: expose the endpointBridge and initialExponentSplit for K(0)=KL(rho_0||pi_0), K(T)=KL(rho_K^eta||pi_T), and the two source exponent factors after t(s)=s/r.", "Cycle 46 lower compiled the initial-exponent scalar/integral split as SALD.discreteForwardKlGronwallCoeffIntervalIntegrable, SALD.discreteForwardKlGronwallCoeffIntegralSubSub, SALD.discreteForwardKlGronwallInitialExponentSplitScalar, and SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces.", "Second lower sub-slice: connect the already compiled collection algebra SALD.discreteForwardKlAlphaComplexityCollectionScalar, SALD.discreteForwardKlDeltaAccumulationScalar, and SALD.discreteForwardKlAccumulatedErrorCollectionScalar to the theorem display, without changing constants.", "Leave residual-exponent monotonicity, barGamma identification, stitched-interval Gronwall, EM conditional-FP, frozen-defect specialization, DV finite-log-mgf, and LSI/KL/FI as obligations unless exact compiled proofs are added.", "Do not work on the general VA-SALD discrete theorem in this lower packet; this cycle stabilizes thm:forward-KL-discrete first." ] reviewerChecklist := [ "SALD.discreteSaldContract lists SALD.cycle46DiscreteForwardKlSkeletonMiddleObligation while remaining contractOnly.", "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle46_middle_route_audit after the theorem skeleton route and before lower EM, defect, DV, and Gronwall nodes.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle46DiscreteForwardKlSkeletonMiddleContract, SALD.cycle46DiscreteForwardKlSkeletonMiddleObligation, and sald.discrete_forward_kl.cycle46_middle_route_audit.", "No analytic backend status is promoted, no theorem statement or source constant changes, and no alternate proof route is introduced.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-46 middle obligation tying the discrete route audit to lower work. -/ def cycle46DiscreteForwardKlSkeletonMiddleObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle46_middle_route_audit" statement := "Cycle 46 middle audits the discrete thm:forward-KL-discrete theorem skeleton after the upper wrapper: main_body.tex:299-323 is kept unchanged, appendix.tex:260-592 is routed through the named EM endpoint/conditional-FP, frozen-defect plus LSI, DV velocity, Gronwall, and accumulated-error interfaces in paper order, and the next lower target is the theorem-display accumulated-error backend sald.discrete_forward_kl.accumulated_error_bridge. All slow analytic interfaces remain obligation or source-cited." source := saldForwardKlDiscreteProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle46DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle46DiscreteForwardKlSkeletonObligation", "SALD.cycle46DiscreteForwardKlSkeletonMiddleContract", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "SALD.frozenDeltaCrossLipSaldContract", "sald.discrete_forward_kl.frozen_delta_cross_lip", "sald.discrete_forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.discreteForwardKlGronwallInstantiationContract", "sald.discrete_forward_kl.gronwall_accumulation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.coefficient_chain_audit", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI" ] note := "This is a middle workflow obligation and source-to-Lean route audit, not an analytic proof. It does not add hidden endpoint, density, absolute-continuity, finite-log-mgf, stitched-interval, or coefficient-regularity assumptions to thm:forward-KL-discrete." /-- Cycle-47 upper packet for the guided residual and continuous general moving-target theorem skeleton. This keeps the cycle-44 slow analytic backend ledger in force and wires the source window `appendix.tex:619-951` into the theorem-level route for `prop:guided_path_residual` and `thm:general-moving-target-SALD`. It is workflow data only: the proposition and theorem remain contract-only, while the guided-density, Fokker--Planck/KL derivative, LSI, DV, and Gronwall backends remain obligations or source-cited interfaces. -/ def cycle47GuidedGeneralSkeletonUpperPacket : GeneralVaSaldUpperPacket where objective := "Main skeleton sprint 4: wire prop:guided_path_residual and thm:general-moving-target-SALD to the already named analytic interfaces, matching appendix.tex:619-951 without changing the paper proposition, theorem statement, constants, or source labels." sourceLabels := [ "prop:guided_path_residual", "proof:prop:guided_path_residual", "thm:general-moving-target-SALD", "eq:general_moving_target_SALD", "eq:general_moving_target_FP", "proof:thm:general-moving-target-SALD:derivative", "proof:thm:general-moving-target-SALD:residual-dv", "proof:thm:general-moving-target-SALD:dv-gronwall", "proof:thm:general-moving-target-SALD:pure-contraction", "proof:thm:unified-forward-KL", "eq:LSI-KL-FI", "lem:dv_variation", "lem:gronwall", "def:alpha-complexity" ] modeDiscipline := [ "faithfulPaper Phase 1: use only appendix.tex:619-951 and the original main_body.tex guided/unified references; sald_version_2.tex remains out of scope.", "Before lower work, the five slow backends are checked as explicit interfaces: Gronwall endpoint calculus, DV common-space/finite-log-mgf, LSI/KL/FI density-test, continuous Fokker-Planck/KL derivative, and EM interpolation Fokker-Planck for downstream discrete reuse.", "The source route is guided normalizer derivative -> centered residual identity -> general moving-target KL derivative -> LSI -> residual DV -> Gronwall -> theorem display -> pure contraction; the unified theorem remains only the one-line c_t<-u_t specialization.", "All missing density, boundary, positivity, finite-KL/FI, finite-log-mgf, endpoint, coefficient-regularity, and divergence-linearity facts remain named obligations rather than hidden theorem assumptions." ] nonGoals := [ "Do not restate prop:guided_path_residual or thm:general-moving-target-SALD, and do not promote SALD.guidedResidualContract or SALD.generalVaSaldContract above contractOnly.", "Do not prove or replace the Gronwall, DV, LSI/KL/FI, Fokker-Planck/KL derivative, guided normalizer, or divergence/integration-by-parts analytic backends in this upper packet.", "Do not replace the paper route with a direct VA-SALD proof, path-space comparison, Girsanov, Pinsker, Talagrand, or PI-based argument.", "Do not start systematic SLT or measure-theory backfill before the guided/general theorem-level route is stable." ] lowerPacket := [ "Middle should synchronize the conversion window and proof-obligation row for this cycle-47 guided/general theorem wrapper.", "Lower should target exactly SALD.generalMovingTargetDerivativeCandidateContract / SALD.generalMovingTargetDerivativeObligation / sald.general_moving_target.kl_derivative after the route wrapper is accepted.", "The lower sub-slice should preserve appendix.tex:765-884: KL differentiation, the general VA-SALD Fokker-Planck equation, target transport by v_t, the residual m_t=v_t-c_t, Young with epsilon=2*dot{t}(s)/sigma_{t(s)}^2, LSI, and time change.", "If that backend is too large, sharpen the source-cited or obligation interface with density/law regularity, integration-by-parts, transport, finite KL/FI, sigma and schedule positivity, and inverse-schedule hypotheses instead of changing the theorem statement." ] reviewerChecklist := [ "SALD.guidedResidualContract and SALD.generalVaSaldContract list the cycle-47 theorem-route obligations while remaining contractOnly.", "SALD.generalVaSaldProofDag contains ASTIS.SALD.guided_general.cycle47_theorem_skeleton_route before the lower guided residual and general moving-target nodes.", "SALD.saldDependenciesForLabel entries for prop:guided_path_residual and thm:general-moving-target-SALD include the cycle-47 packet, obligations, and DAG nodes.", "No analytic backend status is promoted, no source theorem statement or source constant changes, and no alternate proof route is introduced.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-47 obligation tying the guided residual and continuous general moving-target theorem skeleton to the already named analytic interfaces. -/ def cycle47GuidedGeneralSkeletonObligation : ProofObligation where id := "sald.guided_general.cycle47_theorem_skeleton_route" statement := "Cycle 47 upper wires appendix.tex:619-951 into the faithful guided/general theorem skeleton: prop:guided_path_residual uses the guided normalizer derivative and centered residual identity obligations; thm:general-moving-target-SALD uses the general Fokker-Planck/KL derivative interface, eq:LSI-KL-FI, lem:dv_variation with the residual m_t finite-log-mgf witness, lem:gronwall with sigma-weighted endpoint/exponent side conditions, and the pure-contraction residual-zero obligation. All slow analytic interfaces remain obligation or source-cited." source := saldGeneralMovingTargetSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle47GuidedGeneralSkeletonUpperPacket", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "sald.general_moving_target.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "SALD.generalMovingTargetGronwallInstantiationContract", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "This is a theorem-route wrapper for main skeleton sprint 4. It does not add hidden smoothness, density, endpoint, absolute-continuity, finite-log-mgf, sigma-positivity, or interval-integrability assumptions to prop:guided_path_residual or thm:general-moving-target-SALD." /-- Cycle-47 middle audit for the guided residual and continuous general moving-target theorem skeleton. This is the source-to-Lean synchronization layer for the upper route wrapper. It checks the exact appendix proof order, keeps all slow analysis in named interfaces, and selects a lower-ready backend without changing the source proposition or theorem display. -/ def cycle47GuidedGeneralSkeletonMiddleContract : GeneralVaSaldGuidedPathMiddleContract where guidedResidualSource := saldGuidedResidualSource generalContinuousSource := saldGeneralMovingTargetSource unifiedSource := saldUnifiedForwardKlSource discreteSource := saldGeneralMovingTargetDiscreteSource objective := "Middle source-to-Lean audit for main skeleton sprint 4: verify that cycle47GuidedGeneralSkeletonObligation consumes the guided residual identity, continuous general KL derivative, LSI/KL/FI, residual DV, Gronwall, and pure-contraction interfaces in the exact appendix.tex:619-951 order, while prop:guided_path_residual and thm:general-moving-target-SALD remain unchanged." sourceStepMap := [ "appendix.tex:619-704 proves prop:guided_path_residual: compute dot Z_t, use partial_t p_t=-div(p_t*u_t), integrate by parts, differentiate pi_t=Z_t^(-1)*p_t*exp(-f_t), cancel divergence terms, substitute dot Z_t/Z_t, and conclude the centered mean-zero residual.", "appendix.tex:724-744 states thm:general-moving-target-SALD with dynamics dX_s=(dot t(s)*c_{t(s)}+(sigma_{t(s)}^2/2)*nabla log pi_{t(s)})ds+sigma_{t(s)}dW_s, residual m_t=v_t-c_t, finite E_alpha0(pi_t,m_t), alpha in (0,alpha0], and the sigma-weighted KL bound.", "appendix.tex:765-835 differentiates KL(rho_s||pi_{t(s)}), inserts the general moving-target Fokker-Planck equation, and evaluates the two integration-by-parts terms.", "appendix.tex:835-864 combines c_t and v_t into the residual m_t=v_t-c_t and applies Young with epsilon=2*dot{t}(s)/sigma_{t(s)}^2 to get the pre-LSI inequality.", "appendix.tex:865-884 changes from s to t and applies eq:LSI-KL-FI to get the pre-DV K'(t) inequality with coefficient sigma_t^(-2)*dot{s}(t)^(-1) on the residual energy.", "appendix.tex:885-907 applies lem:dv_variation with Z=alpha*||m_t||^2, rewrites the log-mgf as E_alpha(pi_t,m_t), and forms the scalar differential inequality.", "appendix.tex:909-934 applies lem:gronwall with a(t)=(sigma_t^2/2)*dot{s}(t)*C_LSI(t)-sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1) and b(t)=sigma_t^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t), then matches the theorem display.", "appendix.tex:936-945 specializes c_t=v_t so m_t=0 and E_alpha(pi_t,m_t)=0, yielding the pure contraction display; appendix.tex:949-951 is downstream reuse by thm:unified-forward-KL." ] leanStepMap := [ "Route appendix.tex:619-704 through SALD.guidedResidualIdentityContract, sald.guided_path_residual.normalizer_derivative, and sald.guided_path_residual.identity.", "Keep SALD.generalMovingTargetStatementContract and SALD.generalVaSaldContract unchanged; the middle contract only audits the route wrapper.", "Route appendix.tex:765-884 through SALD.generalMovingTargetDerivativeCandidateContract, sald.general_moving_target.kl_derivative, SALD.saldLsiKlFiDensityTestContract, probability.lsi_to_kl_fi, and sald.forward_kl.schedule_time_change.", "Route appendix.tex:885-907 through SALD.generalMovingTargetDvFiniteLogMgfWitnessContract, SALD.generalMovingTargetDvPositiveAlphaScalingContract, sald.general_moving_target.dv_finite_log_mgf_witness, sald.general_moving_target.dv_positive_alpha_scaling, and sald.general_moving_target.dv_m_energy.", "Route appendix.tex:909-945 through SALD.generalMovingTargetGronwallInstantiationContract, SALD.generalMovingTargetGronwallSideConditionContract, sald.general_moving_target.gronwall_application, sald.general_moving_target.gronwall_side_conditions, and sald.general_moving_target.pure_contraction.", "Keep SALD.unifiedForwardKlSpecializationContract visible only as downstream reuse of appendix.tex:949-951; this cycle does not introduce a direct VA-SALD proof.", "Select the next lower target as sald.general_moving_target.kl_derivative, because it is the theorem-level backend that feeds LSI, residual DV, and Gronwall after the skeleton route is wired." ] citedResultInterfaces := [ "lem:dv_variation remains source-cited through the residual finite-log-mgf witness; no entropy-duality theorem is imported or marked formalized.", "lem:gronwall remains the endpoint-safe real-analysis obligation with coefficient regularity, endpoint rewrites, exponent splitting, and residual-exponent side conditions.", "eq:LSI-KL-FI remains the density-test and Fisher-chain obligation; compiled scalar and RN-density helpers are dependencies only.", "The continuous general Fokker-Planck/KL derivative remains a local SDE/measure-analysis obligation; it is not promoted by this middle audit.", "The EM interpolation Fokker-Planck backend remains checked as a downstream slow interface for the later discrete general theorem, not as a continuous theorem assumption." ] obligations := [ "sald.guided_general.cycle47_theorem_skeleton_route", "sald.guided_general.cycle47_middle_route_audit", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "sald.general_moving_target.kl_derivative", "probability.lsi_to_kl_fi", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "sald.unified_forward_kl.specialization", "sald.general_moving_target_discrete.em_interpolation_fp" ] lowerPacket := [ "Target exactly SALD.generalMovingTargetDerivativeCandidateContract / SALD.generalMovingTargetDerivativeObligation / sald.general_moving_target.kl_derivative.", "First lower sub-slice: expose the density/law, Fokker-Planck, mass conservation, and integration-by-parts interfaces for appendix.tex:765-812 without changing the theorem statement.", "Second lower sub-slice: identify the target transport velocity v_t, combine the c_t and v_t terms into m_t=v_t-c_t, and preserve the Young coefficient epsilon=2*dot{t}(s)/sigma_{t(s)}^2 from appendix.tex:835-864.", "Leave LSI density-test, residual DV finite-log-mgf, Gronwall endpoint/exponent side conditions, guided normalizer differentiation, and pure-contraction alpha-complexity as separate obligations unless exact compiled proofs are added.", "Do not work on the discrete general theorem in this lower packet; it remains downstream reuse through the existing EM interpolation and discrete side-condition interfaces." ] reviewerChecklist := [ "Every source step from appendix.tex:619-951 is classified as Lean contract, cited result, or named obligation.", "SALD.guidedResidualContract and SALD.generalVaSaldContract list SALD.cycle47GuidedGeneralSkeletonMiddleObligation while preserving contractOnly status.", "SALD.generalVaSaldProofDag contains ASTIS.SALD.guided_general.cycle47_middle_route_audit after the cycle-47 route wrapper and before lower backend nodes.", "SALD.saldDependenciesForLabel \"prop:guided_path_residual\" and \"thm:general-moving-target-SALD\" include SALD.cycle47GuidedGeneralSkeletonMiddleContract, SALD.cycle47GuidedGeneralSkeletonMiddleObligation, and sald.guided_general.cycle47_middle_route_audit.", "No analytic backend status is promoted, no theorem statement or source constant changes, and no alternate proof route is introduced." ] status := ProofStatus.obligation /-- Cycle-47 middle obligation tying the guided/general route audit to lower work. -/ def cycle47GuidedGeneralSkeletonMiddleObligation : ProofObligation where id := "sald.guided_general.cycle47_middle_route_audit" statement := "Cycle 47 middle audits the guided residual and continuous thm:general-moving-target-SALD theorem skeleton after the upper wrapper: appendix.tex:619-951 is kept in paper order, prop:guided_path_residual and thm:general-moving-target-SALD are unchanged, the route consumes the named guided residual, general KL derivative, LSI/KL/FI, residual DV, Gronwall, and pure-contraction interfaces, and the next lower target is sald.general_moving_target.kl_derivative. All slow analytic interfaces remain obligation or source-cited." source := saldGeneralMovingTargetSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle47GuidedGeneralSkeletonUpperPacket", "SALD.cycle47GuidedGeneralSkeletonObligation", "SALD.cycle47GuidedGeneralSkeletonMiddleContract", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "sald.general_moving_target.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "SALD.unifiedForwardKlSpecializationContract", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI" ] note := "This is a middle workflow obligation and source-to-Lean route audit, not an analytic proof. It does not add hidden endpoint, density, absolute-continuity, finite-log-mgf, sigma-positivity, or coefficient-regularity assumptions to the guided residual proposition or continuous general theorem." /-- Cycle-47 proof-DAG pane for the guided/general theorem route. -/ def cycle47GuidedGeneralSkeletonDag : List ProofDagBlock := [ { id := "ASTIS.SALD.guided_general.cycle47_theorem_skeleton_route" interface := "Guided/general theorem-level route: guided residual normalizer and identity feed the continuous general moving-target statement, whose derivative, LSI, residual DV, Gronwall, and pure-contraction interfaces compose to the exact appendix.tex:724-945 theorem display." source := saldGeneralMovingTargetSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle47GuidedGeneralSkeletonUpperPacket", "SALD.cycle47GuidedGeneralSkeletonObligation", "SALD.cycle47GuidedGeneralSkeletonMiddleContract", "SALD.cycle47GuidedGeneralSkeletonMiddleObligation", "SALD.guidedResidualIdentityContract", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvEnergyCandidateContract", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.saldGronwallCandidateContract", "dvVariationalFormulaInterface saldDvVariationSource" ] reusedBy := ["prop:guided_path_residual", "thm:general-moving-target-SALD", "main skeleton sprint 4"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.guided_general.cycle47_middle_route_audit" interface := "Middle source-to-Lean audit for appendix.tex:619-951: verify the exact guided residual and continuous general theorem route, then select the general KL derivative backend for lower work." source := saldGeneralMovingTargetSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle47GuidedGeneralSkeletonMiddleContract", "SALD.cycle47GuidedGeneralSkeletonMiddleObligation", "SALD.cycle47GuidedGeneralSkeletonObligation", "SALD.guidedResidualIdentityContract", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.kl_derivative" ] reusedBy := ["prop:guided_path_residual", "thm:general-moving-target-SALD", "cycle 47 lower derivative packet"] status := ProofStatus.obligation } ] /-- Cycle-52 upper packet for the guided residual and continuous general moving-target theorem after the forward and discrete forward-KL routes. This is a route-closure check for the current skeleton sprint. It records the five slow analytic backends explicitly, then wires `prop:guided_path_residual` and `thm:general-moving-target-SALD` through the already named interfaces from `appendix.tex:619-951`. It is workflow data only: no theorem statement, source constant, source label, or analytic status is changed. -/ def cycle52GuidedGeneralSkeletonUpperPacket : GeneralVaSaldUpperPacket where objective := "Cycle 52 upper: after the cycle-50 continuous forward-KL route and cycle-51 discrete forward-KL route, close the guided residual and continuous general moving-target theorem skeleton by explicitly consuming the five source-cited analytic interfaces and the already named guided/general obligations over appendix.tex:619-951." sourceLabels := [ "prop:guided_path_residual", "proof:prop:guided_path_residual", "thm:general-moving-target-SALD", "eq:general_moving_target_SALD", "eq:general_moving_target_FP", "proof:thm:general-moving-target-SALD:derivative", "proof:thm:general-moving-target-SALD:residual-dv", "proof:thm:general-moving-target-SALD:dv-gronwall", "proof:thm:general-moving-target-SALD:pure-contraction", "proof:thm:unified-forward-KL", "eq:LSI-KL-FI", "lem:dv_variation", "lem:gronwall", "def:alpha-complexity" ] modeDiscipline := [ "faithfulPaper Phase 1: preserve appendix.tex:619-951 exactly as a theorem-level route; sald_version_2.tex remains out of scope.", "Five-backend check 1, Gronwall: SALD.saldGronwallCandidateContract, SALD.saldGronwallEndpointCalculusContract, SALD.saldGronwallExponentRewriteContract, and SALD.gronwallAnalyticObligation expose endpoint-safe differentiability, FTC, coefficient regularity, and exponent rewrite obligations.", "Five-backend check 2, DV: dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, SALD.generalMovingTargetDvFiniteLogMgfWitnessContract, and SALD.generalMovingTargetDvPositiveAlphaScalingContract expose common-space, absolute-continuity, finite-KL, finite-log-mgf, measurability, and positive-alpha scaling.", "Five-backend check 3, LSI/KL/FI: SALD.saldLsiKlFiDensityTestContract and SALD.lsiKlFiDensityTestObligation expose density, zero-set convention, admissible sqrt-density test, entropy identity, and Fisher chain-rule obligations.", "Five-backend check 4, continuous derivative: SALD.forwardKlDerivativeCandidateContract and SALD.generalMovingTargetDerivativeCandidateContract keep the Fokker-Planck/KL derivative identities source-cited or obligation-level, with density, boundary, transport, finite KL/FI, sigma, and schedule side conditions explicit.", "Five-backend check 5, EM interpolation: SALD.discreteForwardKlEmInterpolationSideConditionContract, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, and SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation keep endpoint laws, conditional drift, conditional-law density, weak Fokker-Planck, and stitched interval interfaces explicit for downstream discrete reuse." ] nonGoals := [ "Do not restate prop:guided_path_residual or thm:general-moving-target-SALD, and do not promote SALD.guidedResidualContract or SALD.generalVaSaldContract above contractOnly.", "Do not prove or replace the Gronwall, DV, LSI/KL/FI, Fokker-Planck/KL derivative, EM interpolation, guided normalizer, divergence-linearity, or integration-by-parts backends in this upper packet.", "Do not introduce a direct VA-SALD proof, path-space comparison, Girsanov, Pinsker, Talagrand, PI, or SLT-based proof route.", "Do not start systematic measure-theory or SDE backfill until the theorem-level guided/general route remains stable under review." ] lowerPacket := [ "Middle should synchronize the conversion window and proof-obligation row for SALD.cycle52GuidedGeneralSkeletonUpperPacket / SALD.cycle52GuidedGeneralSkeletonObligation.", "Preferred lower target: SALD.generalMovingTargetDerivativeCandidateContract / SALD.generalMovingTargetDerivativeObligation / sald.general_moving_target.kl_derivative over appendix.tex:765-884.", "First lower sub-slice: expose the density/law, general Fokker-Planck, mass conservation, target transport velocity, and integration-by-parts interfaces needed before Young, LSI, DV, and Gronwall.", "Alternative lower target: SALD.guidedResidualIdentityContract / sald.guided_path_residual.identity, only for the normalizer derivative, product/quotient differentiation, divergence cancellation, and mean-zero residual from appendix.tex:630-704.", "If a backend is too large, sharpen the named source-cited or obligation interface with the missing regularity, common-space, absolute-continuity, finite-quantity, endpoint, or coefficient hypotheses instead of changing a theorem statement." ] reviewerChecklist := [ "SALD.guidedResidualContract and SALD.generalVaSaldContract list SALD.cycle52GuidedGeneralSkeletonObligation while remaining contractOnly.", "SALD.generalVaSaldProofDag contains ASTIS.SALD.guided_general.cycle52_upper_route after the cycle-47 route and before the downstream unified/discrete reuse nodes.", "SALD.saldDependenciesForLabel entries for prop:guided_path_residual and thm:general-moving-target-SALD include the cycle-52 packet, obligation, and DAG route node.", "The five slow analytic interfaces remain source-cited or obligation-level; no theorem statement, source constant, source label, or external reuse status changes.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-52 obligation tying the guided residual and continuous general moving-target theorem route to the five explicit analytic backends. -/ def cycle52GuidedGeneralSkeletonObligation : ProofObligation where id := "sald.guided_general.cycle52_upper_route" statement := "Cycle 52 upper verifies that the five slow analytic backends are explicit source-cited or obligation interfaces, then routes appendix.tex:619-951 through the named guided residual and continuous general moving-target interfaces: guided normalizer derivative and centered residual identity; general Fokker-Planck/KL derivative with residual m_t=v_t-c_t; eq:LSI-KL-FI; lem:dv_variation with the residual finite-log-mgf and positive-alpha scaling witnesses; lem:gronwall with sigma-weighted endpoint/exponent side conditions; and the pure-contraction residual-zero obligation. The proposition and theorem remain contractOnly." source := saldGeneralMovingTargetSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle49MainSkeletonAnalyticMiddleObligation", "SALD.cycle50ForwardKlSkeletonObligation", "SALD.cycle51DiscreteForwardKlSkeletonObligation", "SALD.cycle47GuidedGeneralSkeletonObligation", "SALD.cycle47GuidedGeneralSkeletonMiddleObligation", "SALD.cycle52GuidedGeneralSkeletonUpperPacket", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "sald.general_moving_target.kl_derivative", "SALD.saldGronwallCandidateContract", "SALD.saldGronwallEndpointCalculusContract", "SALD.saldGronwallExponentRewriteContract", "SALD.gronwallAnalyticObligation", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiDensityTestObligation", "probability.lsi_to_kl_fi", "SALD.forwardKlDerivativeCandidateContract", "sald.forward_kl.kl_derivative", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "sald.general_moving_target_discrete.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "This is an upper workflow obligation and route check. It does not add hidden assumptions, solve a backend, or mark Gronwall, DV, LSI/KL/FI, Fokker-Planck/KL differentiation, or EM interpolation as formalized." /-- Cycle-52 middle audit for the guided residual and continuous general moving-target theorem route. This synchronizes the cycle-52 upper route with the conversion window and proof-obligation ledger. It keeps the theorem-level target on `appendix.tex:619-951`, selects the continuous general KL derivative as the preferred lower backend, and preserves every slow analytic dependency as an explicit source-cited interface or obligation. -/ def cycle52GuidedGeneralSkeletonMiddleContract : GeneralVaSaldGuidedPathMiddleContract where guidedResidualSource := saldGuidedResidualSource generalContinuousSource := saldGeneralMovingTargetSource unifiedSource := saldUnifiedForwardKlSource discreteSource := saldGeneralMovingTargetDiscreteSource objective := "Cycle 52 middle: verify the post-upper guided/general route against appendix.tex:619-951, keep prop:guided_path_residual and thm:general-moving-target-SALD unchanged, and select sald.general_moving_target.kl_derivative over appendix.tex:765-884 as the lower-ready backend while guided residual, LSI, residual DV, Gronwall, pure contraction, and EM reuse stay separate obligations." sourceStepMap := [ "appendix.tex:619-704 proves prop:guided_path_residual by differentiating Z_t, using partial_t p_t=-div(p_t*u_t), integrating by parts, differentiating pi_t=Z_t^(-1)*p_t*exp(-f_t), canceling divergence terms, and concluding the centered mean-zero residual.", "appendix.tex:724-744 states thm:general-moving-target-SALD with dynamics driven by c_t, diffusion sigma_t, residual m_t=v_t-c_t, finite residual alpha0-complexity, alpha in (0,alpha0], and the exact sigma-weighted KL bound.", "appendix.tex:765-835 differentiates KL(rho_s||pi_{t(s)}), uses the general moving-target Fokker-Planck equation, and evaluates the rho_s and target-time terms by integration by parts.", "appendix.tex:835-884 combines the c_t and v_t terms into m_t, applies Young with epsilon=2*dot t(s)/sigma_{t(s)}^2, changes from s to t, and uses eq:LSI-KL-FI to reach the pre-DV residual-energy inequality.", "appendix.tex:885-907 applies lem:dv_variation with Z=alpha*||m_t||^2 and rewrites the log-mgf term as mathfrak E_alpha(pi_t,m_t).", "appendix.tex:908-945 applies lem:gronwall with the source sigma-weighted coefficients, matches the theorem display, and specializes c_t=v_t so m_t=0 for pure contraction.", "appendix.tex:949-951 is downstream reuse by thm:unified-forward-KL through the specialization c_t<-u_t; this middle packet does not introduce a direct VA-SALD proof." ] leanStepMap := [ "Use SALD.cycle52GuidedGeneralSkeletonUpperPacket and SALD.cycle52GuidedGeneralSkeletonObligation as the parent route check after the cycle-50 and cycle-51 theorem-route wrappers.", "Route appendix.tex:619-704 through SALD.guidedResidualIdentityContract, sald.guided_path_residual.normalizer_derivative, and sald.guided_path_residual.identity; keep normalizer positivity, differentiation under the integral, integration by parts, and mean-zero transport as obligations.", "Keep SALD.generalMovingTargetStatementContract and SALD.generalVaSaldContract contractOnly; this middle audit adds no theorem hypotheses.", "Route appendix.tex:765-884 through SALD.generalMovingTargetDerivativeCandidateContract, SALD.generalMovingTargetDerivativeObligation, sald.general_moving_target.kl_derivative, SALD.generalMovingTargetPostYoungDerivativeBoundScalar, SALD.generalMovingTargetLsiDerivativeBoundScalar, SALD.generalMovingTargetTimeChangedDerivativeBoundScalar, and SALD.generalMovingTargetPreDvDerivativeBoundScalar.", "Route appendix.tex:885-907 through SALD.generalMovingTargetDvFiniteLogMgfWitnessContract, SALD.generalMovingTargetDvPositiveAlphaScalingContract, sald.general_moving_target.dv_finite_log_mgf_witness, sald.general_moving_target.dv_positive_alpha_scaling, and sald.general_moving_target.dv_m_energy.", "Route appendix.tex:908-945 through SALD.generalMovingTargetGronwallInstantiationContract, SALD.generalMovingTargetGronwallSideConditionContract, sald.general_moving_target.gronwall_application, sald.general_moving_target.gronwall_side_conditions, and sald.general_moving_target.pure_contraction.", "Keep SALD.discreteForwardKlEmInterpolationSideConditionContract and SALD.generalMovingTargetDiscreteDerivativeSideConditionContract visible only as downstream EM/Fokker-Planck interfaces for later discrete general reuse." ] citedResultInterfaces := [ "lem:gronwall remains the endpoint-safe differentiability/FTC and coefficient-regularity obligation; cycle-52 only consumes it through the sigma-weighted general Gronwall interface.", "lem:dv_variation remains source-cited; the general theorem must still supply common-space, absolute-continuity, finite-KL/log-likelihood, measurability, finite-log-mgf, and positive-alpha witnesses for Z=alpha*||m_t||^2.", "eq:LSI-KL-FI remains the density, zero-set, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule obligation.", "The continuous general Fokker-Planck/KL derivative remains sald.general_moving_target.kl_derivative; compiled Real scalar helpers are handoffs only after analytic inputs are supplied.", "The Euler-Maruyama interpolation Fokker-Planck backend remains a downstream discrete obligation and is not promoted or imported into the continuous theorem statement." ] obligations := [ "sald.guided_general.cycle52_upper_route", "sald.guided_general.cycle52_middle_route_audit", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "sald.general_moving_target.kl_derivative", "probability.lsi_to_kl_fi", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "sald.unified_forward_kl.specialization", "sald.discrete_forward_kl.em_interpolation_fp", "sald.general_moving_target_discrete.em_interpolation_fp" ] lowerPacket := [ "Preferred target: SALD.generalMovingTargetDerivativeCandidateContract / SALD.generalMovingTargetDerivativeObligation / sald.general_moving_target.kl_derivative.", "First lower sub-slice: appendix.tex:765-812, exposing density/law regularity, mass conservation, the general Fokker-Planck equation, KL differentiation under the integral, and integration-by-parts side conditions.", "Second lower sub-slice: appendix.tex:813-864, supplying the target transport velocity v_t, the rescaled transport equation for pi_{t(s)}, the residual identification m_t=v_t-c_t, and the paper Young coefficient epsilon=2*dot t(s)/sigma_{t(s)}^2.", "Third lower sub-slice: appendix.tex:865-884, preserving the LSI handoff and s-to-t schedule/sigma positivity assumptions without adding them to the theorem statement.", "Alternative target only if derivative is blocked: SALD.guidedResidualIdentityContract / sald.guided_path_residual.identity for appendix.tex:630-704, sharpening normalizer, product/quotient differentiation, divergence cancellation, and mean-zero residual obligations.", "Leave residual DV, Gronwall endpoint/exponent, pure contraction, unified specialization, and downstream EM interpolation as separate obligations." ] reviewerChecklist := [ "SALD.guidedResidualContract and SALD.generalVaSaldContract list SALD.cycle52GuidedGeneralSkeletonMiddleObligation while remaining contractOnly.", "SALD.generalVaSaldProofDag contains ASTIS.SALD.guided_general.cycle52_middle_route_audit after ASTIS.SALD.guided_general.cycle52_upper_route.", "SALD.saldDependenciesForLabel for prop:guided_path_residual and thm:general-moving-target-SALD includes the cycle-52 middle contract, middle obligation, and sald.guided_general.cycle52_middle_route_audit.", "No Gronwall, DV, LSI/KL/FI, Fokker-Planck/KL derivative, guided residual, pure-contraction, or EM backend status is promoted.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-52 middle obligation tying the guided/general route audit to lower work. -/ def cycle52GuidedGeneralSkeletonMiddleObligation : ProofObligation where id := "sald.guided_general.cycle52_middle_route_audit" statement := "Cycle 52 middle audits appendix.tex:619-951 after the upper route check: prop:guided_path_residual and thm:general-moving-target-SALD remain unchanged, the proof route consumes the named guided residual, general KL derivative, LSI/KL/FI, residual DV, Gronwall, and pure-contraction interfaces in paper order, and the next lower target is sald.general_moving_target.kl_derivative over appendix.tex:765-884. EM interpolation remains only a downstream source-cited obligation interface." source := saldGeneralMovingTargetSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle52GuidedGeneralSkeletonUpperPacket", "SALD.cycle52GuidedGeneralSkeletonObligation", "SALD.cycle52GuidedGeneralSkeletonMiddleContract", "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle49MainSkeletonAnalyticMiddleObligation", "SALD.cycle50ForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle47GuidedGeneralSkeletonMiddleObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "SALD.generalMovingTargetPostYoungDerivativeBoundScalar", "SALD.generalMovingTargetLsiDerivativeBoundScalar", "SALD.generalMovingTargetTimeChangedDerivativeBoundScalar", "SALD.generalMovingTargetPreDvDerivativeBoundScalar", "SALD.generalMovingTargetPostDvGronwallCoefficientScalar", "SALD.generalMovingTargetPostDvGronwallCoefficientOfSigmaScheduleScalar", "SALD.generalMovingTargetDerivativeDvGronwallCoefficientScalar", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiDensityTestObligation", "probability.lsi_to_kl_fi", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "SALD.unifiedForwardKlSpecializationContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "This is a middle workflow obligation and source-to-Lean route audit, not an analytic proof. It adds no hidden density, boundary, endpoint, absolute-continuity, finite-log-mgf, finite-KL/FI, sigma-positivity, schedule, or coefficient-regularity assumptions to the guided residual proposition or continuous general theorem." /-- Cycle-52 lower obligation for the general moving-target derivative/DV scalar handoff. -/ def cycle52GuidedGeneralDerivativeDvLowerObligation : ProofObligation where id := "sald.general_moving_target.cycle52_derivative_dv_lower" statement := "Cycle 52 lower compiles the source-shaped scalar handoff from appendix.tex lines 765-907: once the general Fokker--Planck/KL derivative backend, residual Young estimate, LSI half-Fisher comparison, inverse-schedule calculus, and selected residual DV estimate are supplied explicitly, SALD.generalMovingTargetDerivativeDvGronwallCoefficientScalar derives the pre-Gronwall differential inequality with coefficient (sigma_t^2/2)*dot{s}(t)*C_LSI(t)-sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1) and residual sigma_t^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t)." source := saldGeneralMovingTargetDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle52GuidedGeneralSkeletonMiddleContract", "SALD.cycle52GuidedGeneralSkeletonMiddleObligation", "SALD.generalMovingTargetDerivativeDvGronwallCoefficientScalar", "SALD.generalMovingTargetPreDvDerivativeBoundScalar", "SALD.generalMovingTargetPostDvGronwallCoefficientScalar", "SALD.generalMovingTargetPostDvGronwallCoefficientOfSigmaScheduleScalar", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "SALD.saldLsiKlFiDensityTestContract", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "sald.general_moving_target.kl_derivative", "probability.lsi_to_kl_fi", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "sald.forward_kl.schedule_time_change" ] note := "The scalar lemmas compile, but the Fokker--Planck/KL derivative identity, integration by parts, residual Young analytic estimate, LSI density-test theorem, schedule calculus, residual finite-log-mgf/common-space witnesses, source-cited DV, Gronwall, and thm:general-moving-target-SALD remain obligations. No theorem statement or source constant is changed." /-- Cycle-52 proof-DAG pane for the guided/general route closure check. -/ def cycle52GuidedGeneralSkeletonDag : List ProofDagBlock := [ { id := "ASTIS.SALD.guided_general.cycle52_upper_route" interface := "Cycle-52 upper route closure: after the forward-KL and discrete forward-KL route wrappers, verify the five analytic backends and connect prop:guided_path_residual plus thm:general-moving-target-SALD to their named guided residual, KL-derivative, LSI, residual-DV, Gronwall, and pure-contraction obligations." source := saldGeneralMovingTargetSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle52GuidedGeneralSkeletonUpperPacket", "SALD.cycle52GuidedGeneralSkeletonObligation", "SALD.cycle52GuidedGeneralSkeletonMiddleContract", "SALD.cycle52GuidedGeneralSkeletonMiddleObligation", "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle50ForwardKlSkeletonObligation", "SALD.cycle51DiscreteForwardKlSkeletonObligation", "SALD.cycle47GuidedGeneralSkeletonObligation", "SALD.cycle47GuidedGeneralSkeletonMiddleObligation", "SALD.guidedResidualIdentityContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvEnergyCandidateContract", "SALD.generalMovingTargetGronwallSideConditionContract", "SALD.saldGronwallEndpointCalculusContract", "SALD.saldLsiKlFiDensityTestContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation" ] reusedBy := ["prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "main skeleton sprint 4 cycle 52"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.guided_general.cycle52_middle_route_audit" interface := "Middle source-to-Lean audit for cycle 52: verify appendix.tex:619-951 in paper order, keep the guided residual and general moving-target statements fixed, and select the continuous general KL derivative backend while guided residual, LSI, DV, Gronwall, pure contraction, and downstream EM interfaces remain explicit obligations." source := saldGeneralMovingTargetSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle52GuidedGeneralSkeletonMiddleContract", "SALD.cycle52GuidedGeneralSkeletonMiddleObligation", "SALD.cycle52GuidedGeneralSkeletonObligation", "SALD.cycle49MainSkeletonAnalyticMiddleObligation", "SALD.cycle50ForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle47GuidedGeneralSkeletonMiddleObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.identity", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract" ] reusedBy := ["prop:guided_path_residual", "thm:general-moving-target-SALD", "cycle52 lower derivative packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target.cycle52_derivative_dv_lower" interface := "Lower proof-producing scalar handoff for appendix.tex:765-907: compose the supplied general KL derivative/LSI pre-DV inequality with the residual DV estimate to obtain the exact sigma-weighted Gronwall coefficient and residual term." source := saldGeneralMovingTargetDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle52GuidedGeneralDerivativeDvLowerObligation", "SALD.generalMovingTargetDerivativeDvGronwallCoefficientScalar", "SALD.generalMovingTargetPreDvDerivativeBoundScalar", "SALD.generalMovingTargetPostDvGronwallCoefficientScalar", "SALD.generalMovingTargetPostDvGronwallCoefficientOfSigmaScheduleScalar", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "sald.general_moving_target.kl_derivative", "probability.lsi_to_kl_fi", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_m_energy" ] reusedBy := ["sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_m_energy", "thm:general-moving-target-SALD", "thm:unified-forward-KL"] status := ProofStatus.obligation } ] /-- Cycle-57 upper packet returning to the guided residual and continuous general moving-target theorem after the cycle-56 discrete forward-KL recovery. This packet records the required upper phase judgment, rechecks the five slow analytic interfaces, and keeps the current sprint on theorem-level route closure for `appendix.tex:619-951`. It is workflow data only: no theorem statement, source label, source coefficient, or analytic backend status is changed. -/ def cycle57GuidedGeneralSkeletonUpperPacket : GeneralVaSaldUpperPacket where objective := "Cycle 57 upper: previous cycle 56 passed reviewer and build gate, so no recovery is needed; Phase 1 is stable for the forward-KL and discrete forward-KL routes but not yet stable enough for broad cited-theory backfill until the guided residual and continuous general moving-target route is rechecked; the single lower packet that best reduces risk is sald.general_moving_target.kl_derivative over appendix.tex:765-884." sourceLabels := [ "prop:guided_path_residual", "proof:prop:guided_path_residual", "thm:general-moving-target-SALD", "eq:general_moving_target_SALD", "eq:general_moving_target_FP", "proof:thm:general-moving-target-SALD:derivative", "proof:thm:general-moving-target-SALD:residual-dv", "proof:thm:general-moving-target-SALD:dv-gronwall", "proof:thm:general-moving-target-SALD:pure-contraction", "proof:thm:unified-forward-KL", "eq:LSI-KL-FI", "lem:dv_variation", "lem:gronwall", "def:alpha-complexity" ] modeDiscipline := [ "Global phase judgment: cycle 56 did not fail and needs no recovery; the route is ready to advance from discrete forward-KL recovery back to guided/general theorem skeleton closure.", "Global phase judgment: do not begin broad cited-theory backfill yet; only after prop:guided_path_residual and thm:general-moving-target-SALD are rechecked against appendix.tex:619-951 should one narrow backend at a time be selected.", "Global phase judgment: the largest current proof risk is the continuous general Fokker-Planck/KL derivative backend, because it feeds the residual Young/LSI/DV/Gronwall chain and later unified/discrete general reuse.", "Five-backend check 1, Gronwall: SALD.saldGronwallCandidateContract, SALD.saldGronwallEndpointCalculusContract, SALD.saldGronwallExponentRewriteContract, and SALD.gronwallAnalyticObligation keep endpoint-safe differentiability, FTC, coefficient regularity, and exponent side conditions explicit.", "Five-backend check 2, DV: dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, SALD.generalMovingTargetDvFiniteLogMgfWitnessContract, and SALD.generalMovingTargetDvPositiveAlphaScalingContract keep common-space, absolute-continuity, finite-KL, finite-log-mgf, measurable residual test, and positive-alpha scaling explicit.", "Five-backend check 3, LSI/KL/FI: SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi keep density, zero-set convention, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule obligations explicit.", "Five-backend check 4, continuous derivative: SALD.forwardKlDerivativeCandidateContract and SALD.generalMovingTargetDerivativeCandidateContract keep the Fokker-Planck/KL derivative identities source-cited or obligation-level, with density, mass, boundary, transport, sigma, and schedule side conditions explicit.", "Five-backend check 5, EM interpolation: SALD.discreteForwardKlEmInterpolationSideConditionContract, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, and sald.general_moving_target_discrete.em_interpolation_fp remain downstream endpoint/conditional-law Fokker-Planck interfaces and are not imported into the continuous theorem statement.", "FaithfulPaper Phase 1: use appendix.tex:619-951 only for this guided/general route; sald_version_2.tex remains excluded." ] nonGoals := [ "Do not restate prop:guided_path_residual, thm:general-moving-target-SALD, or thm:unified-forward-KL.", "Do not promote SALD.guidedResidualContract, SALD.generalVaSaldContract, or any slow analytic interface above contractOnly/sourceCited/obligation.", "Do not replace the source route by a direct VA-SALD proof, path-space comparison, Girsanov, Pinsker, Talagrand, PI, or SLT-based proof.", "Do not start systematic measure-theory or SDE backfill in this upper cycle; keep any missing analytic fact as a precise source-cited interface or obligation." ] lowerPacket := [ "Middle should synchronize this cycle-57 upper route with conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, and the guided/general proof DAG.", "Lower should target exactly SALD.generalMovingTargetDerivativeCandidateContract / SALD.generalMovingTargetDerivativeObligation / sald.general_moving_target.kl_derivative over appendix.tex:765-884.", "First lower sub-slice: appendix.tex:765-812, exposing law regularity, mass conservation, the general moving-target Fokker-Planck equation, KL differentiation under the integral, and integration-by-parts side conditions.", "Second lower sub-slice: appendix.tex:813-864, exposing target transport by v_t, rescaled transport of pi_{t(s)}, residual m_t=v_t-c_t, and Young with epsilon=2*dot t(s)/sigma_{t(s)}^2.", "Third lower sub-slice: appendix.tex:865-884, preserving the LSI handoff and s-to-t schedule/sigma side conditions without adding them to the theorem statement.", "If the derivative backend is blocked, refine the named obligation with the exact density, boundary, common-space, finite-quantity, sigma-positivity, or schedule gap; do not weaken the source theorem." ] reviewerChecklist := [ "SALD.guidedResidualContract and SALD.generalVaSaldContract list SALD.cycle57GuidedGeneralSkeletonObligation while remaining contractOnly.", "SALD.generalVaSaldProofDag contains ASTIS.SALD.guided_general.cycle57_upper_route after the cycle-52 guided/general route and before downstream unified/discrete reuse.", "SALD.saldDependenciesForLabel for prop:guided_path_residual and thm:general-moving-target-SALD includes the cycle-57 packet, obligation, DAG node, and selected lower-packet node.", "The five slow analytic interfaces remain source-cited or obligation-level; no theorem statement, source constant, source label, or external reuse status changes.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-57 obligation for the upper guided/general route recheck. -/ def cycle57GuidedGeneralSkeletonObligation : ProofObligation where id := "sald.guided_general.cycle57_upper_route" statement := "Cycle 57 upper records that cycle 56 passed cleanly and needs no recovery, then rechecks appendix.tex:619-951 at theorem level: prop:guided_path_residual is consumed through the guided normalizer and centered residual identity; thm:general-moving-target-SALD is consumed through the general Fokker-Planck/KL derivative, eq:LSI-KL-FI, residual DV finite-log-mgf and positive-alpha witnesses, lem:gronwall endpoint/exponent side conditions, and the pure-contraction residual-zero obligation. Broad cited-theory backfill remains deferred; the selected lower packet is sald.general_moving_target.kl_derivative over appendix.tex:765-884." source := saldGeneralMovingTargetSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle57GuidedGeneralSkeletonUpperPacket", "SALD.cycle56DiscreteForwardKlSkeletonObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle52GuidedGeneralSkeletonObligation", "SALD.cycle52GuidedGeneralSkeletonMiddleObligation", "SALD.cycle52GuidedGeneralDerivativeDvLowerObligation", "SALD.cycle54MainSkeletonAnalyticInterfaceObligation", "SALD.cycle54MainSkeletonAnalyticMiddleObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "SALD.generalMovingTargetPostYoungDerivativeBoundScalar", "SALD.generalMovingTargetLsiDerivativeBoundScalar", "SALD.generalMovingTargetTimeChangedDerivativeBoundScalar", "SALD.generalMovingTargetPreDvDerivativeBoundScalar", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "SALD.saldGronwallEndpointCalculusContract", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "This is an upper workflow obligation and route check, not analytic backfill. It adds no hidden assumptions and does not promote Gronwall, DV, LSI/KL/FI, Fokker-Planck/KL differentiation, EM interpolation, guided residual, or the continuous general theorem." /-- Cycle-57 middle audit for the guided residual and continuous general moving-target theorem route. This is the middle-role source-to-Lean synchronization layer for the cycle-57 upper route. It verifies the appendix proof order, keeps every slow analytic dependency in an already named interface, and selects the continuous general KL derivative backend for lower work without changing any paper statement. -/ def cycle57GuidedGeneralSkeletonMiddleContract : GeneralVaSaldGuidedPathMiddleContract where guidedResidualSource := saldGuidedResidualSource generalContinuousSource := saldGeneralMovingTargetSource unifiedSource := saldUnifiedForwardKlSource discreteSource := saldGeneralMovingTargetDiscreteSource objective := "Cycle 57 middle: audit the upper guided/general route against appendix.tex:619-951 after the clean cycle-56 recovery, add the middle route obligation to prop:guided_path_residual and thm:general-moving-target-SALD, and keep lower work focused on sald.general_moving_target.kl_derivative over appendix.tex:765-884." sourceStepMap := [ "appendix.tex:619-704 proves prop:guided_path_residual by differentiating the normalizer Z_t, using partial_t p_t=-div(p_t*u_t), integrating by parts, differentiating pi_t=Z_t^(-1)*p_t*exp(-f_t), cancelling divergence terms, substituting dot Z_t/Z_t, and deriving the centered mean-zero residual.", "appendix.tex:724-744 states thm:general-moving-target-SALD with dynamics driven by c_t, diffusion sigma_t, residual m_t=v_t-c_t, finite residual alpha0-complexity, alpha in (0,alpha0], and the exact sigma-weighted KL bound.", "appendix.tex:765-812 differentiates KL(rho_s||pi_{t(s)}), drops the mass term, inserts the general moving-target Fokker-Planck equation, and evaluates the rho_s term as a c_t transport contribution minus sigma^2/2 times FI.", "appendix.tex:814-835 uses that v_t transports pi_t, so dot t(s)*v_{t(s)} transports pi_{t(s)}, and evaluates the target-time term by integration by parts.", "appendix.tex:835-864 combines c_t and v_t into m_t=v_t-c_t and applies Young with epsilon=2*dot t(s)/sigma_{t(s)}^2, preserving the sigma_t^(-2)*dot t(s)^2 residual-energy coefficient.", "appendix.tex:865-884 changes to K(t)=KL(rho_{s(t)}||pi_t), uses dot t(s(t))=dot s(t)^(-1), and applies eq:LSI-KL-FI to produce the pre-DV residual-energy inequality.", "appendix.tex:885-907 applies lem:dv_variation with Z=alpha*||m_t||^2 and rewrites the log-mgf quotient as mathfrak E_alpha(pi_t,m_t).", "appendix.tex:909-945 applies lem:gronwall with the source sigma-weighted a(t) and b(t), matches the theorem display, and proves the c_t=v_t pure-contraction clause by the zero residual alpha-complexity calculation.", "appendix.tex:949-951 is downstream reuse for thm:unified-forward-KL by setting c_t<-u_t; this middle audit does not introduce a direct VA-SALD proof." ] leanStepMap := [ "Use SALD.cycle57GuidedGeneralSkeletonUpperPacket and SALD.cycle57GuidedGeneralSkeletonObligation as the parent route check after cycle 56.", "Route appendix.tex:619-704 through SALD.guidedResidualIdentityContract, sald.guided_path_residual.normalizer_derivative, and sald.guided_path_residual.identity.", "Keep SALD.generalMovingTargetStatementContract, SALD.guidedResidualContract, and SALD.generalVaSaldContract contractOnly; this middle audit adds no theorem hypotheses.", "Route appendix.tex:765-884 through SALD.generalMovingTargetDerivativeCandidateContract, SALD.generalMovingTargetDerivativeObligation, sald.general_moving_target.kl_derivative, the cycle-52 scalar handoffs, SALD.saldLsiKlFiDensityTestContract, probability.lsi_to_kl_fi, and sald.forward_kl.schedule_time_change.", "Route appendix.tex:885-907 through SALD.generalMovingTargetDvFiniteLogMgfWitnessContract, SALD.generalMovingTargetDvPositiveAlphaScalingContract, sald.general_moving_target.dv_finite_log_mgf_witness, sald.general_moving_target.dv_positive_alpha_scaling, and sald.general_moving_target.dv_m_energy.", "Route appendix.tex:909-945 through SALD.generalMovingTargetGronwallInstantiationContract, SALD.generalMovingTargetGronwallSideConditionContract, sald.general_moving_target.gronwall_application, sald.general_moving_target.gronwall_side_conditions, and sald.general_moving_target.pure_contraction.", "Keep SALD.discreteForwardKlEmInterpolationSideConditionContract and SALD.generalMovingTargetDiscreteDerivativeSideConditionContract visible only as downstream EM/Fokker-Planck interfaces for later discrete general reuse." ] citedResultInterfaces := [ "lem:gronwall remains an obligation through endpoint-safe differentiability, FTC/order integration, coefficient regularity, endpoint rewrites, exponent splitting, and residual-exponent side conditions.", "lem:dv_variation remains source-cited; the continuous general theorem still needs common-space, absolute-continuity, finite KL/log-likelihood, measurability, finite-log-mgf, and positive-alpha witnesses for Z=alpha*||m_t||^2.", "eq:LSI-KL-FI remains the density, zero-set, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule obligation.", "The continuous general Fokker-Planck/KL derivative is the selected lower backend; compiled real scalar helpers only consume analytic inputs after they are supplied.", "The Euler-Maruyama interpolation Fokker-Planck backend remains a downstream discrete obligation and is not imported into the continuous theorem statement." ] obligations := [ "sald.guided_general.cycle57_upper_route", "sald.guided_general.cycle57_middle_route_audit", "sald.general_moving_target.cycle57_derivative_split_lower", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "sald.general_moving_target.kl_derivative", "probability.lsi_to_kl_fi", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "sald.unified_forward_kl.specialization", "sald.general_moving_target_discrete.em_interpolation_fp" ] lowerPacket := [ "Target exactly SALD.generalMovingTargetDerivativeCandidateContract / SALD.generalMovingTargetDerivativeObligation / sald.general_moving_target.kl_derivative.", "First lower sub-slice: appendix.tex:765-812, exposing law regularity, mass conservation, Fokker-Planck substitution, KL differentiation under the integral, and integration-by-parts side conditions; SALD.generalMovingTargetKlDerivativeResidualSplitScalar compiles only the scalar residual split after those analytic inputs are supplied.", "Second lower sub-slice: appendix.tex:813-864, exposing target transport by v_t, the rescaled pi_{t(s)} transport equation, residual identification m_t=v_t-c_t, and the Young coefficient epsilon=2*dot t(s)/sigma_{t(s)}^2.", "Third lower sub-slice: appendix.tex:865-884, preserving the LSI handoff plus inverse-schedule and sigma-positivity side conditions without adding them to the theorem statement; SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar composes the split with the existing pre-DV scalar pipeline.", "Leave guided residual calculus, residual DV, Gronwall endpoint/exponent, pure contraction, unified specialization, and downstream EM interpolation as separate obligations unless exact compiled proofs are added." ] reviewerChecklist := [ "SALD.guidedResidualContract and SALD.generalVaSaldContract list SALD.cycle57GuidedGeneralSkeletonMiddleObligation while remaining contractOnly.", "SALD.generalVaSaldProofDag contains ASTIS.SALD.guided_general.cycle57_middle_route_audit between the cycle-57 upper route and the lower derivative packet.", "SALD.saldDependenciesForLabel for prop:guided_path_residual and thm:general-moving-target-SALD includes the cycle-57 middle contract, middle obligation, and sald.guided_general.cycle57_middle_route_audit.", "No Gronwall, DV, LSI/KL/FI, Fokker-Planck/KL derivative, guided residual, pure-contraction, or EM backend status is promoted.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-57 middle obligation tying the guided/general route audit to lower work. -/ def cycle57GuidedGeneralSkeletonMiddleObligation : ProofObligation where id := "sald.guided_general.cycle57_middle_route_audit" statement := "Cycle 57 middle audits appendix.tex:619-951 after the upper route recheck: prop:guided_path_residual and thm:general-moving-target-SALD remain unchanged, the proof route consumes the named guided residual, continuous general KL derivative, LSI/KL/FI, residual DV, Gronwall, pure-contraction, and downstream EM interfaces in paper order, and the lower packet remains sald.general_moving_target.kl_derivative over appendix.tex:765-884. All slow analytic interfaces remain obligation or source-cited." source := saldGeneralMovingTargetSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle57GuidedGeneralSkeletonUpperPacket", "SALD.cycle57GuidedGeneralSkeletonObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleContract", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle52GuidedGeneralSkeletonMiddleObligation", "SALD.cycle52GuidedGeneralDerivativeDvLowerObligation", "SALD.cycle54MainSkeletonAnalyticMiddleObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "SALD.generalMovingTargetPostYoungDerivativeBoundScalar", "SALD.generalMovingTargetLsiDerivativeBoundScalar", "SALD.generalMovingTargetTimeChangedDerivativeBoundScalar", "SALD.generalMovingTargetPreDvDerivativeBoundScalar", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "SALD.saldGronwallEndpointCalculusContract", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "SALD.unifiedForwardKlSpecializationContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "This is a middle workflow obligation and source-to-Lean route audit, not an analytic proof. It adds no hidden density, boundary, endpoint, absolute-continuity, finite-log-mgf, finite-KL/FI, sigma-positivity, schedule, or coefficient-regularity assumptions to prop:guided_path_residual or thm:general-moving-target-SALD." /-- Cycle-57 lower obligation for the first continuous general derivative split. The compiled scalar lemmas here begin the selected lower packet for `sald.general_moving_target.kl_derivative` by reducing the raw derivative split to the residual derivative display once the source analytic identities are supplied. -/ def cycle57GuidedGeneralDerivativeSplitLowerObligation : ProofObligation where id := "sald.general_moving_target.cycle57_derivative_split_lower" statement := "Cycle 57 lower compiles the source-shaped scalar handoff for appendix.tex:765-884: from the raw KL derivative split with a mass-conservation term, the general VA-SALD Fokker-Planck first-term evaluation, the target-transport second-term evaluation, and the residual identity m_t=v_t-c_t, SALD.generalMovingTargetKlDerivativeResidualSplitScalar derives the residual derivative display; SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar then feeds that display into the existing Young, LSI, and time-change scalar pipeline. The analytic mass conservation, KL differentiation, Fokker-Planck, integration-by-parts, target transport, residual Young, LSI density-test, sigma positivity, and inverse-schedule backends remain obligations." source := saldGeneralMovingTargetDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle57GuidedGeneralSkeletonMiddleContract", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "SALD.generalMovingTargetKlDerivativeResidualSplitScalar", "SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar", "SALD.generalMovingTargetPostYoungDerivativeBoundScalar", "SALD.generalMovingTargetLsiDerivativeBoundScalar", "SALD.generalMovingTargetTimeChangedDerivativeBoundScalar", "SALD.generalMovingTargetPreDvDerivativeBoundScalar", "sald.general_moving_target.kl_derivative", "probability.lsi_to_kl_fi", "sald.forward_kl.schedule_time_change" ] note := "Proof-producing scalar/order core only. It does not promote sald.general_moving_target.kl_derivative, the general Fokker-Planck/KL derivative identity, LSI/KL/FI, schedule calculus, residual DV, Gronwall, or thm:general-moving-target-SALD." /-- Cycle-57 proof-DAG pane for the guided/general upper route and selected lower backend. -/ def cycle57GuidedGeneralSkeletonDag : List ProofDagBlock := [ { id := "ASTIS.SALD.guided_general.cycle57_upper_route" interface := "Cycle-57 upper route check: after the clean cycle-56 discrete forward-KL recovery, recheck appendix.tex:619-951 and route prop:guided_path_residual plus thm:general-moving-target-SALD through the already named guided residual, KL derivative, LSI, residual-DV, Gronwall, pure-contraction, and downstream EM interfaces." source := saldGeneralMovingTargetSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle57GuidedGeneralSkeletonUpperPacket", "SALD.cycle57GuidedGeneralSkeletonObligation", "SALD.cycle56DiscreteForwardKlSkeletonObligation", "SALD.cycle52GuidedGeneralSkeletonObligation", "SALD.cycle52GuidedGeneralSkeletonMiddleObligation", "SALD.guidedResidualIdentityContract", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetGronwallSideConditionContract", "SALD.saldLsiKlFiDensityTestContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldGronwallEndpointCalculusContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract" ] reusedBy := ["prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "ASTIS-SALD-001 cycle 57"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.guided_general.cycle57_middle_route_audit" interface := "Cycle-57 middle audit: verify appendix.tex:619-951 in paper order, keep guided residual and continuous general theorem statements fixed, and route lower work to sald.general_moving_target.kl_derivative while guided residual, LSI, residual DV, Gronwall, pure contraction, unified specialization, and EM reuse remain separate obligations." source := saldGeneralMovingTargetSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle57GuidedGeneralSkeletonMiddleContract", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralSkeletonObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle52GuidedGeneralSkeletonMiddleObligation", "SALD.guidedResidualIdentityContract", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetGronwallSideConditionContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.saldGronwallEndpointCalculusContract" ] reusedBy := ["prop:guided_path_residual", "thm:general-moving-target-SALD", "cycle57 lower derivative packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.guided_general.cycle57_lower_packet.general_derivative" interface := "Selected lower packet for cycle 57: refine sald.general_moving_target.kl_derivative over appendix.tex:765-884, starting with law regularity, mass conservation, Fokker-Planck substitution, KL differentiation, integration by parts, target transport, residual Young, LSI, and schedule side conditions. The first scalar split now compiles through SALD.generalMovingTargetKlDerivativeResidualSplitScalar and SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar." source := saldGeneralMovingTargetDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle57GuidedGeneralSkeletonObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralDerivativeSplitLowerObligation", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "SALD.generalMovingTargetKlDerivativeResidualSplitScalar", "SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar", "SALD.generalMovingTargetPostYoungDerivativeBoundScalar", "SALD.generalMovingTargetLsiDerivativeBoundScalar", "SALD.generalMovingTargetTimeChangedDerivativeBoundScalar", "SALD.generalMovingTargetPreDvDerivativeBoundScalar", "SALD.saldLsiKlFiDensityTestContract", "sald.forward_kl.schedule_time_change" ] reusedBy := ["sald.general_moving_target.kl_derivative", "thm:general-moving-target-SALD", "thm:unified-forward-KL"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target.cycle57_derivative_split_lower" interface := "Lower proof-producing scalar handoff for appendix.tex:765-884: consume the supplied mass-conservation drop, general Fokker-Planck first-term evaluation, target-transport term, residual identification, Young, LSI, and schedule inputs to reach the existing pre-DV derivative inequality." source := saldGeneralMovingTargetDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle57GuidedGeneralDerivativeSplitLowerObligation", "SALD.generalMovingTargetKlDerivativeResidualSplitScalar", "SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "probability.lsi_to_kl_fi", "sald.forward_kl.schedule_time_change" ] reusedBy := ["sald.general_moving_target.kl_derivative", "thm:general-moving-target-SALD"] status := ProofStatus.obligation } ] /-- Cycle-48 upper packet for closing the unified and discrete general theorem skeleton route. This keeps the cycle-44 slow analytic backend ledger in force and wires the last two theorem-level nodes requested by the task contract: `thm:unified-forward-KL` and `thm:general-moving-target-SALD-discrete`. It is workflow data only. The source proof still goes through the guided residual, the continuous general theorem, and the general EM interpolation interfaces; no theorem statement or analytic backend is promoted. -/ def cycle48UnifiedDiscreteSkeletonUpperPacket : GeneralVaSaldUpperPacket where objective := "Main skeleton sprint 5: wire thm:unified-forward-KL and thm:general-moving-target-SALD-discrete through the continuous/general theorem skeletons and the five explicit source-cited analytic interfaces, preserving all source statements, constants, and labels." sourceLabels := [ "thm:unified-forward-KL", "eq:residual-term", "eq:poisson-eq", "eq:SALD_Ito", "proof:thm:unified-forward-KL", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:general-moving-target-SALD-discrete", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "eq:general_discrete_delta_def", "lem:frozen_delta_cross_lip", "lem:dv_variation", "lem:gronwall", "eq:LSI-KL-FI", "def:alpha-complexity" ] modeDiscipline := [ "faithfulPaper Phase 1: use only main_body.tex, appendix.tex, and iteration_complexity.tex from the original SALD paper; sald_version_2.tex remains out of scope.", "Before lower work, all five slow analytic backends must stay explicit: Gronwall endpoint calculus, DV common-space/finite-log-mgf, LSI/KL/FI density-test, continuous Fokker-Planck/KL derivative, and EM endpoint/conditional-law Fokker-Planck.", "The unified theorem route is only the paper specialization c_t <- u_t after the transport bridge from prop:guided_path_residual and eq:poisson-eq; no direct VA-SALD KL proof is introduced.", "The discrete general theorem route is EM interpolation -> frozen-delta -> KL derivative/LSI -> residual DV -> Gronwall/stitching -> guided specialization, with doubled residual-energy and Gamma/Delta coefficients unchanged.", "All missing density, boundary, endpoint, absolute-continuity, finite-KL/FI, finite-log-mgf, conditional-law, coefficient-regularity, and divergence-linearity facts remain named obligations." ] nonGoals := [ "Do not restate thm:unified-forward-KL or thm:general-moving-target-SALD-discrete, and do not promote SALD.unifiedForwardKlContract or SALD.generalVaSaldDiscreteContract above contractOnly.", "Do not prove or replace Gronwall, DV, LSI/KL/FI, Fokker-Planck/KL derivative, EM conditional-FP, frozen-delta, or guided residual analytic backends in this upper packet.", "Do not add a path-space, Girsanov, Pinsker, Talagrand, PI, or direct theorem proof route.", "Do not import or mark an SLT theorem formalized; local SLT material may only guide a later narrow backend audit after this route wrapper is accepted.", "Do not run a broad source-index rebaseline except for the acceptance gate." ] lowerPacket := [ "Middle should synchronize the conversion window and proof-obligation rows for SALD.cycle48UnifiedDiscreteSkeletonUpperPacket and SALD.cycle48UnifiedDiscreteSkeletonObligation.", "Lower should target exactly one backend after the route wrapper is accepted: prefer SALD.generalMovingTargetDiscreteDerivativeCandidateContract / SALD.generalMovingTargetDiscreteDerivativeObligation / sald.general_moving_target_discrete.kl_derivative.", "First lower sub-slice: expose appendix.tex:1354-1387 EM endpoint, conditional-law density, and Fokker-Planck interfaces needed before differentiating KL(hat rho_s||tilde pi_s).", "Second lower sub-slice: preserve appendix.tex:1469-1511 frozen/residual algebra and the two sigma_eta^2/8 Young shares already isolated by the cycle-28 middle/lower packets.", "If the backend is too large, sharpen only the source-cited interface with common-space, absolute-continuity, finite-quantity, endpoint, conditional-law, and stitched-interval hypotheses; do not change the theorem statement.", "Only after this skeleton route is stable should lower backfill a narrow measure-theory detail, guided by local SLT material as reference-only and without importing SLT as a dependency." ] reviewerChecklist := [ "SALD.unifiedForwardKlContract and SALD.generalVaSaldDiscreteContract list SALD.cycle48UnifiedDiscreteSkeletonObligation while remaining contractOnly.", "SALD.generalVaSaldProofDag and SALD.generalVaSaldDiscreteProofDag expose cycle-48 route nodes for thm:unified-forward-KL and thm:general-moving-target-SALD-discrete.", "SALD.saldDependenciesForLabel entries for thm:unified-forward-KL and thm:general-moving-target-SALD-discrete include the cycle-48 packet, obligation, and DAG route nodes.", "The five slow analytic interfaces remain source-cited or obligation-level; no theorem statement, source constant, source label, or external reuse status changes.", "No axiom, sorry, admit, Prop := True, := trivial, hidden assumption, or alternate proof route is introduced.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-48 obligation tying the unified and discrete general theorem skeletons to the already named continuous/general and EM analytic interfaces. -/ def cycle48UnifiedDiscreteSkeletonObligation : ProofObligation where id := "sald.unified_discrete_general.cycle48_theorem_skeleton_route" statement := "Cycle 48 upper wires the final two main-skeleton theorem nodes. thm:unified-forward-KL is routed through prop:guided_path_residual, eq:poisson-eq, the transport-velocity bridge, and thm:general-moving-target-SALD with c_t=u_t and m_t=w_t. thm:general-moving-target-SALD-discrete is routed through the general EM interpolation Fokker-Planck backend, frozen-delta lemma, discrete KL derivative side conditions, eq:LSI-KL-FI, lem:dv_variation with the residual m_t finite-log-mgf witness, lem:gronwall with constant-schedule stitching, and the discrete guided specialization. All slow analytic backends remain obligation or source-cited." source := saldGeneralMovingTargetDiscreteSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle47GuidedGeneralSkeletonObligation", "SALD.cycle47GuidedGeneralSkeletonMiddleObligation", "SALD.cycle48UnifiedDiscreteSkeletonUpperPacket", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.identity", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_bridge_middle", "sald.unified_forward_kl.transport_bridge_lower", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "SALD.generalMovingTargetStatementContract", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "SALD.generalMovingTargetDiscreteStatementContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.constant_schedule_stitching", "sald.general_moving_target_discrete.gronwall_side_conditions", "sald.general_moving_target_discrete.unified_specialization", "probability.lsi_to_kl_fi", "probability.dv_variational_formula", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "This is an upper workflow obligation and theorem-route wrapper, not an analytic proof. It keeps the unified theorem as a specialization of the continuous general theorem and the discrete general theorem as the paper's EM/frozen-delta/DV/Gronwall route." /-- Cycle-48 middle audit for the unified and discrete general theorem route. This source-to-Lean synchronization layer checks the upper route wrapper against the exact TeX paragraphs and selects the next discrete KL-derivative backend without changing the source theorem statements. -/ def cycle48UnifiedDiscreteSkeletonMiddleContract : GeneralVaSaldGuidedPathMiddleContract where guidedResidualSource := saldGuidedResidualSource generalContinuousSource := saldGeneralMovingTargetSource unifiedSource := saldUnifiedForwardKlSource discreteSource := saldGeneralMovingTargetDiscreteSource objective := "Middle source-to-Lean audit for main skeleton sprint 5: verify that thm:unified-forward-KL is only the paper specialization of thm:general-moving-target-SALD, and that thm:general-moving-target-SALD-discrete consumes the general EM endpoint/conditional-FP, frozen-delta, LSI, residual DV, and Gronwall interfaces in appendix.tex:1313-1603 without changing constants." sourceStepMap := [ "main_body.tex:359-368 uses prop:guided_path_residual and eq:poisson-eq to show that u_t+w_t transports pi_t.", "main_body.tex:372-395 states thm:unified-forward-KL with the correction-field complexity E_alpha(pi_t,w_t) and the exact sigma_t^{-2} dot{s}(t)^{-1} coefficients.", "appendix.tex:949-951 proves thm:unified-forward-KL by setting c_t <- u_t in thm:general-moving-target-SALD and identifying m_t=w_t.", "appendix.tex:1313-1347 states thm:general-moving-target-SALD-discrete with the doubled residual coefficient, Gamma coefficient, Delta coefficient, alpha ranges, and constant inverse-schedule assumption.", "appendix.tex:1354-1387 fixes k, defines hat rho_s, uses endpoint laws, defines the frozen conditional drift, invokes the general EM conditional Fokker-Planck equation, and starts the KL derivative identity.", "appendix.tex:1389-1467 splits the Laplacian relative to tilde pi_s and combines the slowed target transport velocity with the conditional drift.", "appendix.tex:1469-1511 rewrites the cross field as delta_pi^VA+dot t(s)*m_{t(s)} and applies the two sigma_eta^2/8 Young splits plus lem:frozen_delta_cross_lip.", "appendix.tex:1513-1570 applies eq:LSI-KL-FI and lem:dv_variation to obtain the pre-Gronwall differential inequality.", "appendix.tex:1573-1600 defines K(t), changes variables from s to t, uses dot t(s(t))=dot s(t)^{-1}, and applies lem:gronwall to match the theorem display.", "appendix.tex:1603 records the discrete guided VA-SALD specialization c <- u." ] leanStepMap := [ "Route main_body.tex:359-395 through SALD.unifiedForwardKlSpecializationContract, sald.unified_forward_kl.transport_bridge_middle, sald.unified_forward_kl.transport_bridge_lower, sald.unified_forward_kl.transport_velocity_bridge, and sald.unified_forward_kl.specialization.", "Route appendix.tex:949-951 through SALD.generalMovingTargetStatementContract, SALD.generalVaSaldContract, and the continuous general derivative/DV/Gronwall obligations already checked in cycles 47 and 24.", "Route appendix.tex:1313-1347 through SALD.generalMovingTargetDiscreteStatementContract and SALD.generalVaSaldDiscreteContract; the theorem remains contractOnly.", "Route appendix.tex:1354-1387 through SALD.generalVaSaldEulerMaruyamaContract, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation, sald.general_moving_target_discrete.em_interpolation_fp, and the cycle-48 EM endpoint/conditional-law audit.", "Route appendix.tex:1469-1511 through SALD.generalMovingTargetDiscreteFrozenResidualAlgebraVector, SALD.generalMovingTargetDiscreteYoungFisherShareScalar, SALD.generalMovingTargetDiscreteTwoYoungFisherBudgetScalar, SALD.generalMovingTargetDiscreteResidualYoungCoefficientScalar, and sald.general_moving_target_discrete.derivative_side_conditions.", "Route appendix.tex:1513-1570 through probability.lsi_to_kl_fi, SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract, sald.general_moving_target_discrete.dv_finite_log_mgf_witness, and sald.general_moving_target_discrete.dv_m_energy.", "Route appendix.tex:1573-1600 through SALD.generalMovingTargetDiscreteGronwallInstantiationContract, SALD.generalMovingTargetDiscreteGronwallSideConditionContract, sald.general_moving_target_discrete.gronwall_application, sald.general_moving_target_discrete.constant_schedule_stitching, and sald.general_moving_target_discrete.gronwall_side_conditions.", "Select the next lower target as sald.general_moving_target_discrete.kl_derivative, beginning with the appendix.tex:1354-1387 endpoint/conditional-law/Fokker-Planck interface." ] citedResultInterfaces := [ "lem:gronwall remains the endpoint-safe real-analysis obligation; the discrete theorem uses it only through sald.general_moving_target_discrete.gronwall_application and gronwall_side_conditions.", "lem:dv_variation remains source-cited; the discrete residual witness must expose common-space, absolute-continuity, finite-KL, finite-log-mgf, and alpha scaling for the EM interpolation law.", "eq:LSI-KL-FI remains the density-test/Fisher-chain obligation and is not promoted by the two sigma_eta^2/8 Young bookkeeping helpers.", "The continuous general Fokker-Planck/KL derivative remains sald.general_moving_target.kl_derivative and is used upstream for the unified specialization.", "The EM interpolation Fokker-Planck backend remains sald.general_moving_target_discrete.em_interpolation_fp with endpoint laws, conditional-law density, and regular conditional expectation side conditions explicit; the named-interpolation endpoint-law pair is compiled but is not the conditional-FP proof.", "Local SLT material is reference-only for possible disintegration or one-step patterns; no SLT theorem is imported or marked formalized." ] obligations := [ "sald.unified_discrete_general.cycle48_theorem_skeleton_route", "sald.unified_discrete_general.cycle48_middle_route_audit", "sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions", "sald.general_moving_target_discrete.unified_specialization" ] lowerPacket := [ "Target exactly SALD.generalMovingTargetDiscreteDerivativeCandidateContract / SALD.generalMovingTargetDiscreteDerivativeObligation / sald.general_moving_target_discrete.kl_derivative.", "First lower sub-slice: sharpen appendix.tex:1354-1387 into law-level endpoint equalities via SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation, common-space and absolute-continuity assumptions for hat rho_s and tilde pi_s, conditional-law measurability/integrability for bar b_{k,s}, and the weak Fokker-Planck equation consumed by KL differentiation.", "Second lower sub-slice: reuse the cycle-28 compiled frozen/residual algebra only after the conditional drift, score, slowed transport, and m_t=v_t-c_t identifications are supplied.", "Do not restate the discrete theorem or add hypotheses to it; if a measure-theory fact is missing, refine sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit or sald.general_moving_target_discrete.derivative_side_conditions.", "Treat local SLT one-step and disintegration material as reference-only until a theorem is ported and built locally." ] reviewerChecklist := [ "The source-to-Lean map covers main_body.tex:359-395, appendix.tex:949-951, and appendix.tex:1313-1603 in paper order.", "SALD.unifiedForwardKlContract and SALD.generalVaSaldDiscreteContract list SALD.cycle48UnifiedDiscreteSkeletonMiddleObligation while remaining contractOnly.", "SALD.generalVaSaldProofDag and SALD.generalVaSaldDiscreteProofDag contain ASTIS.SALD.unified_discrete_general.cycle48_middle_route_audit.", "SALD.saldDependenciesForLabel entries for thm:unified-forward-KL and thm:general-moving-target-SALD-discrete include the cycle-48 middle contract, obligation, and DAG route-audit node.", "The EM endpoint/conditional-law audit remains an obligation; no conditional Fokker-Planck, regular conditional law, density, DV, LSI, or Gronwall backend is marked formalized.", "No theorem statement, source label, source constant, or source-file scope changes." ] status := ProofStatus.obligation /-- Cycle-48 narrow measure-theory audit for the discrete general EM endpoint and conditional-law Fokker--Planck backend. -/ def cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit" statement := "Sharpen the appendix.tex:1354-1387 backend for sald.general_moving_target_discrete.kl_derivative: hat X_s, X_k^eta, and X_{k+1}^eta live on a common filtered probability space; the compiled named-interpolation handoff proves hat rho_s endpoint laws rho_k^eta and rho_{k+1}^eta from pointwise endpoint identities and law representations; bar b_{k,s}(x) is a measurable integrable regular-conditional expectation given hat X_s=x; hat rho_s admits a density and is absolutely continuous with tilde pi_s where KL and FI are evaluated; and the weak conditional-drift Fokker-Planck equation holds before the KL derivative identity is differentiated." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.generalVaSaldEulerMaruyamaContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.derivative_side_conditions", "EulerMaruyamaContract", "FokkerPlanckContract", "KLContract", "FIContract" ] note := "This is the promised narrow middle backfill after the theorem-level route is wired. It is a source-dependency audit of local SDE/measure assumptions, guided only by local SLT one-step/disintegration patterns as reference; no SLT theorem is imported or marked formalized." /-- Cycle-48 middle obligation tying the route audit and the narrow EM endpoint/conditional-law interface to lower work. -/ def cycle48UnifiedDiscreteSkeletonMiddleObligation : ProofObligation where id := "sald.unified_discrete_general.cycle48_middle_route_audit" statement := "Cycle 48 middle audits the final main-skeleton route after the upper wrapper: thm:unified-forward-KL remains the guided-residual/correction-field specialization of thm:general-moving-target-SALD, thm:general-moving-target-SALD-discrete remains the paper route through general EM endpoint/conditional-FP, frozen-delta, LSI, residual DV, and Gronwall, and the next lower target is sald.general_moving_target_discrete.kl_derivative with appendix.tex:1354-1387 as the first endpoint/conditional-law/Fokker-Planck sub-slice." source := saldGeneralMovingTargetDiscreteSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle47GuidedGeneralSkeletonMiddleObligation", "SALD.cycle48UnifiedDiscreteSkeletonUpperPacket", "SALD.cycle48UnifiedDiscreteSkeletonObligation", "SALD.cycle48UnifiedDiscreteSkeletonMiddleContract", "sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit", "SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "SALD.generalMovingTargetDiscreteStatementContract", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI" ] note := "This is middle-role synchronization data and an interface refinement, not an analytic proof. It does not add hidden endpoint, density, regular conditional law, absolute-continuity, finite-log-mgf, or coefficient-regularity assumptions to either theorem." /-- Cycle-48 proof-DAG pane for the unified and discrete general theorem route. -/ def cycle48UnifiedDiscreteSkeletonDag : List ProofDagBlock := [ { id := "ASTIS.SALD.unified_forward_KL.cycle48_theorem_skeleton_route" interface := "Unified theorem-level route: combine the guided residual identity and correction-field equation to obtain the transport velocity u_t+w_t, then specialize thm:general-moving-target-SALD with c_t=u_t and m_t=w_t." source := saldUnifiedForwardKlSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle48UnifiedDiscreteSkeletonUpperPacket", "SALD.cycle48UnifiedDiscreteSkeletonObligation", "SALD.cycle48UnifiedDiscreteSkeletonMiddleContract", "SALD.cycle48UnifiedDiscreteSkeletonMiddleObligation", "SALD.cycle47GuidedGeneralSkeletonObligation", "SALD.cycle47GuidedGeneralSkeletonMiddleObligation", "SALD.guidedResidualIdentityContract", "SALD.generalMovingTargetStatementContract", "SALD.unifiedForwardKlSpecializationContract", "sald.guided_path_residual.identity", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_side_conditions" ] reusedBy := ["thm:unified-forward-KL", "main skeleton sprint 5"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.cycle48_theorem_skeleton_route" interface := "Discrete general theorem-level route: compose the source statement, general EM endpoint/conditional-FP backend, frozen-delta bound, KL derivative/LSI block, residual DV witness, Gronwall/stitching side conditions, and discrete guided specialization without changing coefficients." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle48UnifiedDiscreteSkeletonUpperPacket", "SALD.cycle48UnifiedDiscreteSkeletonObligation", "SALD.cycle48UnifiedDiscreteSkeletonMiddleContract", "SALD.cycle48UnifiedDiscreteSkeletonMiddleObligation", "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle47GuidedGeneralSkeletonObligation", "SALD.cycle28GeneralVaSaldUpperPacket", "SALD.cycle28GeneralVaSaldMiddleContract", "sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit", "SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "SALD.cycle20GeneralVaSaldUpperPacket", "SALD.cycle20GeneralVaSaldMiddleContract", "SALD.generalMovingTargetDiscreteStatementContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalFrozenDeltaCrossLipContract", "SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDiscreteGronwallInstantiationContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "probability.lsi_to_kl_fi", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions", "sald.general_moving_target_discrete.unified_specialization" ] reusedBy := ["thm:general-moving-target-SALD-discrete", "main skeleton sprint 5"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.unified_discrete_general.cycle48_middle_route_audit" interface := "Middle source-to-Lean audit for the final skeleton route: verify the unified specialization and discrete general EM/Fokker-Planck/frozen-delta/LSI/DV/Gronwall chain, then select the discrete KL derivative backend beginning with the endpoint and conditional-law interface." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle48UnifiedDiscreteSkeletonMiddleContract", "SALD.cycle48UnifiedDiscreteSkeletonMiddleObligation", "SALD.cycle48UnifiedDiscreteSkeletonObligation", "sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit", "SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_velocity_bridge", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_side_conditions" ] reusedBy := ["thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "cycle48 lower packet"] status := ProofStatus.obligation } ] /-- Cycle-53 upper packet for the unified and discrete general theorem route. This consumes the cycle-52 continuous guided/general route and the cycle-48 unified/discrete route, then records the first narrow measure-level backfill for EM endpoint laws. It remains an upper workflow packet: the source theorems stay fixed and all slow analytic backends stay below formalized. -/ def cycle53UnifiedDiscreteGeneralUpperPacket : GeneralVaSaldUpperPacket where objective := "Cycle 53 upper: close main skeleton sprint 5 by wiring thm:unified-forward-KL through the cycle-52 continuous general theorem route and wiring thm:general-moving-target-SALD-discrete through the explicit source-cited EM, derivative, LSI, residual-DV, and Gronwall interfaces; after that route check, add one narrow Measure.map endpoint-law backfill for the discrete general EM interpolation." sourceLabels := [ "thm:unified-forward-KL", "proof:thm:unified-forward-KL", "eq:SALD_Ito", "eq:poisson-eq", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:general-moving-target-SALD-discrete", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "proof:thm:general-moving-target-SALD-discrete:derivative", "proof:thm:general-moving-target-SALD-discrete:residual-dv", "proof:thm:general-moving-target-SALD-discrete:gronwall", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI" ] modeDiscipline := [ "faithfulPaper Phase 1: use main_body.tex:359-395, appendix.tex:949-951, and appendix.tex:1313-1603 only; sald_version_2.tex remains excluded.", "Five-backend check 1, Gronwall: use SALD.saldGronwallEndpointCalculusContract and the general/discrete Gronwall side-condition obligations for endpoint-safe differentiability, FTC, coefficient regularity, stitching, and display matching.", "Five-backend check 2, DV: use dvVariationalFormulaInterface saldDvVariationSource plus residual finite-log-mgf/common-space witnesses for continuous and EM-interpolated residual fields; the Boucheron equality remains sourceCited.", "Five-backend check 3, LSI/KL/FI: use SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, and probability.lsi_to_kl_fi for density, zero-set, admissible sqrt test, entropy, and Fisher-chain assumptions.", "Five-backend check 4, continuous derivative: consume SALD.generalMovingTargetDerivativeCandidateContract, SALD.generalMovingTargetDerivativeObligation, and the cycle-52 scalar derivative/DV handoff without promoting the Fokker-Planck or integration-by-parts backend.", "Five-backend check 5, EM interpolation: consume SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation, and the endpoint-law handoffs while keeping conditional drift, density/absolute-continuity, and weak Fokker-Planck as obligations.", "Only after the theorem route is wired, the measure-theory backfill is limited to AutoSamplingTheory.lawMapEqOfAEEq and SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation." ] nonGoals := [ "Do not restate thm:unified-forward-KL, thm:general-moving-target-SALD, or thm:general-moving-target-SALD-discrete.", "Do not introduce a direct unified VA-SALD KL proof; the source route remains c_t=u_t specialization of the continuous general theorem.", "Do not prove or promote Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL derivative, regular conditional drift, density/absolute-continuity, conditional Fokker-Planck, frozen-delta, or theorem-level statements.", "Do not import or mark any SLT theorem as formalized; local SLT pushforward-law patterns are reference-only for this narrow Measure.map congruence backfill.", "Do not change the doubled residual coefficient, Gamma/Delta terms, alpha ranges, sigma_eta factors, endpoint labels, or source theorem displays." ] lowerPacket := [ "Middle should synchronize the conversion window, proof-obligation ledger, and SLT audit for SALD.cycle53UnifiedDiscreteGeneralUpperPacket / SALD.cycle53UnifiedDiscreteGeneralSkeletonObligation.", "Preferred lower target remains SALD.generalMovingTargetDiscreteDerivativeCandidateContract / SALD.generalMovingTargetDiscreteDerivativeObligation / sald.general_moving_target_discrete.kl_derivative over appendix.tex:1354-1387.", "First lower sub-slice after the compiled endpoint backfill: connect the Measure.map endpoint-law equality to the named hat rho_s/rho_k^eta representation, then expose common-space, density/absolute-continuity, regular conditional drift, and weak conditional Fokker-Planck interfaces.", "Alternative lower target only if discrete KL derivative is blocked: sald.unified_forward_kl.transport_velocity_bridge, preserving the residual/correction signs from main_body.tex:359-368.", "If any analytic backend is too large, refine its named source-cited interface rather than adding hidden assumptions to the theorem contract." ] reviewerChecklist := [ "SALD.unifiedForwardKlContract and SALD.generalVaSaldDiscreteContract list SALD.cycle53UnifiedDiscreteGeneralSkeletonObligation while remaining contractOnly.", "SALD.generalVaSaldProofDag and SALD.generalVaSaldDiscreteProofDag include ASTIS.SALD.unified_discrete_general.cycle53_upper_route.", "SALD.saldDependenciesForLabel for thm:unified-forward-KL and thm:general-moving-target-SALD-discrete includes the cycle-53 packet, obligation, DAG route, and the compiled Measure.map endpoint-law backfill.", "AutoSamplingTheory.lawMapEqOfAEEq and SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation are the only new formalized backfill lemmas; all large analytic backends remain obligation or source-cited.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-53 obligation tying the final unified/discrete general theorem route to explicit source-cited interfaces and the narrow Measure.map endpoint backfill. -/ def cycle53UnifiedDiscreteGeneralSkeletonObligation : ProofObligation where id := "sald.unified_discrete_general.cycle53_upper_route" statement := "Cycle 53 upper wires thm:unified-forward-KL through the cycle-52 continuous general route, the guided residual/correction-field transport bridge, and the source specialization c_t=u_t, m_t=w_t; it wires thm:general-moving-target-SALD-discrete through the general EM endpoint/conditional-Fokker-Planck interface, frozen-delta side conditions, discrete KL derivative side conditions, eq:LSI-KL-FI, lem:dv_variation residual witness, and lem:gronwall side conditions. After that theorem-level route check, it backfills only the Measure.map a.e.-equality endpoint-law handoff for the discrete general EM interpolation." source := saldGeneralMovingTargetDiscreteSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle49MainSkeletonAnalyticMiddleObligation", "SALD.cycle52GuidedGeneralSkeletonObligation", "SALD.cycle52GuidedGeneralSkeletonMiddleObligation", "SALD.cycle52GuidedGeneralDerivativeDvLowerObligation", "SALD.cycle48UnifiedDiscreteSkeletonObligation", "SALD.cycle48UnifiedDiscreteSkeletonMiddleObligation", "SALD.cycle53UnifiedDiscreteGeneralUpperPacket", "SALD.unifiedForwardKlSpecializationContract", "SALD.cycle16UnifiedForwardKlTransportBridgeMiddleContract", "SALD.cycle16UnifiedForwardKlTransportBridgeLowerContract", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_side_conditions", "SALD.generalMovingTargetDiscreteStatementContract", "SALD.generalVaSaldEulerMaruyamaContract", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "AutoSamplingTheory.lawMapEqOfAEEq", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI" ] note := "This is an upper route check plus one narrow compiled measure-level endpoint-law lemma. It does not prove the conditional drift, density/absolute-continuity, Fokker-Planck, KL derivative, LSI, DV, Gronwall, unified theorem, or discrete general theorem." /-- Cycle-53 middle audit for the final unified/discrete general route. This synchronizes the upper route packet with the conversion window and proof-obligation ledger. It keeps the source proof order fixed and leaves the next lower target on the discrete general KL derivative backend over `appendix.tex:1354-1387`. -/ def cycle53UnifiedDiscreteGeneralMiddleContract : GeneralVaSaldGuidedPathMiddleContract where guidedResidualSource := saldGuidedResidualSource generalContinuousSource := saldGeneralMovingTargetSource unifiedSource := saldUnifiedForwardKlSource discreteSource := saldGeneralMovingTargetDiscreteSource objective := "Cycle 53 middle: audit the post-upper sprint-5 route in paper order, verify that thm:unified-forward-KL consumes the cycle-52 continuous general skeleton and correction-field specialization, verify that thm:general-moving-target-SALD-discrete consumes the EM endpoint/conditional-FP, frozen-delta, LSI, residual-DV, and Gronwall interfaces, and keep sald.general_moving_target_discrete.kl_derivative over appendix.tex:1354-1387 as the lower target." sourceStepMap := [ "main_body.tex:359-368 uses prop:guided_path_residual and eq:poisson-eq to turn u_t+w_t into a transport velocity for pi_t.", "main_body.tex:372-395 states thm:unified-forward-KL with correction-field complexity E_alpha(pi_t,w_t) and the source sigma_t^{-2} dot{s}(t)^{-1} coefficients.", "appendix.tex:949-951 proves thm:unified-forward-KL only by specializing thm:general-moving-target-SALD with c_t=u_t and m_t=w_t.", "appendix.tex:1313-1347 states thm:general-moving-target-SALD-discrete with the doubled residual coefficient, Gamma coefficient, Delta coefficient, alpha ranges, sigma_eta factors, and constant inverse-schedule assumption.", "appendix.tex:1354-1387 defines hat rho_s, records endpoint laws, defines the frozen conditional drift, invokes the general EM conditional Fokker-Planck equation, and begins KL differentiation.", "appendix.tex:1389-1511 performs the Laplacian split, frozen/residual field identification, and two sigma_eta^2/8 Young splits before the frozen-delta bound.", "appendix.tex:1513-1552 applies eq:LSI-KL-FI and lem:dv_variation to the residual m_t on the EM interpolation law.", "appendix.tex:1573-1600 changes variables to K(t), uses the constant schedule, and applies lem:gronwall to match the theorem display.", "appendix.tex:1603 records the discrete guided VA-SALD specialization c_t=u_t." ] leanStepMap := [ "Use SALD.cycle53UnifiedDiscreteGeneralUpperPacket and SALD.cycle53UnifiedDiscreteGeneralSkeletonObligation as parent route data; do not replace the cycle-52 continuous general theorem route.", "Route unified VA-SALD through SALD.unifiedForwardKlSpecializationContract, sald.unified_forward_kl.transport_velocity_bridge, sald.unified_forward_kl.specialization, SALD.generalVaSaldContract, and the cycle-52 guided/general middle and lower obligations.", "Route appendix.tex:1313-1347 through SALD.generalMovingTargetDiscreteStatementContract and SALD.generalVaSaldDiscreteContract; both remain contractOnly.", "Route appendix.tex:1354-1387 through SALD.generalVaSaldEulerMaruyamaContract, SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation, SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation, SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation, and sald.general_moving_target_discrete.em_interpolation_fp.", "Route appendix.tex:1469-1511 through SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.generalMovingTargetDiscreteFrozenResidualAlgebraVector, the three cycle-28 Young/Fisher scalar bookkeeping helpers, and sald.general_moving_target_discrete.derivative_side_conditions.", "Route appendix.tex:1513-1552 through SALD.saldLsiKlFiDensityTestContract, probability.lsi_to_kl_fi, dvVariationalFormulaInterface saldDvVariationSource, SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract, and sald.general_moving_target_discrete.dv_m_energy.", "Route appendix.tex:1573-1600 through SALD.generalMovingTargetDiscreteGronwallInstantiationContract, SALD.generalMovingTargetDiscreteGronwallSideConditionContract, sald.general_moving_target_discrete.gronwall_application, and sald.general_moving_target_discrete.gronwall_side_conditions.", "Keep the lower packet at SALD.generalMovingTargetDiscreteDerivativeCandidateContract / SALD.generalMovingTargetDiscreteDerivativeObligation / sald.general_moving_target_discrete.kl_derivative, with the first sub-slice still the common-space, density/AC, regular conditional drift, weak conditional-FP, and KL differentiation backend." ] citedResultInterfaces := [ "lem:gronwall remains an endpoint-safe differentiability, FTC, coefficient-regularity, stitching, and display-matching obligation.", "lem:dv_variation remains source-cited through common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and alpha-scaling witnesses.", "eq:LSI-KL-FI remains the density, zero-set, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher-chain obligation.", "The continuous Fokker-Planck/KL derivative is consumed upstream through sald.general_moving_target.kl_derivative and the cycle-52 scalar derivative/DV handoff, not reproved for the unified theorem.", "The EM interpolation backend remains sald.general_moving_target_discrete.em_interpolation_fp: the cycle-53 Measure.map lemma proves only an endpoint-law congruence after named endpoint identities are supplied." ] obligations := [ "sald.unified_discrete_general.cycle53_upper_route", "sald.unified_discrete_general.cycle53_middle_route_audit", "sald.guided_general.cycle52_middle_route_audit", "sald.general_moving_target.cycle52_derivative_dv_lower", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions" ] lowerPacket := [ "Target exactly SALD.generalMovingTargetDiscreteDerivativeCandidateContract / SALD.generalMovingTargetDiscreteDerivativeObligation / sald.general_moving_target_discrete.kl_derivative.", "First lower sub-slice: appendix.tex:1354-1387, after the compiled Measure.map endpoint-law handoff, expose the common probability space, hat rho_s and tilde pi_s density/absolute-continuity, regular conditional drift bar b_{k,s}, weak conditional Fokker-Planck equation, KL differentiation under the integral, and integration-by-parts side conditions.", "Second lower sub-slice: appendix.tex:1389-1467, Laplacian split relative to tilde pi_s and slowed target transport equation.", "Third lower sub-slice: appendix.tex:1469-1511, reuse the cycle-28 frozen/residual algebra only after the conditional drift, score, slowed transport, and m_t=v_t-c_t identifications are supplied.", "Do not restate either theorem, change constants, or promote the Measure.map endpoint backfill into conditional drift, density, Fokker-Planck, or KL derivative formalization." ] reviewerChecklist := [ "SALD.unifiedForwardKlContract and SALD.generalVaSaldDiscreteContract list SALD.cycle53UnifiedDiscreteGeneralMiddleObligation while remaining contractOnly.", "SALD.generalVaSaldProofDag and SALD.generalVaSaldDiscreteProofDag include ASTIS.SALD.unified_discrete_general.cycle53_middle_route_audit.", "SALD.saldDependenciesForLabel entries for thm:unified-forward-KL and thm:general-moving-target-SALD-discrete include the cycle-53 middle contract, middle obligation, and sald.unified_discrete_general.cycle53_middle_route_audit.", "Only AutoSamplingTheory.lawMapEqOfAEEq and SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation are formalized in this cycle; analytic backends remain sourceCited or obligation-level.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-53 middle obligation tying the final route audit to lower work. -/ def cycle53UnifiedDiscreteGeneralMiddleObligation : ProofObligation where id := "sald.unified_discrete_general.cycle53_middle_route_audit" statement := "Cycle 53 middle audits the final unified/discrete route after the upper packet: thm:unified-forward-KL remains the paper specialization of the continuous general theorem through the correction-field transport bridge, thm:general-moving-target-SALD-discrete remains the source route through EM endpoint/conditional-FP, frozen-delta, LSI, residual DV, and Gronwall, the cycle-53 Measure.map endpoint-law lemma is only a narrow endpoint handoff, and the next lower target remains sald.general_moving_target_discrete.kl_derivative over appendix.tex:1354-1387." source := saldGeneralMovingTargetDiscreteSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle53UnifiedDiscreteGeneralUpperPacket", "SALD.cycle53UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle53UnifiedDiscreteGeneralMiddleContract", "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle49MainSkeletonAnalyticMiddleObligation", "SALD.cycle52GuidedGeneralSkeletonMiddleObligation", "SALD.cycle52GuidedGeneralDerivativeDvLowerObligation", "SALD.cycle48UnifiedDiscreteSkeletonMiddleObligation", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "SALD.generalMovingTargetStatementContract", "SALD.generalVaSaldContract", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_side_conditions", "SALD.generalMovingTargetDiscreteStatementContract", "SALD.generalVaSaldDiscreteContract", "SALD.generalVaSaldEulerMaruyamaContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "sald.general_moving_target_discrete.em_interpolation_fp", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "SALD.generalMovingTargetDiscreteGronwallInstantiationContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI" ] note := "This is a middle source-to-Lean audit and lower packet, not a theorem proof. It adds no hidden endpoint, density, absolute-continuity, conditional-drift, finite-log-mgf, finite-KL/FI, schedule, sigma-positivity, or coefficient-regularity assumptions to the paper statements." /-- Cycle-53 proof-DAG pane for the final unified/discrete general route. -/ def cycle53UnifiedDiscreteGeneralDag : List ProofDagBlock := [ { id := "ASTIS.SALD.unified_discrete_general.cycle53_upper_route" interface := "Final upper route check for skeleton sprint 5: consume the cycle-52 continuous general route for thm:unified-forward-KL and the cycle-48 discrete general route for thm:general-moving-target-SALD-discrete, with all five slow analytic interfaces explicit." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle53UnifiedDiscreteGeneralUpperPacket", "SALD.cycle53UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle52GuidedGeneralSkeletonObligation", "SALD.cycle48UnifiedDiscreteSkeletonObligation", "sald.unified_forward_kl.specialization", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_side_conditions" ] reusedBy := ["thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "cycle53 lower packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.unified_discrete_general.cycle53_middle_route_audit" interface := "Middle route audit for cycle 53: check the unified specialization and discrete general EM/Fokker-Planck/frozen-delta/LSI/DV/Gronwall chain against the source order, keep the Measure.map endpoint backfill scoped to endpoint laws, and hand off the discrete KL derivative backend." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle53UnifiedDiscreteGeneralMiddleContract", "SALD.cycle53UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle53UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle52GuidedGeneralSkeletonMiddleObligation", "SALD.cycle48UnifiedDiscreteSkeletonMiddleObligation", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "sald.unified_forward_kl.specialization", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_side_conditions" ] reusedBy := ["thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "cycle53 lower packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.cycle53_measure_map_endpoint_backfill" interface := "Narrow measure-theory backfill: a.e. equality of endpoint process representatives implies equality of their Measure.map pushforward laws, giving a concrete endpoint-law handoff below the abstract law interface." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "AutoSamplingTheory.lawMapEqOfAEEq", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit" ] reusedBy := ["sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative"] status := ProofStatus.formalized } ] /-- Cycle-54 upper packet for the repeated analytic-interface sprint. Cycle 54 returns to the sprint-1 focus after the full theorem route has been threaded once. The objective is to check that the five slow analytic backends are still precise source-cited or obligation-level interfaces, and to hand the next lower work to a theorem-useful backend rather than to another isolated scalar lemma. -/ def cycle54MainSkeletonAnalyticInterfaceLedger : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteDerivativeSource objective := "Cycle 54 upper: re-run the main skeleton sprint-1 analytic interface ledger after cycles 50-53, check the five slow source-cited interfaces against all six theorem skeletons, and keep the next lower packet on the discrete general EM conditional-law/Fokker-Planck and KL-derivative backend." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "proof:thm:forward-KL:derivative", "proof:thm:forward-KL-discrete:conditional-fp", "proof:thm:general-moving-target-SALD:derivative", "proof:thm:general-moving-target-SALD-discrete:derivative", "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Gronwall interface: SALD.saldGronwallCandidateContract, SALD.saldGronwallEndpointCalculusContract, SALD.saldGronwallExponentRewriteContract, the cycle-36/41 assembly wrappers, and theorem-specific Gronwall side-condition contracts expose closed-interval differentiability semantics, FTC/order integration, coefficient interval-integrability, endpoint evaluation, and exponent rewrites; lem:gronwall remains ProofStatus.obligation.", "DV interface: dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, and theorem-specific velocity/residual finite-log-mgf witness contracts expose same-space probability measures, absolute continuity, finite KL/log-likelihood, selected-test measurability, finite log-mgf, alpha-range and positive-alpha scaling; the Boucheron variational equality remains ProofStatus.sourceCited.", "LSI/KL/FI interface: SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, and the cycle-43 density/entropy plus integral Fisher-chain helpers expose rho << pi, Radon-Nikodym density, zero-set convention, admissible sqrt-density test or approximation, entropy identity, finite KL/FI, and Fisher chain rule; probability.lsi_to_kl_fi remains ProofStatus.obligation.", "Continuous forward-KL/general Fokker-Planck/KL derivative interface: SALD.forwardKlDerivativeCandidateContract, SALD.forwardKlDerivativeSideConditionContract, SALD.generalMovingTargetDerivativeCandidateContract, and the cycle-50/52 scalar handoffs expose mass conservation, KL differentiation under the integral, SALD/general Fokker-Planck equations, target transport, integration by parts, LSI handoff, and inverse-schedule calculus; the analytic derivative backends remain ProofStatus.obligation.", "Euler-Maruyama interpolation Fokker-Planck interface: SALD.discreteForwardKlEmInterpolationSideConditionContract, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation, SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation, SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation, and SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff expose endpoint laws, named-law pushforward bookkeeping, regular conditional drift, density/absolute-continuity, weak conditional Fokker-Planck, Laplacian split, stitched intervals, and common-space assumptions; only endpoint-law congruence and the sigma-weighted divergence regrouping are formalized." ] theoremRoute := [ "1. thm:forward-KL consumes the continuous KL derivative, LSI/KL/FI, DV velocity witness, endpoint schedule, and Gronwall side-condition interfaces from appendix.tex:168-252 and main_body.tex:238-247.", "2. thm:forward-KL-discrete consumes the EM endpoint/conditional-FP backend, frozen-defect estimate, LSI, DV velocity witness, stitched Gronwall, and accumulated-error bridge from appendix.tex:260-592 and main_body.tex:299-323.", "3. prop:guided_path_residual consumes only the normalizer derivative, guided-path differentiation, divergence cancellation, and mean-zero residual interfaces from appendix.tex:619-704; it remains the upstream input to the unified specialization.", "4. thm:general-moving-target-SALD consumes the continuous general derivative, residual Young/LSI, residual DV, sigma-weighted Gronwall, endpoint/exponent side conditions, and pure-contraction obligations from appendix.tex:724-949.", "5. thm:unified-forward-KL remains the paper specialization of thm:general-moving-target-SALD using prop:guided_path_residual and eq:poisson-eq, with no direct alternate KL proof.", "6. thm:general-moving-target-SALD-discrete consumes the general EM endpoint/conditional-FP backend, frozen-delta lemma, discrete KL derivative/LSI, residual DV, constant-schedule time change, and Gronwall stitching from appendix.tex:1313-1603." ] modeDiscipline := [ "faithfulPaper Phase 1: preserve main_body.tex, appendix.tex, and iteration_complexity.tex as the only source files; sald_version_2.tex remains excluded.", "Do not change theorem statements, constants, alpha ranges, sigma factors, endpoint laws, source labels, or proof routes.", "All unproved analytic backends stay obligation or sourceCited; only already compiled local scalar algebra and endpoint-law congruence helpers may be marked formalized.", "The next lower packet must improve a theorem-consumed backend interface, not start broad SLT/SDE backfill or a detached scalar lemma." ] nonGoals := [ "No source-index rebaseline beyond the acceptance command.", "No imported SLT theorem is marked formalized for DV, LSI, concentration, disintegration, or one-step EM analysis.", "No endpoint, density, absolute-continuity, finite-KL/FI, finite-log-mgf, boundary, smoothness, conditional-law, or stitched-interval hypothesis is added to a paper theorem as a hidden assumption.", "No replacement of derivative -> LSI -> DV -> Gronwall, the correction-field specialization, or the paper EM/frozen-delta path." ] lowerPacket := [ "Target exactly SALD.generalMovingTargetDiscreteDerivativeCandidateContract / SALD.generalMovingTargetDiscreteDerivativeObligation / sald.general_moving_target_discrete.kl_derivative.", "First lower sub-slice remains appendix.tex:1354-1387: after the compiled Measure.map endpoint-law handoff, expose the common probability space, hat rho_s and tilde pi_s density/absolute-continuity, regular conditional drift bar b_{k,s}, weak conditional Fokker-Planck equation, KL differentiation under the integral, and integration-by-parts side conditions.", "Use SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation only as endpoint-law bookkeeping and SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff only as sigma-weighted divergence algebra; do not classify either as the Brownian construction, regular conditional drift, density/absolute-continuity, weak Fokker-Planck, or KL derivative proof.", "If the EM conditional-FP backend is blocked, sharpen the source-cited interface by naming the exact common-space, absolute-continuity, finite-KL/FI, measurability, endpoint, and conditional-law hypotheses rather than changing theorem statements." ] reviewerChecklist := [ "SALD.cycle54MainSkeletonAnalyticInterfaceObligation is listed by all six theorem contracts while they remain contractOnly.", "SALD.saldDependenciesForLabel for the first proof-DAG labels includes the cycle-54 ledger, obligation, and DAG pane.", "The conversion window and proof-obligation ledger contain the cycle-54 upper analytic-interface check and the lower packet over appendix.tex:1354-1387.", "Gronwall, DV, LSI/KL/FI, continuous derivative, EM conditional-FP, and theorem statements are not promoted beyond their current statuses.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle44MainSkeletonAnalyticInterfaceLedger", "SALD.cycle49MainSkeletonAnalyticReadinessLedger", "SALD.cycle50ForwardKlSkeletonObligation", "SALD.cycle51DiscreteForwardKlSkeletonObligation", "SALD.cycle52GuidedGeneralSkeletonObligation", "SALD.cycle53UnifiedDiscreteGeneralSkeletonObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "SALD.cycle54GeneralMovingTargetDiscreteEmFpLowerObligation", "sald.general_moving_target_discrete.kl_derivative" ] status := ProofStatus.obligation /-- Cycle-54 upper obligation for the repeated analytic-interface ledger. -/ def cycle54MainSkeletonAnalyticInterfaceObligation : ProofObligation where id := "sald.main_skeleton.cycle54_analytic_interface_ledger" statement := "Cycle 54 upper re-checks the five slow analytic interfaces after the theorem route has been threaded: Gronwall endpoint-safe differentiability/FTC, Donsker-Varadhan common-space/absolute-continuity/finite-KL/finite-log-mgf, LSI-to-KL/FI density/zero-set/admissible-test/entropy/Fisher-chain assumptions, continuous forward-KL and general Fokker-Planck/KL derivative identities, and Euler-Maruyama interpolation endpoint/conditional-law Fokker-Planck. It wires those interfaces back into thm:forward-KL, thm:forward-KL-discrete, prop:guided_path_residual, thm:general-moving-target-SALD, thm:unified-forward-KL, and thm:general-moving-target-SALD-discrete while keeping every unproved backend below formalized status; the cycle-54 lower sigma-weighted divergence regrouping is formalized only as algebra under explicit FP hypotheses." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle54MainSkeletonAnalyticInterfaceLedger", "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle50ForwardKlSkeletonObligation", "SALD.cycle51DiscreteForwardKlSkeletonObligation", "SALD.cycle52GuidedGeneralSkeletonObligation", "SALD.cycle53UnifiedDiscreteGeneralSkeletonObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "SALD.cycle54GeneralMovingTargetDiscreteEmFpLowerObligation", "SALD.forwardKlProofDag", "SALD.discreteForwardKlProofDag", "SALD.generalVaSaldProofDag", "SALD.generalVaSaldDiscreteProofDag" ] note := "This is an upper workflow obligation and route audit, not a new analytic theorem. It adds no hidden assumptions and does not promote Gronwall, DV, LSI/KL/FI, Fokker-Planck/KL derivative, EM conditional-FP, or theorem statements." /-- Cycle-54 middle audit of the repeated analytic-interface sprint. This is source-to-Lean synchronization for the middle role. It checks that the upper ledger's five interfaces are actually consumed by the theorem DAGs in paper order and keeps the lower packet focused on the theorem-useful Euler-Maruyama conditional-law/Fokker-Planck backend. -/ def cycle54MainSkeletonAnalyticMiddleContract : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteDerivativeSource objective := "Cycle 54 middle: audit the upper analytic-interface re-check against the six theorem consumers, sharpen the lower packet for appendix.tex:1354-1387, and keep all slow analytic backends below formalized status." sourceLabels := [ "appendix.tex:47-79", "main_body.tex:202-215", "appendix.tex:168-252", "appendix.tex:724-951", "main_body.tex:359-395", "appendix.tex:1313-1603", "appendix.tex:1354-1387" ] analyticInterfaces := [ "Gronwall: theorem consumers use SALD.saldGronwallEndpointCalculusContract plus cycle-36/41 assembly wrappers; the source-level endpoint-safe differentiability/FTC bridge and theorem-specific coefficient regularity stay obligations.", "DV: theorem consumers use dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, and theorem-specific finite-log-mgf witnesses; common space, absolute continuity, finite KL/log-likelihood, measurability, and finite log-mgf stay source-cited or obligations.", "LSI/KL/FI: theorem consumers use SALD.saldLsiKlFiDensityTestContract and cycle-43 density/entropy/Fisher-chain helpers; zero-set handling, admissible sqrt-density test or approximation, vector Fisher chain rule, and finite theorem-level KL/FI stay obligations.", "Continuous Fokker-Planck/KL derivative: forward-KL and continuous general moving-target consumers use SALD.forwardKlDerivativeSideConditionContract, SALD.generalMovingTargetDerivativeCandidateContract, and the cycle-50/52 scalar handoffs; density, boundary, integration-by-parts, transport, and schedule calculus stay obligations.", "Euler-Maruyama interpolation Fokker-Planck: discrete theorem consumers use SALD.discreteForwardKlEmInterpolationSideConditionContract, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, the cycle-48/53 endpoint-law handoffs, and SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff for the cycle-54 sigma-weighted divergence regrouping; conditional drift, density/absolute-continuity, weak FP, KL differentiation, and integration by parts stay obligations." ] theoremRoute := [ "1. thm:forward-KL remains routed through continuous derivative, LSI, DV velocity, endpoint schedule, and Gronwall side conditions.", "2. thm:forward-KL-discrete remains routed through EM conditional-FP, frozen defect, LSI, DV velocity, stitched Gronwall, and accumulated-error interfaces.", "3. prop:guided_path_residual remains routed through normalizer differentiation, guided-density differentiation, divergence cancellation, and mean-zero residual obligations.", "4. thm:general-moving-target-SALD remains routed through continuous general derivative, residual LSI/DV, sigma-weighted Gronwall, endpoint/exponent, and pure-contraction obligations.", "5. thm:unified-forward-KL remains the source specialization of the continuous general theorem via guided residual and the correction-field transport bridge.", "6. thm:general-moving-target-SALD-discrete remains routed through general EM endpoint/conditional-FP, frozen delta, discrete KL derivative/LSI, residual DV, constant schedule, and Gronwall stitching." ] modeDiscipline := [ "faithfulPaper Phase 1: use only original main_body.tex, appendix.tex, and iteration_complexity.tex; sald_version_2.tex remains excluded.", "Do not change theorem statements, constants, alpha ranges, sigma factors, source labels, or proof routes.", "Do not classify endpoint-law handoffs as Brownian construction, regular conditional drift, density, weak Fokker-Planck, or KL derivative proofs.", "Keep systematic SLT/SDE backfill deferred until theorem skeleton interfaces remain green." ] nonGoals := [ "No new scalar-only lemma unless it directly discharges appendix.tex:1354-1387 endpoint or conditional-law obligations.", "No imported SLT or external theorem is promoted for DV, LSI, disintegration, concentration, or EM one-step analysis.", "No hidden endpoint, density, absolute-continuity, finite-KL/FI, finite-log-mgf, boundary, smoothness, conditional-law, or stitched-interval assumptions are added to a paper theorem.", "No alternate proof route replaces derivative -> LSI -> DV -> Gronwall or the paper EM/frozen-delta route." ] lowerPacket := [ "Target exactly SALD.generalMovingTargetDiscreteDerivativeCandidateContract / SALD.generalMovingTargetDiscreteDerivativeObligation / sald.general_moving_target_discrete.kl_derivative.", "First lower sub-slice remains appendix.tex:1354-1387 after the endpoint-law handoff: common probability space, hat rho_s and tilde pi_s density/absolute-continuity, regular conditional drift bar b_{k,s}, weak conditional Fokker-Planck equation, KL differentiation under the integral, and integration-by-parts side conditions.", "Use SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation only as pushforward endpoint-law bookkeeping from named interpolation identities.", "If conditional-FP is blocked, sharpen the source-cited interface by naming the common-space, absolute-continuity, finite-KL/FI, measurability, endpoint, conditional-law, and weak-FP hypotheses rather than modifying theorem statements." ] reviewerChecklist := [ "SALD.cycle54MainSkeletonAnalyticMiddleObligation is listed by all six theorem contracts and in saldDependenciesForLabel through the cycle-54 dependency list.", "SALD.cycle54MainSkeletonAnalyticInterfaceDag contains ASTIS.SALD.cycle54.middle_interface_audit between the upper re-check and lower packet.", "The conversion window and proof-obligation ledger contain the cycle-54 middle audit and keep appendix.tex:1354-1387 as the lower target.", "Gronwall, DV, LSI/KL/FI, continuous derivative, EM conditional-FP, and theorem statements are not promoted beyond their current statuses.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle54MainSkeletonAnalyticInterfaceLedger", "SALD.cycle54MainSkeletonAnalyticInterfaceObligation", "SALD.cycle49MainSkeletonAnalyticMiddleObligation", "SALD.cycle50ForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle52GuidedGeneralSkeletonMiddleObligation", "SALD.cycle53UnifiedDiscreteGeneralMiddleObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "SALD.cycle54GeneralMovingTargetDiscreteEmFpLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] status := ProofStatus.obligation /-- Cycle-54 middle obligation tying the analytic-interface audit to the six theorem contracts and the lower EM conditional-FP packet. -/ def cycle54MainSkeletonAnalyticMiddleObligation : ProofObligation where id := "sald.main_skeleton.cycle54_middle_interface_audit" statement := "Cycle 54 middle verifies that the upper analytic-interface ledger is consumed in source order by thm:forward-KL, thm:forward-KL-discrete, prop:guided_path_residual, thm:general-moving-target-SALD, thm:unified-forward-KL, and thm:general-moving-target-SALD-discrete. It sharpens the lower target to appendix.tex:1354-1387 for common-space, density/absolute-continuity, regular conditional drift, weak EM Fokker-Planck, KL differentiation, and integration-by-parts inputs, while keeping all unproved analytic backends obligation or sourceCited." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle54MainSkeletonAnalyticMiddleContract", "SALD.cycle54MainSkeletonAnalyticInterfaceObligation", "SALD.cycle49MainSkeletonAnalyticMiddleObligation", "SALD.cycle50ForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle52GuidedGeneralSkeletonMiddleObligation", "SALD.cycle53UnifiedDiscreteGeneralMiddleObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Middle workflow audit only. It does not add theorem hypotheses, change source constants, or promote Gronwall, DV, LSI/KL/FI, continuous FP/KL derivative, EM conditional-FP, or theorem statements." /-- Cycle-54 proof-DAG pane for the repeated analytic-interface ledger. -/ def cycle54MainSkeletonAnalyticInterfaceDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle54.analytic_interface_recheck" interface := "Upper re-check of the five slow source-cited or obligation interfaces after the theorem skeleton route is wired: Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL derivative, and EM interpolation Fokker-Planck." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle54MainSkeletonAnalyticInterfaceLedger", "SALD.cycle54MainSkeletonAnalyticInterfaceObligation", "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle53UnifiedDiscreteGeneralSkeletonObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle54.middle_interface_audit" interface := "Middle audit of the upper cycle-54 analytic ledger against the six theorem consumers, with the lower packet narrowed to appendix.tex:1354-1387 EM conditional-law/Fokker-Planck and KL-derivative inputs." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle54MainSkeletonAnalyticMiddleContract", "SALD.cycle54MainSkeletonAnalyticMiddleObligation", "SALD.cycle54MainSkeletonAnalyticInterfaceObligation", "SALD.cycle49MainSkeletonAnalyticMiddleObligation", "SALD.cycle53UnifiedDiscreteGeneralMiddleObligation", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "sald.general_moving_target_discrete.kl_derivative" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.cycle54_em_fp_sigma_split" interface := "Lower proof-producing algebra for appendix.tex:1380-1387: under the supplied weak FP identity, Laplacian split, and abstract divergence linearity, regroup the sigma_eta^2/2 Laplacian contribution into the source conditional-FP divergence form." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "SALD.cycle54GeneralMovingTargetDiscreteEmFpLowerObligation", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] reusedBy := ["sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle54.lower_packet.general_discrete_em_fp" interface := "Lower packet remains on appendix.tex:1354-1387 for the discrete general EM conditional-law/Fokker-Planck and KL-derivative backend, using the cycle-53 Measure.map endpoint-law handoff only as endpoint bookkeeping and the cycle-54 sigma split only as divergence algebra." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeObligation", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "SALD.cycle54GeneralMovingTargetDiscreteEmFpLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] reusedBy := ["thm:general-moving-target-SALD-discrete", "cycle54 lower work"] status := ProofStatus.obligation } ] /-- Cycle-55 upper packet for returning the global analytic-interface audit to the continuous forward-KL skeleton. The cycle focus is deliberately narrow: consume the cycle-54 five-backend check, then re-wire `thm:forward-KL` through the already named source-cited interfaces in the exact paper order. No analytic backend or theorem statement is promoted by this packet. -/ def cycle55ForwardKlSkeletonUpperPacket : ForwardKlUpperPacket where objective := "Cycle 55 upper: after the cycle-54 analytic-interface re-check, wire those five source-cited or obligation-level interfaces back into the faithful continuous thm:forward-KL skeleton, matching main_body.tex:238-247 and appendix.tex:164-252 without changing constants, source labels, theorem status, or proof route." sourceLabels := [ "thm:forward-KL", "proof:thm:forward-KL", "proof:thm:forward-KL:derivative", "proof:thm:forward-KL:dv-energy", "proof:thm:forward-KL:gronwall", "eq:SALD", "eq:FP-eq", "eq:LSI-KL-FI", "lem:dv_variation", "lem:gronwall", "def:alpha-complexity", "proof:thm:forward-KL-discrete:conditional-fp" ] modeDiscipline := [ "faithfulPaper Phase 1: keep the original theorem statement from main_body.tex:238-247 and the appendix proof route from appendix.tex:164-252 fixed; sald_version_2.tex remains excluded.", "Before assigning lower work, explicitly check the five slow backends: endpoint-safe Gronwall, common-space DV with finite KL/log-mgf, LSI/KL/FI density-test and Fisher-chain bridge, continuous Fokker-Planck/KL derivative identity, and downstream EM interpolation endpoint/conditional-law Fokker-Planck.", "Route the proof only as derivative -> LSI -> inverse-schedule time change -> DV velocity energy -> Gronwall, preserving K(t), a(t), b(t), both 1/2 coefficients, alpha in (0,alpha0], and the two source exponent displays.", "Endpoint, density, absolute-continuity, finite-KL/FI, finite-log-mgf, boundary, schedule, coefficient-regularity, and interval-integrability facts remain named obligations rather than hidden theorem assumptions." ] nonGoals := [ "Do not restate thm:forward-KL, change SALD.continuousForwardKlStatementContract, or promote SALD.continuousSaldContract above contractOnly.", "Do not prove or replace Gronwall, DV, LSI/KL/FI, the continuous Fokker-Planck/KL derivative, or the EM interpolation backend in this upper packet.", "Do not merge the source exponent factors, change the coefficient (1/2)*dot{s}(t)^(-1)*alpha^(-1), or substitute an alternate entropy/transport proof route.", "Do not begin broad SLT, concentration, measure-theory, or SDE backfill before the theorem-level route remains synchronized under review." ] lowerPacket := [ "Middle should synchronize this cycle-55 upper route with conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, and the thm:forward-KL proof DAG.", "Preferred lower target remains SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative over appendix.tex:168-228.", "First lower sub-slice: mass conservation, KL differentiation under the integral, SALD Fokker-Planck substitution, boundary/no-flux integration by parts, and the -FI identification from appendix.tex:168-185.", "Second lower sub-slice: target transport velocity, target integration by parts, Cauchy-Schwarz/Young with the exact 1/2 share, and the L2 velocity term from appendix.tex:187-208.", "Third lower sub-slice: inverse-schedule chain rule, slowed-velocity square scaling, and dot{s}(t)*dot t(s(t))^2 = dot{s}(t)^(-1) from appendix.tex:218-228.", "Keep LSI/KL/FI, DV finite-log-mgf, Gronwall endpoint side conditions, and EM interpolation as separate named interfaces unless their exact analytic backends compile locally." ] reviewerChecklist := [ "SALD.continuousSaldContract lists SALD.cycle55ForwardKlSkeletonObligation while remaining contractOnly.", "SALD.forwardKlProofDag contains ASTIS.SALD.forward_KL.cycle55_continuous_skeleton_route after the cycle-54 analytic re-check and cycle-50 route nodes.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes the cycle-55 packet, obligation, DAG node, and lower-packet node.", "All five slow analytic backends remain ProofStatus.obligation or ProofStatus.sourceCited unless their full analytic dependencies build locally.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-55 obligation tying the re-checked analytic interfaces to the continuous forward-KL theorem route. -/ def cycle55ForwardKlSkeletonObligation : ProofObligation where id := "sald.forward_kl.cycle55_continuous_skeleton_route" statement := "Cycle 55 upper consumes the cycle-54 five-backend analytic-interface re-check and re-wires thm:forward-KL specifically: main_body.tex:238-247 stays fixed, appendix.tex:168-228 routes through the continuous KL derivative/Fokker-Planck interface and eq:LSI-KL-FI, appendix.tex:230-241 routes through the source-cited DV finite-log-mgf witness and alpha-complexity interface, and appendix.tex:244-252 routes through lem:gronwall with endpoint and coefficient side conditions. The downstream EM interpolation interface stays visible only as a sibling slow backend for discrete theorem reuse." source := saldForwardKlProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle55ForwardKlSkeletonUpperPacket", "SALD.cycle54MainSkeletonAnalyticInterfaceLedger", "SALD.cycle54MainSkeletonAnalyticInterfaceObligation", "SALD.cycle54MainSkeletonAnalyticMiddleObligation", "SALD.cycle50ForwardKlSkeletonObligation", "SALD.cycle50ForwardKlSkeletonMiddleObligation", "SALD.cycle50ForwardKlDerivativeLowerObligation", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "sald.forward_kl.endpoint_schedule_identities", "sald.forward_kl.gronwall_application", "sald.forward_kl.gronwall_side_conditions", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "This is an upper workflow obligation and theorem-route synchronization record, not an analytic proof. It adds no hidden assumptions to thm:forward-KL and does not promote Gronwall, DV, LSI/KL/FI, Fokker-Planck/KL differentiation, EM interpolation, or the theorem statement." /-- Cycle-55 middle audit for the focused continuous forward-KL route. This synchronizes the cycle-55 upper route with the conversion window, proof-obligation ledger, and lower packet. It keeps `thm:forward-KL` on the paper route derivative -> LSI -> DV -> Gronwall and selects the continuous KL derivative/Fokker--Planck backend for lower work. -/ def cycle55ForwardKlSkeletonMiddleContract : ForwardKlMiddleSourceToLeanContract where sourceStatement := saldForwardKlSource sourceProof := saldForwardKlProofSource derivativeSource := saldForwardKlDerivativeSource dvSource := saldForwardKlDvEnergySource gronwallSource := saldForwardKlGronwallSource objective := "Cycle 55 middle: synchronize the cycle-55 upper forward-KL route with the source-to-Lean ledger, verify main_body.tex:238-247 and appendix.tex:164-252 in paper order after the cycle-54 five-backend re-check, and keep sald.forward_kl.kl_derivative over appendix.tex:168-228 as the lower-ready backend without changing constants or theorem status." sourceStepMap := [ "main_body.tex:238-247 fixes the theorem statement: C_LSI(t)>=0, finite E_alpha0(pi_t,v_t), alpha in (0,alpha0], rho_s is the SALD law, and the final bound has the two source exponent factors plus the residual alpha-complexity integral.", "appendix.tex:168-185 differentiates KL(rho_s||tilde pi_s), uses mass conservation, substitutes the SALD Fokker--Planck equation, integrates by parts, and identifies the first term as -FI.", "appendix.tex:187-208 builds the slowed target velocity tilde v_s=dot t(s)*v_{t(s)}, evaluates the target-time term by integration by parts, and applies Cauchy--Schwarz/Young with the exact 1/2 and 1/2 split.", "appendix.tex:210-228 applies eq:LSI-KL-FI and the inverse-schedule chain rule, preserving the t-time velocity coefficient (1/2)*dot{s}(t)^(-1).", "appendix.tex:230-241 applies lem:dv_variation with Z=alpha*||v_t||^2, uses the alpha0-to-alpha finite-log-mgf witness, and rewrites the log-mgf quotient as mathfrak E_alpha(pi_t,v_t).", "appendix.tex:244-252 applies lem:gronwall with a(t)=dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1) and b(t)=(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t), then performs only the paper exponent split and residual-exponent drop." ] leanStepMap := [ "Use SALD.cycle54MainSkeletonAnalyticInterfaceObligation, SALD.cycle54MainSkeletonAnalyticMiddleObligation, SALD.cycle50ForwardKlSkeletonMiddleObligation, and SALD.cycle55ForwardKlSkeletonObligation as parent route checks.", "Keep SALD.continuousForwardKlStatementContract and SALD.continuousSaldContract contractOnly; this middle audit adds no theorem hypotheses.", "Route appendix.tex:168-228 through SALD.forwardKlDerivativeCandidateContract, SALD.forwardKlDerivativeSideConditionContract, SALD.forwardKlDerivativeObligation, sald.forward_kl.density_boundary_regular, sald.forward_kl.schedule_time_change, and sald.forward_kl.kl_derivative.", "Route appendix.tex:210-217 through SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi; density, zero-set, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher-chain assumptions remain obligations.", "Route appendix.tex:230-241 through dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, SALD.forwardKlDvFiniteLogMgfWitnessContract, sald.forward_kl.dv_finite_log_mgf_witness, and sald.forward_kl.dv_energy_bound.", "Route appendix.tex:244-252 through SALD.saldGronwallEndpointCalculusContract, SALD.forwardKlGronwallInstantiationContract, SALD.forwardKlGronwallSideConditionContract, sald.forward_kl.gronwall_side_conditions, and sald.forward_kl.gronwall_application.", "Keep SALD.discreteForwardKlEmInterpolationSideConditionContract and sald.discrete_forward_kl.em_interpolation_fp visible only as the downstream EM/Fokker--Planck slow backend for discrete reuse.", "Select the next lower target as SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative over appendix.tex:168-228." ] citedResultInterfaces := [ "lem:gronwall remains the endpoint-safe differentiability/FTC and coefficient-regularity obligation; cycle 55 middle only records the theorem-specific instantiation.", "lem:dv_variation remains source-cited; the forward-KL use must still supply common-space, absolute-continuity, finite KL/log-likelihood, measurability, finite log-mgf, and positive-alpha witnesses.", "eq:LSI-KL-FI remains the density-test, zero-set, admissibility, entropy-identity, finite KL/FI, and Fisher-chain obligation.", "The continuous Fokker--Planck/KL derivative is the selected local SDE/measure-analysis backend and is not promoted by this middle audit.", "The Euler--Maruyama interpolation Fokker--Planck interface remains a downstream discrete obligation, not a hidden continuous forward-KL assumption." ] obligations := [ "sald.forward_kl.cycle55_continuous_skeleton_route", "sald.forward_kl.cycle55_middle_route_audit", "sald.forward_kl.kl_derivative", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "probability.lsi_to_kl_fi", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "sald.forward_kl.gronwall_side_conditions", "sald.forward_kl.gronwall_application", "sald.discrete_forward_kl.em_interpolation_fp" ] lowerPacket := [ "Target exactly SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative.", "First lower sub-slice: appendix.tex:168-185 mass conservation, KL differentiation under the integral, SALD Fokker--Planck substitution, boundary/no-flux integration by parts, and -FI identification.", "Second lower sub-slice: appendix.tex:187-208 target transport velocity, target integration by parts, Cauchy--Schwarz/Young with the exact 1/2 share, and the L2 velocity term.", "Third lower sub-slice: appendix.tex:218-228 inverse-schedule chain rule, slowed-velocity square scaling, and dot{s}(t)*dot t(s(t))^2=dot{s}(t)^(-1).", "Leave LSI/KL/FI, DV finite-log-mgf, Gronwall endpoint/exponent conditions, and EM interpolation as separate named interfaces unless their exact analytic backends compile locally." ] reviewerChecklist := [ "SALD.continuousSaldContract lists SALD.cycle55ForwardKlSkeletonMiddleObligation while remaining contractOnly.", "SALD.forwardKlProofDag contains ASTIS.SALD.forward_KL.cycle55_middle_route_audit after ASTIS.SALD.forward_KL.cycle55_continuous_skeleton_route.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes SALD.cycle55ForwardKlSkeletonMiddleContract, SALD.cycle55ForwardKlSkeletonMiddleObligation, and sald.forward_kl.cycle55_middle_route_audit.", "No theorem statement, source coefficient, source label, source-file selection, or analytic backend status is changed.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-55 middle obligation tying the continuous forward-KL route audit to the lower derivative/Fokker--Planck packet. -/ def cycle55ForwardKlSkeletonMiddleObligation : ProofObligation where id := "sald.forward_kl.cycle55_middle_route_audit" statement := "Cycle 55 middle audits the focused continuous thm:forward-KL skeleton after the upper route: main_body.tex:238-247 remains unchanged, appendix.tex:168-252 is routed through the named continuous KL derivative/Fokker--Planck, LSI/KL/FI, DV finite-log-mgf, and Gronwall endpoint/exponent interfaces in paper order, and the next lower target is sald.forward_kl.kl_derivative over appendix.tex:168-228. The downstream EM interpolation backend remains a visible sibling obligation for discrete reuse only." source := saldForwardKlProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle55ForwardKlSkeletonUpperPacket", "SALD.cycle55ForwardKlSkeletonObligation", "SALD.cycle55ForwardKlSkeletonMiddleContract", "SALD.cycle54MainSkeletonAnalyticInterfaceObligation", "SALD.cycle54MainSkeletonAnalyticMiddleObligation", "SALD.cycle50ForwardKlSkeletonObligation", "SALD.cycle50ForwardKlSkeletonMiddleObligation", "SALD.cycle50ForwardKlDerivativeLowerObligation", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "sald.forward_kl.gronwall_side_conditions", "sald.forward_kl.gronwall_application", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp" ] note := "This is a middle workflow obligation and source-to-Lean route audit, not an analytic proof. It adds no hidden density, boundary, endpoint, absolute-continuity, finite-log-mgf, finite-KL/FI, schedule, or coefficient-regularity assumptions to thm:forward-KL." /-- Cycle-55 lower obligation for the first continuous derivative scalar slice. -/ def cycle55ForwardKlDerivativeMassLowerObligation : ProofObligation where id := "sald.forward_kl.cycle55_derivative_mass_lower" statement := "Cycle 55 lower compiles the first source-shaped scalar handoff in appendix.tex:168-185: after the KL derivative backend supplies the raw derivative split with the mass term, mass conservation supplies that term as zero, and the SALD Fokker-Planck/integration-by-parts backend supplies the first term as -FI, SALD.forwardKlMassConservationFirstTermFisherScalar derives the reduced derivative display dK = -FI + targetTerm. The analytic mass conservation, differentiation-under-integral, Fokker-Planck, boundary, and FI-identification facts remain obligations." source := saldForwardKlDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle55ForwardKlSkeletonMiddleContract", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlMassConservationDropScalar", "SALD.forwardKlMassConservationFirstTermFisherScalar", "SALD.forwardKlFirstTermFisherSubstitutionScalar", "sald.forward_kl.kl_derivative", "sald.forward_kl.density_boundary_regular", "eq:SALD", "eq:FP-eq" ] note := "This is only theorem-independent Real equality bookkeeping for the first derivative slice. It does not close sald.forward_kl.kl_derivative, prove the Fokker-Planck equation, or promote thm:forward-KL." /-- Cycle-55 proof-DAG pane for the focused continuous forward-KL route. -/ def cycle55ForwardKlSkeletonDag : List ProofDagBlock := [ { id := "ASTIS.SALD.forward_KL.cycle55_continuous_skeleton_route" interface := "Cycle-55 focused route: consume the cycle-54 five-backend re-check and cycle-50 forward-KL route, then wire main_body.tex:238-247 and appendix.tex:164-252 through derivative/Fokker-Planck, LSI/KL/FI, DV finite-log-mgf, Gronwall endpoint/exponent, and downstream EM-interface visibility without changing the theorem display." source := saldForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle55ForwardKlSkeletonUpperPacket", "SALD.cycle55ForwardKlSkeletonObligation", "SALD.cycle55ForwardKlSkeletonMiddleContract", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle54MainSkeletonAnalyticInterfaceObligation", "SALD.cycle54MainSkeletonAnalyticMiddleObligation", "SALD.cycle50ForwardKlSkeletonObligation", "SALD.cycle50ForwardKlSkeletonMiddleObligation", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract" ] reusedBy := ["thm:forward-KL", "ASTIS-SALD-001 cycle 55"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle55_middle_route_audit" interface := "Cycle-55 middle audit: verify the upper continuous forward-KL route in source order, keep the theorem display and constants fixed, and select appendix.tex:168-228 sald.forward_kl.kl_derivative as the lower backend while LSI, DV, Gronwall, and EM interfaces remain separate obligations." source := saldForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle55ForwardKlSkeletonMiddleContract", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle55ForwardKlSkeletonObligation", "SALD.cycle54MainSkeletonAnalyticMiddleObligation", "SALD.cycle50ForwardKlSkeletonMiddleObligation", "SALD.cycle50ForwardKlDerivativeLowerObligation", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "sald.forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract" ] reusedBy := ["thm:forward-KL", "cycle 55 lower derivative packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle55_lower_packet.kl_derivative" interface := "Lower packet selected by cycle 55: keep proof-producing work on appendix.tex:168-228 for sald.forward_kl.kl_derivative, starting with mass conservation, KL differentiation, SALD Fokker-Planck substitution, integration by parts, and inverse-schedule calculus while LSI, DV, Gronwall, and EM backends stay separate." source := saldForwardKlDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle55ForwardKlSkeletonObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "SALD.cycle55ForwardKlDerivativeMassLowerObligation", "SALD.forwardKlMassConservationDropScalar", "SALD.forwardKlMassConservationFirstTermFisherScalar", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar", "SALD.forwardKlDerivativeDvGronwallCoefficientOfKlFiVelocityScalingScalar" ] reusedBy := ["sald.forward_kl.kl_derivative", "thm:forward-KL"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle55_derivative_mass_lower" interface := "Cycle-55 lower proof-producing scalar slice for appendix.tex:168-185: drop the mass-conservation term from the raw KL derivative split, then compose with the supplied first-term=-FI identity to recover dK=-FI+targetTerm before the target-transport and LSI steps." source := saldForwardKlDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle55ForwardKlDerivativeMassLowerObligation", "SALD.forwardKlMassConservationDropScalar", "SALD.forwardKlMassConservationFirstTermFisherScalar", "SALD.forwardKlFirstTermFisherSubstitutionScalar", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.kl_derivative" ] reusedBy := ["sald.forward_kl.kl_derivative", "thm:forward-KL", "cycle 55 lower handoff"] status := ProofStatus.obligation } ] /-- Cycle-56 upper packet for returning the main skeleton sprint to the discrete forward-KL theorem. The focus is the theorem route, not a new analytic proof: consume the existing source-cited EM/Fokker-Planck interfaces, the cycle-51 derivative scalar handoff, and the cycle-55 continuous forward-KL rewire, then keep the discrete Gronwall/accumulated-error bridge as the next lower theorem-level backend. -/ def cycle56DiscreteForwardKlSkeletonUpperPacket : DiscreteForwardKlUpperPacket where objective := "Cycle 56 upper: wire thm:forward-KL-discrete through the existing source-cited EM endpoint/conditional-Fokker-Planck interfaces, cycle-51 derivative/LSI scalar handoff, DV velocity witness, and appendix.tex:526-592 Gronwall/accumulated-error bridge, matching main_body.tex:299-323 without changing constants or theorem status." sourceLabels := [ "thm:forward-KL-discrete", "proof:thm:forward-KL-discrete", "eq:frozen_interp_terminal_disc_prop_additive_final", "proof:thm:forward-KL-discrete:conditional-fp", "proof:thm:forward-KL-discrete:derivative", "proof:thm:forward-KL-discrete:dv-velocity", "proof:thm:forward-KL-discrete:gronwall", "proof:thm:forward-KL-discrete:accumulated-error", "thm:forward-KL", "eq:LSI-KL-FI", "lem:dv_variation", "lem:gronwall" ] modeDiscipline := [ "faithfulPaper Phase 1: keep main_body.tex:299-323 and appendix.tex:260-592 fixed; sald_version_2.tex remains excluded.", "Five-backend check 1, Gronwall: SALD.saldGronwallEndpointCalculusContract and SALD.discreteForwardKlGronwallInstantiationContract expose endpoint-safe differentiability/FTC, stitched intervals, coefficient regularity, and exponent rewrite obligations.", "Five-backend check 2, DV: dvVariationalFormulaInterface saldDvVariationSource and SALD.discreteForwardKlDvFiniteLogMgfWitnessContract expose common-space, absolute-continuity, finite KL, finite log-mgf, measurability, and positive-alpha scaling for nu=hat rho_s, mu=tilde pi_s.", "Five-backend check 3, LSI/KL/FI: SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi expose density, zero-set convention, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule obligations.", "Five-backend check 4, continuous derivative: SALD.forwardKlDerivativeCandidateContract, SALD.forwardKlDerivativeSideConditionContract, and the cycle-55 mass handoff keep the continuous Fokker-Planck/KL derivative backend visible for reused transport and schedule side conditions without promoting it.", "Five-backend check 5, EM interpolation: SALD.discreteForwardKlEmInterpolationSideConditionContract exposes endpoint laws, conditional drift, conditional-law density, conditional Fokker-Planck, Laplacian split, and stitched interval interfaces as source-cited obligations." ] nonGoals := [ "Do not restate thm:forward-KL-discrete, change SALD.discreteForwardKlStatementContract, or promote SALD.discreteSaldContract above contractOnly.", "Do not prove or replace the EM conditional-Fokker-Planck backend, frozen-defect lemma, LSI/KL/FI density-test, source-cited DV formula, Gronwall, endpoint stitching, residual exponent, or accumulated-error bridge in this upper packet.", "Do not add hidden density, absolute-continuity, endpoint, finite-log-mgf, boundary, coefficient-regularity, inverse-schedule, or stitched-interval assumptions to the theorem statement.", "Do not start broad SLT, concentration, measure-theory, or SDE backfill until this theorem-level route remains stable under review." ] lowerPacket := [ "Middle should synchronize this cycle-56 upper route with conversion-windows/ASTIS-SALD-001.md and proof-obligations/ASTIS-SALD-001.md, preserving the two-way Lean/Markdown/source map.", "Lower should target exactly SALD.discreteForwardKlGronwallInstantiationContract / SALD.discreteForwardKlGronwallAccumulationObligation / sald.discrete_forward_kl.gronwall_accumulation over appendix.tex:526-592.", "First lower sub-slice: consume the existing sald.discrete_forward_kl.kl_derivative and sald.discrete_forward_kl.dv_velocity_bound interfaces to instantiate the source a(t) and b(t) in Eq. KL-derivative-7-discrete.", "Second lower sub-slice: keep endpoint law stitching and K(T)=KL(rho_K^eta||pi_T), K(0)=KL(rho_0||pi_0) as named side conditions rather than new theorem assumptions.", "Third lower sub-slice: route the linear slowdown t(s)=s/r, residual exponent, barGamma, and barDelta collection to SALD.discreteForwardKlAccumulatedErrorBridgeContract without changing the main-body coefficients." ] reviewerChecklist := [ "SALD.discreteSaldContract lists SALD.cycle56DiscreteForwardKlSkeletonObligation while remaining contractOnly.", "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle56_theorem_interface_route after the cycle-51 derivative lower route and before the reusable EM/defect/DV/Gronwall nodes.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes the cycle-56 upper packet, obligation, DAG node, and the explicit EM/Fokker-Planck interface names.", "All five slow analytic backends remain ProofStatus.obligation or ProofStatus.sourceCited unless their full analytic dependencies build locally.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-56 obligation tying the discrete theorem route to the existing source-cited EM/Fokker-Planck and Gronwall/accumulated-error interfaces. -/ def cycle56DiscreteForwardKlSkeletonObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle56_theorem_interface_route" statement := "Cycle 56 upper wires thm:forward-KL-discrete at theorem level: main_body.tex:299-323 stays fixed; appendix.tex:260-385 is consumed through explicit EM endpoint, conditional-drift density, conditional-Fokker-Planck, Laplacian split, and stitched-interval interfaces; appendix.tex:388-491 uses the cycle-51 derivative/LSI scalar handoff; appendix.tex:493-523 uses the discrete DV velocity witness; and appendix.tex:526-592 remains the selected Gronwall/linear-slowdown accumulated-error backend. The EM/Fokker-Planck backend is source-cited obligation data, not a proof in this cycle." source := saldForwardKlDiscreteProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle56DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle54MainSkeletonAnalyticInterfaceObligation", "SALD.cycle54MainSkeletonAnalyticMiddleObligation", "SALD.cycle55ForwardKlSkeletonObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlSkeletonObligation", "SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "sald.discrete_forward_kl.kl_derivative", "SALD.discreteForwardKlPostLsiDerivativeBoundScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfKlFiScalar", "SALD.frozenDeltaCrossLipSaldContract", "sald.discrete_forward_kl.frozen_delta_cross_lip", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "sald.discrete_forward_kl.gronwall_accumulation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "thm:forward-KL" ] note := "This is an upper workflow obligation and route check. It adds no hidden assumptions, proves no EM/Fokker-Planck theorem, and does not promote Gronwall, DV, LSI/KL/FI, KL differentiation, endpoint stitching, accumulated-error collection, or thm:forward-KL-discrete." /-- Cycle-56 middle audit for the discrete forward-KL theorem route. This synchronizes the upper cycle-56 route with the source-to-Lean ledger. It keeps the already wired derivative and DV interfaces as inputs and selects the appendix Gronwall/accumulated-error backend as the next lower packet. -/ def cycle56DiscreteForwardKlSkeletonMiddleContract : DiscreteForwardKlEmDefectAccumulationMiddleContract where sourceStatement := saldForwardKlDiscreteSource interpolationSource := saldForwardKlDiscreteInterpolationSource derivativeSource := saldForwardKlDiscreteDerivativeSource gronwallSource := saldForwardKlDiscreteGronwallSource accumulatedSource := saldForwardKlDiscreteAccumulatedErrorSource objective := "Cycle 56 middle: audit the upper thm:forward-KL-discrete route against main_body.tex:299-323 and appendix.tex:260-592, verify that EM/Fokker-Planck is consumed only through named source-cited interfaces, and select sald.discrete_forward_kl.gronwall_accumulation over appendix.tex:526-592 as the next lower backend." sourceStepMap := [ "main_body.tex:299-323 fixes the theorem display: linear slowdown t(s)=s/r, inherited forward-KL assumptions, score Lipschitz assumptions, alpha0' finite score and 1+M complexities, the 4*eta^2*L_space^2<1/2 step-size condition, and the exact T/(r*alpha), 2*r*eta^2*barGamma/alpha', r^(-1)*A_alpha, and 2*r*eta*barDelta constants.", "appendix.tex:260-385 supplies the EM interpolation, endpoint law identities, conditional frozen drift, conditional-drift Fokker-Planck equation, Laplacian split, and stitched interval inputs; all remain named obligations or source-cited interfaces.", "appendix.tex:388-491 is already routed through the cycle-51 derivative/LSI scalar handoff: EM-FP derivative input, frozen-cross quarter-FI bound, moving Young quarter-FI bound, and eq:LSI-KL-FI.", "appendix.tex:493-523 is already routed through the discrete DV velocity witness with nu=hat rho_s, mu=tilde pi_s, and Z=alpha*||v_{t(s)}||^2.", "appendix.tex:526-553 changes from s to t and produces the exact source coefficient dot{s}(t)*C_LSI(t)-dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t).", "appendix.tex:557-592 applies lem:gronwall to the general-schedule inequality and is the selected lower backend before the downstream linear-slowdown accumulated-error bridge to the main-body display." ] leanStepMap := [ "Keep SALD.discreteForwardKlStatementContract and SALD.discreteSaldContract unchanged and contractOnly.", "Use SALD.cycle54MainSkeletonAnalyticMiddleObligation, SALD.cycle55ForwardKlSkeletonMiddleObligation, SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation, and SALD.cycle56DiscreteForwardKlSkeletonObligation as parent route checks.", "Route appendix.tex:260-385 through SALD.discreteForwardKlEmInterpolationSideConditionContract, sald.discrete_forward_kl.em_endpoint_laws, sald.discrete_forward_kl.conditional_drift_density, sald.discrete_forward_kl.em_conditional_fokker_planck, sald.discrete_forward_kl.stitched_interval_regularity, and sald.discrete_forward_kl.em_interpolation_fp.", "Route appendix.tex:388-491 through SALD.discreteForwardKlDerivativeCandidateContract, SALD.cycle51DiscreteForwardKlDerivativeLowerObligation, SALD.discreteForwardKlPostLsiDerivativeBoundScalar, SALD.discreteForwardKlPostLsiDerivativeBoundOfKlFiScalar, SALD.frozenDeltaCrossLipSaldContract, and SALD.saldLsiKlFiDensityTestContract.", "Route appendix.tex:493-523 through dvVariationalFormulaInterface saldDvVariationSource, SALD.discreteForwardKlDvFiniteLogMgfWitnessContract, sald.discrete_forward_kl.dv_finite_log_mgf_witness, and sald.discrete_forward_kl.dv_velocity_bound.", "Route appendix.tex:526-592 through SALD.discreteForwardKlGronwallInstantiationContract, SALD.discreteForwardKlGronwallAccumulationObligation, sald.discrete_forward_kl.gronwall_accumulation, SALD.discreteForwardKlAccumulatedErrorBridgeContract, and sald.discrete_forward_kl.accumulated_error_bridge.", "Select the next lower target as SALD.discreteForwardKlGronwallInstantiationContract / SALD.discreteForwardKlGronwallAccumulationObligation / sald.discrete_forward_kl.gronwall_accumulation, using the derivative and DV outputs as explicit hypotheses." ] citedResultInterfaces := [ "EM endpoint/conditional-law Fokker-Planck remains a source-cited obligation interface; this middle audit does not prove Brownian construction, disintegration, density/AC, weak FP, or endpoint stitching.", "eq:LSI-KL-FI remains the density-test, zero-set, admissibility, entropy-identity, finite KL/FI, and Fisher-chain obligation consumed by the cycle-51 scalar handoff.", "lem:dv_variation remains source-cited and theorem-specific common-space, absolute-continuity, finite-KL/log-mgf, measurability, and positive-alpha witnesses remain obligations.", "lem:gronwall remains the endpoint-safe differentiability/FTC, stitched-interval, coefficient-regularity, and exponent-rewrite obligation selected for lower work.", "The continuous forward-KL cycle-55 mass handoff is route context only; it is not used to promote the discrete KL derivative or theorem statement." ] obligations := [ "sald.discrete_forward_kl.cycle56_theorem_interface_route", "sald.discrete_forward_kl.cycle56_middle_route_audit", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.kl_derivative", "sald.discrete_forward_kl.frozen_delta_cross_lip", "probability.lsi_to_kl_fi", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge" ] lowerPacket := [ "Target exactly SALD.discreteForwardKlGronwallInstantiationContract / SALD.discreteForwardKlGronwallAccumulationObligation / sald.discrete_forward_kl.gronwall_accumulation.", "First lower sub-slice: consume sald.discrete_forward_kl.kl_derivative and sald.discrete_forward_kl.dv_velocity_bound through SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar and SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged to instantiate the source K'(t) <= -a(t)*K(t)+b(t) inequality from appendix.tex:526-553.", "Second lower sub-slice: expose stitched endpoint side conditions K(0)=KL(rho_0||pi_0), K(T)=KL(rho_K^eta||pi_T), and interval compatibility across EM steps as named assumptions, not theorem hypotheses.", "Third lower sub-slice: apply the existing Gronwall endpoint/FTC interface to obtain the appendix general-schedule display before the linear-slowdown collection.", "Leave linear slowdown, residual exponent, barGamma/barDelta collection, EM conditional-FP, frozen-defect specialization, LSI/KL/FI, and DV finite-log-mgf as separate named interfaces unless their exact backends compile locally." ] reviewerChecklist := [ "SALD.discreteSaldContract lists SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation while remaining contractOnly.", "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle56_middle_route_audit immediately after the cycle-56 theorem route.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle56DiscreteForwardKlSkeletonMiddleContract, SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation, and sald.discrete_forward_kl.cycle56_middle_route_audit.", "No EM/Fokker-Planck, KL derivative, frozen-defect, LSI, DV, Gronwall, accumulated-error, or theorem statement status is promoted.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-56 middle obligation tying the discrete route audit to the selected Gronwall lower packet. -/ def cycle56DiscreteForwardKlSkeletonMiddleObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle56_middle_route_audit" statement := "Cycle 56 middle audits the discrete thm:forward-KL-discrete route after the upper wrapper: main_body.tex:299-323 remains unchanged; appendix.tex:260-385 EM endpoint/conditional-Fokker-Planck interfaces are consumed only as named obligations; appendix.tex:388-491 uses the cycle-51 derivative/LSI scalar handoff; appendix.tex:493-523 uses the discrete DV velocity witness; and appendix.tex:526-592 is selected as the next lower Gronwall accumulation backend. The accumulated-error bridge stays downstream and all slow analytic backends remain obligation or source-cited." source := saldForwardKlDiscreteProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle56DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle56DiscreteForwardKlSkeletonObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleContract", "SALD.cycle54MainSkeletonAnalyticInterfaceObligation", "SALD.cycle54MainSkeletonAnalyticMiddleObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "sald.discrete_forward_kl.kl_derivative", "SALD.discreteForwardKlPostLsiDerivativeBoundOfKlFiScalar", "SALD.frozenDeltaCrossLipSaldContract", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "sald.discrete_forward_kl.gronwall_accumulation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "sald.discrete_forward_kl.accumulated_error_bridge", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI" ] note := "This is a middle workflow obligation and source-to-Lean route audit, not an analytic proof. It adds no hidden endpoint, density, absolute-continuity, finite-log-mgf, conditional-law, boundary, stitched-interval, or coefficient assumptions to thm:forward-KL-discrete." /-- Cycle-56 lower obligation for the discrete Gronwall accumulation backend. -/ def cycle56DiscreteForwardKlGronwallLowerObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle56_gronwall_lower" statement := "Cycle 56 lower recovers the interrupted discrete forward-KL Gronwall route by compiling the scalar post-DV time-change handoff in appendix.tex:526-553. Once the EM/KL derivative backend, frozen-defect bound, LSI comparison, and DV velocity estimate supply the s-time inequality, SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar produces the exact t-time coefficient a(t)=dot{s}(t)*C_LSI(t)-dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t) and residual b(t)=dot{s}(t)^(-1)*E_alpha(pi_t,v_t)+2*dot{s}(t)*eta*Delta(t), while SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged packages that result pointwise in t for lem:gronwall." source := saldForwardKlDiscreteGronwallSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle56DiscreteForwardKlSkeletonMiddleContract", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged", "SALD.discreteForwardKlTimeChangeSquareCoefficientRewriteScalar", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlGronwallAccumulationObligation", "sald.discrete_forward_kl.gronwall_accumulation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "sald.discrete_forward_kl.kl_derivative", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_velocity_bound", "sald.forward_kl.schedule_time_change", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI" ] note := "The compiled scalar theorem is real-order coefficient bookkeeping only. It does not prove EM conditional Fokker-Planck, KL differentiation, frozen-defect specialization, LSI/KL/FI, DV, Gronwall, stitched endpoint regularity, residual-exponent monotonicity, accumulated-error collection, or thm:forward-KL-discrete." /-- Cycle-58 upper packet for the final unified/discrete general theorem refresh. This packet consumes the now-clean cycle-56 discrete forward-KL route and the cycle-57 guided/general route. It keeps the cycle focus on theorem-level closure for `thm:unified-forward-KL` and `thm:general-moving-target-SALD-discrete`; broad cited-theory backfill remains deferred until the final theorem skeleton route is accepted. -/ def cycle58UnifiedDiscreteGeneralUpperPacket : GeneralVaSaldUpperPacket where objective := "Cycle 58 upper: no recovery is needed after the cycle-57 reviewer/build pass; Phase 1 is stable enough to re-close the unified and discrete general theorem route but not enough for broad cited-theory backfill; the single lower packet that reduces the largest remaining proof risk is sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600, using the cycle-53 derivative/DV handoff and cycle-56 Gronwall recovery pattern as inputs." sourceLabels := [ "thm:unified-forward-KL", "proof:thm:unified-forward-KL", "eq:residual-term", "eq:poisson-eq", "eq:SALD_Ito", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:general-moving-target-SALD-discrete", "eq:general_moving_target_KL_bound_discrete", "eq:general_KL_derivative_7_discrete", "eq:general_KL_derivative_8_discrete", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "eq:general_discrete_delta_def", "lem:frozen_delta_cross_lip", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] modeDiscipline := [ "Global phase judgment: cycle 57 passed reviewer and build gate, so there is no failed cycle to recover.", "Global phase judgment: Phase 1 theorem-skeleton translation is stable for a final unified/discrete general route refresh, but broad measure-theory or SDE backfill must still wait for this route to pass review.", "Global phase judgment: the largest remaining theorem-level proof risk is the discrete general Gronwall/display backend, because it stitches the EM intervals, applies the constant inverse-schedule rewrite, and matches the displayed theorem bound.", "Five-backend check 1, Gronwall: SALD.saldGronwallEndpointCalculusContract, SALD.generalMovingTargetDiscreteGronwallInstantiationContract, SALD.generalMovingTargetDiscreteGronwallSideConditionContract, and SALD.gronwallAnalyticObligation expose endpoint-safe differentiability, FTC/order integration, coefficient regularity, stitched endpoint laws, and display matching.", "Five-backend check 2, DV: dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, SALD.generalMovingTargetDvFiniteLogMgfWitnessContract, and SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract expose common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and alpha-scaling witnesses.", "Five-backend check 3, LSI/KL/FI: SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi keep density, zero-set convention, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule assumptions explicit.", "Five-backend check 4, continuous derivative: SALD.generalMovingTargetDerivativeCandidateContract, SALD.generalMovingTargetDerivativeObligation, and the cycle-57 derivative split keep the Fokker-Planck/KL derivative route source-cited or obligation-level for thm:general-moving-target-SALD and thm:unified-forward-KL.", "Five-backend check 5, EM interpolation: SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation, SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff, and sald.general_moving_target_discrete.em_interpolation_fp expose endpoint laws, conditional-law density, regular conditional drift, weak Fokker-Planck, and KL-differentiation side conditions without promoting the backend.", "FaithfulPaper Phase 1: use main_body.tex:359-395, appendix.tex:949-951, and appendix.tex:1313-1603; sald_version_2.tex remains excluded." ] nonGoals := [ "Do not restate thm:unified-forward-KL, thm:general-moving-target-SALD, or thm:general-moving-target-SALD-discrete.", "Do not promote SALD.unifiedForwardKlContract, SALD.generalVaSaldContract, SALD.generalVaSaldDiscreteContract, Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL differentiation, or EM interpolation above their existing contractOnly/sourceCited/obligation statuses.", "Do not replace the source route by a direct VA-SALD proof, Girsanov/path-space comparison, Pinsker/Talagrand/PI argument, or broad SLT import.", "Do not change the sigma_eta factors, doubled residual coefficient, Gamma/Delta terms, alpha ranges, constant inverse-schedule assumption, source labels, or theorem displays.", "Do not start systematic measure-theory or SDE backfill in this upper packet; if a backend is too large, sharpen the named source-cited interface or proof obligation." ] lowerPacket := [ "Middle should synchronize this cycle-58 upper route with conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, SALD.generalVaSaldProofDag, and SALD.generalVaSaldDiscreteProofDag.", "Lower should target exactly SALD.generalMovingTargetDiscreteGronwallSideConditionContract / SALD.generalMovingTargetDiscreteGronwallSideConditionObligation / sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600.", "First lower sub-slice: endpoint and stitching interface for K(t)=KL(hat rho_{s(t)}||pi_t), including K(0)=KL(rho_0||pi_0), K(T)=KL(rho_K^eta||pi_T), interval compatibility, and constant schedule admissibility.", "Second lower sub-slice: coefficient rewrites from eq:general_KL_derivative_8_discrete to the theorem display, reusing SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar, SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar, SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar, and SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar.", "Third lower sub-slice: Gronwall regularity/display matching for the exact a(t) and b(t), using the cycle-56 recovered Gronwall-accumulation route only as a pattern and not as a theorem substitution.", "If narrow backfill is attempted after the route check, keep it to one measure-theory detail around endpoint-law stitching or Measure.map-style law equality, guided by local SLT material as reference-only unless the imported theorem builds locally." ] reviewerChecklist := [ "SALD.unifiedForwardKlContract and SALD.generalVaSaldDiscreteContract list SALD.cycle58UnifiedDiscreteGeneralSkeletonObligation while remaining contractOnly.", "SALD.generalVaSaldProofDag and SALD.generalVaSaldDiscreteProofDag include ASTIS.SALD.unified_discrete_general.cycle58_upper_route and ASTIS.SALD.general_moving_target_discrete.cycle58_lower_packet.gronwall_display.", "SALD.saldDependenciesForLabel for thm:unified-forward-KL and thm:general-moving-target-SALD-discrete includes the cycle-58 packet, obligation, DAG, and lower-packet node.", "The five slow analytic interfaces remain ProofStatus.obligation or ProofStatus.sourceCited; no theorem contract or large analytic backend is marked formalized.", "The lower packet is theorem-level Gronwall/display stitching, not a new transcript expansion or isolated scalar sublemma.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-58 obligation tying the final unified/discrete general theorem refresh to explicit source-cited interfaces. -/ def cycle58UnifiedDiscreteGeneralSkeletonObligation : ProofObligation where id := "sald.unified_discrete_general.cycle58_upper_route" statement := "Cycle 58 upper records that cycle 57 needs no recovery, rechecks the five slow analytic backends, wires thm:unified-forward-KL through prop:guided_path_residual, the correction-field transport bridge, and thm:general-moving-target-SALD, and wires thm:general-moving-target-SALD-discrete through the general EM endpoint/conditional-Fokker-Planck interface, frozen-delta side conditions, discrete KL derivative/DV handoff, eq:LSI-KL-FI, lem:dv_variation, and lem:gronwall. The selected lower packet is sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600; all theorem statuses and slow analytic interfaces remain below formalized." source := saldGeneralMovingTargetDiscreteSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle58UnifiedDiscreteGeneralUpperPacket", "SALD.cycle54MainSkeletonAnalyticInterfaceObligation", "SALD.cycle54MainSkeletonAnalyticMiddleObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle57GuidedGeneralSkeletonObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralDerivativeSplitLowerObligation", "SALD.cycle53UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle53UnifiedDiscreteGeneralMiddleObligation", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "SALD.generalMovingTargetStatementContract", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "SALD.generalMovingTargetDiscreteStatementContract", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "SALD.generalMovingTargetDiscreteGronwallInstantiationContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.constant_schedule_stitching", "sald.general_moving_target_discrete.cycle20_gronwall_middle", "sald.general_moving_target_discrete.gronwall_side_conditions", "SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar", "SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar", "sald.general_moving_target_discrete.unified_specialization", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "This is an upper workflow obligation and theorem-route refresh, not an analytic proof. It adds no hidden endpoint, density, absolute-continuity, finite-log-mgf, finite-KL/FI, conditional-law, schedule, sigma-positivity, or coefficient-regularity assumptions to the source theorem statements." /-- Cycle-58 middle audit for the unified/discrete general route. This source-to-Lean synchronization layer checks the upper route against the paper order and hands lower work to the discrete general Gronwall/display backend over `appendix.tex:1573-1600`. -/ def cycle58UnifiedDiscreteGeneralMiddleContract : GeneralVaSaldGuidedPathMiddleContract where guidedResidualSource := saldGuidedResidualSource generalContinuousSource := saldGeneralMovingTargetSource unifiedSource := saldUnifiedForwardKlSource discreteSource := saldGeneralMovingTargetDiscreteSource objective := "Cycle 58 middle: audit the upper unified/discrete general route in source order, verify that thm:unified-forward-KL remains only the c_t=u_t, m_t=w_t specialization of thm:general-moving-target-SALD, verify that thm:general-moving-target-SALD-discrete consumes the EM, frozen-delta, derivative/DV, LSI, and Gronwall interfaces, and select sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600 as the lower packet." sourceStepMap := [ "main_body.tex:359-368 uses prop:guided_path_residual and eq:poisson-eq to turn u_t+w_t into a transport velocity for pi_t.", "main_body.tex:372-395 states thm:unified-forward-KL with the correction-field complexity E_alpha(pi_t,w_t) and the exact sigma_t^{-2} dot{s}(t)^{-1} coefficients.", "appendix.tex:949-951 proves thm:unified-forward-KL only by specializing thm:general-moving-target-SALD with c_t=u_t and m_t=w_t.", "appendix.tex:1313-1347 states thm:general-moving-target-SALD-discrete with sigma_eta factors, the doubled residual coefficient, Gamma/Delta terms, alpha ranges, and the constant inverse-schedule assumption.", "appendix.tex:1354-1511 supplies the general EM endpoint/conditional-FP, frozen residual field split, and two sigma_eta^2/8 Young shares through existing obligations and scalar handoffs.", "appendix.tex:1513-1570 applies eq:LSI-KL-FI and lem:dv_variation to the residual m_t under the EM interpolation law.", "appendix.tex:1573-1583 defines K(t), stitches the EM endpoint laws, differentiates through s(t), and uses dot t(s(t))=dot s(t)^{-1}.", "appendix.tex:1584-1597 gives the t-time Gronwall input with the theorem-display coefficients.", "appendix.tex:1600 applies lem:gronwall; appendix.tex:1603 records the discrete guided VA-SALD specialization c_t=u_t." ] leanStepMap := [ "Use SALD.cycle58UnifiedDiscreteGeneralUpperPacket and SALD.cycle58UnifiedDiscreteGeneralSkeletonObligation as parent route data; do not replace the cycle-57 continuous general route.", "Route unified VA-SALD through SALD.unifiedForwardKlSpecializationContract, sald.unified_forward_kl.transport_velocity_bridge, sald.unified_forward_kl.specialization, SALD.generalVaSaldContract, and the cycle-57 guided/general middle and lower obligations.", "Route appendix.tex:1313-1347 through SALD.generalMovingTargetDiscreteStatementContract and SALD.generalVaSaldDiscreteContract; both remain contractOnly.", "Route appendix.tex:1354-1511 through SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, sald.general_moving_target_discrete.em_interpolation_fp, sald.general_moving_target_discrete.kl_derivative, cycle-53 derivative/DV scalar handoffs, and the frozen-delta obligations.", "Route appendix.tex:1513-1570 through SALD.saldLsiKlFiDensityTestContract, probability.lsi_to_kl_fi, dvVariationalFormulaInterface saldDvVariationSource, SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract, and sald.general_moving_target_discrete.dv_m_energy.", "Route appendix.tex:1573-1600 through SALD.generalMovingTargetDiscreteGronwallInstantiationContract, SALD.generalMovingTargetDiscreteGronwallSideConditionContract, SALD.cycle20GeneralVaSaldDiscreteGronwallMiddleObligation, sald.general_moving_target_discrete.gronwall_application, and sald.general_moving_target_discrete.gronwall_side_conditions.", "Select lower work exactly at SALD.generalMovingTargetDiscreteGronwallSideConditionContract / SALD.generalMovingTargetDiscreteGronwallSideConditionObligation / sald.general_moving_target_discrete.gronwall_side_conditions.", "Keep local SLT one-step and disintegration material reference-only; this cycle performs no broad SLT import and marks no SLT theorem formalized." ] citedResultInterfaces := [ "lem:gronwall remains an endpoint-safe differentiability, FTC, coefficient-regularity, endpoint-stitching, and display-matching obligation.", "lem:dv_variation remains source-cited through common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and alpha-scaling witnesses.", "eq:LSI-KL-FI remains the density, zero-set, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher-chain obligation.", "The continuous Fokker-Planck/KL derivative is consumed upstream through sald.general_moving_target.kl_derivative and the cycle-57 derivative split, not reproved for the unified theorem.", "The EM interpolation backend remains sald.general_moving_target_discrete.em_interpolation_fp; endpoint law and sigma-regrouping helpers compile only under explicit hypotheses and do not prove the conditional-FP backend." ] obligations := [ "sald.unified_discrete_general.cycle58_upper_route", "sald.unified_discrete_general.cycle58_middle_route_audit", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions", "sald.general_moving_target_discrete.unified_specialization" ] lowerPacket := [ "Target exactly SALD.generalMovingTargetDiscreteGronwallSideConditionContract / SALD.generalMovingTargetDiscreteGronwallSideConditionObligation / sald.general_moving_target_discrete.gronwall_side_conditions.", "First sub-slice: appendix.tex:1573-1583 endpoint stitching for K(t), including K(0)=KL(rho_0||pi_0), K(T)=KL(rho_K^eta||pi_T), interval compatibility, and constant-schedule admissibility.", "Second sub-slice: appendix.tex:1584-1597 coefficient rewrites from eq:general_KL_derivative_8_discrete to the theorem display, reusing SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar, SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar, SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar, and SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar.", "Third sub-slice: appendix.tex:1600 Gronwall regularity and exact display matching for the source a(t) and b(t).", "Do not restate either theorem, change coefficients, or promote Gronwall, DV, LSI/KL/FI, EM interpolation, KL derivative, frozen-delta, or endpoint stitching beyond the existing statuses." ] reviewerChecklist := [ "SALD.unifiedForwardKlContract and SALD.generalVaSaldDiscreteContract list SALD.cycle58UnifiedDiscreteGeneralMiddleObligation while remaining contractOnly.", "SALD.generalVaSaldProofDag and SALD.generalVaSaldDiscreteProofDag contain ASTIS.SALD.unified_discrete_general.cycle58_middle_route_audit.", "SALD.saldDependenciesForLabel entries for thm:unified-forward-KL and thm:general-moving-target-SALD-discrete include the cycle-58 middle contract, middle obligation, and route-audit node.", "The lower packet is sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600, not a new KL derivative, DV, LSI, or EM proof.", "No theorem statement, source constant, source label, source-file scope, SLT reuse status, or analytic backend status changes.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-58 middle obligation tying the route audit to lower Gronwall/display work. -/ def cycle58UnifiedDiscreteGeneralMiddleObligation : ProofObligation where id := "sald.unified_discrete_general.cycle58_middle_route_audit" statement := "Cycle 58 middle audits the final unified/discrete general route after the upper refresh: thm:unified-forward-KL remains the paper specialization of the continuous general theorem through the correction-field transport bridge, thm:general-moving-target-SALD-discrete remains the source route through EM interpolation, frozen-delta, LSI, residual DV, and Gronwall, and the next lower target is sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600 for endpoint stitching, constant-schedule coefficient rewrites, coefficient regularity, and exact theorem-display matching." source := saldGeneralMovingTargetDiscreteGronwallSideConditionSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle58UnifiedDiscreteGeneralUpperPacket", "SALD.cycle58UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle58UnifiedDiscreteGeneralMiddleContract", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralDerivativeSplitLowerObligation", "SALD.cycle53UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle53GeneralMovingTargetDiscreteDerivativeDvLowerObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "SALD.generalMovingTargetStatementContract", "SALD.generalVaSaldContract", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.gronwall_side_conditions", "SALD.generalMovingTargetDiscreteStatementContract", "SALD.generalVaSaldDiscreteContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "SALD.generalMovingTargetDiscreteGronwallInstantiationContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionObligation", "SALD.cycle20GeneralVaSaldDiscreteGronwallMiddleObligation", "SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar", "SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar", "sald.general_moving_target_discrete.constant_schedule_stitching", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI" ] note := "This is middle-role synchronization data and lower-packet selection, not a theorem proof. It adds no hidden endpoint, density, absolute-continuity, regular conditional law, finite-log-mgf, finite-KL/FI, schedule, sigma-positivity, coefficient-regularity, or Gronwall assumptions to the paper statements." /-- Cycle-58 proof-DAG pane for the final unified/discrete general route refresh and selected lower packet. -/ def cycle58UnifiedDiscreteGeneralDag : List ProofDagBlock := [ { id := "ASTIS.SALD.unified_discrete_general.cycle58_upper_route" interface := "Cycle-58 upper route refresh: consume the cycle-57 guided/general route and cycle-53 unified/discrete route, recheck the five slow analytic interfaces, and keep thm:unified-forward-KL plus thm:general-moving-target-SALD-discrete wired through the paper specialization and EM/frozen-delta/DV/Gronwall chain." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle58UnifiedDiscreteGeneralUpperPacket", "SALD.cycle58UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.cycle53UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "sald.unified_forward_kl.specialization", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions" ] reusedBy := ["thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "cycle58 lower Gronwall/display packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.unified_discrete_general.cycle58_middle_route_audit" interface := "Cycle-58 middle route audit: verify the unified specialization and discrete general EM/frozen-delta/LSI/DV/Gronwall chain against the paper order, then hand off appendix.tex:1573-1600 to the discrete general Gronwall/display side-condition backend." source := saldGeneralMovingTargetDiscreteGronwallSideConditionSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle58UnifiedDiscreteGeneralMiddleContract", "SALD.cycle58UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle58UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.cycle53UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle53GeneralMovingTargetDiscreteDerivativeDvLowerObligation", "sald.unified_forward_kl.specialization", "sald.general_moving_target.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions" ] reusedBy := ["thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "cycle58 lower Gronwall/display packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.cycle58_lower_packet.gronwall_display" interface := "Selected lower packet for cycle 58: refine sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600, covering endpoint stitching for K(t), constant inverse-schedule coefficient rewrites, Gronwall regularity, and exact matching of eq:general_moving_target_KL_bound_discrete." source := saldGeneralMovingTargetDiscreteGronwallSideConditionSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle58UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle58UnifiedDiscreteGeneralMiddleObligation", "SALD.generalMovingTargetDiscreteGronwallInstantiationContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionObligation", "SALD.cycle20GeneralVaSaldDiscreteGronwallMiddleObligation", "SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar", "SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.constant_schedule_stitching", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "sald.gronwall.integrating_factor" ] reusedBy := ["sald.general_moving_target_discrete.gronwall_side_conditions", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation } ] /-- Cycle-59 upper ledger for the post-cycle-58 analytic-interface sprint. The previous cycle closed the unified/discrete general route through reviewer and build. This ledger records the upper-level phase judgment required before any cited-theory backfill: the five slow analytic backends are explicit enough for theorem-skeleton routing, but only one narrow backend should be assigned next. -/ def cycle59MainSkeletonAnalyticInterfaceLedger : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteGronwallSideConditionSource objective := "Cycle 59 upper: after the accepted cycle-58 reviewer/build gate, re-check the five source-cited analytic interfaces and wire the post-cycle-58 theorem route back into all six SALD theorem consumers. Phase 1 is stable enough for one narrow cited-theory/SDE backfill, but not for broad reusable API work; the single lower packet remains sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "proof:thm:forward-KL:derivative", "proof:thm:forward-KL-discrete:conditional-fp", "proof:thm:general-moving-target-SALD:derivative", "proof:thm:general-moving-target-SALD-discrete:derivative", "proof:thm:general-moving-target-SALD-discrete:gronwall-side-conditions", "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Global phase judgment: cycle 58 passed reviewer and build, so no failed previous cycle must be recovered before cycle 59 work.", "Global phase judgment: Phase 1 theorem-skeleton translation is stable enough to begin exactly one narrow cited-theory/SDE backend after this ledger, but broad measure-theory, SLT, or reusable API reorganization remains deferred.", "Global phase judgment: the lower packet that best reduces remaining proof risk is sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600, because it is the last theorem-display step after the EM derivative/DV handoff.", "Gronwall: SALD.saldGronwallCandidateContract, SALD.saldGronwallEndpointCalculusContract, SALD.saldGronwallExponentRewriteContract, theorem-specific Gronwall instantiation contracts, and the cycle-36/41 wrappers expose endpoint-safe differentiability/FTC, interval integrability, endpoint evaluation, exponent algebra, stitched regularity, and coefficient/display side conditions; lem:gronwall remains ProofStatus.obligation.", "Donsker-Varadhan: dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, and theorem-specific finite-log-mgf witnesses expose common-space probability measures, absolute continuity, finite KL/log-likelihood, selected-test measurability, finite log-mgf, alpha monotonicity, and positive-alpha scaling; the Boucheron variational equality remains ProofStatus.sourceCited.", "LSI/KL/FI: SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, and the cycle-43 density/entropy/Fisher-chain helpers expose rho << pi, Radon-Nikodym density, zero-density convention, admissible sqrt-density test or approximation, entropy identity, finite KL/FI, and Fisher chain rule; probability.lsi_to_kl_fi remains ProofStatus.obligation.", "Continuous Fokker-Planck/KL derivative: SALD.forwardKlDerivativeCandidateContract, SALD.forwardKlDerivativeSideConditionContract, SALD.generalMovingTargetDerivativeCandidateContract, and the cycle-50/52/57 scalar handoffs expose mass conservation, KL differentiation under the integral, SALD/general Fokker-Planck equations, target transport, integration by parts, LSI handoff, and inverse-schedule calculus; analytic derivative backends remain ProofStatus.obligation.", "Euler-Maruyama interpolation Fokker-Planck: SALD.discreteForwardKlEmInterpolationSideConditionContract, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, the cycle-48/53 endpoint-law handoffs, SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff, and the cycle-58 pointwise Gronwall-input wrapper expose endpoint laws, named-law pushforward bookkeeping, regular conditional drift, density/absolute-continuity, weak conditional FP, Laplacian split, stitched intervals, time change, and common-space assumptions without promoting the EM backend." ] theoremRoute := [ "1. thm:forward-KL remains wired through continuous KL derivative/Fokker-Planck, eq:LSI-KL-FI, DV velocity witness, endpoint schedule, and Gronwall side conditions.", "2. thm:forward-KL-discrete remains wired through EM endpoint/conditional-FP, frozen defect, LSI, DV velocity, stitched Gronwall, and accumulated-error interfaces, including the recovered cycle-56 Gronwall input wrapper.", "3. prop:guided_path_residual remains wired through normalizer differentiation, guided-density differentiation, divergence cancellation, and mean-zero residual obligations.", "4. thm:general-moving-target-SALD remains wired through the continuous general derivative split, residual LSI/DV, sigma-weighted Gronwall, endpoint/exponent side conditions, and pure-contraction obligation.", "5. thm:unified-forward-KL remains the appendix specialization of thm:general-moving-target-SALD through prop:guided_path_residual, the correction-field transport bridge, and c_t=u_t, m_t=w_t.", "6. thm:general-moving-target-SALD-discrete remains wired through general EM endpoint/conditional-FP, frozen-delta, discrete KL derivative/LSI, residual DV, constant-schedule time change, the cycle-58 pointwise Gronwall input, and final Gronwall/display stitching." ] modeDiscipline := [ "faithfulPaper Phase 1: use only main_body.tex, appendix.tex, and iteration_complexity.tex; sald_version_2.tex remains excluded.", "Keep source theorem statements, constants, alpha ranges, sigma factors, endpoint laws, source labels, and proof order fixed.", "Keep every unproved analytic backend below formalized status; already compiled scalar or endpoint-law helpers are dependencies only.", "Systematic cited-theory backfill may begin only one backend at a time after this ledger and must remain tied to a theorem route." ] nonGoals := [ "No source-index rebaseline beyond the acceptance command unless a reviewer reports a source-anchor defect.", "No broad SLT import, concentration port, disintegration port, or reusable teaching API reorganization in this upper packet.", "No hidden endpoint, density, absolute-continuity, finite-KL/FI, finite-log-mgf, boundary, smoothness, conditional-law, sigma-positivity, schedule, coefficient-regularity, or stitched-interval assumption is added to a paper theorem.", "No alternate proof route replaces derivative -> LSI -> DV -> Gronwall, the correction-field specialization, or the paper EM/frozen-delta route." ] lowerPacket := [ "Middle should synchronize this cycle-59 analytic-interface ledger with conversion-windows/ASTIS-SALD-001.md and proof-obligations/ASTIS-SALD-001.md, then keep the theorem route in the order forward-KL, discrete forward-KL, guided residual, general moving-target, unified forward-KL, discrete general moving-target.", "Lower should target exactly SALD.generalMovingTargetDiscreteGronwallSideConditionContract / SALD.generalMovingTargetDiscreteGronwallSideConditionObligation / sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600.", "First lower sub-slice: endpoint stitching for K(t)=KL(hat rho_{s(t)}||pi_t), including K(0)=KL(rho_0||pi_0), K(T)=KL(rho_K^eta||pi_T), interval compatibility, and constant inverse-schedule admissibility.", "Second lower sub-slice: coefficient regularity and exact display matching for a(t) and b(t), reusing SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged, SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput, and the constant-schedule coefficient rewrite scalars only as local inputs.", "If proof-producing work is blocked, sharpen the source-cited interface by naming endpoint, common-space, absolute-continuity, finite-KL/FI, measurability, schedule, and coefficient hypotheses rather than modifying theorem statements." ] reviewerChecklist := [ "SALD.cycle59MainSkeletonAnalyticInterfaceObligation is listed by all six theorem contracts while they remain contractOnly.", "All four theorem proof DAGs include SALD.cycle59MainSkeletonAnalyticInterfaceDag, and saldDependenciesForLabel includes the cycle-59 dependency names for the five slow interfaces and six theorem labels.", "The conversion window and proof-obligation ledger contain the cycle-59 global phase judgment, five-interface check, theorem route, lower packet, and reviewer checklist.", "Gronwall, DV, LSI/KL/FI, continuous derivative, EM conditional-FP, theorem contracts, and SLT reuse statuses are not promoted beyond their current statuses.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle54MainSkeletonAnalyticInterfaceObligation", "SALD.cycle54MainSkeletonAnalyticMiddleObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.cycle58UnifiedDiscreteGeneralMiddleObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput", "SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "sald.general_moving_target_discrete.gronwall_side_conditions" ] status := ProofStatus.obligation /-- Cycle-59 upper obligation for the analytic-interface recheck. -/ def cycle59MainSkeletonAnalyticInterfaceObligation : ProofObligation where id := "sald.main_skeleton.cycle59_analytic_interface_ledger" statement := "Cycle 59 upper records the global phase judgment after the accepted cycle-58 reviewer/build pass: no recovery is needed; Phase 1 theorem-skeleton translation is stable enough for exactly one narrow cited-theory/SDE backend, not broad backfill; and the lower packet that best reduces remaining proof risk is sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600. It rechecks the five slow analytic interfaces and wires them into thm:forward-KL, thm:forward-KL-discrete, prop:guided_path_residual, thm:general-moving-target-SALD, thm:unified-forward-KL, and thm:general-moving-target-SALD-discrete while keeping every unproved backend below formalized." source := saldGeneralMovingTargetDiscreteGronwallSideConditionSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle59MainSkeletonAnalyticInterfaceLedger", "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle54MainSkeletonAnalyticInterfaceObligation", "SALD.cycle54MainSkeletonAnalyticMiddleObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.cycle58UnifiedDiscreteGeneralMiddleObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput", "SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "SALD.forwardKlProofDag", "SALD.discreteForwardKlProofDag", "SALD.generalVaSaldProofDag", "SALD.generalVaSaldDiscreteProofDag" ] note := "This is upper workflow and source-to-Lean route data, not a proof of any analytic backend. It does not add theorem hypotheses or promote Gronwall, DV, LSI/KL/FI, Fokker-Planck/KL differentiation, EM conditional-FP, Gronwall stitching, theorem contracts, or SLT reuse statuses." /-- Cycle-59 middle audit for the analytic-interface ledger. This source-to-Lean synchronization layer checks the upper cycle-59 ledger against the six theorem consumers and keeps lower work on the theorem-level discrete general Gronwall/display side conditions. -/ def cycle59MainSkeletonAnalyticMiddleContract : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteGronwallSideConditionSource objective := "Cycle 59 middle: audit the upper analytic-interface ledger against the source proof order, verify that all six theorem skeletons consume the five slow interfaces explicitly, and keep the lower packet exactly on sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600." sourceLabels := [ "appendix.tex:47-79", "main_body.tex:202-215", "appendix.tex:168-252", "appendix.tex:260-592", "appendix.tex:619-951", "main_body.tex:359-395", "appendix.tex:1313-1603", "appendix.tex:1573-1600" ] analyticInterfaces := [ "Gronwall: theorem consumers use SALD.saldGronwallEndpointCalculusContract, SALD.saldGronwallExponentRewriteContract, theorem-specific side-condition contracts, and cycle-36/41/56/58 pointwise wrappers. Endpoint-safe differentiability, FTC/order integration, stitched endpoint laws, coefficient regularity, and final display matching remain obligations.", "Donsker-Varadhan: theorem consumers use dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, and theorem-specific finite-log-mgf witnesses. Common space, absolute continuity, finite KL/log-likelihood, selected-test measurability, finite log-mgf, alpha monotonicity, and positive-alpha scaling remain source-cited or obligations.", "LSI/KL/FI: theorem consumers use SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, probability.lsi_to_kl_fi, and the cycle-43 density/entropy/Fisher-chain helpers. Zero-set handling, admissible sqrt-density test or approximation, finite KL/FI, and vector Fisher chain rule remain obligations.", "Continuous Fokker-Planck/KL derivative: thm:forward-KL and thm:general-moving-target-SALD consume SALD.forwardKlDerivativeSideConditionContract, SALD.generalMovingTargetDerivativeCandidateContract, and the cycle-50/52/55/57 scalar handoffs. Density, boundary, mass conservation, target transport, integration by parts, and schedule calculus remain obligations.", "Euler-Maruyama interpolation Fokker-Planck: discrete theorem consumers use SALD.discreteForwardKlEmInterpolationSideConditionContract, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, the cycle-48/53 endpoint-law handoffs, SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff, and the cycle-58 pointwise Gronwall-input wrapper. Conditional drift, density/absolute-continuity, weak FP, KL differentiation, and integration by parts remain obligations." ] theoremRoute := [ "1. thm:forward-KL is still derivative -> LSI -> DV velocity -> endpoint schedule -> Gronwall, with source display main_body.tex:238-247 and proof appendix.tex:168-252.", "2. thm:forward-KL-discrete is still EM endpoint/conditional-FP -> frozen defect -> LSI -> DV velocity -> stitched Gronwall -> accumulated error, with source display main_body.tex:299-323 and proof appendix.tex:260-592.", "3. prop:guided_path_residual is still normalizer derivative -> guided-density differentiation -> divergence cancellation -> mean-zero residual over appendix.tex:619-704.", "4. thm:general-moving-target-SALD is still continuous general derivative split -> residual LSI/DV -> sigma-weighted Gronwall -> pure contraction over appendix.tex:724-951.", "5. thm:unified-forward-KL is still the paper specialization of thm:general-moving-target-SALD via prop:guided_path_residual, eq:poisson-eq, c_t=u_t, and m_t=w_t.", "6. thm:general-moving-target-SALD-discrete is still general EM endpoint/conditional-FP -> frozen delta -> discrete KL derivative/LSI -> residual DV -> constant schedule -> cycle-58 pointwise Gronwall input -> final display stitching." ] modeDiscipline := [ "faithfulPaper Phase 1: use only main_body.tex, appendix.tex, and iteration_complexity.tex; sald_version_2.tex remains excluded.", "Do not change theorem statements, constants, alpha ranges, sigma factors, endpoint laws, source labels, or proof order.", "Keep every unproved analytic backend below formalized status; local compiled scalar or endpoint-law helpers stay dependencies only.", "The next lower packet must be a theorem-consumed backend over appendix.tex:1573-1600, not broad SLT/SDE backfill." ] nonGoals := [ "No source-index rebaseline beyond the acceptance command.", "No imported SLT theorem is marked formalized for DV, LSI, concentration, disintegration, or one-step EM analysis.", "No hidden endpoint, density, absolute-continuity, finite-KL/FI, finite-log-mgf, boundary, smoothness, conditional-law, sigma-positivity, schedule, coefficient-regularity, or stitched-interval assumption is added to a theorem.", "No alternate proof route replaces derivative -> LSI -> DV -> Gronwall, the correction-field specialization, or the paper EM/frozen-delta route." ] lowerPacket := [ "Target exactly SALD.generalMovingTargetDiscreteGronwallSideConditionContract / SALD.generalMovingTargetDiscreteGronwallSideConditionObligation / sald.general_moving_target_discrete.gronwall_side_conditions.", "First sub-slice: appendix.tex:1573-1583 endpoint stitching for K(t), including K(0)=KL(rho_0||pi_0), K(T)=KL(rho_K^eta||pi_T), interval compatibility, and constant inverse-schedule admissibility.", "Second sub-slice: appendix.tex:1584-1597 coefficient regularity and exact a(t), b(t) display matching, reusing SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged, SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput, and the constant-schedule coefficient rewrite scalars only as local inputs.", "Third sub-slice: appendix.tex:1600 endpoint-safe Gronwall application and final theorem-display matching against appendix.tex:1316-1347.", "If proof-producing work blocks, sharpen the endpoint, common-space, absolute-continuity, finite-KL/FI, measurability, schedule, coefficient, and stitched-interval interfaces rather than changing theorem statements." ] reviewerChecklist := [ "SALD.cycle59MainSkeletonAnalyticMiddleObligation is listed by all six theorem contracts while they remain contractOnly.", "SALD.cycle59MainSkeletonAnalyticInterfaceDag contains ASTIS.SALD.cycle59.middle_interface_audit between the upper route rewire and lower packet.", "SALD.saldDependenciesForLabel includes the cycle-59 middle contract, middle obligation, and middle DAG node through cycle59MainSkeletonDependencyNames.", "The conversion window, proof-obligation ledger, and SLT reuse audit contain the cycle-59 middle audit and keep the lower packet on appendix.tex:1573-1600.", "Gronwall, DV, LSI/KL/FI, continuous derivative, EM conditional-FP, theorem contracts, and SLT reuse statuses are not promoted.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle59MainSkeletonAnalyticInterfaceLedger", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation", "SALD.cycle58UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput", "SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "sald.general_moving_target_discrete.gronwall_side_conditions" ] status := ProofStatus.obligation /-- Cycle-59 middle obligation tying the analytic-interface audit to the six theorem consumers and the selected lower Gronwall/display packet. -/ def cycle59MainSkeletonAnalyticMiddleObligation : ProofObligation where id := "sald.main_skeleton.cycle59_middle_interface_audit" statement := "Cycle 59 middle verifies that the upper analytic-interface ledger is consumed in paper order by thm:forward-KL, thm:forward-KL-discrete, prop:guided_path_residual, thm:general-moving-target-SALD, thm:unified-forward-KL, and thm:general-moving-target-SALD-discrete. It keeps the next lower target exactly on sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600 for endpoint stitching, coefficient regularity, endpoint-safe Gronwall use, and exact display matching, while keeping all unproved analytic backends obligation or sourceCited." source := saldGeneralMovingTargetDiscreteGronwallSideConditionSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle59MainSkeletonAnalyticMiddleContract", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation", "SALD.cycle58UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput", "SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionObligation", "sald.general_moving_target_discrete.gronwall_side_conditions", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI" ] note := "Middle workflow audit only. It adds no theorem hypotheses, changes no source constants, and does not promote Gronwall, DV, LSI/KL/FI, continuous FP/KL derivative, EM conditional-FP, discrete Gronwall stitching, theorem contracts, or SLT reuse statuses." /-- Cycle-59 lower obligation for the discrete general Gronwall/display side-condition packet. The proof-producing part of this lower packet is local: it introduces named Gronwall coefficients from the cycle-58 pointwise derivative input and rewrites the final Gronwall endpoint inequality to the theorem's KL endpoints. The analytic endpoint laws, coefficient regularity, stitched absolute continuity, and `lem:gronwall` application remain obligations. -/ def cycle59GeneralMovingTargetDiscreteGronwallLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle59_gronwall_lower" statement := "Cycle 59 lower compiles two local theorem-display wrappers for appendix.tex lines 1573-1600. SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput turns the cycle-58 pointwise t-time derivative inequality into named Gronwall coefficient functions a(t) and b(t) matching eq:general_KL_derivative_8_discrete. SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar rewrites a supplied Gronwall bound from K(T) and K(0) to KL(rho_K^eta||pi_T) and KL(rho_0||pi_0). Stitched endpoint laws, coefficient regularity, endpoint-safe Gronwall, and exact theorem-display instantiation remain in sald.general_moving_target_discrete.gronwall_side_conditions." source := saldGeneralMovingTargetDiscreteGronwallSideConditionSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle59MainSkeletonAnalyticMiddleObligation", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput", "SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionObligation", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.constant_schedule_stitching", "sald.gronwall.integrating_factor" ] note := "Proof-producing local Real/logical wrappers only. This does not prove EM endpoint stitching, coefficient continuity/integrability, the closed-interval Gronwall theorem, or any DV, LSI/KL/FI, KL derivative, conditional-Fokker-Planck, or theorem-level statement." /-- Cycle-59 proof-DAG pane for the post-cycle-58 analytic-interface check. -/ def cycle59MainSkeletonAnalyticInterfaceDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle59.analytic_interface_ledger" interface := "Upper global judgment and five-backend recheck after cycle 58: no recovery, narrow one-backend backfill only, and discrete general Gronwall/display side conditions remain the largest theorem-level risk." source := saldGeneralMovingTargetDiscreteGronwallSideConditionSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle59MainSkeletonAnalyticInterfaceLedger", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation", "SALD.cycle58UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle59.theorem_route_rewire" interface := "Route the five interfaces through the six faithful theorem skeletons in paper order, keeping every theorem contractOnly and every slow backend sourceCited or obligation unless already compiled locally." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.continuousSaldContract", "SALD.discreteSaldContract", "SALD.guidedResidualContract", "SALD.generalVaSaldContract", "SALD.unifiedForwardKlContract", "SALD.generalVaSaldDiscreteContract", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.cycle58UnifiedDiscreteGeneralMiddleObligation" ] reusedBy := ["ASTIS-SALD-001 cycle 59 middle audit"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle59.middle_interface_audit" interface := "Middle audit of the cycle-59 analytic ledger against the six theorem consumers, with the lower packet kept on appendix.tex:1573-1600 endpoint stitching, coefficient regularity, Gronwall, and display matching." source := saldGeneralMovingTargetDiscreteGronwallSideConditionSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle59MainSkeletonAnalyticMiddleContract", "SALD.cycle59MainSkeletonAnalyticMiddleObligation", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation", "SALD.cycle58UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "sald.general_moving_target_discrete.gronwall_side_conditions" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle59.lower_packet.general_discrete_gronwall_side_conditions" interface := "Selected lower packet for cycle 59: continue sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600, focusing endpoint stitching, coefficient regularity, and exact Gronwall-display matching after the cycle-58 pointwise Gronwall input wrapper." source := saldGeneralMovingTargetDiscreteGronwallSideConditionSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle59MainSkeletonAnalyticMiddleObligation", "SALD.generalMovingTargetDiscreteGronwallInstantiationContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionObligation", "SALD.cycle59GeneralMovingTargetDiscreteGronwallLowerObligation", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput", "SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar", "SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar", "SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.constant_schedule_stitching", "sald.general_moving_target_discrete.gronwall_side_conditions", "sald.gronwall.integrating_factor" ] reusedBy := ["sald.general_moving_target_discrete.gronwall_side_conditions", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation } ] /-- Cycle-60 upper packet for the continuous forward-KL skeleton after the accepted cycle-59 route audit. The previous reviewer/build gate accepted the cycle-59 ledger. This packet therefore returns to the cycle focus: make the continuous theorem route consume the source-cited analytic interfaces explicitly, and assign the next lower work to the continuous KL derivative/Fokker-Planck backend rather than to a new source-index rebaseline or a broad cited-theory port. -/ def cycle60ForwardKlSkeletonUpperPacket : ForwardKlUpperPacket where objective := "Cycle 60 upper: after the accepted cycle-59 reviewer/build gate, wire the source-cited analytic interfaces into the faithful continuous thm:forward-KL proof skeleton, matching main_body.tex:238-247 and appendix.tex:164-252 without changing constants, theorem status, proof order, or source labels." sourceLabels := [ "thm:forward-KL", "proof:thm:forward-KL", "proof:thm:forward-KL:derivative", "proof:thm:forward-KL:dv-energy", "proof:thm:forward-KL:gronwall", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "eq:SALD", "eq:FP-eq", "def:alpha-complexity", "proof:thm:forward-KL-discrete:conditional-fp" ] modeDiscipline := [ "Global phase judgment: cycle 59 passed reviewer and build, so no failed previous cycle must be recovered before cycle 60 work.", "Global phase judgment: Phase 1 theorem-skeleton translation is stable enough for focused theorem-level interface wiring, but not for broad cited-theory or reusable API backfill.", "Global phase judgment: the single lower packet that best reduces remaining continuous forward-KL risk is SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative over appendix.tex:168-228.", "Five-backend check: Gronwall has endpoint-safe differentiability/FTC and coefficient-side-condition contracts; DV has common-space, absolute-continuity, finite-KL, finite-log-mgf, measurability, and positive-alpha interfaces; LSI/KL/FI has density, zero-set, admissible-test, entropy-identity, and Fisher-chain interfaces; the continuous Fokker-Planck/KL derivative has mass-conservation, boundary, target-transport, and time-change interfaces; EM interpolation FP remains a downstream discrete sibling backend.", "Route thm:forward-KL in the paper order derivative/Fokker-Planck -> LSI/KL/FI -> inverse schedule -> DV velocity energy -> Gronwall, preserving both 1/2 coefficients, alpha in (0,alpha0], dot{s}(t)^(-1), and the theorem's two exponent displays.", "Keep the six theorem-route order from the cycle-59 ledger visible: forward-KL, discrete forward-KL, guided residual, general moving-target, unified forward-KL, and discrete general moving-target." ] nonGoals := [ "Do not restate thm:forward-KL, change SALD.continuousForwardKlStatementContract, or promote SALD.continuousSaldContract above contractOnly.", "Do not prove or replace Gronwall, DV, LSI/KL/FI, the continuous Fokker-Planck/KL derivative, or the EM interpolation backend in this upper packet.", "Do not change source constants, merge exponent factors, alter the alpha range, add hidden endpoint/density/regularity assumptions, or use sald_version_2.tex.", "Do not start broad SLT, concentration, disintegration, measure-theory, or SDE backfill before this theorem route is reviewed cleanly." ] lowerPacket := [ "Middle should synchronize this cycle-60 upper route with conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, and the thm:forward-KL proof DAG.", "Lower should target exactly SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative over appendix.tex:168-228.", "First sub-slice: appendix.tex:168-185 mass conservation, KL differentiation under the integral, SALD Fokker-Planck substitution, boundary/no-flux integration by parts, and the -FI identification.", "Second sub-slice: appendix.tex:187-208 slowed-target transport velocity, target integration by parts, Cauchy-Schwarz/Young with the exact 1/2 share, and the L2 velocity term.", "Third sub-slice: appendix.tex:218-228 inverse-schedule chain rule, slowed-velocity square scaling, and dot{s}(t)*dot t(s(t))^2 = dot{s}(t)^(-1).", "Leave LSI/KL/FI, DV finite-log-mgf, Gronwall endpoint/exponent side conditions, and EM interpolation as separate named interfaces unless their exact analytic backends compile locally." ] reviewerChecklist := [ "SALD.continuousSaldContract lists SALD.cycle60ForwardKlSkeletonObligation while remaining contractOnly.", "SALD.forwardKlProofDag contains ASTIS.SALD.forward_KL.cycle60_post_cycle59_route and ASTIS.SALD.forward_KL.cycle60_lower_packet.kl_derivative after the cycle-59 and cycle-55 route nodes.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes SALD.cycle60ForwardKlSkeletonUpperPacket, SALD.cycle60ForwardKlSkeletonObligation, and the cycle-60 DAG nodes.", "All five slow analytic backends remain ProofStatus.obligation or ProofStatus.sourceCited unless their full analytic dependencies build locally.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-60 upper obligation tying the accepted cycle-59 analytic ledger back to the focused continuous forward-KL theorem route. -/ def cycle60ForwardKlSkeletonObligation : ProofObligation where id := "sald.forward_kl.cycle60_post_cycle59_route" statement := "Cycle 60 upper records the post-cycle-59 global phase judgment and wires thm:forward-KL through the source-cited analytic interfaces in the paper order: appendix.tex:168-228 continuous KL derivative/Fokker-Planck plus LSI/time change, appendix.tex:230-241 DV velocity-energy with finite-log-mgf witness, and appendix.tex:244-252 Gronwall endpoint/exponent display matching. main_body.tex:238-247 remains the unchanged theorem statement, and the next lower packet is sald.forward_kl.kl_derivative over appendix.tex:168-228." source := saldForwardKlProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle60ForwardKlSkeletonUpperPacket", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation", "SALD.cycle59MainSkeletonAnalyticMiddleObligation", "SALD.cycle55ForwardKlSkeletonObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle55ForwardKlDerivativeMassLowerObligation", "SALD.cycle50ForwardKlSkeletonObligation", "SALD.cycle50ForwardKlSkeletonMiddleObligation", "SALD.cycle50ForwardKlDerivativeLowerObligation", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "sald.forward_kl.endpoint_schedule_identities", "sald.forward_kl.gronwall_side_conditions", "sald.forward_kl.gronwall_application", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "Upper workflow obligation only. It adds no theorem hypotheses and does not promote Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL differentiation, EM interpolation, SLT reuse, or thm:forward-KL." /-- Cycle-60 middle audit for the post-cycle-59 continuous forward-KL route. This source-to-Lean synchronization layer checks that the upper cycle-60 route is consumed by the continuous theorem contract in the same order as `appendix.tex:164-252`, while keeping the lower packet on the continuous KL derivative/Fokker--Planck backend. -/ def cycle60ForwardKlSkeletonMiddleContract : ForwardKlMiddleSourceToLeanContract where sourceStatement := saldForwardKlSource sourceProof := saldForwardKlProofSource derivativeSource := saldForwardKlDerivativeSource dvSource := saldForwardKlDvEnergySource gronwallSource := saldForwardKlGronwallSource objective := "Cycle 60 middle: audit the post-cycle-59 continuous thm:forward-KL route against main_body.tex:238-247 and appendix.tex:164-252, verify that the five source-cited analytic interfaces are consumed in the paper order, and keep SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative over appendix.tex:168-228 as the lower packet without changing theorem constants, statements, labels, or statuses." sourceStepMap := [ "main_body.tex:238-247 fixes the theorem statement: pi_t satisfies LSI with C_LSI(t)>=0, E_alpha0(pi_t,v_t)<+infty for the transport velocity field, alpha lies in (0,alpha0], rho_s is the SALD law, and the terminal bound keeps the two exponent factors and residual alpha-complexity integral.", "appendix.tex:168-185 differentiates KL(rho_s||tilde pi_s), drops the mass term by int partial_s rho_s=0, substitutes the SALD Fokker--Planck equation, integrates by parts, and identifies the first term as -FI.", "appendix.tex:187-208 builds the slowed target velocity tilde v_s=dot t(s)*v_{t(s)}, rewrites partial_s tilde pi_s as -div(tilde v_s tilde pi_s), integrates by parts, and applies Cauchy--Schwarz/Young with the exact one-half split.", "appendix.tex:210-228 applies eq:LSI-KL-FI and the inverse-schedule chain rule to obtain the t-time derivative inequality with coefficient (1/2)*dot{s}(t)^(-1)*||v_t||^2.", "appendix.tex:230-241 applies lem:dv_variation to Z=alpha*||v_t||^2 under the finite-log-mgf witness, producing the exact pre-Gronwall coefficient dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1).", "appendix.tex:244-252 applies lem:gronwall, then performs only the source exponent split and residual exponent drop that yield the main-body theorem display." ] leanStepMap := [ "Use SALD.cycle60ForwardKlSkeletonUpperPacket and SALD.cycle60ForwardKlSkeletonObligation as parent route data after the accepted cycle-59 analytic ledger.", "Keep SALD.continuousForwardKlStatementContract and SALD.continuousSaldContract contractOnly; this middle audit adds no theorem hypotheses.", "Route appendix.tex:168-228 through SALD.forwardKlDerivativeCandidateContract, SALD.forwardKlDerivativeSideConditionContract, SALD.forwardKlDerivativeObligation, SALD.forwardKlDensityBoundaryObligation, SALD.forwardKlScheduleTimeChangeObligation, sald.forward_kl.density_boundary_regular, sald.forward_kl.schedule_time_change, and sald.forward_kl.kl_derivative.", "Route appendix.tex:210-217 through SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi, preserving density, zero-set convention, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher-chain obligations.", "Route appendix.tex:230-241 through dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, SALD.forwardKlDvFiniteLogMgfWitnessContract, sald.forward_kl.dv_finite_log_mgf_witness, and sald.forward_kl.dv_energy_bound.", "Route appendix.tex:244-252 through SALD.saldGronwallEndpointCalculusContract, SALD.forwardKlGronwallInstantiationContract, SALD.forwardKlGronwallSideConditionContract, sald.forward_kl.gronwall_side_conditions, and sald.forward_kl.gronwall_application.", "Keep SALD.discreteForwardKlEmInterpolationSideConditionContract and sald.discrete_forward_kl.em_interpolation_fp visible only as the downstream EM/Fokker--Planck sibling backend for discrete theorem reuse.", "Select the next lower target exactly as SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative over appendix.tex:168-228." ] citedResultInterfaces := [ "lem:gronwall remains the endpoint-safe differentiability/FTC, coefficient-regularity, endpoint-rewrite, interval-integrability, and exponent-display obligation; cycle 60 middle only checks the theorem-specific instantiation.", "lem:dv_variation remains source-cited with common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and alpha-scaling witnesses.", "eq:LSI-KL-FI remains the density-test, zero-set convention, admissible-test, entropy-identity, finite KL/FI, and Fisher-chain-rule obligation.", "The continuous Fokker--Planck/KL derivative identity remains the selected local SDE/measure-analysis backend and is not promoted by this middle audit.", "The Euler--Maruyama interpolation Fokker--Planck endpoint/conditional-law backend remains a downstream discrete obligation, not a hidden assumption of thm:forward-KL." ] obligations := [ "sald.forward_kl.cycle60_post_cycle59_route", "sald.forward_kl.cycle60_middle_route_audit", "sald.forward_kl.kl_derivative", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "probability.lsi_to_kl_fi", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "sald.forward_kl.gronwall_side_conditions", "sald.forward_kl.gronwall_application", "sald.discrete_forward_kl.em_interpolation_fp" ] lowerPacket := [ "Target exactly SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative.", "First lower sub-slice: appendix.tex:168-185 mass conservation, KL differentiation under the integral, SALD Fokker--Planck substitution, boundary/no-flux integration by parts, and -FI identification.", "Second lower sub-slice: appendix.tex:187-208 slowed target transport velocity, target integration by parts, Cauchy--Schwarz/Young with the exact 1/2 share, and the L2 velocity term.", "Third lower sub-slice: appendix.tex:218-228 inverse-schedule chain rule, slowed-velocity square scaling, and dot{s}(t)*dot t(s(t))^2=dot{s}(t)^(-1).", "Leave LSI/KL/FI, DV finite-log-mgf, Gronwall endpoint/exponent conditions, and EM interpolation as separate named interfaces unless their exact analytic backends compile locally." ] reviewerChecklist := [ "SALD.continuousSaldContract lists SALD.cycle60ForwardKlSkeletonMiddleObligation while remaining contractOnly.", "SALD.forwardKlProofDag contains ASTIS.SALD.forward_KL.cycle60_middle_route_audit after ASTIS.SALD.forward_KL.cycle60_post_cycle59_route.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes SALD.cycle60ForwardKlSkeletonMiddleContract, SALD.cycle60ForwardKlSkeletonMiddleObligation, and sald.forward_kl.cycle60_middle_route_audit.", "No theorem statement, source coefficient, source label, source-file selection, SLT reuse entry, or analytic backend status is changed.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-60 middle obligation tying the post-cycle-59 route audit to the selected continuous derivative/Fokker--Planck lower packet. -/ def cycle60ForwardKlSkeletonMiddleObligation : ProofObligation where id := "sald.forward_kl.cycle60_middle_route_audit" statement := "Cycle 60 middle audits the post-cycle-59 continuous thm:forward-KL route: main_body.tex:238-247 remains unchanged; appendix.tex:168-252 is consumed in paper order through the continuous KL derivative/Fokker-Planck interface, eq:LSI-KL-FI, lem:dv_variation, and lem:gronwall; the downstream EM interpolation backend remains visible only for discrete reuse; and the next lower packet stays exactly sald.forward_kl.kl_derivative over appendix.tex:168-228. All theorem statuses and slow analytic interfaces remain below formalized." source := saldForwardKlProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle60ForwardKlSkeletonUpperPacket", "SALD.cycle60ForwardKlSkeletonObligation", "SALD.cycle60ForwardKlSkeletonMiddleContract", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation", "SALD.cycle59MainSkeletonAnalyticMiddleObligation", "SALD.cycle55ForwardKlSkeletonObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle55ForwardKlDerivativeMassLowerObligation", "SALD.cycle50ForwardKlSkeletonMiddleObligation", "SALD.cycle50ForwardKlDerivativeLowerObligation", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "sald.forward_kl.gronwall_side_conditions", "sald.forward_kl.gronwall_application", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "This is a middle workflow obligation and source-to-Lean route audit, not an analytic proof. It adds no hidden density, boundary, endpoint, absolute-continuity, finite-log-mgf, finite-KL/FI, schedule, coefficient-regularity, or EM conditional-law assumptions to thm:forward-KL." /-- Cycle-60 lower obligation for the raw continuous derivative scalar wrapper. -/ def cycle60ForwardKlDerivativeRawLowerObligation : ProofObligation where id := "sald.forward_kl.cycle60_derivative_raw_lower" statement := "Cycle 60 lower compiles the source-shaped scalar handoff for appendix.tex:168-228 starting from the raw KL derivative split with the mass term: after the analytic backend supplies the differentiated KL display, mass conservation, SALD Fokker-Planck/integration-by-parts first-term identity, target Cauchy estimate, KL/FI comparison, slowed-velocity scaling, and inverse-schedule product identity, SALD.forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar derives the t-time pre-DV inequality dK/dt <= -dot{s}(t)*C_LSI(t)*K(t)+(1/2)*dot{s}(t)^(-1)*||v_t||^2." source := saldForwardKlDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle60ForwardKlSkeletonMiddleContract", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar", "SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar", "SALD.forwardKlMassConservationDropScalar", "SALD.forwardKlMassConservationFirstTermFisherScalar", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "SALD.saldLsiKlFiDensityTestContract", "sald.forward_kl.kl_derivative", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "probability.lsi_to_kl_fi" ] note := "This is formalized only as Real/order/equality bookkeeping. It keeps mass conservation, KL differentiation, Fokker-Planck, boundary integration by parts, target transport, LSI density-test, and inverse-function calculus as explicit obligations, and it does not promote thm:forward-KL or sald.forward_kl.kl_derivative." /-- Cycle-60 proof-DAG pane for the post-cycle-59 continuous forward-KL skeleton route. -/ def cycle60ForwardKlSkeletonDag : List ProofDagBlock := [ { id := "ASTIS.SALD.forward_KL.cycle60_post_cycle59_route" interface := "Post-cycle-59 upper route: consume the accepted five-backend analytic ledger and cycle-55 continuous route, then wire main_body.tex:238-247 and appendix.tex:164-252 through derivative/Fokker-Planck, LSI/KL/FI, DV finite-log-mgf, Gronwall endpoint/exponent, and downstream EM-interface visibility without changing the theorem display." source := saldForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle60ForwardKlSkeletonUpperPacket", "SALD.cycle60ForwardKlSkeletonObligation", "SALD.cycle60ForwardKlSkeletonMiddleContract", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation", "SALD.cycle59MainSkeletonAnalyticMiddleObligation", "SALD.cycle55ForwardKlSkeletonObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract" ] reusedBy := ["thm:forward-KL", "ASTIS-SALD-001 cycle 60"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle60_middle_route_audit" interface := "Cycle-60 middle audit: verify the post-cycle-59 continuous forward-KL route in source order, keep the theorem display and constants fixed, and select appendix.tex:168-228 sald.forward_kl.kl_derivative as the lower backend while LSI, DV, Gronwall, and EM interfaces remain separate obligations." source := saldForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle60ForwardKlSkeletonMiddleContract", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle60ForwardKlSkeletonObligation", "SALD.cycle59MainSkeletonAnalyticMiddleObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle55ForwardKlDerivativeMassLowerObligation", "SALD.cycle50ForwardKlDerivativeLowerObligation", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "sald.forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract" ] reusedBy := ["thm:forward-KL", "cycle 60 lower derivative packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle60_five_backend_check" interface := "Explicit cycle-60 check of the five slow interfaces before lower work: endpoint-safe Gronwall, common-space DV with finite log-mgf, LSI/KL/FI density-test bridge, continuous Fokker-Planck/KL derivative, and EM interpolation Fokker-Planck for downstream discrete reuse." source := saldForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.saldGronwallEndpointCalculusContract", "SALD.forwardKlGronwallSideConditionContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle60_lower_packet.kl_derivative" interface := "Selected lower packet for cycle 60: return proof-producing work to sald.forward_kl.kl_derivative over appendix.tex:168-228, starting with mass conservation, KL differentiation, SALD Fokker-Planck substitution, target transport, boundary integration by parts, and inverse-schedule calculus." source := saldForwardKlDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle60ForwardKlSkeletonObligation", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "SALD.cycle60ForwardKlDerivativeRawLowerObligation", "SALD.cycle55ForwardKlDerivativeMassLowerObligation", "SALD.forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar", "SALD.forwardKlMassConservationDropScalar", "SALD.forwardKlMassConservationFirstTermFisherScalar", "SALD.forwardKlDerivativeDvGronwallCoefficientOfKlFiVelocityScalingScalar", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative" ] reusedBy := ["sald.forward_kl.kl_derivative", "thm:forward-KL", "cycle 60 lower handoff"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle60_derivative_raw_lower" interface := "Cycle-60 lower proof-producing scalar wrapper for appendix.tex:168-228: start from the raw KL derivative split with the mass term, drop that term by supplied mass conservation, and feed the existing first-term, target Young, LSI, slowed-velocity, and inverse-schedule pipeline to obtain the t-time pre-DV derivative inequality." source := saldForwardKlDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle60ForwardKlDerivativeRawLowerObligation", "SALD.forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar", "SALD.forwardKlMassConservationDropScalar", "SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "probability.lsi_to_kl_fi", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative" ] reusedBy := ["sald.forward_kl.kl_derivative", "thm:forward-KL", "cycle 60 reviewer"] status := ProofStatus.obligation } ] /-- Cycle-61 upper packet for the recovered discrete forward-KL skeleton. Cycle 60 passed the continuous forward-KL reviewer/build gate. This packet returns to the interrupted cycle-56 discrete route and keeps the next work at the theorem skeleton level: EM/Fokker--Planck, frozen defect, LSI, DV, and Gronwall/accumulated-error interfaces are consumed explicitly without promoting any slow analytic backend. -/ def cycle61DiscreteForwardKlSkeletonUpperPacket : DiscreteForwardKlUpperPacket where objective := "Cycle 61 upper: recover the interrupted cycle-56 thm:forward-KL-discrete route after the accepted cycle-60 continuous forward-KL gate, reuse the existing cycle-51/cycle-56 discrete route, and wire main_body.tex:299-323 plus appendix.tex:260-592 through the source-cited EM/Fokker-Planck, LSI/KL/FI, DV, Gronwall, and accumulated-error interfaces without changing statements, constants, labels, or statuses." sourceLabels := [ "thm:forward-KL-discrete", "proof:thm:forward-KL-discrete", "eq:frozen_interp_terminal_disc_prop_additive_final", "lem:frozen_delta_cross_lip_sald", "eq:KL-derivative-0-discrete", "eq:KL-derivative-1-discrete", "eq:KL-derivative-2-discrete", "eq:KL-derivative-3-discrete", "eq:frozen-cross-bound-discrete", "eq:moving-cross-bound-discrete", "eq:KL-derivative-5-discrete", "eq:dv-v-term-discrete", "eq:KL-derivative-6-discrete", "eq:KL-derivative-7-discrete", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "thm:forward-KL" ] modeDiscipline := [ "Global phase judgment: cycle 60 passed reviewer/build, so no previous-cycle failure needs recovery before cycle 61 work; nevertheless the older interrupted cycle-56 discrete route is the required recovery target before any new source theorem.", "Global phase judgment: Phase 1 theorem-skeleton translation is stable enough to continue theorem-level discrete wiring, but not stable enough for broad cited-theory, SLT, disintegration, or reusable API backfill.", "Global phase judgment: the single lower packet that best reduces remaining proof risk is SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge, reusing the cycle-56 pointwise Gronwall input over appendix.tex:526-592.", "Five-backend check: lem:gronwall is represented by SALD.saldGronwallEndpointCalculusContract and theorem-specific Gronwall side conditions with endpoint-safe differentiability/FTC assumptions; lem:dv_variation is represented by dvVariationalFormulaInterface saldDvVariationSource plus finite-log-mgf/common-space witnesses; eq:LSI-KL-FI is represented by SALD.saldLsiKlFiDensityTestContract with density, zero-set, admissible-test, entropy, finite KL/FI, and Fisher-chain obligations; the continuous FP/KL derivative is represented by SALD.forwardKlDerivativeSideConditionContract and the cycle-60 lower scalar wrapper; the EM interpolation FP backend is represented by SALD.discreteForwardKlEmInterpolationSideConditionContract, endpoint laws, conditional drift, conditional-FP, and stitched-interval obligations.", "Route order remains EM endpoint/conditional-FP -> KL derivative and frozen defect -> LSI/KL/FI -> DV velocity -> s-to-t time change -> lem:gronwall -> linear-slowdown accumulated-error bridge.", "The six theorem skeleton consumers remain in the paper order from the analytic ledger: thm:forward-KL, thm:forward-KL-discrete, prop:guided_path_residual, thm:general-moving-target-SALD, thm:unified-forward-KL, and thm:general-moving-target-SALD-discrete." ] nonGoals := [ "Do not restate thm:forward-KL-discrete, change SALD.discreteForwardKlStatementContract, or promote SALD.discreteSaldContract above contractOnly.", "Do not prove or replace EM conditional-Fokker-Planck, the frozen-defect lemma, LSI/KL/FI, source-cited DV, Gronwall, endpoint stitching, or the accumulated-error bridge in this upper packet.", "Do not add hidden density, absolute-continuity, endpoint, finite-log-mgf, coefficient-regularity, stitched-interval, or inverse-schedule assumptions to the source theorem.", "Do not spend this cycle on source-index rebaseline, broad SLT import, or isolated scalar sublemmas unless they are already part of the selected Gronwall/accumulated-error backend." ] lowerPacket := [ "Middle should synchronize this cycle-61 upper route with conversion-windows/ASTIS-SALD-001.md and proof-obligations/ASTIS-SALD-001.md before lower work.", "Lower should target exactly SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.", "First lower sub-slice: appendix.tex:557-590 endpoint and Gronwall-output stitching, using SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged and SALD.discreteForwardKlGronwallInstantiationContract as supplied inputs.", "Second lower sub-slice: linear slowdown t(s)=s/r and dot{s}=r coefficient collection, preserving the source constants T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'}.", "If proof-producing work blocks, sharpen endpoint, interval-integrability, coefficient-regularity, barGamma/barDelta, finite-log-mgf, or stitched-law obligations rather than changing theorem statements." ] reviewerChecklist := [ "SALD.discreteSaldContract lists SALD.cycle61DiscreteForwardKlSkeletonObligation while remaining contractOnly.", "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle61_recovered_theorem_route and ASTIS.SALD.forward_KL_discrete.cycle61_lower_packet.gronwall_accumulated after the cycle-56 nodes.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle61DiscreteForwardKlSkeletonUpperPacket, SALD.cycle61DiscreteForwardKlSkeletonObligation, and the cycle-61 DAG nodes.", "The conversion window and proof-obligation ledger contain the cycle-61 global phase judgment, five-interface check, lower packet, and reviewer checklist.", "Gronwall, DV, LSI/KL/FI, continuous FP/KL derivative, EM conditional-FP, theorem contracts, and SLT reuse statuses are not promoted.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-61 upper obligation tying the recovered discrete route to the next Gronwall/accumulated-error lower packet. -/ def cycle61DiscreteForwardKlSkeletonObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle61_recovered_theorem_route" statement := "Cycle 61 upper records that cycle 60 passed and recovers the interrupted cycle-56 thm:forward-KL-discrete route. It wires main_body.tex:299-323 and appendix.tex:260-592 through the explicit EM endpoint/conditional-Fokker-Planck interface, discrete KL derivative with frozen defect and LSI, DV velocity witness, cycle-56 t-time Gronwall input, and the linear-slowdown accumulated-error bridge. The selected lower packet is sald.discrete_forward_kl.accumulated_error_bridge over appendix.tex:557-590 and main_body.tex:309-323, with all theorem statuses and slow analytic interfaces kept below formalized." source := saldForwardKlDiscreteProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle61DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle60ForwardKlDerivativeRawLowerObligation", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation", "SALD.cycle59MainSkeletonAnalyticMiddleObligation", "SALD.cycle56DiscreteForwardKlSkeletonObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle51DiscreteForwardKlSkeletonObligation", "SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "SALD.discreteForwardKlDerivativeObligation", "sald.discrete_forward_kl.kl_derivative", "SALD.frozenDeltaCrossLipSaldContract", "sald.discrete_forward_kl.frozen_delta_cross_lip", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged", "sald.discrete_forward_kl.gronwall_accumulation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "SALD.discreteForwardKlAccumulatedErrorBridgeObligation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.coefficient_chain_audit", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity", "thm:forward-KL" ] note := "Upper workflow and source-to-Lean route data only. It adds no theorem hypotheses, changes no source constants, and does not promote EM/Fokker-Planck, frozen-delta, LSI/KL/FI, DV, Gronwall, accumulated-error, continuous derivative, SLT reuse, or thm:forward-KL-discrete." /-- Cycle-61 middle audit for the recovered discrete forward-KL route. This synchronizes the upper packet with the source transcript and moves lower work from the cycle-56 pointwise Gronwall input to the accumulated-error bridge that matches the main-body linear-slowdown theorem display. -/ def cycle61DiscreteForwardKlSkeletonMiddleContract : DiscreteForwardKlEmDefectAccumulationMiddleContract where sourceStatement := saldForwardKlDiscreteSource interpolationSource := saldForwardKlDiscreteInterpolationSource derivativeSource := saldForwardKlDiscreteDerivativeSource gronwallSource := saldForwardKlDiscreteGronwallSource accumulatedSource := saldForwardKlDiscreteAccumulatedErrorSource objective := "Cycle 61 middle: audit the recovered thm:forward-KL-discrete route against main_body.tex:299-323 and appendix.tex:260-592 after the accepted cycle-60 continuous route, verify that the theorem consumes source-cited EM/Fokker-Planck, LSI/KL/FI, DV, Gronwall, and accumulated-error interfaces in paper order, and select sald.discrete_forward_kl.accumulated_error_bridge over appendix.tex:557-590 plus main_body.tex:309-323 as the lower backend." sourceStepMap := [ "main_body.tex:299-323 fixes the discrete theorem: inherited thm:forward-KL assumptions, score Lipschitz hypotheses, alpha0' finite score and 1+M complexities, 4*eta^2*L_space^2<1/2, linear slowdown t(s)=s/r, and the exact terminal constants T/(r*alpha), 2*r*eta^2*barGamma/alpha', r^(-1)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'}.", "appendix.tex:260-385 supplies the EM interpolation, endpoint laws, conditional drift, conditional-Fokker-Planck equation, Laplacian split, and stitched interval setup; cycle 61 continues to use these only through named obligation interfaces.", "appendix.tex:388-491 is already routed through the cycle-51 derivative and LSI scalar handoff, including the frozen-cross quarter-FI estimate, moving Young quarter-FI estimate, and eq:LSI-KL-FI substitution.", "appendix.tex:493-523 is already routed through the discrete DV velocity witness for nu=hat rho_s, mu=tilde pi_s, and Z=alpha*||v_{t(s)}||^2.", "appendix.tex:526-553 is already routed through the cycle-56 pointwise t-time Gronwall input with coefficient dot{s}(t)*C_LSI(t)-dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t).", "appendix.tex:557-590 gives the general-schedule Gronwall output, and main_body.tex:309-323 is the linear-slowdown display that must be reached by endpoint stitching, exponent splitting, barGamma/barDelta collection, and alpha-complexity collection." ] leanStepMap := [ "Keep SALD.discreteForwardKlStatementContract and SALD.discreteSaldContract unchanged and contractOnly.", "Use SALD.cycle61DiscreteForwardKlSkeletonUpperPacket, SALD.cycle61DiscreteForwardKlSkeletonObligation, SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation, SALD.cycle56DiscreteForwardKlGronwallLowerObligation, and SALD.cycle51DiscreteForwardKlDerivativeLowerObligation as parent route checks.", "Route appendix.tex:260-385 through SALD.discreteForwardKlEmInterpolationSideConditionContract, sald.discrete_forward_kl.em_endpoint_laws, sald.discrete_forward_kl.conditional_drift_density, sald.discrete_forward_kl.em_conditional_fokker_planck, sald.discrete_forward_kl.stitched_interval_regularity, and sald.discrete_forward_kl.em_interpolation_fp.", "Route appendix.tex:388-491 through SALD.discreteForwardKlDerivativeCandidateContract, SALD.discreteForwardKlDerivativeObligation, SALD.cycle51DiscreteForwardKlDerivativeLowerObligation, SALD.frozenDeltaCrossLipSaldContract, sald.discrete_forward_kl.frozen_delta_cross_lip, SALD.saldLsiKlFiDensityTestContract, and probability.lsi_to_kl_fi.", "Route appendix.tex:493-523 through dvVariationalFormulaInterface saldDvVariationSource, SALD.discreteForwardKlDvFiniteLogMgfWitnessContract, sald.discrete_forward_kl.dv_finite_log_mgf_witness, and sald.discrete_forward_kl.dv_velocity_bound.", "Route appendix.tex:526-553 through SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar, SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged, SALD.discreteForwardKlGronwallInstantiationContract, and sald.discrete_forward_kl.gronwall_accumulation.", "Route appendix.tex:557-590 and main_body.tex:309-323 through SALD.discreteForwardKlAccumulatedErrorBridgeContract, SALD.discreteForwardKlAccumulatedErrorBridgeObligation, SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces, SALD.discreteForwardKlResidualExponentBoundScalar, SALD.discreteForwardKlResidualExpBoundScalar, SALD.discreteForwardKlAlphaComplexityCollectionScalar, SALD.discreteForwardKlDeltaAccumulationScalar, SALD.discreteForwardKlAccumulatedErrorCollectionScalar, SALD.discreteForwardKlResidualIntegralDisplayBoundScalar, sald.discrete_forward_kl.linear_slowdown_specialization, sald.discrete_forward_kl.residual_exponent_bound, and sald.discrete_forward_kl.accumulated_error_bridge.", "Select lower work on the accumulated-error bridge, not on EM conditional-FP, KL differentiation, LSI/KL/FI, DV, or the already compiled cycle-56 pointwise time-change wrapper." ] citedResultInterfaces := [ "EM endpoint/conditional-law Fokker-Planck remains a source-cited obligation interface; cycle 61 middle does not prove Brownian construction, disintegration, density/AC, weak FP, boundary integration by parts, or endpoint stitching.", "eq:LSI-KL-FI remains the density-test, zero-set, admissible-test or approximation, entropy-identity, finite KL/FI, and Fisher-chain obligation consumed by the derivative scalar handoff.", "lem:dv_variation remains source-cited with theorem-specific common-space, absolute-continuity, finite KL, finite-log-mgf, measurability, and positive-alpha witnesses.", "lem:gronwall remains the endpoint-safe differentiability/FTC, coefficient-regularity, interval-integrability, and exponent-rewrite obligation; the cycle-56 pointwise input is only the scalar source-shaped derivative handoff.", "The continuous forward-KL derivative route from cycle 60 is reused only as route context for inherited assumptions and does not close any discrete EM or accumulated-error backend." ] obligations := [ "sald.discrete_forward_kl.cycle61_recovered_theorem_route", "sald.discrete_forward_kl.cycle61_middle_route_audit", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.kl_derivative", "sald.discrete_forward_kl.frozen_delta_cross_lip", "probability.lsi_to_kl_fi", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge" ] lowerPacket := [ "Target exactly SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.", "First lower sub-slice: appendix.tex:557-590 endpoint and Gronwall-output stitching from the supplied cycle-56 pointwise derivative input and SALD.discreteForwardKlGronwallInstantiationContract.", "Second lower sub-slice: linear slowdown t(s)=s/r, dot{s}=r, and exponent collection, preserving T/(r*alpha) and 2*r*eta^2*barGamma/alpha'.", "Third lower sub-slice: collect the residual integral into r^(-1)*A_alpha(pi,v) plus 2*r*eta*barDelta_{alpha'} using SALD.discreteForwardKlAlphaComplexityCollectionScalar, SALD.discreteForwardKlDeltaAccumulationScalar, SALD.discreteForwardKlAccumulatedErrorCollectionScalar, and SALD.discreteForwardKlResidualIntegralDisplayBoundScalar.", "If blocked, sharpen endpoint-law stitching, coefficient regularity, interval-integrability, barGamma/barDelta definitions, or finite-log-mgf obligations rather than changing thm:forward-KL-discrete." ] reviewerChecklist := [ "SALD.discreteSaldContract lists SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation while remaining contractOnly.", "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle61_middle_route_audit between the recovered theorem route and the lower accumulated-error packet.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle61DiscreteForwardKlSkeletonMiddleContract, SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation, and sald.discrete_forward_kl.cycle61_middle_route_audit.", "No theorem statement, source constant, source label, theorem status, SLT reuse status, or analytic backend status is changed.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-61 middle obligation tying the recovered theorem route to the accumulated-error bridge lower packet. -/ def cycle61DiscreteForwardKlSkeletonMiddleObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle61_middle_route_audit" statement := "Cycle 61 middle audits the recovered thm:forward-KL-discrete route after the upper wrapper: main_body.tex:299-323 remains unchanged; appendix.tex:260-385 EM endpoint/conditional-Fokker-Planck interfaces stay explicit source-cited obligations; appendix.tex:388-491 reuses the cycle-51 derivative/LSI scalar handoff; appendix.tex:493-523 reuses the discrete DV velocity witness; appendix.tex:526-553 reuses the cycle-56 pointwise Gronwall input; and appendix.tex:557-590 plus main_body.tex:309-323 are selected as the accumulated-error bridge lower backend. All theorem statuses and slow analytic interfaces remain below formalized." source := saldForwardKlDiscreteProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle61DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle61DiscreteForwardKlSkeletonObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleContract", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle60ForwardKlDerivativeRawLowerObligation", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation", "SALD.cycle59MainSkeletonAnalyticMiddleObligation", "SALD.cycle56DiscreteForwardKlSkeletonObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "SALD.discreteForwardKlDerivativeObligation", "sald.discrete_forward_kl.kl_derivative", "SALD.frozenDeltaCrossLipSaldContract", "sald.discrete_forward_kl.frozen_delta_cross_lip", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged", "sald.discrete_forward_kl.gronwall_accumulation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "SALD.discreteForwardKlAccumulatedErrorBridgeObligation", "SALD.discreteForwardKlGronwallCoeffIntervalIntegrable", "SALD.discreteForwardKlGronwallCoeffIntegralSubSub", "SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces", "SALD.discreteForwardKlResidualExponentBoundScalar", "SALD.discreteForwardKlResidualExpBoundScalar", "SALD.discreteForwardKlAlphaComplexityCollectionScalar", "SALD.discreteForwardKlDeltaAccumulationScalar", "SALD.discreteForwardKlAccumulatedErrorCollectionScalar", "SALD.discreteForwardKlResidualIntegralDisplayBoundScalar", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity", "thm:forward-KL" ] note := "Middle workflow and source-to-Lean audit only. It adds no theorem hypotheses, changes no source constants, and does not promote EM/Fokker-Planck, frozen-delta, LSI/KL/FI, DV, Gronwall, accumulated-error collection, continuous derivative, SLT reuse, or thm:forward-KL-discrete." /-- Cycle-61 lower obligation for the residual integral display wrapper. -/ def cycle61DiscreteForwardKlAccumulatedErrorLowerObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle61_accumulated_error_lower" statement := "Cycle 61 lower compiles the scalar wrapper for the final residual-integral display in thm:forward-KL-discrete: after a separate residual-exponent argument supplies the common positive exponential factor, SALD.discreteForwardKlResidualIntegralDisplayBoundScalar uses the A_alpha and barDelta interval-collection cores to obtain exactly exp(T/(r*alpha)+2*r*eta^2*barGamma/alpha')*((1/r)*A_alpha(pi,v)+2*r*eta*barDelta_{alpha'}). EM/Fokker-Planck, endpoint stitching, residual-exponent monotonicity, barGamma/barDelta identifications, and the full accumulated-error bridge remain obligations." source := saldForwardKlDiscreteAccumulatedErrorSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "SALD.discreteForwardKlAccumulatedErrorBridgeObligation", "SALD.discreteForwardKlAlphaComplexityCollectionScalar", "SALD.discreteForwardKlDeltaAccumulationScalar", "SALD.discreteForwardKlAccumulatedErrorCollectionScalar", "SALD.discreteForwardKlResidualIntegralDisplayBoundScalar", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge", "def:alpha-complexity" ] note := "Proof-producing scalar wrapper plus obligation only. It preserves the source constants from main_body.tex:309-323 and does not promote the theorem, Gronwall, EM/Fokker-Planck, residual exponent, endpoint stitching, or barGamma/barDelta backends." /-- Cycle-61 proof-DAG pane for the recovered discrete forward-KL route. -/ def cycle61DiscreteForwardKlSkeletonDag : List ProofDagBlock := [ { id := "ASTIS.SALD.forward_KL_discrete.cycle61_global_phase_judgment" interface := "Upper judgment for cycle 61: cycle 60 passed and needs no recovery; Phase 1 is stable only for theorem-level discrete route recovery, not broad backfill; the largest remaining risk is the Gronwall-to-accumulated-error bridge for thm:forward-KL-discrete." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle61DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation" ] reusedBy := ["thm:forward-KL-discrete", "ASTIS-SALD-001 cycle 61"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle61_five_backend_check" interface := "Explicit cycle-61 check before assigning lower work: endpoint-safe Gronwall, common-space DV with finite log-mgf, LSI/KL/FI density-test bridge, continuous FP/KL derivative reuse, and EM endpoint/conditional-law Fokker-Planck are all named source-cited or obligation interfaces." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.saldGronwallEndpointCalculusContract", "SALD.discreteForwardKlGronwallInstantiationContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.cycle60ForwardKlDerivativeRawLowerObligation", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle61_recovered_theorem_route" interface := "Recovered discrete theorem route: consume EM endpoint/conditional-FP, frozen defect, LSI, DV velocity, cycle-56 time-changed Gronwall input, and accumulated-error bridge in the source order of appendix.tex:260-592 and main_body.tex:299-323." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle61DiscreteForwardKlSkeletonObligation", "SALD.cycle56DiscreteForwardKlSkeletonObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "sald.discrete_forward_kl.kl_derivative", "SALD.frozenDeltaCrossLipSaldContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.discreteForwardKlGronwallInstantiationContract", "sald.discrete_forward_kl.gronwall_accumulation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "sald.discrete_forward_kl.accumulated_error_bridge" ] reusedBy := ["thm:forward-KL-discrete", "cycle61 middle route audit"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle61_middle_route_audit" interface := "Middle audit for the recovered discrete route: verify source order from EM interpolation through derivative, LSI, DV, cycle-56 pointwise Gronwall, and select the accumulated-error bridge to the main-body linear-slowdown display as the lower packet." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle61DiscreteForwardKlSkeletonMiddleContract", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle61DiscreteForwardKlSkeletonObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "sald.discrete_forward_kl.kl_derivative", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "sald.discrete_forward_kl.accumulated_error_bridge" ] reusedBy := ["thm:forward-KL-discrete", "cycle61 lower accumulated-error packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle61_lower_packet.gronwall_accumulated" interface := "Selected lower packet for cycle 61: refine sald.discrete_forward_kl.accumulated_error_bridge over appendix.tex:557-590 and main_body.tex:309-323, using the supplied cycle-56 pointwise Gronwall input and preserving the linear-slowdown constants." source := saldForwardKlDiscreteAccumulatedErrorSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle61DiscreteForwardKlSkeletonObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "SALD.discreteForwardKlAccumulatedErrorBridgeObligation", "SALD.discreteForwardKlGronwallCoeffIntervalIntegrable", "SALD.discreteForwardKlGronwallCoeffIntegralSubSub", "SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces", "SALD.discreteForwardKlResidualExponentBoundScalar", "SALD.discreteForwardKlResidualExpBoundScalar", "SALD.discreteForwardKlAlphaComplexityCollectionScalar", "SALD.discreteForwardKlDeltaAccumulationScalar", "SALD.discreteForwardKlAccumulatedErrorCollectionScalar", "SALD.discreteForwardKlResidualIntegralDisplayBoundScalar", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.gronwall.integrating_factor" ] reusedBy := ["sald.discrete_forward_kl.accumulated_error_bridge", "thm:forward-KL-discrete"] status := ProofStatus.obligation } ] /-- Cycle-62 upper packet for the guided residual and continuous general moving-target theorem route after the accepted cycle-61 discrete recovery. This is workflow data only. It records the upper phase judgment, checks the five slow analytic interfaces before assigning follow-up work, and keeps the source route `appendix.tex:619-951` fixed while selecting one lower packet. -/ def cycle62GuidedGeneralSkeletonUpperPacket : GeneralVaSaldUpperPacket where objective := "Cycle 62 upper: cycle 61 passed reviewer/build and needs no recovery; Phase 1 is not stable enough for cited-theory backfill until prop:guided_path_residual and thm:general-moving-target-SALD are rechecked after the discrete route recovery; the single lower packet is a guided residual to general moving-target route audit over appendix.tex:619-951 that narrows proof search back to sald.general_moving_target.kl_derivative." sourceLabels := [ "prop:guided_path_residual", "proof:prop:guided_path_residual", "thm:general-moving-target-SALD", "eq:general_moving_target_SALD", "eq:general_moving_target_FP", "proof:thm:general-moving-target-SALD:derivative", "proof:thm:general-moving-target-SALD:residual-dv", "proof:thm:general-moving-target-SALD:dv-gronwall", "proof:thm:general-moving-target-SALD:pure-contraction", "proof:thm:unified-forward-KL", "eq:LSI-KL-FI", "lem:dv_variation", "lem:gronwall", "def:alpha-complexity" ] modeDiscipline := [ "Global phase judgment: cycle 61 did not fail and needs no recovery; it accepted the recovered thm:forward-KL-discrete route and accumulated-error bridge while keeping theorem and backend statuses below formalized.", "Global phase judgment: Phase 1 theorem-skeleton translation is not yet stable enough for cited-theory backfill, because the guided residual and continuous general moving-target theorem route must be rechecked after the discrete recovery.", "Global phase judgment: the lower packet that reduces the largest current proof risk is the appendix.tex:619-951 guided residual to general moving-target route audit, with lower proof search narrowed to sald.general_moving_target.kl_derivative once the audit is synchronized.", "Five-backend check 1, Gronwall: SALD.saldGronwallCandidateContract, SALD.saldGronwallEndpointCalculusContract, SALD.saldGronwallExponentRewriteContract, and SALD.gronwallAnalyticObligation keep endpoint-safe differentiability, FTC/order integration, coefficient regularity, endpoint rewrites, exponent splitting, and residual-exponent side conditions explicit.", "Five-backend check 2, DV: dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, SALD.generalMovingTargetDvFiniteLogMgfWitnessContract, and SALD.generalMovingTargetDvPositiveAlphaScalingContract keep common-space, absolute-continuity, finite KL, finite log-mgf, measurability of alpha*||m_t||^2, and positive-alpha scaling explicit.", "Five-backend check 3, LSI/KL/FI: SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi keep density, zero-set convention, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule assumptions explicit.", "Five-backend check 4, continuous Fokker-Planck/KL derivative: SALD.forwardKlDerivativeCandidateContract, SALD.forwardKlDerivativeSideConditionContract, SALD.generalMovingTargetDerivativeCandidateContract, and sald.general_moving_target.kl_derivative keep density/law regularity, mass conservation, integration by parts, target transport, sigma positivity, and inverse-schedule side conditions explicit.", "Five-backend check 5, EM interpolation Fokker-Planck: SALD.discreteForwardKlEmInterpolationSideConditionContract, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation, and sald.general_moving_target_discrete.em_interpolation_fp expose endpoint laws, conditional drift, conditional-law density/absolute-continuity, weak Fokker-Planck, Laplacian split, and stitched-interval interfaces for downstream discrete reuse.", "The theorem route check remains in paper order: forward-KL through cycle 60, discrete forward-KL through cycle 61, guided residual through the normalizer and identity obligations, general moving-target through derivative/LSI/DV/Gronwall/pure-contraction interfaces, unified forward-KL as the c_t<-u_t specialization, and discrete general moving-target through the existing EM/frozen-delta/LSI/DV/Gronwall interfaces.", "FaithfulPaper Phase 1: use appendix.tex:619-951 and the original main_body.tex unified reference only; sald_version_2.tex remains excluded." ] nonGoals := [ "Do not restate prop:guided_path_residual, thm:general-moving-target-SALD, thm:unified-forward-KL, or any theorem display.", "Do not add hidden density, endpoint, common-space, finite-log-mgf, sigma-positivity, schedule, or coefficient-regularity assumptions to the source theorem statements.", "Do not promote guided residual calculus, Fokker-Planck/KL differentiation, LSI/KL/FI, DV, Gronwall, pure contraction, EM interpolation, or any theorem status above contractOnly/sourceCited/obligation.", "Do not replace the source route by a direct VA-SALD proof, path-space comparison, Girsanov, Pinsker, Talagrand, PI, or SLT-based proof.", "Do not begin systematic measure-theory or SDE backfill in this upper cycle; missing analytic facts must stay as source-cited interfaces or obligations." ] lowerPacket := [ "Middle should synchronize this cycle-62 upper route with conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, and the guided/general proof DAG before lower proof search.", "Lower route audit target: appendix.tex:619-951, checking that every guided residual and general moving-target proof step maps to SALD.guidedResidualIdentityContract, SALD.generalMovingTargetDerivativeCandidateContract, SALD.saldLsiKlFiDensityTestContract, SALD.generalMovingTargetDvFiniteLogMgfWitnessContract, SALD.generalMovingTargetGronwallSideConditionContract, or a named obligation.", "After the route audit is accepted, proof-producing lower work should target exactly SALD.generalMovingTargetDerivativeCandidateContract / SALD.generalMovingTargetDerivativeObligation / sald.general_moving_target.kl_derivative over appendix.tex:765-884.", "First derivative sub-slice: appendix.tex:765-812, exposing law regularity, mass conservation, KL differentiation under the integral, the general moving-target Fokker-Planck equation, and integration-by-parts side conditions.", "Second derivative sub-slice: appendix.tex:813-864, exposing target transport by v_t, rescaled transport of pi_{t(s)}, residual m_t=v_t-c_t, and Young with epsilon=2*dot t(s)/sigma_{t(s)}^2.", "Third derivative sub-slice: appendix.tex:865-884, preserving the LSI handoff and s-to-t schedule/sigma side conditions without adding them to the theorem statement.", "Leave guided normalizer differentiation, residual DV finite-log-mgf, Gronwall endpoint/exponent, pure contraction, unified specialization, and downstream EM interpolation as separate obligations unless exact compiled proofs are added." ] reviewerChecklist := [ "The cycle-62 route uses cycle 61 as the recovered discrete forward-KL prerequisite and does not reopen the cycle-56 failure.", "SALD.guidedResidualContract and SALD.generalVaSaldContract list SALD.cycle62GuidedGeneralSkeletonObligation while remaining contractOnly.", "SALD.generalVaSaldProofDag contains ASTIS.SALD.guided_general.cycle62_global_phase_judgment, cycle62_five_backend_check, cycle62_upper_route, and the selected lower-packet node.", "SALD.saldDependenciesForLabel for prop:guided_path_residual and thm:general-moving-target-SALD includes the cycle-62 packet, obligation, and DAG nodes.", "The five slow analytic interfaces remain source-cited or obligation-level; no source theorem statement, source constant, source label, SLT reuse status, or analytic backend status changes.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-62 upper obligation tying the guided/general theorem route to the accepted cycle-61 recovery and the five explicit analytic backend interfaces. -/ def cycle62GuidedGeneralSkeletonObligation : ProofObligation where id := "sald.guided_general.cycle62_upper_route" statement := "Cycle 62 upper records that cycle 61 passed and needs no recovery, defers cited-theory backfill until the guided/general route is synchronized, checks the five slow analytic backends, and wires appendix.tex:619-951 through the named guided residual and continuous general moving-target interfaces. prop:guided_path_residual uses the guided normalizer derivative and centered residual identity obligations; thm:general-moving-target-SALD uses the general Fokker-Planck/KL derivative, eq:LSI-KL-FI, residual DV finite-log-mgf and positive-alpha witnesses, lem:gronwall endpoint/exponent side conditions, and pure-contraction residual-zero obligation. The selected lower packet is the guided/general route audit that narrows later proof search to sald.general_moving_target.kl_derivative." source := saldGeneralMovingTargetSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle62GuidedGeneralSkeletonUpperPacket", "SALD.cycle61DiscreteForwardKlSkeletonObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation", "SALD.cycle60ForwardKlSkeletonObligation", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation", "SALD.cycle59MainSkeletonAnalyticMiddleObligation", "SALD.cycle57GuidedGeneralSkeletonObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralDerivativeSplitLowerObligation", "SALD.cycle52GuidedGeneralSkeletonObligation", "SALD.cycle52GuidedGeneralSkeletonMiddleObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "SALD.generalMovingTargetPostYoungDerivativeBoundScalar", "SALD.generalMovingTargetLsiDerivativeBoundScalar", "SALD.generalMovingTargetTimeChangedDerivativeBoundScalar", "SALD.generalMovingTargetPreDvDerivativeBoundScalar", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "SALD.saldGronwallEndpointCalculusContract", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "SALD.unifiedForwardKlSpecializationContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL", "thm:forward-KL-discrete", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "Upper workflow and source-to-Lean route data only. It adds no theorem hypotheses, changes no source constants, and does not promote guided residual calculus, Fokker-Planck/KL differentiation, LSI/KL/FI, DV, Gronwall, pure contraction, EM interpolation, SLT reuse, or theorem status." /-- Cycle-62 middle audit for the guided residual and continuous general moving-target theorem route. This synchronizes the upper route wrapper with the conversion window and proof-obligation ledger. It keeps the source route `appendix.tex:619-951` fixed, checks that every proof step has a named Lean-facing consumer, and hands later proof-producing work back to the continuous general KL derivative backend. -/ def cycle62GuidedGeneralSkeletonMiddleContract : GeneralVaSaldGuidedPathMiddleContract where guidedResidualSource := saldGuidedResidualSource generalContinuousSource := saldGeneralMovingTargetSource unifiedSource := saldUnifiedForwardKlSource discreteSource := saldGeneralMovingTargetDiscreteSource objective := "Cycle 62 middle: audit the cycle-62 upper guided/general route against appendix.tex:619-951 after the accepted cycle-61 discrete forward-KL recovery, add the middle route obligation to prop:guided_path_residual and thm:general-moving-target-SALD, and keep lower proof search narrowed to sald.general_moving_target.kl_derivative over appendix.tex:765-884." sourceStepMap := [ "appendix.tex:619-704 proves prop:guided_path_residual by differentiating Z_t, using partial_t p_t=-div(p_t*u_t), integrating by parts, differentiating pi_t=Z_t^(-1)*p_t*exp(-f_t), cancelling divergence terms, substituting dot Z_t/Z_t, and deriving the centered mean-zero residual.", "appendix.tex:724-744 states thm:general-moving-target-SALD with dynamics driven by c_t, residual m_t=v_t-c_t, finite residual alpha0-complexity, alpha in (0,alpha0], and the exact sigma-weighted KL bound.", "appendix.tex:765-812 differentiates KL(rho_s||pi_{t(s)}), uses mass conservation, inserts the general moving-target Fokker-Planck equation, and evaluates the rho_s term by integration by parts.", "appendix.tex:813-835 uses the transport equation for pi_t, rescales v_t by dot t(s), and evaluates the target-time contribution by integration by parts.", "appendix.tex:835-884 combines c_t and v_t into m_t=v_t-c_t, applies Young with epsilon=2*dot t(s)/sigma_{t(s)}^2, changes from s to t, and uses eq:LSI-KL-FI to reach the pre-DV residual-energy inequality.", "appendix.tex:885-907 applies lem:dv_variation with Z=alpha*||m_t||^2 and rewrites the log-mgf quotient as mathfrak E_alpha(pi_t,m_t).", "appendix.tex:908-945 applies lem:gronwall with the source sigma-weighted coefficients, matches the theorem display, and proves the c_t=v_t pure-contraction clause by the zero residual alpha-complexity calculation.", "appendix.tex:949-951 reuses the continuous general theorem for thm:unified-forward-KL by setting c_t<-u_t; this middle audit does not introduce an alternate VA-SALD proof." ] leanStepMap := [ "Use SALD.cycle62GuidedGeneralSkeletonUpperPacket and SALD.cycle62GuidedGeneralSkeletonObligation as the parent route wrapper after the cycle-61 discrete recovery.", "Route appendix.tex:619-704 through SALD.guidedResidualIdentityContract, sald.guided_path_residual.normalizer_derivative, and sald.guided_path_residual.identity; positive finite normalizer, differentiation under the integral, boundary integration by parts, quotient/product differentiation, and centering stay obligations.", "Keep SALD.generalMovingTargetStatementContract, SALD.guidedResidualContract, and SALD.generalVaSaldContract contractOnly; this middle audit adds no theorem hypotheses.", "Route appendix.tex:765-884 through SALD.generalMovingTargetDerivativeCandidateContract, SALD.generalMovingTargetDerivativeObligation, sald.general_moving_target.kl_derivative, the cycle-57 residual split scalar handoff, SALD.saldLsiKlFiDensityTestContract, probability.lsi_to_kl_fi, and sald.forward_kl.schedule_time_change.", "Route appendix.tex:885-907 through SALD.generalMovingTargetDvFiniteLogMgfWitnessContract, SALD.generalMovingTargetDvPositiveAlphaScalingContract, sald.general_moving_target.dv_finite_log_mgf_witness, sald.general_moving_target.dv_positive_alpha_scaling, and sald.general_moving_target.dv_m_energy.", "Route appendix.tex:908-945 through SALD.generalMovingTargetGronwallInstantiationContract, SALD.generalMovingTargetGronwallSideConditionContract, sald.general_moving_target.gronwall_application, sald.general_moving_target.gronwall_side_conditions, and sald.general_moving_target.pure_contraction.", "Keep SALD.unifiedForwardKlSpecializationContract as downstream reuse and keep SALD.discreteForwardKlEmInterpolationSideConditionContract plus SALD.generalMovingTargetDiscreteDerivativeSideConditionContract visible only for later discrete general reuse." ] citedResultInterfaces := [ "lem:gronwall remains an obligation through endpoint-safe differentiability, FTC/order integration, coefficient regularity, endpoint rewrites, exponent splitting, and residual-exponent side conditions.", "lem:dv_variation remains source-cited; the continuous general theorem still needs common-space, absolute-continuity, finite KL/log-likelihood, measurability, finite-log-mgf, and positive-alpha witnesses for Z=alpha*||m_t||^2.", "eq:LSI-KL-FI remains the density, zero-set, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule obligation.", "The continuous general Fokker-Planck/KL derivative remains sald.general_moving_target.kl_derivative; compiled scalar helpers only consume analytic inputs after they are supplied.", "The Euler-Maruyama interpolation Fokker-Planck backend remains a downstream discrete obligation and is not imported into the continuous theorem statement." ] obligations := [ "sald.guided_general.cycle62_upper_route", "sald.guided_general.cycle62_middle_route_audit", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.cycle57_derivative_split_lower", "probability.lsi_to_kl_fi", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "sald.unified_forward_kl.specialization", "sald.general_moving_target_discrete.em_interpolation_fp" ] lowerPacket := [ "Target exactly SALD.generalMovingTargetDerivativeCandidateContract / SALD.generalMovingTargetDerivativeObligation / sald.general_moving_target.kl_derivative.", "First lower sub-slice: appendix.tex:765-812, expose density/law regularity, mass conservation, Fokker-Planck substitution, KL differentiation under the integral, and integration-by-parts side conditions.", "Second lower sub-slice: appendix.tex:813-864, expose target transport by v_t, rescaled transport of pi_{t(s)}, residual identification m_t=v_t-c_t, and the Young coefficient epsilon=2*dot t(s)/sigma_{t(s)}^2.", "Third lower sub-slice: appendix.tex:865-884, preserve the LSI handoff plus inverse-schedule and sigma-positivity side conditions without adding them to the theorem statement.", "Leave guided residual calculus, residual DV, Gronwall endpoint/exponent, pure contraction, unified specialization, and downstream EM interpolation as separate obligations unless exact compiled proofs are added." ] reviewerChecklist := [ "SALD.guidedResidualContract and SALD.generalVaSaldContract list SALD.cycle62GuidedGeneralSkeletonMiddleObligation while remaining contractOnly.", "SALD.generalVaSaldProofDag contains ASTIS.SALD.guided_general.cycle62_middle_route_audit between the cycle-62 upper route and the lower-packet node.", "SALD.saldDependenciesForLabel for prop:guided_path_residual and thm:general-moving-target-SALD includes the cycle-62 middle contract, middle obligation, and sald.guided_general.cycle62_middle_route_audit.", "No Gronwall, DV, LSI/KL/FI, Fokker-Planck/KL derivative, guided residual, pure-contraction, EM backend, theorem status, source label, source constant, or SLT reuse status is promoted.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-62 middle obligation tying the guided/general route audit to the next lower backend. -/ def cycle62GuidedGeneralSkeletonMiddleObligation : ProofObligation where id := "sald.guided_general.cycle62_middle_route_audit" statement := "Cycle 62 middle audits appendix.tex:619-951 after the upper route wrapper and accepted cycle-61 discrete recovery: prop:guided_path_residual and thm:general-moving-target-SALD remain unchanged, the proof route consumes the named guided residual, continuous general KL derivative, LSI/KL/FI, residual DV, Gronwall, pure-contraction, unified-specialization, and downstream EM interfaces in paper order, and the next proof-producing lower packet remains sald.general_moving_target.kl_derivative over appendix.tex:765-884. All slow analytic interfaces remain obligation or source-cited." source := saldGeneralMovingTargetSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle62GuidedGeneralSkeletonUpperPacket", "SALD.cycle62GuidedGeneralSkeletonObligation", "SALD.cycle62GuidedGeneralSkeletonMiddleContract", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation", "SALD.cycle59MainSkeletonAnalyticMiddleObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralDerivativeSplitLowerObligation", "SALD.cycle52GuidedGeneralSkeletonMiddleObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "SALD.generalMovingTargetKlDerivativeResidualSplitScalar", "SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar", "SALD.generalMovingTargetPostYoungDerivativeBoundScalar", "SALD.generalMovingTargetLsiDerivativeBoundScalar", "SALD.generalMovingTargetTimeChangedDerivativeBoundScalar", "SALD.generalMovingTargetPreDvDerivativeBoundScalar", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "SALD.saldGronwallEndpointCalculusContract", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "SALD.unifiedForwardKlSpecializationContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "This is a middle workflow obligation and source-to-Lean audit, not an analytic proof. It adds no hidden density, boundary, endpoint, absolute-continuity, finite-log-mgf, finite-KL/FI, sigma-positivity, schedule, or coefficient-regularity assumptions to prop:guided_path_residual or thm:general-moving-target-SALD." /-- Cycle-62 lower scalar handoff for the continuous general KL derivative. -/ def cycle62GuidedGeneralScaledResidualLowerObligation : ProofObligation where id := "sald.general_moving_target.cycle62_scaled_residual_lower" statement := "Cycle 62 lower compiles the appendix.tex:813-835 scalar sign/scale handoff inside sald.general_moving_target.kl_derivative: after the target transport equation supplies tilde v_s=dot{t}(s)*v_{t(s)}, the c_t drift and v_t target-transport pairings, and the residual pairing m_t=v_t-c_t, SALD.generalMovingTargetKlDerivativeScaledResidualDisplayScalar rewrites the KL derivative display as -(sigma_{t(s)}^2/2)*FI - dot{t}(s)*int rho_s . The analytic target-transport, integration-by-parts, density, mass-conservation, LSI, Young, and schedule backends remain obligations." source := saldGeneralMovingTargetDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle62GuidedGeneralSkeletonMiddleContract", "SALD.cycle62GuidedGeneralSkeletonMiddleObligation", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "SALD.generalMovingTargetKlDerivativeScaledResidualDisplayScalar", "SALD.generalMovingTargetKlDerivativeResidualSplitScalar", "sald.general_moving_target.kl_derivative", "TransportVelocityContract", "FokkerPlanckContract", "KLContract", "FIContract", "sald.forward_kl.density_boundary_regular" ] note := "Formalized local Real algebra only. It preserves the source residual sign and dot{t}(s) scaling, and does not promote the continuous Fokker-Planck/KL derivative backend or thm:general-moving-target-SALD." /-- Cycle-62 proof-DAG pane for the guided/general upper route. -/ def cycle62GuidedGeneralSkeletonDag : List ProofDagBlock := [ { id := "ASTIS.SALD.guided_general.cycle62_global_phase_judgment" interface := "Upper judgment for cycle 62: cycle 61 passed and needs no recovery; Phase 1 is not ready for broad cited-theory backfill until the guided residual and continuous general moving-target route is synchronized; the selected lower packet is the appendix.tex:619-951 guided/general route audit narrowing to sald.general_moving_target.kl_derivative." source := saldGeneralMovingTargetSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle62GuidedGeneralSkeletonUpperPacket", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation" ] reusedBy := ["prop:guided_path_residual", "thm:general-moving-target-SALD", "ASTIS-SALD-001 cycle 62"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.guided_general.cycle62_five_backend_check" interface := "Explicit cycle-62 check before middle/lower work: endpoint-safe Gronwall, common-space DV with finite log-mgf, LSI/KL/FI density-test bridge, continuous Fokker-Planck/KL derivative, and EM endpoint/conditional-law Fokker-Planck are all named source-cited or obligation interfaces." source := saldGeneralMovingTargetSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.saldGronwallEndpointCalculusContract", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.guided_general.cycle62_upper_route" interface := "Upper route for appendix.tex:619-951 after cycle 61: route guided residual through normalizer/centered identity, route continuous general SALD through derivative, LSI, residual DV, Gronwall, and pure-contraction interfaces, and keep unified reuse as the source specialization." source := saldGeneralMovingTargetSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle62GuidedGeneralSkeletonObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.identity", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "sald.general_moving_target.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.pure_contraction", "SALD.unifiedForwardKlSpecializationContract" ] reusedBy := ["prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.guided_general.cycle62_middle_route_audit" interface := "Middle source-to-Lean audit for cycle 62: verify appendix.tex:619-951 in paper order after the upper wrapper, keep guided residual and continuous general theorem statements fixed, and route proof-producing lower work to sald.general_moving_target.kl_derivative while guided residual, LSI, residual DV, Gronwall, pure contraction, unified specialization, and EM reuse remain separate obligations." source := saldGeneralMovingTargetSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle62GuidedGeneralSkeletonMiddleContract", "SALD.cycle62GuidedGeneralSkeletonMiddleObligation", "SALD.cycle62GuidedGeneralSkeletonObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.guidedResidualIdentityContract", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetGronwallSideConditionContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.saldGronwallEndpointCalculusContract", "SALD.unifiedForwardKlSpecializationContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract" ] reusedBy := ["prop:guided_path_residual", "thm:general-moving-target-SALD", "cycle62 lower derivative packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.guided_general.cycle62_lower_packet.route_audit" interface := "Selected lower packet for cycle 62 after the middle audit: narrow proof-producing work to the continuous general Fokker-Planck/KL derivative backend over appendix.tex:765-884." source := saldGeneralMovingTargetDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle62GuidedGeneralSkeletonObligation", "SALD.cycle62GuidedGeneralSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralDerivativeSplitLowerObligation", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "SALD.generalMovingTargetKlDerivativeResidualSplitScalar", "SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar", "SALD.saldLsiKlFiDensityTestContract", "sald.forward_kl.schedule_time_change" ] reusedBy := ["sald.general_moving_target.kl_derivative", "thm:general-moving-target-SALD", "thm:unified-forward-KL"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target.cycle62_scaled_residual_lower" interface := "Lower proof-producing scalar core for appendix.tex:813-835: preserve the tilde-v dot{t}(s) scaling and rewrite c_t-v_t into the source residual sign -dot{t}(s)*m_t before the Young step." source := saldGeneralMovingTargetDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle62GuidedGeneralScaledResidualLowerObligation", "SALD.generalMovingTargetKlDerivativeScaledResidualDisplayScalar", "SALD.generalMovingTargetKlDerivativeResidualSplitScalar", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "TransportVelocityContract" ] reusedBy := ["sald.general_moving_target.kl_derivative", "thm:general-moving-target-SALD", "thm:unified-forward-KL"] status := ProofStatus.obligation } ] /-- Cycle-63 upper packet for the unified and discrete general theorem route. This is workflow data only. It records the required upper phase judgment, rechecks the five slow analytic interfaces before assigning follow-up work, wires `thm:unified-forward-KL` and `thm:general-moving-target-SALD-discrete` through the accepted continuous and general skeletons, and permits exactly one narrow measure-theory backfill below the EM endpoint-law interface. -/ def cycle63UnifiedDiscreteGeneralSkeletonUpperPacket : GeneralVaSaldUpperPacket where objective := "Cycle 63 upper: cycle 62 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough to rewire thm:unified-forward-KL and thm:general-moving-target-SALD-discrete through the continuous/general skeletons and then begin exactly one narrow measure-theory backfill; the single lower packet is a paired Measure.map endpoint-law helper for the discrete general EM interpolation over appendix.tex:1354-1387." sourceLabels := [ "thm:unified-forward-KL", "proof:thm:unified-forward-KL", "eq:residual-term", "eq:poisson-eq", "eq:SALD_Ito", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:general-moving-target-SALD-discrete", "eq:general_moving_target_KL_bound_discrete", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "eq:general_discrete_delta_def", "proof:thm:general-moving-target-SALD-discrete:em-endpoints", "proof:thm:general-moving-target-SALD-discrete:derivative", "proof:thm:general-moving-target-SALD-discrete:gronwall", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] modeDiscipline := [ "Global phase judgment: cycle 62 did not fail and needs no recovery; it accepted the guided residual / continuous general route and the appendix.tex:813-835 scaled residual scalar handoff while keeping all theorem statuses and slow analytic interfaces below formalized.", "Global phase judgment: Phase 1 theorem-skeleton translation is stable enough for this unified/discrete-general route refresh and one narrow endpoint-law measure backfill, but still not for broad cited-theory, disintegration, Fokker-Planck, or reusable API reorganization.", "Global phase judgment: the lower packet that best reduces remaining proof risk is the discrete general EM endpoint/common-space layer, narrowed to a paired pushforward-law equality from componentwise a.e. endpoint identities and marginal extraction from that joint law; this supports later stitched-law and conditional-law work without touching the theorem statement.", "Five-backend check 1, Gronwall: SALD.saldGronwallCandidateContract, SALD.saldGronwallEndpointCalculusContract, SALD.saldGronwallExponentRewriteContract, SALD.generalMovingTargetGronwallSideConditionContract, and SALD.generalMovingTargetDiscreteGronwallSideConditionContract keep endpoint-safe differentiability/FTC, interval integrability, coefficient regularity, endpoint rewrites, exponent splitting, and display matching explicit.", "Five-backend check 2, DV: dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, SALD.generalMovingTargetDvFiniteLogMgfWitnessContract, and SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract keep common-space, absolute-continuity, finite KL/log-likelihood, selected-test measurability, finite log-mgf, and positive-alpha scaling witnesses explicit.", "Five-backend check 3, LSI/KL/FI: SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi keep density, zero-set convention, admissible sqrt-density test or approximation, entropy identity, finite KL/FI, and Fisher chain-rule assumptions explicit.", "Five-backend check 4, continuous Fokker-Planck/KL derivative: SALD.forwardKlDerivativeCandidateContract, SALD.generalMovingTargetDerivativeCandidateContract, SALD.cycle62GuidedGeneralScaledResidualLowerObligation, and sald.general_moving_target.kl_derivative keep mass conservation, KL differentiation, Fokker-Planck substitution, integration by parts, target transport, residual scaling, LSI, and schedule side conditions explicit.", "Five-backend check 5, EM interpolation Fokker-Planck: SALD.discreteForwardKlEmInterpolationSideConditionContract, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation, AutoSamplingTheory.lawMapEqOfAEEq, and sald.general_moving_target_discrete.em_interpolation_fp expose endpoint laws, common-space Measure.map bookkeeping, conditional drift, density/absolute-continuity, weak conditional Fokker-Planck, Laplacian splitting, and stitched-interval side conditions without promoting the backend.", "The theorem route remains paper ordered: forward-KL through cycle 60, discrete forward-KL through cycle 61, guided residual and continuous general through cycle 62, unified forward-KL as the c_t<-u_t specialization using prop:guided_path_residual and eq:poisson-eq, and discrete general moving-target through general EM, frozen-delta, derivative/LSI, residual DV, constant-schedule Gronwall, and theorem-display stitching.", "FaithfulPaper Phase 1: use original main_body.tex:359-395, appendix.tex:949-951, and appendix.tex:1313-1603; sald_version_2.tex remains excluded." ] nonGoals := [ "Do not restate thm:unified-forward-KL, thm:general-moving-target-SALD, or thm:general-moving-target-SALD-discrete.", "Do not change sigma_eta factors, doubled residual coefficients, Gamma/Delta terms, alpha ranges, constant inverse-schedule assumptions, endpoint laws, source labels, or theorem displays.", "Do not promote Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL differentiation, EM conditional Fokker-Planck, conditional laws, theorem contracts, or SLT reuse status above their current sourceCited/obligation/contractOnly level.", "Do not replace the source derivative -> LSI -> DV -> Gronwall route, the correction-field specialization, or the paper EM/frozen-delta route by a direct VA-SALD, path-space, Girsanov, Pinsker, Talagrand, PI, or broad SLT proof.", "Do not start a broad measure-theory or SDE port; the permitted backfill is exactly one endpoint-law Measure.map helper." ] lowerPacket := [ "Middle should synchronize this cycle-63 upper route with conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, SALD.generalVaSaldProofDag, and SALD.generalVaSaldDiscreteProofDag.", "Lower should keep the narrow measure-theory backfill at paired endpoint Measure.map bookkeeping: AutoSamplingTheory.lawMapProdEqOfAEEq for the joint law, AutoSamplingTheory.lawMapProdFst/Snd for marginal extraction, then record it under SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation / sald.general_moving_target_discrete.cycle63_joint_endpoint_law_backfill.", "Source slice: appendix.tex:1354-1387, endpoint/common-space bookkeeping for the general EM interpolation before conditional drift and weak Fokker-Planck are used.", "Use local SLT material only as a style guide for Measure.map and a.e. equality patterns, especially SLT/SmallBallProb.lean and SLT/GaussianMeasure.lean; no SLT theorem is imported or marked as a SALD backend.", "The paired pushforward and marginal helpers may be used later to keep two endpoint representatives on one common probability space, but they do not prove Brownian construction, regular conditional drift, density/AC, weak Fokker-Planck, KL differentiation, or stitched Gronwall regularity." ] reviewerChecklist := [ "SALD.unifiedForwardKlContract and SALD.generalVaSaldDiscreteContract list SALD.cycle63UnifiedDiscreteGeneralSkeletonObligation while remaining contractOnly.", "SALD.generalVaSaldProofDag and SALD.generalVaSaldDiscreteProofDag include SALD.cycle63UnifiedDiscreteGeneralDag with ASTIS.SALD.unified_discrete_general.cycle63_upper_route and ASTIS.SALD.general_moving_target_discrete.cycle63_joint_endpoint_law_backfill.", "SALD.saldDependenciesForLabel for thm:unified-forward-KL and thm:general-moving-target-SALD-discrete includes the cycle-63 packet, obligation, DAG, and paired Measure.map backfill.", "AutoSamplingTheory.lawMapProdEqOfAEEq and AutoSamplingTheory.lawMapProdFst/Snd compile as local measure helpers, but the EM interpolation Fokker-Planck backend and theorem statuses remain below formalized.", "The conversion window, proof-obligation ledger, and SLT reuse audit record the global phase judgment, five-backend check, theorem route, lower packet, and no-SLT-import status.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-63 upper obligation for the unified/discrete general route refresh. -/ def cycle63UnifiedDiscreteGeneralSkeletonObligation : ProofObligation where id := "sald.unified_discrete_general.cycle63_upper_route" statement := "Cycle 63 upper records that cycle 62 passed and needs no recovery, that Phase 1 theorem-skeleton translation is stable enough for one narrow endpoint-law backfill after route wiring, and that thm:unified-forward-KL and thm:general-moving-target-SALD-discrete remain wired through the accepted continuous/general skeletons and explicit source-cited interfaces. The route consumes prop:guided_path_residual, the correction-field transport bridge, thm:general-moving-target-SALD, general EM endpoint/conditional-Fokker-Planck interfaces, frozen-delta side conditions, discrete KL derivative/LSI, residual DV, lem:gronwall, and theorem-display stitching. All theorem statuses and slow analytic interfaces remain below formalized." source := saldGeneralMovingTargetDiscreteSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle63UnifiedDiscreteGeneralSkeletonUpperPacket", "SALD.cycle62GuidedGeneralSkeletonObligation", "SALD.cycle62GuidedGeneralSkeletonMiddleObligation", "SALD.cycle62GuidedGeneralScaledResidualLowerObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation", "SALD.cycle59MainSkeletonAnalyticMiddleObligation", "SALD.cycle58UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle58UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle53UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle53UnifiedDiscreteGeneralMiddleObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.identity", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_side_conditions", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "SALD.generalMovingTargetDiscreteStatementContract", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions", "AutoSamplingTheory.lawMapEqOfAEEq", "AutoSamplingTheory.lawMapProdEqOfAEEq", "AutoSamplingTheory.lawMapProdFst", "AutoSamplingTheory.lawMapProdSnd", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "This is an upper workflow obligation and theorem-route refresh. It does not add hidden endpoint, common-space, density, absolute-continuity, finite-KL/FI, finite-log-mgf, conditional-law, schedule, sigma-positivity, or coefficient-regularity assumptions to the source theorem statements." /-- Cycle-63 narrow measure-theory backfill below the discrete general EM endpoint-law interface. -/ def cycle63UnifiedDiscreteGeneralMeasureBackfillObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle63_joint_endpoint_law_backfill" statement := "Cycle 63 backfills one narrow measure-theory detail for appendix.tex:1354-1387: AutoSamplingTheory.lawMapProdEqOfAEEq proves that componentwise almost-everywhere equality of two endpoint representatives implies equality of their paired Measure.map pushforward laws, AutoSamplingTheory.lawMapProdFst/Snd project the paired pushforward back to the two endpoint laws under explicit measurability hypotheses, and SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation plus SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation package the named EM endpoint identities into those helpers. This supports later common-space endpoint stitching for the general EM interpolation but does not construct conditional laws, densities, absolute continuity, weak Fokker-Planck, KL differentiation, or Gronwall regularity." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.formalized dependsOn := [ "AutoSamplingTheory.lawMapEqOfAEEq", "AutoSamplingTheory.lawMapProdEqOfAEEq", "AutoSamplingTheory.lawMapProdFst", "AutoSamplingTheory.lawMapProdSnd", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Formalized local Measure.map/a.e.-equality and measurable projection bookkeeping only. SLT Measure.map and a.e. patterns were used as reference style; no SLT theorem is imported or marked as a SALD analytic backend." /-- Cycle-63 middle audit for the unified/discrete general route. This source-to-Lean synchronization layer checks the upper route and the local paired endpoint-law backfill against the paper order, then narrows lower work to the first conditional-law/Fokker--Planck interface that the discrete general theorem still needs. -/ def cycle63UnifiedDiscreteGeneralMiddleContract : GeneralVaSaldGuidedPathMiddleContract where guidedResidualSource := saldGuidedResidualSource generalContinuousSource := saldGeneralMovingTargetSource unifiedSource := saldUnifiedForwardKlSource discreteSource := saldGeneralMovingTargetDiscreteSource objective := "Cycle 63 middle: audit the upper unified/discrete-general route in paper order, confirm that thm:unified-forward-KL remains only the continuous general specialization with c_t=u_t and m_t=w_t, confirm that thm:general-moving-target-SALD-discrete consumes the general EM endpoint/conditional-FP, frozen-delta, LSI, residual DV, and Gronwall interfaces, and keep the next lower packet on the conditional-law/Fokker-Planck half of appendix.tex:1354-1387 after the paired endpoint-law Measure.map helper." sourceStepMap := [ "main_body.tex:359-368 uses prop:guided_path_residual and eq:poisson-eq to make u_t+w_t a transport velocity for pi_t.", "main_body.tex:372-395 states thm:unified-forward-KL with correction-field complexity E_alpha(pi_t,w_t) and the source sigma_t^{-2} dot{s}(t)^{-1} coefficients.", "appendix.tex:949-951 proves thm:unified-forward-KL by specializing thm:general-moving-target-SALD with c_t=u_t and m_t=w_t.", "appendix.tex:1313-1347 states thm:general-moving-target-SALD-discrete with the doubled residual coefficient, Gamma/Delta terms, sigma_eta factors, alpha ranges, and constant inverse-schedule assumption.", "appendix.tex:1354-1357 fixes k and records the endpoint laws hat rho_{s_k}=rho_k^eta and hat rho_{s_{k+1}}=rho_{k+1}^eta.", "appendix.tex:1358-1387 differentiates KL, defines the frozen conditional drift bar b_{k,s}, invokes the conditional Fokker-Planck equation, and starts the derivative route.", "appendix.tex:1389-1511 splits the Laplacian, identifies delta_pi^VA+dot t(s)m_t, and applies the two sigma_eta^2/8 Young splits plus the frozen-delta bound.", "appendix.tex:1513-1570 applies eq:LSI-KL-FI and lem:dv_variation to the residual m_t under the EM interpolation law.", "appendix.tex:1573-1600 changes variables to K(t), uses dot t(s(t))=dot s(t)^{-1}, and applies lem:gronwall to match the theorem display.", "appendix.tex:1603 records the discrete guided VA-SALD specialization c_t=u_t." ] leanStepMap := [ "Use SALD.cycle63UnifiedDiscreteGeneralSkeletonUpperPacket and SALD.cycle63UnifiedDiscreteGeneralSkeletonObligation as parent route data; do not replace the cycle-62 continuous/general route.", "Route thm:unified-forward-KL through SALD.unifiedForwardKlSpecializationContract, sald.unified_forward_kl.transport_velocity_bridge, sald.unified_forward_kl.specialization, SALD.generalVaSaldContract, and the accepted cycle-62 guided/general obligations.", "Route appendix.tex:1313-1347 through SALD.generalMovingTargetDiscreteStatementContract and SALD.generalVaSaldDiscreteContract; both remain contractOnly.", "Route appendix.tex:1354-1357 through SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation, AutoSamplingTheory.lawMapProdEqOfAEEq, AutoSamplingTheory.lawMapProdFst/Snd, SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation, and SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation as endpoint-law bookkeeping only.", "Route appendix.tex:1358-1387 through SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, and sald.general_moving_target_discrete.em_interpolation_fp for common-space, density/AC, regular conditional drift, weak conditional-FP, and KL differentiation obligations.", "Route appendix.tex:1389-1511 through SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.generalMovingTargetDiscreteFrozenResidualAlgebraVector, the cycle-28 Young/Fisher scalar bookkeeping helpers, and sald.general_moving_target_discrete.derivative_side_conditions.", "Route appendix.tex:1513-1570 through SALD.saldLsiKlFiDensityTestContract, probability.lsi_to_kl_fi, dvVariationalFormulaInterface saldDvVariationSource, SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract, and sald.general_moving_target_discrete.dv_m_energy.", "Route appendix.tex:1573-1600 through SALD.generalMovingTargetDiscreteGronwallInstantiationContract, SALD.generalMovingTargetDiscreteGronwallSideConditionContract, cycle-58/cycle-59 Gronwall display handoffs, sald.general_moving_target_discrete.gronwall_application, and sald.general_moving_target_discrete.gronwall_side_conditions.", "Select lower work exactly at sald.general_moving_target_discrete.em_interpolation_fp / sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit for the conditional-law Fokker-Planck backend after the endpoint Measure.map layer." ] citedResultInterfaces := [ "lem:gronwall remains an endpoint-safe differentiability, FTC, coefficient-regularity, endpoint-stitching, and display-matching obligation.", "lem:dv_variation remains source-cited through common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and alpha-scaling witnesses.", "eq:LSI-KL-FI remains the density, zero-set convention, admissible sqrt-density test or approximation, entropy identity, finite KL/FI, and Fisher-chain obligation.", "The continuous Fokker-Planck/KL derivative is consumed upstream through sald.general_moving_target.kl_derivative and the cycle-62 scaled residual scalar handoff, not reproved for the unified theorem.", "The EM interpolation backend remains sald.general_moving_target_discrete.em_interpolation_fp; AutoSamplingTheory.lawMapProdEqOfAEEq, AutoSamplingTheory.lawMapProdFst/Snd, and the two SALD joint endpoint wrappers prove only paired endpoint-law congruence and marginal extraction on a common space.", "Local SLT material is reference-only for Measure.map and a.e.-equality style; no SLT theorem is imported or marked formalized." ] obligations := [ "sald.unified_discrete_general.cycle63_upper_route", "sald.unified_discrete_general.cycle63_middle_route_audit", "sald.general_moving_target_discrete.cycle63_joint_endpoint_law_backfill", "sald.guided_general.cycle62_middle_route_audit", "sald.general_moving_target.cycle62_scaled_residual_lower", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions" ] lowerPacket := [ "Target exactly SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation / SALD.generalMovingTargetDiscreteDerivativeSideConditionContract / sald.general_moving_target_discrete.em_interpolation_fp.", "First sub-slice: after the paired endpoint-law helper, expose common probability space data, hat rho_s and tilde pi_s density/absolute-continuity, and endpoint-law stitching assumptions needed before KL is differentiated.", "Second sub-slice: define and typecheck the regular conditional drift bar b_{k,s}(x), including measurability and integrability hypotheses for the conditional expectation in appendix.tex:1364-1371.", "Third sub-slice: state the weak conditional Fokker-Planck equation from appendix.tex:1372-1387 with conditional drift and Laplacian terms in the exact source signs.", "Do not restate either theorem, change coefficients, or promote endpoint Measure.map bookkeeping into conditional drift, density, weak Fokker-Planck, KL derivative, DV, LSI/KL/FI, or Gronwall formalization." ] reviewerChecklist := [ "SALD.unifiedForwardKlContract and SALD.generalVaSaldDiscreteContract list SALD.cycle63UnifiedDiscreteGeneralMiddleObligation while remaining contractOnly.", "SALD.generalVaSaldProofDag and SALD.generalVaSaldDiscreteProofDag contain ASTIS.SALD.unified_discrete_general.cycle63_middle_route_audit and the paired endpoint-law backfill row.", "SALD.saldDependenciesForLabel entries for thm:unified-forward-KL and thm:general-moving-target-SALD-discrete include the cycle-63 middle contract, middle obligation, and route-audit node.", "Only AutoSamplingTheory.lawMapProdEqOfAEEq, AutoSamplingTheory.lawMapProdFst/Snd, SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation, and SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation are added as formalized local endpoint-law bookkeeping in this cycle; analytic backends remain sourceCited or obligation-level.", "The lower packet is the conditional-law/Fokker-Planck backend over appendix.tex:1358-1387, not a new theorem statement or broad SLT/SDE port.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-63 middle obligation tying the route audit to the next lower conditional-law/Fokker--Planck packet. -/ def cycle63UnifiedDiscreteGeneralMiddleObligation : ProofObligation where id := "sald.unified_discrete_general.cycle63_middle_route_audit" statement := "Cycle 63 middle audits the unified/discrete general route after the upper refresh: thm:unified-forward-KL remains the paper specialization of the continuous general theorem through the correction-field transport bridge, thm:general-moving-target-SALD-discrete remains the source route through EM endpoint/conditional-FP, frozen-delta, LSI, residual DV, and Gronwall, the paired Measure.map helper is only common-space endpoint-law bookkeeping, and the next lower packet is the conditional-law/Fokker-Planck backend over appendix.tex:1358-1387." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle63UnifiedDiscreteGeneralSkeletonUpperPacket", "SALD.cycle63UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle63UnifiedDiscreteGeneralMiddleContract", "SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation", "SALD.cycle62GuidedGeneralSkeletonMiddleObligation", "SALD.cycle62GuidedGeneralScaledResidualLowerObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle58UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle53UnifiedDiscreteGeneralMiddleObligation", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "SALD.generalMovingTargetStatementContract", "SALD.generalVaSaldContract", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_side_conditions", "SALD.generalMovingTargetDiscreteStatementContract", "SALD.generalVaSaldDiscreteContract", "SALD.generalVaSaldEulerMaruyamaContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "AutoSamplingTheory.lawMapProdEqOfAEEq", "AutoSamplingTheory.lawMapProdFst", "AutoSamplingTheory.lawMapProdSnd", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "SALD.generalMovingTargetDiscreteGronwallInstantiationContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI" ] note := "This is middle source-to-Lean synchronization and lower-packet selection, not a theorem proof. It adds no hidden endpoint, density, absolute-continuity, regular conditional law, finite-log-mgf, finite-KL/FI, schedule, sigma-positivity, coefficient-regularity, or Fokker-Planck assumptions to the paper statements." /-- Cycle-63 proof-DAG pane for the unified/discrete general route and one narrow endpoint-law backfill. -/ def cycle63UnifiedDiscreteGeneralDag : List ProofDagBlock := [ { id := "ASTIS.SALD.unified_discrete_general.cycle63_global_phase_judgment" interface := "Upper judgment for cycle 63: cycle 62 passed and needs no recovery; Phase 1 is stable enough for unified/discrete-general route wiring and one narrow endpoint-law backfill; the largest remaining proof risk is the discrete general EM endpoint/common-space layer before conditional-law and Fokker-Planck work." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle63UnifiedDiscreteGeneralSkeletonUpperPacket", "SALD.cycle62GuidedGeneralSkeletonMiddleObligation", "SALD.cycle62GuidedGeneralScaledResidualLowerObligation", "SALD.cycle58UnifiedDiscreteGeneralMiddleObligation" ] reusedBy := ["thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "ASTIS-SALD-001 cycle 63"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.unified_discrete_general.cycle63_five_backend_check" interface := "Explicit cycle-63 check before lower work: endpoint-safe Gronwall, common-space DV with finite log-mgf, LSI/KL/FI density-test bridge, continuous Fokker-Planck/KL derivative, and EM endpoint/conditional-law Fokker-Planck all have named source-cited or obligation interfaces." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.saldGronwallEndpointCalculusContract", "SALD.generalMovingTargetGronwallSideConditionContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.unified_discrete_general.cycle63_upper_route" interface := "Route thm:unified-forward-KL through guided residual, correction-field transport, and thm:general-moving-target-SALD; route thm:general-moving-target-SALD-discrete through general EM endpoint/conditional-FP, frozen delta, KL derivative/LSI, residual DV, Gronwall, and display stitching, with statements and constants unchanged." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle63UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle62GuidedGeneralSkeletonMiddleObligation", "SALD.guidedResidualIdentityContract", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "SALD.generalMovingTargetDiscreteStatementContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_side_conditions" ] reusedBy := ["thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.unified_discrete_general.cycle63_middle_route_audit" interface := "Middle route audit for cycle 63: check the unified specialization and discrete general EM/frozen-delta/LSI/DV/Gronwall chain against the source order, keep the paired Measure.map endpoint backfill scoped to endpoint laws, and hand off the conditional-law/Fokker-Planck backend over appendix.tex:1358-1387." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle63UnifiedDiscreteGeneralMiddleContract", "SALD.cycle63UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle63UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "sald.unified_forward_kl.specialization", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_side_conditions" ] reusedBy := ["thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "cycle63 lower conditional-FP packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.cycle63_joint_endpoint_law_backfill" interface := "Narrow measure-theory backfill: from componentwise a.e. endpoint identities, derive equality of paired Measure.map pushforward laws and project that joint law to each marginal endpoint law for common-space EM endpoint bookkeeping." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/Probability.lean" dependsOn := [ "AutoSamplingTheory.lawMapEqOfAEEq", "AutoSamplingTheory.lawMapProdEqOfAEEq", "AutoSamplingTheory.lawMapProdFst", "AutoSamplingTheory.lawMapProdSnd", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := ["sald.general_moving_target_discrete.em_interpolation_fp", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.formalized } ] /-! ### Cycle 64: analytic interface ledger refresh -/ /-- Cycle-64 upper ledger for the analytic-interface sprint. This is route data only. It records the required upper phase judgment, re-checks the five slow analytic backends, and routes them through the six faithful theorem skeletons in paper order while keeping every unproved backend below `formalized`. -/ def cycle64MainSkeletonAnalyticInterfaceLedger : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteDerivativeSource objective := "Cycle 64 upper: cycle 63 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for exactly one narrow cited-theory/SDE backend backfill, not broad reusable API work; the single lower packet that best reduces risk is the EM interpolation conditional-law/Fokker-Planck backend over appendix.tex:1358-1387, because it supports both thm:forward-KL-discrete and thm:general-moving-target-SALD-discrete." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "proof:thm:forward-KL:derivative", "proof:thm:forward-KL-discrete:conditional-fp", "proof:thm:general-moving-target-SALD:derivative", "proof:thm:general-moving-target-SALD-discrete:derivative", "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Gronwall: SALD.saldGronwallCandidateContract, SALD.saldGronwallEndpointCalculusContract, SALD.saldGronwallExponentRewriteContract, and the cycle-36/41 assembled wrappers expose continuous a,b, endpoint-safe differentiability or right-derivative/FTC semantics, interval-integrability, endpoint evaluation, exponent algebra, and theorem-specific coefficient/display side conditions; lem:gronwall remains ProofStatus.obligation.", "Donsker-Varadhan: dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, and theorem-specific DV witness contracts expose common-space probability measures, absolute continuity, finite KL/log-likelihood, selected-test measurability, finite log-mgf, positive-alpha scaling, and E_alpha rewriting; the Boucheron variational equality remains ProofStatus.sourceCited.", "LSI/KL/FI: SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, probability.lsi_to_kl_fi, and the cycle-43 density/entropy/Fisher helpers expose rho << pi, Radon-Nikodym density, zero-set convention, admissible sqrt-density test or approximation, entropy identity, finite KL/FI, and the Fisher chain rule; eq:LSI-KL-FI remains ProofStatus.obligation.", "Continuous Fokker-Planck/KL derivative: SALD.forwardKlDerivativeCandidateContract, SALD.forwardKlDerivativeSideConditionContract, SALD.generalMovingTargetDerivativeCandidateContract, and cycle-60/62 scalar handoffs expose mass conservation, KL differentiation under the integral, SALD/general Fokker-Planck equations, integration by parts, target transport, residual scaling, LSI, and inverse-schedule calculus; analytic derivative backends remain ProofStatus.obligation.", "Euler-Maruyama interpolation Fokker-Planck: SALD.discreteForwardKlEmInterpolationSideConditionContract, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation, cycle-63 endpoint-law Measure.map helpers, and sald.general_moving_target_discrete.em_interpolation_fp expose endpoint laws, common-space bookkeeping, regular conditional drift, density/absolute-continuity, weak conditional Fokker-Planck signs, Laplacian split, and stitched intervals; the conditional-law/FP backend remains ProofStatus.obligation." ] theoremRoute := [ "1. thm:forward-KL is wired through continuous KL derivative/Fokker-Planck, eq:LSI-KL-FI, DV velocity-energy, endpoint schedule identities, and Gronwall side conditions.", "2. thm:forward-KL-discrete is wired through EM endpoint/conditional-FP, frozen defect, LSI, DV velocity, Gronwall accumulation, and accumulated-error display interfaces.", "3. prop:guided_path_residual is wired through normalizer differentiation, guided-density differentiation, divergence cancellation, and mean-zero residual obligations; it is not promoted by this ledger.", "4. thm:general-moving-target-SALD is wired through continuous general KL derivative/Fokker-Planck, residual Young/LSI, residual DV, sigma-weighted Gronwall, endpoint/exponent side conditions, and pure-contraction specialization.", "5. thm:unified-forward-KL remains the appendix specialization of thm:general-moving-target-SALD through prop:guided_path_residual, the correction-field transport bridge, c_t=u_t, and m_t=w_t.", "6. thm:general-moving-target-SALD-discrete is wired through general EM endpoint/conditional-FP, frozen-delta, KL derivative/LSI, residual DV, constant-schedule time change, Gronwall/display stitching, and the guided discrete specialization." ] modeDiscipline := [ "faithfulPaper Phase 1: use only main_body.tex, appendix.tex, and iteration_complexity.tex under the original SALD paper root; sald_version_2.tex remains excluded.", "Keep all source theorem statements, constants, alpha ranges, sigma factors, endpoint laws, source labels, and proof order fixed.", "Every unproved analytic backend remains ProofStatus.obligation or ProofStatus.sourceCited; already compiled scalar and endpoint-law helpers are local dependencies only.", "Only one cited-theory/SDE backend may be pursued after this upper ledger, and it must be theorem-route useful." ] nonGoals := [ "No source-index rebaseline beyond the required acceptance command unless a reviewer reports a blocking anchor defect.", "No broad SLT import, Gaussian concentration port, entropy-duality library reorganization, disintegration project, or teaching API rewrite.", "No hidden endpoint, density, absolute-continuity, finite-KL/FI, finite-log-mgf, boundary, smoothness, conditional-law, sigma-positivity, schedule, coefficient-regularity, or stitched-interval assumption is added to a source theorem statement.", "No alternate proof route replaces derivative -> LSI -> DV -> Gronwall, the correction-field specialization, or the paper EM/frozen-delta route." ] lowerPacket := [ "Middle should synchronize this cycle-64 analytic-interface ledger with conversion-windows/ASTIS-SALD-001.md and proof-obligations/ASTIS-SALD-001.md, preserving the theorem route order forward-KL, discrete forward-KL, guided residual, general moving-target, unified forward-KL, discrete general moving-target.", "Lower should target exactly SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation / SALD.generalMovingTargetDiscreteDerivativeSideConditionContract / sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387.", "First sub-slice: after the cycle-63 endpoint-law helpers, expose common probability space data, hat rho_s and tilde pi_s density/absolute-continuity, and endpoint-law stitching assumptions needed before KL differentiation.", "Second sub-slice: define and typecheck the regular conditional drift bar b_{k,s}(x), including measurability and integrability hypotheses for the conditional expectation in appendix.tex:1364-1371.", "Third sub-slice: state the weak conditional Fokker-Planck equation from appendix.tex:1372-1387 with the exact source signs for the drift divergence and sigma_eta^2/2 Laplacian.", "Do not promote endpoint Measure.map bookkeeping into conditional drift, density, weak Fokker-Planck, KL derivative, DV, LSI/KL/FI, or Gronwall formalization." ] reviewerChecklist := [ "SALD.cycle64MainSkeletonAnalyticInterfaceObligation is listed by all six theorem contracts while those contracts remain ProofStatus.contractOnly.", "SALD.forwardKlProofDag, SALD.discreteForwardKlProofDag, SALD.generalVaSaldProofDag, and SALD.generalVaSaldDiscreteProofDag include SALD.cycle64MainSkeletonAnalyticInterfaceDag.", "SALD.saldDependenciesForLabel includes cycle64MainSkeletonDependencyNames for the five slow interfaces and the six theorem-route labels.", "The lower packet is the EM interpolation conditional-law/Fokker-Planck backend over appendix.tex:1358-1387, not a new theorem statement, theorem status promotion, or broad SLT/SDE port.", "Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL differentiation, EM conditional-FP, theorem contracts, and SLT reuse statuses all remain below formalized unless already compiled locally.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle63UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation" ] status := ProofStatus.obligation /-- Cycle-64 upper obligation for the refreshed analytic-interface ledger. -/ def cycle64MainSkeletonAnalyticInterfaceObligation : ProofObligation where id := "sald.main_skeleton.cycle64_analytic_interface_ledger" statement := "Cycle 64 upper records that cycle 63 passed and needs no recovery, that Phase 1 is stable enough for exactly one narrow backend backfill, and that the five slow analytic interfaces remain precise and below formalized while being wired through thm:forward-KL, thm:forward-KL-discrete, prop:guided_path_residual, thm:general-moving-target-SALD, thm:unified-forward-KL, and thm:general-moving-target-SALD-discrete. The selected lower packet is the EM interpolation conditional-law/Fokker-Planck backend over appendix.tex:1358-1387." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle64MainSkeletonAnalyticInterfaceLedger", "SALD.cycle63UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation", "SALD.saldGronwallCandidateContract", "SALD.saldGronwallEndpointCalculusContract", "SALD.saldGronwallExponentRewriteContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "sald.general_moving_target_discrete.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI" ] note := "Upper workflow obligation only. It adds no theorem hypotheses and does not promote Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL differentiation, EM conditional Fokker-Planck, theorem contracts, or SLT reuse status." /-- Cycle-64 middle audit for the analytic-interface ledger. This synchronizes the upper ledger with the source transcript and narrows the lower packet to the EM interpolation conditional-law/Fokker--Planck backend that remains after the cycle-63 endpoint-law bookkeeping. -/ def cycle64MainSkeletonAnalyticMiddleContract : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteDerivativeSource objective := "Cycle 64 middle: audit the upper analytic-interface ledger against the source theorem route, verify that the five slow analytic interfaces are consumed by the six faithful theorem skeletons in paper order, and narrow lower work to sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387 after the cycle-63 endpoint-law Measure.map helpers." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "proof:thm:forward-KL:derivative", "proof:thm:forward-KL-discrete:conditional-fp", "proof:thm:general-moving-target-SALD:derivative", "proof:thm:general-moving-target-SALD-discrete:em-conditional-fp", "eq:general_KL_derivative_0_discrete", "eq:general_moving_target_SALD_frozen_interp", "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Middle five-backend check, Gronwall: all theorem routes still consume SALD.saldGronwallEndpointCalculusContract and theorem-specific side-condition obligations; no endpoint-safe differentiability, FTC, coefficient-regularity, or display-matching fact is promoted.", "Middle five-backend check, DV: all DV uses still pass through dvVariationalFormulaInterface saldDvVariationSource plus theorem-specific common-space, absolute-continuity, finite-KL, measurability, positive-alpha, and finite-log-mgf witnesses.", "Middle five-backend check, LSI/KL/FI: all LSI uses still pass through SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi with density, zero-set convention, admissible-test, entropy identity, finite KL/FI, and Fisher-chain assumptions explicit.", "Middle five-backend check, continuous Fokker-Planck/KL derivative: thm:forward-KL and thm:general-moving-target-SALD consume the existing derivative candidate and side-condition interfaces; cycle-60 and cycle-62 scalar handoffs do not close the analytic backend.", "Middle five-backend check, EM interpolation Fokker-Planck: discrete theorem routes consume SALD.discreteForwardKlEmInterpolationSideConditionContract and SALD.generalMovingTargetDiscreteDerivativeSideConditionContract; cycle-63 endpoint-law helpers prove only Measure.map bookkeeping, leaving conditional drift, density/AC, weak FP, KL differentiation, and stitching open." ] theoremRoute := [ "1. thm:forward-KL: appendix.tex:168-252 remains derivative/Fokker-Planck -> LSI/KL/FI -> DV -> Gronwall.", "2. thm:forward-KL-discrete: appendix.tex:260-592 remains EM conditional-FP -> KL derivative/frozen defect/LSI -> DV -> Gronwall -> accumulated errors.", "3. prop:guided_path_residual: appendix.tex:619-704 remains normalizer differentiation, residual identity, and mean-zero cancellation.", "4. thm:general-moving-target-SALD: appendix.tex:724-949 remains continuous general KL derivative -> residual LSI/DV -> sigma-weighted Gronwall.", "5. thm:unified-forward-KL: appendix.tex:949-951 remains only the c_t=u_t, m_t=w_t specialization through prop:guided_path_residual and thm:general-moving-target-SALD.", "6. thm:general-moving-target-SALD-discrete: appendix.tex:1313-1603 remains general EM endpoint/conditional-FP -> frozen delta -> KL derivative/LSI -> residual DV -> constant-schedule Gronwall/display stitching." ] modeDiscipline := [ "faithfulPaper Phase 1: this is source-to-Lean synchronization only, using the original main_body.tex, appendix.tex, and iteration_complexity.tex; sald_version_2.tex remains excluded.", "The cycle-63 endpoint-law Measure.map helpers may be reused only as common-space endpoint bookkeeping; they do not supply conditional laws, densities, weak Fokker-Planck, KL differentiation, or Gronwall stitching.", "All theorem contracts remain ProofStatus.contractOnly and all slow analytic interfaces remain ProofStatus.obligation or ProofStatus.sourceCited unless already compiled locally.", "The selected lower packet is theorem-route useful and narrow: appendix.tex:1358-1387 for the general discrete EM conditional-law/Fokker-Planck backend." ] nonGoals := [ "Do not restate any theorem display, alter alpha ranges, sigma_eta factors, endpoint laws, Gamma/Delta terms, residual coefficients, or source labels.", "Do not replace the source derivative -> LSI -> DV -> Gronwall proof route or the paper EM/frozen-delta route.", "Do not begin broad SLT, disintegration, SDE, Gaussian concentration, entropy-duality, or reusable API backfill.", "Do not mark conditional drift, density/absolute-continuity, weak Fokker-Planck, KL differentiation, DV, LSI/KL/FI, Gronwall, theorem contracts, or SLT reuse as formalized." ] lowerPacket := [ "Target exactly SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation / SALD.generalMovingTargetDiscreteDerivativeSideConditionContract / sald.general_moving_target_discrete.em_interpolation_fp.", "Source slice appendix.tex:1358-1366: expose endpoint-safe KL differentiation, mass conservation int partial_s hat rho_s dx=0, density and absolute-continuity requirements for hat rho_s and tilde pi_s, and finite KL on the interpolation interval.", "Source slice appendix.tex:1368-1377: define the frozen conditional drift bar b_{k,s}(x) from the regular conditional law of X_k^eta given hat X_s=x, with measurability and integrability obligations for both drift summands; cycle 64 lower compiles only the abstract linear-combination algebra once conditional-expectation linearity is supplied.", "Source slice appendix.tex:1379-1387: state the weak conditional Fokker-Planck equation for hat rho_s with drift term -div(hat rho_s bar b_{k,s}) and Laplacian +(sigma_eta^2/2) Delta hat rho_s, preserving the source signs.", "Keep the Laplacian split after appendix.tex:1389, frozen-delta algebra, LSI, DV, and Gronwall as separate downstream obligations." ] reviewerChecklist := [ "SALD.cycle64MainSkeletonAnalyticMiddleObligation is listed by all six theorem contracts while those contracts remain ProofStatus.contractOnly.", "SALD.cycle64MainSkeletonAnalyticInterfaceDag contains ASTIS.SALD.cycle64.middle_interface_audit before the lower-packet node.", "SALD.saldDependenciesForLabel includes SALD.cycle64MainSkeletonAnalyticMiddleContract, SALD.cycle64MainSkeletonAnalyticMiddleObligation, and sald.main_skeleton.cycle64_middle_interface_audit through cycle64MainSkeletonDependencyNames.", "The conversion window and proof-obligation ledger record the source-to-Lean split for appendix.tex:1358-1387.", "No theorem statement, source constant, source label, source-file scope, SLT reuse entry, or analytic backend status changes.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle64MainSkeletonAnalyticInterfaceLedger", "SALD.cycle64MainSkeletonAnalyticInterfaceObligation", "SALD.cycle63UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation", "SALD.cycle59MainSkeletonAnalyticMiddleObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination", "SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination", "SALD.cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation /-- Cycle-64 middle obligation tying the analytic-interface audit to the selected EM conditional-law/Fokker--Planck lower packet. -/ def cycle64MainSkeletonAnalyticMiddleObligation : ProofObligation where id := "sald.main_skeleton.cycle64_middle_interface_audit" statement := "Cycle 64 middle audits the upper analytic-interface ledger against the paper route: the five slow interfaces are consumed by the six theorem skeletons in source order, the cycle-63 endpoint-law Measure.map helpers are kept as endpoint/common-space bookkeeping only, and the selected lower packet remains sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387 for KL differentiation prerequisites, regular conditional drift, and the weak conditional Fokker-Planck equation with source signs. All theorem statuses and slow analytic interfaces remain below formalized." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle64MainSkeletonAnalyticMiddleContract", "SALD.cycle64MainSkeletonAnalyticInterfaceObligation", "SALD.cycle63UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle59MainSkeletonAnalyticMiddleObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination", "SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination", "sald.general_moving_target_discrete.cycle64_conditional_drift_lower", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] note := "Middle workflow obligation only. It adds no hidden density, endpoint, common-space, regular conditional law, finite-KL/FI, finite-log-mgf, weak Fokker-Planck, schedule, sigma-positivity, coefficient-regularity, or stitched-interval assumption to any source theorem." /-- Cycle-64 proof-DAG pane for the analytic-interface ledger and selected conditional-law/Fokker--Planck lower packet. -/ def cycle64MainSkeletonAnalyticInterfaceDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle64.global_phase_judgment" interface := "Upper judgment for cycle 64: cycle 63 passed; Phase 1 supports one narrow backend backfill only; the largest remaining proof risk is the EM interpolation conditional-law/Fokker-Planck backend over appendix.tex:1358-1387." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle64MainSkeletonAnalyticInterfaceLedger", "SALD.cycle63UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation" ] reusedBy := ["ASTIS-SALD-001 cycle 64", "sald.general_moving_target_discrete.em_interpolation_fp"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle64.five_backend_check" interface := "Recheck the five slow analytic interfaces: endpoint-safe Gronwall, common-space DV with finite log-mgf, LSI/KL/FI density-test bridge, continuous Fokker-Planck/KL derivative, and EM endpoint/conditional-law Fokker-Planck." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle64.theorem_route_rewire" interface := "Wire the five analytic interfaces into the six faithful theorem skeletons in paper order without changing source statements, constants, labels, theorem statuses, or backend statuses." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle64MainSkeletonAnalyticInterfaceObligation", "SALD.continuousSaldContract", "SALD.discreteSaldContract", "SALD.guidedResidualContract", "SALD.generalVaSaldContract", "SALD.unifiedForwardKlContract", "SALD.generalVaSaldDiscreteContract", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle62GuidedGeneralSkeletonMiddleObligation", "SALD.cycle63UnifiedDiscreteGeneralMiddleObligation" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle64.middle_interface_audit" interface := "Middle audit for cycle 64: synchronize the five-backend analytic ledger with the source-to-Lean theorem route, keep cycle-63 endpoint-law helpers scoped to endpoint bookkeeping, and narrow lower work to appendix.tex:1358-1387 general EM conditional-law/Fokker-Planck prerequisites." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle64MainSkeletonAnalyticMiddleContract", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.cycle64MainSkeletonAnalyticInterfaceObligation", "SALD.cycle63UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "sald.general_moving_target_discrete.cycle64_conditional_drift_lower", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "cycle64 lower conditional-FP packet" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle64.lower_packet.general_discrete_em_conditional_fp" interface := "Selected lower packet for cycle 64: refine sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387 after endpoint-law bookkeeping, exposing common-space/density assumptions, regular conditional drift, and the weak conditional Fokker-Planck equation with source signs." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle64MainSkeletonAnalyticInterfaceObligation", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination", "SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination", "sald.general_moving_target_discrete.cycle64_conditional_drift_lower", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] /-! ### Cycle 65: continuous forward-KL post-cycle64 route -/ /-- Cycle-65 upper packet for the continuous forward-KL skeleton after the accepted cycle-64 analytic-interface and conditional-drift pass. This is upper-route data only. It returns the main skeleton sprint to `thm:forward-KL`, rechecks the five slow interfaces, and assigns the next lower packet to the continuous Fokker--Planck/KL derivative backend without changing the paper statement, constants, labels, or proof statuses. -/ def cycle65ForwardKlSkeletonUpperPacket : ForwardKlUpperPacket where objective := "Cycle 65 upper: after the accepted cycle-64 reviewer/build gate, rewire the source-cited analytic interfaces specifically into the faithful continuous thm:forward-KL proof skeleton, matching main_body.tex:238-247 and appendix.tex:164-252 without changing constants, theorem status, proof order, or source labels." sourceLabels := [ "thm:forward-KL", "proof:thm:forward-KL", "proof:thm:forward-KL:derivative", "proof:thm:forward-KL:dv-energy", "proof:thm:forward-KL:gronwall", "eq:KL-derivative-0", "eq:KL-derivative-1", "eq:KL-derivative-2", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "eq:SALD", "eq:FP-eq", "def:alpha-complexity", "proof:thm:forward-KL-discrete:conditional-fp" ] modeDiscipline := [ "Global phase judgment: cycle 64 passed reviewer/build, so no failed previous cycle must be recovered before cycle 65 work.", "Global phase judgment: Phase 1 theorem-skeleton translation is stable enough for this narrow continuous forward-KL route audit, but not for broad cited-theory, SDE, disintegration, or reusable API backfill.", "Global phase judgment: the single lower packet that best reduces the remaining continuous theorem risk is SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative over appendix.tex:168-228.", "Five-backend check: Gronwall keeps endpoint-safe differentiability/FTC, interval-integrability, endpoint, and exponent side-condition obligations; DV keeps common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and positive-alpha interfaces; LSI/KL/FI keeps density, zero-set, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule obligations; continuous Fokker-Planck/KL derivative keeps mass-conservation, boundary, target-transport, and schedule-calculus obligations; EM interpolation FP remains a downstream discrete sibling backend.", "Route thm:forward-KL in the paper order appendix.tex:168-228 derivative/Fokker-Planck and time change, appendix.tex:230-241 DV velocity-energy, and appendix.tex:244-252 Gronwall endpoint/exponent display matching.", "Preserve the exact source coefficients: the one-half Young split, dot{s}(t)^(-1), alpha in (0, alpha0], the two theorem exponent factors, and the residual alpha-complexity integral." ] nonGoals := [ "Do not restate thm:forward-KL, change SALD.continuousForwardKlStatementContract, or promote SALD.continuousSaldContract above contractOnly.", "Do not prove or replace Gronwall, DV, LSI/KL/FI, the continuous Fokker-Planck/KL derivative, or the EM interpolation backend in this upper packet.", "Do not merge exponent factors, alter source constants, add hidden endpoint/density/regularity assumptions, or use sald_version_2.tex.", "Do not begin broad SLT, Gaussian concentration, entropy-duality, disintegration, measure-theory, or SDE backfill before this continuous route is reviewed cleanly." ] lowerPacket := [ "Middle should synchronize this cycle-65 upper route with conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, and the thm:forward-KL proof DAG.", "Lower should target exactly SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative over appendix.tex:168-228.", "First lower sub-slice: appendix.tex:168-185 mass conservation, differentiating KL under the integral, SALD Fokker-Planck substitution, boundary/no-flux integration by parts, and the -FI identification.", "Second lower sub-slice: appendix.tex:187-208 slowed target velocity, target integration by parts, Cauchy-Schwarz/Young with the exact 1/2 share, and the L2 velocity term.", "Third lower sub-slice: appendix.tex:218-228 inverse-schedule chain rule, slowed-velocity square scaling, and dot{s}(t)*dot t(s(t))^2 = dot{s}(t)^(-1).", "Leave LSI/KL/FI, DV finite-log-mgf, Gronwall endpoint/exponent side conditions, and EM interpolation as separate named interfaces unless their exact analytic backends compile locally." ] reviewerChecklist := [ "SALD.continuousSaldContract lists SALD.cycle65ForwardKlSkeletonObligation while remaining contractOnly.", "SALD.forwardKlProofDag contains ASTIS.SALD.forward_KL.cycle65_continuous_route and ASTIS.SALD.forward_KL.cycle65_lower_packet.kl_derivative after the cycle-64 and cycle-60 route data.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes SALD.cycle65ForwardKlSkeletonUpperPacket, SALD.cycle65ForwardKlSkeletonObligation, and the cycle-65 DAG nodes.", "All five slow analytic backends remain ProofStatus.obligation or ProofStatus.sourceCited unless their full analytic dependencies build locally.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-65 upper obligation tying the accepted cycle-64 analytic ledger back to the focused continuous forward-KL theorem route. -/ def cycle65ForwardKlSkeletonObligation : ProofObligation where id := "sald.forward_kl.cycle65_continuous_route" statement := "Cycle 65 upper records the post-cycle-64 global phase judgment and wires thm:forward-KL through the five source-cited analytic interfaces in the paper order: appendix.tex:168-228 continuous KL derivative/Fokker-Planck plus LSI/time change, appendix.tex:230-241 DV velocity-energy with finite-log-mgf witness, and appendix.tex:244-252 Gronwall endpoint/exponent display matching. main_body.tex:238-247 remains the unchanged theorem statement, and the selected lower packet is sald.forward_kl.kl_derivative over appendix.tex:168-228." source := saldForwardKlProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle65ForwardKlSkeletonUpperPacket", "SALD.cycle64MainSkeletonAnalyticInterfaceObligation", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation", "SALD.cycle60ForwardKlSkeletonObligation", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle60ForwardKlDerivativeRawLowerObligation", "SALD.cycle55ForwardKlSkeletonObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle50ForwardKlSkeletonObligation", "SALD.cycle50ForwardKlSkeletonMiddleObligation", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "SALD.forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar", "SALD.forwardKlPointwisePreDvDerivativeBoundOfRawKlFiVelocityScaling", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "sald.forward_kl.endpoint_schedule_identities", "sald.forward_kl.gronwall_side_conditions", "sald.forward_kl.gronwall_application", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "Upper workflow obligation only. It adds no theorem hypotheses and does not promote Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL differentiation, EM interpolation, theorem status, or SLT reuse status." /-- Cycle-65 middle audit for the continuous forward-KL route after the post-cycle-64 upper packet. This source-to-Lean synchronization layer checks the exact theorem statement and proof order used by `thm:forward-KL`, then hands lower work back to the continuous KL derivative/Fokker--Planck backend. It is workflow data only. -/ def cycle65ForwardKlSkeletonMiddleContract : ForwardKlMiddleSourceToLeanContract where sourceStatement := saldForwardKlSource sourceProof := saldForwardKlProofSource derivativeSource := saldForwardKlDerivativeSource dvSource := saldForwardKlDvEnergySource gronwallSource := saldForwardKlGronwallSource objective := "Cycle 65 middle: synchronize the post-cycle-64 continuous thm:forward-KL route against main_body.tex:238-247 and appendix.tex:164-252, verify that the upper packet consumes the five source-cited analytic interfaces in paper order, and keep the lower packet exactly on SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative over appendix.tex:168-228 without changing constants, statements, source labels, or backend statuses." sourceStepMap := [ "main_body.tex:238-247 fixes the theorem display: LSI constants C_LSI(t)>=0, finite alpha0-complexity for the transport velocity v_t, alpha in (0,alpha0], SALD law rho_s, the initial KL factor exp(-int dot{s} C_LSI)*exp(int (1/2)*dot{s}^(-1)*alpha^(-1)), and the residual alpha-complexity integral.", "appendix.tex:168-185 differentiates KL(rho_s||tilde pi_s), uses int partial_s rho_s dx=0, substitutes the SALD Fokker--Planck equation, integrates by parts, and identifies the first term with -FI.", "appendix.tex:187-208 defines tilde v_s=dot t(s)*v_{t(s)}, proves it transports tilde pi_s, evaluates the target-time term by integration by parts, and applies Cauchy--Schwarz/Young with the exact one-half coefficient.", "appendix.tex:210-228 combines the derivative identity with eq:LSI-KL-FI and the inverse-schedule chain rule to obtain the t-time inequality with residual coefficient (1/2)*dot{s}(t)^(-1)*||v_t||^2.", "appendix.tex:230-241 applies lem:dv_variation to Z=alpha*||v_t||^2, using the theorem's finite-log-mgf assumption through the alpha-complexity interface, and produces the pre-Gronwall coefficient dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1).", "appendix.tex:244-252 applies lem:gronwall and then performs the source exponent split and nonnegative-LSI residual exponent drop that match the theorem display." ] leanStepMap := [ "Use SALD.cycle65ForwardKlSkeletonUpperPacket and SALD.cycle65ForwardKlSkeletonObligation as parent route data after the accepted cycle-64 analytic-interface pass.", "Keep SALD.continuousForwardKlStatementContract and SALD.continuousSaldContract contractOnly; this middle audit adds no theorem hypotheses and changes no theorem display.", "Route appendix.tex:168-228 through SALD.forwardKlDerivativeCandidateContract, SALD.forwardKlDerivativeSideConditionContract, SALD.forwardKlDerivativeObligation, SALD.forwardKlDensityBoundaryObligation, SALD.forwardKlScheduleTimeChangeObligation, sald.forward_kl.density_boundary_regular, sald.forward_kl.schedule_time_change, and sald.forward_kl.kl_derivative.", "Consume SALD.forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar and the pointwise wrapper SALD.forwardKlPointwisePreDvDerivativeBoundOfRawKlFiVelocityScaling only after the raw derivative split, mass-conservation input, LSI comparison, slowed-velocity scaling, and inverse-schedule identity are supplied explicitly.", "Route appendix.tex:210-217 through SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi, preserving density, zero-set convention, admissible sqrt-density test or approximation, entropy identity, finite KL/FI, and Fisher chain-rule obligations.", "Route appendix.tex:230-241 through dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, SALD.forwardKlDvFiniteLogMgfWitnessContract, sald.forward_kl.dv_finite_log_mgf_witness, and sald.forward_kl.dv_energy_bound.", "Route appendix.tex:244-252 through SALD.saldGronwallEndpointCalculusContract, SALD.forwardKlGronwallInstantiationContract, SALD.forwardKlGronwallSideConditionContract, sald.forward_kl.endpoint_schedule_identities, sald.forward_kl.gronwall_side_conditions, and sald.forward_kl.gronwall_application.", "Keep SALD.discreteForwardKlEmInterpolationSideConditionContract and sald.discrete_forward_kl.em_interpolation_fp visible only as downstream discrete sibling interfaces, not hidden assumptions of thm:forward-KL." ] citedResultInterfaces := [ "lem:gronwall remains an obligation with endpoint-safe differentiability/FTC, interval-integrability, coefficient regularity, endpoint rewrites, exponent splitting, and theorem-display matching.", "lem:dv_variation remains source-cited with common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and positive-alpha scaling witnesses.", "eq:LSI-KL-FI remains an obligation for density, zero-set convention, admissible sqrt-density test or approximation, entropy identity, finite KL/FI, and Fisher chain rule.", "The continuous Fokker--Planck/KL derivative identity remains the selected local SDE/measure-analysis backend, with mass conservation, boundary/no-flux, target transport, and schedule calculus still open.", "The Euler--Maruyama interpolation Fokker--Planck endpoint/conditional-law backend remains a downstream discrete obligation and is not imported into the continuous theorem." ] obligations := [ "sald.forward_kl.cycle65_continuous_route", "sald.forward_kl.cycle65_middle_route_audit", "sald.forward_kl.cycle65_derivative_pointwise_lower", "sald.forward_kl.kl_derivative", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "probability.lsi_to_kl_fi", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "sald.forward_kl.endpoint_schedule_identities", "sald.forward_kl.gronwall_side_conditions", "sald.forward_kl.gronwall_application", "sald.discrete_forward_kl.em_interpolation_fp" ] lowerPacket := [ "Target exactly SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative.", "First lower sub-slice: appendix.tex:168-185 mass conservation, differentiating KL under the integral, SALD Fokker--Planck substitution, boundary/no-flux integration by parts, and -FI identification.", "Second lower sub-slice: appendix.tex:187-208 slowed target velocity, target integration by parts, Cauchy--Schwarz/Young with the exact 1/2 share, and the L2 velocity term.", "Third lower sub-slice: appendix.tex:218-228 inverse-schedule chain rule, slowed-velocity square scaling, and dot{s}(t)*dot t(s(t))^2=dot{s}(t)^(-1), with the compiled pointwise wrapper producing the t-indexed pre-DV inequality.", "Leave LSI/KL/FI, DV finite-log-mgf, Gronwall endpoint/exponent conditions, and EM interpolation as separate named interfaces unless their exact analytic backends compile locally." ] reviewerChecklist := [ "SALD.continuousSaldContract lists SALD.cycle65ForwardKlSkeletonMiddleObligation while remaining contractOnly.", "SALD.forwardKlProofDag contains ASTIS.SALD.forward_KL.cycle65_middle_route_audit after ASTIS.SALD.forward_KL.cycle65_continuous_route and before the selected lower-packet node.", "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes SALD.cycle65ForwardKlSkeletonMiddleContract, SALD.cycle65ForwardKlSkeletonMiddleObligation, and sald.forward_kl.cycle65_middle_route_audit.", "No theorem statement, source coefficient, source label, source-file selection, SLT reuse entry, or analytic backend status is changed.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-65 middle obligation tying the post-cycle-64 forward-KL route audit to the selected continuous derivative/Fokker--Planck lower packet. -/ def cycle65ForwardKlSkeletonMiddleObligation : ProofObligation where id := "sald.forward_kl.cycle65_middle_route_audit" statement := "Cycle 65 middle audits the post-cycle-64 continuous thm:forward-KL skeleton: main_body.tex:238-247 remains unchanged, appendix.tex:168-252 is consumed in paper order through the continuous KL derivative/Fokker-Planck interface, eq:LSI-KL-FI, lem:dv_variation, and lem:gronwall, the downstream EM interpolation backend remains visible only for discrete reuse, and the next proof-producing lower packet stays exactly sald.forward_kl.kl_derivative over appendix.tex:168-228. All theorem statuses and slow analytic interfaces remain below formalized." source := saldForwardKlProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle65ForwardKlSkeletonUpperPacket", "SALD.cycle65ForwardKlSkeletonObligation", "SALD.cycle65ForwardKlSkeletonMiddleContract", "SALD.cycle64MainSkeletonAnalyticInterfaceObligation", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation", "SALD.cycle60ForwardKlSkeletonObligation", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle60ForwardKlDerivativeRawLowerObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle50ForwardKlDerivativeLowerObligation", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "SALD.forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar", "SALD.forwardKlPointwisePreDvDerivativeBoundOfRawKlFiVelocityScaling", "SALD.cycle65ForwardKlDerivativePointwiseLowerObligation", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "sald.forward_kl.endpoint_schedule_identities", "sald.forward_kl.gronwall_side_conditions", "sald.forward_kl.gronwall_application", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "This is a middle workflow obligation and source-to-Lean route audit, not an analytic proof. It adds no hidden density, boundary, endpoint, absolute-continuity, finite-log-mgf, finite-KL/FI, schedule, coefficient-regularity, or EM conditional-law assumptions to thm:forward-KL." /-- Cycle-65 lower proof-producing obligation for the pointwise continuous derivative wrapper. -/ def cycle65ForwardKlDerivativePointwiseLowerObligation : ProofObligation where id := "sald.forward_kl.cycle65_derivative_pointwise_lower" statement := "Cycle 65 lower compiles the pointwise wrapper for appendix.tex:168-228: after the analytic backend supplies the raw KL derivative split, mass conservation, SALD Fokker-Planck/integration-by-parts first-term identity, target Cauchy estimate, LSI/KL/FI comparison, slowed-velocity scaling, and inverse-schedule product identity at each t, SALD.forwardKlPointwisePreDvDerivativeBoundOfRawKlFiVelocityScaling derives the pointwise t-time pre-DV inequality K'(t) <= -dot{s}(t)*C_LSI(t)*K(t)+(1/2)*dot{s}(t)^(-1)*||v_t||^2." source := saldForwardKlDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle65ForwardKlSkeletonMiddleContract", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.cycle60ForwardKlDerivativeRawLowerObligation", "SALD.forwardKlPointwisePreDvDerivativeBoundOfRawKlFiVelocityScaling", "SALD.forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative" ] note := "The new Lean theorem is a pointwise Real/order wrapper only. It does not prove mass conservation, differentiation under the KL integral, Fokker-Planck, integration by parts, target transport, Cauchy-Schwarz, LSI/KL/FI, inverse-function calculus, DV, Gronwall, or thm:forward-KL." /-- Cycle-65 proof-DAG pane for the post-cycle-64 continuous forward-KL skeleton route. -/ def cycle65ForwardKlSkeletonDag : List ProofDagBlock := [ { id := "ASTIS.SALD.forward_KL.cycle65_global_phase_judgment" interface := "Upper judgment for cycle 65: cycle 64 passed and needs no recovery; Phase 1 supports this narrow continuous forward-KL route audit only; the largest remaining continuous proof risk is sald.forward_kl.kl_derivative over appendix.tex:168-228." source := saldForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle65ForwardKlSkeletonUpperPacket", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation" ] reusedBy := ["ASTIS-SALD-001 cycle 65", "thm:forward-KL"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle65_five_backend_check" interface := "Explicit cycle-65 check of the five slow interfaces before lower work: endpoint-safe Gronwall, common-space DV with finite log-mgf, LSI/KL/FI density-test bridge, continuous Fokker-Planck/KL derivative, and EM interpolation Fokker-Planck for downstream discrete reuse." source := saldForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.saldGronwallEndpointCalculusContract", "SALD.forwardKlGronwallSideConditionContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.cycle64MainSkeletonAnalyticInterfaceObligation" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle65_continuous_route" interface := "Post-cycle-64 upper route: consume the five-backend analytic ledger and existing continuous forward-KL route data, then wire main_body.tex:238-247 and appendix.tex:164-252 through derivative/Fokker-Planck, LSI/KL/FI, DV finite-log-mgf, Gronwall endpoint/exponent, and downstream EM-interface visibility without changing the theorem display." source := saldForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle65ForwardKlSkeletonUpperPacket", "SALD.cycle65ForwardKlSkeletonObligation", "SALD.cycle65ForwardKlSkeletonMiddleContract", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.cycle64MainSkeletonAnalyticInterfaceObligation", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.cycle60ForwardKlSkeletonObligation", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract" ] reusedBy := ["thm:forward-KL", "cycle 65 middle route audit"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle65_middle_route_audit" interface := "Cycle-65 middle audit: verify the post-cycle-64 continuous forward-KL route in source order, keep main_body.tex:238-247 and all constants fixed, and select appendix.tex:168-228 sald.forward_kl.kl_derivative as the lower backend while LSI, DV, Gronwall, and EM interfaces remain separate obligations." source := saldForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle65ForwardKlSkeletonMiddleContract", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.cycle65ForwardKlSkeletonObligation", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle60ForwardKlDerivativeRawLowerObligation", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "sald.forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract" ] reusedBy := ["thm:forward-KL", "cycle 65 lower derivative packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle65_lower_packet.kl_derivative" interface := "Selected lower packet for cycle 65: keep proof-producing work on sald.forward_kl.kl_derivative over appendix.tex:168-228, with the source split into mass/KL differentiation and -FI, target-velocity Young, then inverse-schedule square scaling." source := saldForwardKlDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle65ForwardKlSkeletonObligation", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.cycle60ForwardKlDerivativeRawLowerObligation", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "SALD.cycle65ForwardKlDerivativePointwiseLowerObligation", "SALD.forwardKlPointwisePreDvDerivativeBoundOfRawKlFiVelocityScaling", "SALD.forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar", "SALD.forwardKlMassConservationDropScalar", "SALD.forwardKlMassConservationFirstTermFisherScalar", "SALD.forwardKlDerivativeDvGronwallCoefficientOfKlFiVelocityScalingScalar", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative" ] reusedBy := ["sald.forward_kl.kl_derivative", "thm:forward-KL", "cycle 65 lower handoff"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle65_derivative_pointwise_lower" interface := "Cycle-65 lower proof-producing wrapper for appendix.tex:168-228: lift the compiled raw-derivative scalar handoff to the pointwise t-indexed pre-DV inequality used by thm:forward-KL before DV and Gronwall." source := saldForwardKlDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle65ForwardKlDerivativePointwiseLowerObligation", "SALD.forwardKlPointwisePreDvDerivativeBoundOfRawKlFiVelocityScaling", "SALD.forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative" ] reusedBy := ["sald.forward_kl.kl_derivative", "thm:forward-KL", "cycle 65 lower handoff"] status := ProofStatus.obligation } ] /-! ### Cycle 66: discrete forward-KL post-cycle65 route -/ /-- Cycle-66 upper packet for the discrete forward-KL skeleton after the accepted cycle-65 continuous route. This is upper-route data only. It returns the main skeleton sprint to `thm:forward-KL-discrete`, checks that the five slow analytic interfaces are still explicit, and assigns the next lower packet to the Gronwall/output stitching and accumulated-error bridge without changing the paper statement, constants, labels, or proof statuses. -/ def cycle66DiscreteForwardKlSkeletonUpperPacket : DiscreteForwardKlUpperPacket where objective := "Cycle 66 upper: after the accepted cycle-65 reviewer/build gate, rewire the theorem-level interfaces into thm:forward-KL-discrete, matching main_body.tex:299-323 and appendix.tex:260-592 while using the source-cited EM/Fokker-Planck, LSI/KL/FI, DV, Gronwall, and accumulated-error interfaces explicitly instead of proving them from scratch." sourceLabels := [ "thm:forward-KL-discrete", "proof:thm:forward-KL-discrete", "eq:frozen_interp_terminal_disc_prop_additive_final", "eq:discrete_delta_def", "lem:frozen_delta_cross_lip_sald", "eq:KL-derivative-0-discrete", "eq:KL-derivative-1-discrete", "eq:KL-derivative-2-discrete", "eq:KL-derivative-3-discrete", "eq:frozen-cross-bound-discrete", "eq:moving-cross-bound-discrete", "eq:KL-derivative-5-discrete", "eq:dv-v-term-discrete", "eq:KL-derivative-6-discrete", "eq:KL-derivative-7-discrete", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity", "thm:forward-KL" ] modeDiscipline := [ "Global phase judgment: cycle 65 passed reviewer/build, so no failed previous cycle must be recovered before cycle 66 work.", "Global phase judgment: Phase 1 theorem-skeleton translation is stable enough for this narrow discrete forward-KL route audit, but not for broad cited-theory, SLT, SDE, disintegration, or reusable API backfill.", "Global phase judgment: the single lower packet that best reduces the remaining discrete theorem risk is SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge over appendix.tex:557-592 and main_body.tex:309-323.", "Five-backend check: lem:gronwall keeps endpoint-safe differentiability/FTC, interval-integrability, coefficient-regularity, endpoint, and exponent side-condition obligations; lem:dv_variation keeps common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and positive-alpha interfaces; eq:LSI-KL-FI keeps density, zero-set, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule obligations; the continuous Fokker-Planck/KL derivative is visible through the cycle-65 continuous route but remains a local analytic obligation; the EM interpolation endpoint/conditional-law Fokker-Planck backend is represented by SALD.discreteForwardKlEmInterpolationSideConditionContract and its endpoint, conditional drift, conditional-FP, and stitched-interval obligations.", "Route thm:forward-KL-discrete in the paper order: appendix.tex:260-385 EM interpolation and conditional-FP, appendix.tex:388-491 KL derivative with frozen defect and LSI, appendix.tex:493-523 DV velocity, appendix.tex:526-553 time-changed Gronwall input, and appendix.tex:557-592 Gronwall output plus accumulated-error display.", "Preserve the exact source coefficients and constants: 4*eta^2*L_space^2<1/2, alpha in (0,alpha0), alpha' in (0,alpha0'], T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'}." ] nonGoals := [ "Do not restate thm:forward-KL-discrete, change SALD.discreteForwardKlStatementContract, or promote SALD.discreteSaldContract above contractOnly.", "Do not prove or replace the EM conditional-Fokker-Planck backend, frozen-delta lemma, LSI/KL/FI, source-cited DV, Gronwall lemma, endpoint stitching, or accumulated-error bridge in this upper packet.", "Do not add hidden density, absolute-continuity, endpoint, finite-log-mgf, coefficient-regularity, stitched-interval, or inverse-schedule assumptions to the theorem statement.", "Do not spend this cycle on source-index rebaseline, broad SLT import, general theorem polishing, or isolated scalar sublemmas outside the selected Gronwall/accumulated-error backend." ] lowerPacket := [ "Middle should synchronize this cycle-66 route with conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, research-wiki/cited-results/SLT_reuse_audit.md, and the thm:forward-KL-discrete proof DAG.", "Lower should target exactly SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.", "First lower sub-slice: appendix.tex:557-592 endpoint and Gronwall-output stitching, using SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged and SALD.discreteForwardKlGronwallInstantiationContract as supplied inputs.", "Second lower sub-slice: linear slowdown t(s)=s/r, dot{s}=r, and exponent splitting, preserving the source factors exp(-r*int C_LSI), exp(T/(r*alpha)+2*r*eta^2*barGamma/alpha'), and the terminal endpoint KL(rho_K^eta||pi_T).", "Third lower sub-slice: collect the residual integral into (1/r)*A_alpha(pi,v)+2*r*eta*barDelta_{alpha'} using the existing alpha-complexity and Delta collection scalar wrappers.", "If blocked, sharpen endpoint-law stitching, interval-integrability, coefficient regularity, barGamma/barDelta definitions, finite-log-mgf witnesses, or residual-exponent obligations rather than changing thm:forward-KL-discrete." ] reviewerChecklist := [ "SALD.discreteSaldContract lists SALD.cycle66DiscreteForwardKlSkeletonObligation while remaining contractOnly.", "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle66_discrete_route and ASTIS.SALD.forward_KL_discrete.cycle66_lower_packet.accumulated_error after the cycle-61 recovered route.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle66DiscreteForwardKlSkeletonUpperPacket, SALD.cycle66DiscreteForwardKlSkeletonObligation, and the cycle-66 DAG nodes.", "All five slow analytic backends remain ProofStatus.obligation or ProofStatus.sourceCited unless every analytic dependency is compiled locally.", "No theorem statement, source coefficient, source label, source-file selection, SLT reuse entry, or analytic backend status is changed.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-66 upper obligation tying the accepted cycle-65 continuous route back to the focused discrete forward-KL theorem route. -/ def cycle66DiscreteForwardKlSkeletonObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle66_discrete_route" statement := "Cycle 66 upper records the post-cycle-65 global phase judgment and wires thm:forward-KL-discrete through the five source-cited analytic interfaces in the paper order: appendix.tex:260-385 EM interpolation endpoint/conditional-Fokker-Planck setup, appendix.tex:388-491 discrete KL derivative with frozen defect and LSI, appendix.tex:493-523 DV velocity-energy, appendix.tex:526-553 time-changed Gronwall input, and appendix.tex:557-592 Gronwall output/accumulated-error collection to main_body.tex:309-323. The selected lower packet is sald.discrete_forward_kl.accumulated_error_bridge." source := saldForwardKlDiscreteProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle66DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.cycle65ForwardKlDerivativePointwiseLowerObligation", "SALD.cycle64MainSkeletonAnalyticInterfaceObligation", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.cycle61DiscreteForwardKlSkeletonObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation", "SALD.cycle56DiscreteForwardKlSkeletonObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "SALD.discreteForwardKlDerivativeObligation", "sald.discrete_forward_kl.kl_derivative", "SALD.frozenDeltaCrossLipSaldContract", "sald.discrete_forward_kl.frozen_delta_cross_lip", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged", "sald.discrete_forward_kl.gronwall_accumulation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "SALD.discreteForwardKlAccumulatedErrorBridgeObligation", "SALD.discreteForwardKlGronwallCoeffIntervalIntegrable", "SALD.discreteForwardKlGronwallCoeffIntegralSubSub", "SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces", "SALD.discreteForwardKlResidualExponentBoundScalar", "SALD.discreteForwardKlResidualExpBoundScalar", "SALD.discreteForwardKlAlphaComplexityCollectionScalar", "SALD.discreteForwardKlDeltaAccumulationScalar", "SALD.discreteForwardKlAccumulatedErrorCollectionScalar", "SALD.discreteForwardKlResidualIntegralDisplayBoundScalar", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.coefficient_chain_audit", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity", "thm:forward-KL" ] note := "Upper workflow obligation only. It adds no theorem hypotheses and does not promote EM/Fokker-Planck, frozen-delta, LSI/KL/FI, DV, Gronwall, accumulated-error collection, continuous derivative, theorem status, or SLT reuse status." /-- Cycle-66 middle audit for the post-cycle-65 discrete forward-KL route. This keeps the theorem-level transcript fixed while moving lower work to the Gronwall output and accumulated-error bridge selected by the upper packet. -/ def cycle66DiscreteForwardKlSkeletonMiddleContract : DiscreteForwardKlEmDefectAccumulationMiddleContract where sourceStatement := saldForwardKlDiscreteSource interpolationSource := saldForwardKlDiscreteInterpolationSource derivativeSource := saldForwardKlDiscreteDerivativeSource gronwallSource := saldForwardKlDiscreteGronwallSource accumulatedSource := saldForwardKlDiscreteAccumulatedErrorSource objective := "Cycle 66 middle: audit the post-cycle-65 thm:forward-KL-discrete route against main_body.tex:299-323 and appendix.tex:260-592, verify that the theorem consumes explicit source-cited EM/Fokker-Planck, LSI/KL/FI, DV, Gronwall, and accumulated-error interfaces in paper order, and keep lower work on sald.discrete_forward_kl.accumulated_error_bridge over appendix.tex:557-592 plus main_body.tex:309-323." sourceStepMap := [ "main_body.tex:299-323 fixes the linear-slowdown statement: inherited thm:forward-KL assumptions, score Lipschitz hypotheses, alpha0' finite score and 1+M complexities, 4*eta^2*L_space^2<1/2, alpha in (0,alpha0), alpha' in (0,alpha0'], and the exact terminal constants.", "appendix.tex:260-385 supplies the frozen EM interpolation, endpoint laws, conditional drift, conditional Fokker-Planck equation, Laplacian split, and stitched interval setup; cycle 66 middle keeps these as named obligations and does not attempt the EM backend.", "appendix.tex:388-491 is consumed through the existing discrete KL derivative, frozen-cross, moving Young, and LSI/KL/FI scalar handoff route.", "appendix.tex:493-523 is consumed through the discrete DV witness with nu=hat rho_s, mu=tilde pi_s, and Z=alpha*||v_{t(s)}||^2, preserving the dot t(s)^2 coefficient.", "appendix.tex:526-553 is consumed through the cycle-56 post-DV s-to-t derivative wrapper and pointwise Gronwall input.", "appendix.tex:557-592 gives the Gronwall output; main_body.tex:309-323 is reached only after endpoint stitching, linear-slowdown exponent splitting, barGamma/barDelta identification, and A_alpha collection." ] leanStepMap := [ "Keep SALD.discreteForwardKlStatementContract and SALD.discreteSaldContract unchanged and contractOnly.", "Use SALD.cycle66DiscreteForwardKlSkeletonUpperPacket, SALD.cycle66DiscreteForwardKlSkeletonObligation, SALD.cycle65ForwardKlSkeletonMiddleObligation, SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation, SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation, and SALD.cycle56DiscreteForwardKlGronwallLowerObligation as parent route checks.", "Route appendix.tex:260-385 through SALD.discreteForwardKlEmInterpolationSideConditionContract, sald.discrete_forward_kl.em_endpoint_laws, sald.discrete_forward_kl.conditional_drift_density, sald.discrete_forward_kl.em_conditional_fokker_planck, sald.discrete_forward_kl.stitched_interval_regularity, and sald.discrete_forward_kl.em_interpolation_fp.", "Route appendix.tex:388-491 through SALD.discreteForwardKlDerivativeCandidateContract, SALD.discreteForwardKlDerivativeObligation, SALD.cycle51DiscreteForwardKlDerivativeLowerObligation, SALD.frozenDeltaCrossLipSaldContract, sald.discrete_forward_kl.frozen_delta_cross_lip, SALD.saldLsiKlFiDensityTestContract, and probability.lsi_to_kl_fi.", "Route appendix.tex:493-523 through dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, SALD.discreteForwardKlDvFiniteLogMgfWitnessContract, sald.discrete_forward_kl.dv_finite_log_mgf_witness, and sald.discrete_forward_kl.dv_velocity_bound.", "Route appendix.tex:526-553 through SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar, SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged, SALD.discreteForwardKlGronwallInstantiationContract, and sald.discrete_forward_kl.gronwall_accumulation.", "Route appendix.tex:557-592 and main_body.tex:309-323 through SALD.discreteForwardKlAccumulatedErrorBridgeContract, SALD.discreteForwardKlAccumulatedErrorBridgeObligation, SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces, SALD.discreteForwardKlResidualExponentBoundScalar, SALD.discreteForwardKlResidualExpBoundScalar, SALD.discreteForwardKlAlphaComplexityCollectionScalar, SALD.discreteForwardKlDeltaAccumulationScalar, SALD.discreteForwardKlAccumulatedErrorCollectionScalar, SALD.discreteForwardKlResidualIntegralDisplayBoundScalar, SALD.discreteForwardKlMainDisplayBoundScalar, sald.discrete_forward_kl.linear_slowdown_specialization, sald.discrete_forward_kl.residual_exponent_bound, and sald.discrete_forward_kl.accumulated_error_bridge.", "Select lower work on the accumulated-error bridge, not on EM conditional-FP, KL differentiation, LSI/KL/FI, DV, or a new scalar side lemma." ] citedResultInterfaces := [ "EM endpoint/conditional-law Fokker-Planck remains a source-cited obligation interface; cycle 66 middle does not prove Brownian construction, regular conditional laws, density/AC, weak FP, boundary integration by parts, or endpoint stitching.", "eq:LSI-KL-FI remains the density-test, zero-set convention, admissible sqrt-density test or approximation, entropy-identity, finite KL/FI, and Fisher-chain obligation.", "lem:dv_variation remains source-cited with common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and positive-alpha witnesses.", "lem:gronwall remains the endpoint-safe differentiability/FTC, coefficient-regularity, interval-integrability, endpoint-evaluation, and exponent-rewrite obligation; cycle 56 only supplies the pointwise source-shaped input.", "The continuous forward-KL derivative route from cycle 65 remains sibling context for inherited assumptions and does not close any discrete EM or accumulated-error backend." ] obligations := [ "sald.discrete_forward_kl.cycle66_discrete_route", "sald.discrete_forward_kl.cycle66_middle_route_audit", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.kl_derivative", "sald.discrete_forward_kl.frozen_delta_cross_lip", "probability.lsi_to_kl_fi", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge" ] lowerPacket := [ "Target exactly SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.", "First lower sub-slice: appendix.tex:557-592 endpoint and Gronwall-output stitching from the supplied cycle-56 pointwise derivative input and SALD.discreteForwardKlGronwallInstantiationContract.", "Second lower sub-slice: linear slowdown t(s)=s/r, dot{s}=r, and exponent collection, preserving exp(-r*int C_LSI), T/(r*alpha), and 2*r*eta^2*barGamma/alpha'.", "Third lower sub-slice: collect the residual integral into (1/r)*A_alpha(pi,v) plus 2*r*eta*barDelta_{alpha'} using SALD.discreteForwardKlAlphaComplexityCollectionScalar, SALD.discreteForwardKlDeltaAccumulationScalar, SALD.discreteForwardKlAccumulatedErrorCollectionScalar, and SALD.discreteForwardKlResidualIntegralDisplayBoundScalar.", "Cycle 66 lower sub-slice: combine the supplied initial Gronwall split and supplied residual display into the exact main-body two-term bound using SALD.discreteForwardKlMainDisplayBoundScalar.", "If blocked, sharpen endpoint-law stitching, coefficient regularity, interval-integrability, barGamma/barDelta definitions, finite-log-mgf witnesses, or residual-exponent obligations rather than changing thm:forward-KL-discrete." ] reviewerChecklist := [ "SALD.discreteSaldContract lists SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation while remaining contractOnly.", "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle66_middle_route_audit between the cycle-66 route and lower accumulated-error packet.", "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle66DiscreteForwardKlSkeletonMiddleContract, SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation, and sald.discrete_forward_kl.cycle66_middle_route_audit.", "No theorem statement, source constant, source label, source-file selection, theorem status, SLT reuse status, or analytic backend status is changed.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-66 middle obligation tying the post-cycle-65 discrete route to the accumulated-error bridge lower packet. -/ def cycle66DiscreteForwardKlSkeletonMiddleObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle66_middle_route_audit" statement := "Cycle 66 middle audits the post-cycle-65 thm:forward-KL-discrete route: main_body.tex:299-323 remains unchanged; appendix.tex:260-385 EM endpoint/conditional-Fokker-Planck interfaces stay explicit source-cited obligations; appendix.tex:388-491 reuses the derivative, frozen-defect, and LSI/KL/FI handoff; appendix.tex:493-523 reuses the discrete DV velocity witness; appendix.tex:526-553 reuses the cycle-56 pointwise Gronwall input; and appendix.tex:557-592 plus main_body.tex:309-323 remain the selected accumulated-error bridge lower backend. All theorem statuses and slow analytic interfaces remain below formalized." source := saldForwardKlDiscreteProofSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle66DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle66DiscreteForwardKlSkeletonObligation", "SALD.cycle66DiscreteForwardKlSkeletonMiddleContract", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.cycle65ForwardKlDerivativePointwiseLowerObligation", "SALD.cycle64MainSkeletonAnalyticInterfaceObligation", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.cycle61DiscreteForwardKlSkeletonObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation", "SALD.cycle56DiscreteForwardKlSkeletonObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "SALD.discreteForwardKlDerivativeObligation", "sald.discrete_forward_kl.kl_derivative", "SALD.frozenDeltaCrossLipSaldContract", "sald.discrete_forward_kl.frozen_delta_cross_lip", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged", "sald.discrete_forward_kl.gronwall_accumulation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "SALD.discreteForwardKlAccumulatedErrorBridgeObligation", "SALD.discreteForwardKlGronwallCoeffIntervalIntegrable", "SALD.discreteForwardKlGronwallCoeffIntegralSubSub", "SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces", "SALD.discreteForwardKlResidualExponentBoundScalar", "SALD.discreteForwardKlResidualExpBoundScalar", "SALD.discreteForwardKlAlphaComplexityCollectionScalar", "SALD.discreteForwardKlDeltaAccumulationScalar", "SALD.discreteForwardKlAccumulatedErrorCollectionScalar", "SALD.discreteForwardKlResidualIntegralDisplayBoundScalar", "SALD.discreteForwardKlMainDisplayBoundScalar", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity", "thm:forward-KL" ] note := "Middle workflow and source-to-Lean audit only. It adds no theorem hypotheses, changes no source constants, and does not promote EM/Fokker-Planck, frozen-delta, LSI/KL/FI, DV, Gronwall, accumulated-error collection, continuous derivative, SLT reuse, or thm:forward-KL-discrete." /-- Cycle-66 lower obligation for the final scalar display wrapper in the discrete forward-KL accumulated-error bridge. -/ def cycle66DiscreteForwardKlAccumulatedDisplayLowerObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle66_accumulated_display_lower" statement := "Cycle 66 lower compiles SALD.discreteForwardKlMainDisplayBoundScalar for the final scalar/order wrapper in thm:forward-KL-discrete: once Gronwall supplies the initial term plus residual bound, the initial exponent split and residual-display wrapper produce exactly the main_body.tex:309-323 two-term display with exp(-r*int C_LSI)*exp(T/(r*alpha)+2*r*eta^2*barGamma/alpha')*KL(rho_0||pi_0) plus the same positive exponential times ((1/r)*A_alpha(pi,v)+2*r*eta*barDelta_{alpha'})." source := saldForwardKlDiscreteAccumulatedErrorSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "SALD.discreteForwardKlAccumulatedErrorBridgeObligation", "SALD.discreteForwardKlGronwallInitialExponentSplitScalar", "SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces", "SALD.discreteForwardKlResidualIntegralDisplayBoundScalar", "SALD.discreteForwardKlMainDisplayBoundScalar", "SALD.discreteForwardKlAccumulatedErrorCollectionScalar", "SALD.discreteForwardKlAlphaComplexityCollectionScalar", "SALD.discreteForwardKlDeltaAccumulationScalar", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge" ] note := "Proof-producing scalar wrapper plus obligation only. It preserves the theorem target, source constants, and source labels. It does not prove EM/Fokker-Planck, endpoint stitching, Gronwall, residual-exponent monotonicity, barGamma/barDelta source identifications, or thm:forward-KL-discrete." /-- Cycle-66 proof-DAG pane for the post-cycle-65 discrete forward-KL skeleton route. -/ def cycle66DiscreteForwardKlSkeletonDag : List ProofDagBlock := [ { id := "ASTIS.SALD.forward_KL_discrete.cycle66_global_phase_judgment" interface := "Upper judgment for cycle 66: cycle 65 passed and needs no recovery; Phase 1 supports this narrow discrete forward-KL route audit only; the largest remaining discrete proof risk is sald.discrete_forward_kl.accumulated_error_bridge over appendix.tex:557-592 and main_body.tex:309-323." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle66DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation" ] reusedBy := ["ASTIS-SALD-001 cycle 66", "thm:forward-KL-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle66_five_backend_check" interface := "Explicit cycle-66 check of the five slow interfaces before lower work: endpoint-safe Gronwall, common-space DV with finite log-mgf, LSI/KL/FI density-test bridge, continuous Fokker-Planck/KL derivative, and EM interpolation endpoint/conditional-law Fokker-Planck." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.saldGronwallEndpointCalculusContract", "SALD.discreteForwardKlGronwallInstantiationContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle66_discrete_route" interface := "Post-cycle-65 upper route: consume the five-backend analytic ledger and existing cycle-51/56/61 discrete route data, then wire main_body.tex:299-323 and appendix.tex:260-592 through EM conditional-FP, KL derivative/frozen-defect/LSI, DV velocity, time-changed Gronwall, and accumulated-error display interfaces without changing the theorem display." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle66DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle66DiscreteForwardKlSkeletonObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.discreteForwardKlDerivativeCandidateContract", "SALD.frozenDeltaCrossLipSaldContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlAccumulatedErrorBridgeContract" ] reusedBy := ["thm:forward-KL-discrete", "cycle 66 middle route audit"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle66_middle_route_audit" interface := "Middle audit for cycle 66: verify the post-cycle-65 discrete route in paper order, keep the EM/Fokker-Planck, derivative/LSI, DV, and Gronwall interfaces explicit, and keep lower work on the accumulated-error bridge to the main-body linear-slowdown display." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle66DiscreteForwardKlSkeletonMiddleContract", "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle66DiscreteForwardKlSkeletonObligation", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "sald.discrete_forward_kl.kl_derivative", "SALD.frozenDeltaCrossLipSaldContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "sald.discrete_forward_kl.accumulated_error_bridge" ] reusedBy := ["thm:forward-KL-discrete", "cycle 66 lower accumulated-error packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle66_lower_packet.accumulated_error" interface := "Selected lower packet for cycle 66: keep proof-producing work on sald.discrete_forward_kl.accumulated_error_bridge over appendix.tex:557-592 and main_body.tex:309-323, using the supplied cycle-56 pointwise Gronwall input, the cycle-61 residual-display wrapper, and the cycle-66 final main-display scalar wrapper." source := saldForwardKlDiscreteAccumulatedErrorSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle66DiscreteForwardKlSkeletonObligation", "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle66DiscreteForwardKlAccumulatedDisplayLowerObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "SALD.discreteForwardKlAccumulatedErrorBridgeObligation", "SALD.discreteForwardKlGronwallCoeffIntervalIntegrable", "SALD.discreteForwardKlGronwallCoeffIntegralSubSub", "SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces", "SALD.discreteForwardKlResidualExponentBoundScalar", "SALD.discreteForwardKlResidualExpBoundScalar", "SALD.discreteForwardKlAlphaComplexityCollectionScalar", "SALD.discreteForwardKlDeltaAccumulationScalar", "SALD.discreteForwardKlAccumulatedErrorCollectionScalar", "SALD.discreteForwardKlResidualIntegralDisplayBoundScalar", "SALD.discreteForwardKlMainDisplayBoundScalar", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge" ] reusedBy := ["sald.discrete_forward_kl.accumulated_error_bridge", "thm:forward-KL-discrete", "cycle 66 lower handoff"] status := ProofStatus.obligation } ] /-! ### Cycle 67: guided residual and continuous general moving-target route -/ /-- Cycle-67 upper packet for returning to the guided residual and continuous general moving-target theorem after the accepted cycle-66 discrete route. This is a theorem-skeleton route packet only. It rechecks the five slow interfaces, wires `appendix.tex:619-951` through the named guided/general interfaces, and selects one lower packet for the residual-to-Gronwall bridge without changing any source theorem statement or analytic backend status. -/ def cycle67GuidedGeneralSkeletonUpperPacket : GeneralVaSaldUpperPacket where objective := "Cycle 67 upper: after the accepted cycle-66 discrete forward-KL route, wire prop:guided_path_residual and thm:general-moving-target-SALD through the already named guided residual, continuous derivative, LSI/KL/FI, residual DV, and sigma-weighted Gronwall interfaces over appendix.tex:619-951, preserving source statements and exposing the residual-to-Gronwall bridge as the single lower packet." sourceLabels := [ "prop:guided_path_residual", "eq:guided_path_residual", "eq:logZ_derivative", "thm:general-moving-target-SALD", "eq:general_moving_target_SALD", "eq:general_moving_target_KL_bound", "eq:general_moving_target_pure_contraction", "eq:general_KL_derivative_0", "eq:general_moving_target_FP", "eq:general_KL_derivative_1", "eq:general_KL_derivative_2", "eq:general_KL_derivative_3", "proof:thm:unified-forward-KL", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] modeDiscipline := [ "faithfulPaper Phase 1: use appendix.tex:619-951 exactly; sald_version_2.tex remains out of scope.", "Global phase judgment: cycle 66 passed reviewer/build and does not need recovery; Phase 1 is not stable enough for cited-theory backfill until this guided/general theorem route is rechecked; the selected lower packet is the residual-to-Gronwall bridge from appendix.tex:765-945.", "Five-backend check 1, Gronwall: consume SALD.saldGronwallEndpointCalculusContract, SALD.generalMovingTargetGronwallInstantiationContract, SALD.generalMovingTargetGronwallSideConditionContract, sald.general_moving_target.gronwall_application, and sald.general_moving_target.gronwall_side_conditions only as obligation-level endpoint/FTC/coefficient interfaces.", "Five-backend check 2, DV: consume dvVariationalFormulaInterface saldDvVariationSource, SALD.generalMovingTargetDvFiniteLogMgfWitnessContract, SALD.generalMovingTargetDvPositiveAlphaScalingContract, and sald.general_moving_target.dv_m_energy with common-space, absolute-continuity, finite-KL, finite-log-mgf, measurability, and positive-alpha side conditions explicit.", "Five-backend check 3, LSI/KL/FI: consume SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, and probability.lsi_to_kl_fi while preserving density, zero-set, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule obligations.", "Five-backend check 4, continuous derivative: consume SALD.generalMovingTargetDerivativeCandidateContract, SALD.generalMovingTargetDerivativeObligation, sald.general_moving_target.kl_derivative, and cycle-57/62 scalar handoffs as local SDE/measure-analysis obligations.", "Five-backend check 5, EM interpolation: keep SALD.discreteForwardKlEmInterpolationSideConditionContract, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, and sald.general_moving_target_discrete.em_interpolation_fp visible only as downstream discrete interfaces." ] nonGoals := [ "Do not restate prop:guided_path_residual, thm:general-moving-target-SALD, or thm:unified-forward-KL.", "Do not prove or promote Gronwall, DV, LSI/KL/FI, Fokker-Planck/KL differentiation, guided normalizer calculus, pure contraction, or EM interpolation.", "Do not add hidden smoothness, density, boundary, endpoint, finite-log-mgf, finite-KL/FI, coefficient-regularity, sigma-positivity, or schedule hypotheses to the paper theorem statements.", "Do not introduce Girsanov, Pinsker, Talagrand, path-space, PI, or direct VA-SALD proof routes.", "Do not start systematic SLT or SDE backfill in this upper packet; local SLT material remains reference-only." ] lowerPacket := [ "Middle should synchronize SALD.cycle67GuidedGeneralSkeletonUpperPacket, SALD.cycle67GuidedGeneralSkeletonObligation, and SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation into the conversion window, proof-obligation ledger, and SLT audit.", "Target exactly SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation / sald.general_moving_target.cycle67_residual_to_gronwall_bridge.", "First lower sub-slice: verify appendix.tex:765-884 is consumed in order by sald.general_moving_target.kl_derivative, the residual split/scaled-residual handoffs, SALD.saldLsiKlFiDensityTestContract, and the source schedule/sigma side-condition obligations.", "Second lower sub-slice: verify appendix.tex:885-907 is consumed by the residual DV finite-log-mgf and positive-alpha interfaces with Z=alpha*||m_t||^2.", "Third lower sub-slice: verify appendix.tex:908-945 is consumed by the sigma-weighted Gronwall instantiation, endpoint/exponent side-condition obligations, and pure-contraction zero-residual interface.", "If any analytic step is too large, sharpen only the named source-cited or obligation interface; do not change theorem statements, constants, labels, or proof route." ] reviewerChecklist := [ "SALD.guidedResidualContract and SALD.generalVaSaldContract list SALD.cycle67GuidedGeneralSkeletonObligation and SALD.cycle67GuidedGeneralSkeletonMiddleObligation while remaining contractOnly.", "SALD.generalVaSaldContract lists SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation while thm:general-moving-target-SALD remains contractOnly.", "SALD.generalVaSaldProofDag contains ASTIS.SALD.guided_general.cycle67_guided_residual_route, ASTIS.SALD.guided_general.cycle67_general_moving_target_route, ASTIS.SALD.guided_general.cycle67_middle_route_audit, and ASTIS.SALD.guided_general.cycle67_lower_packet.residual_to_gronwall_bridge.", "SALD.saldDependenciesForLabel for prop:guided_path_residual and thm:general-moving-target-SALD includes the cycle-67 packet, upper and middle obligations, bridge, and DAG nodes.", "The five slow analytic interfaces remain source-cited or obligation-level; no theorem statement, source constant, source label, or external reuse status changes.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-67 upper obligation for the guided residual and continuous general moving-target theorem route. -/ def cycle67GuidedGeneralSkeletonObligation : ProofObligation where id := "sald.guided_general.cycle67_upper_route" statement := "Cycle 67 upper records that cycle 66 passed and then rewires appendix.tex:619-951 through the named guided/general theorem skeleton interfaces: prop:guided_path_residual uses the normalizer derivative and centered residual identity; thm:general-moving-target-SALD uses the continuous general KL derivative, LSI/KL/FI bridge, residual DV finite-log-mgf and positive-alpha witnesses, sigma-weighted Gronwall side conditions, and pure-contraction residual-zero obligation. The selected lower packet is the residual-to-Gronwall bridge over appendix.tex:765-945. The proposition and theorem remain contractOnly." source := saldGeneralMovingTargetSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle67GuidedGeneralSkeletonUpperPacket", "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle66DiscreteForwardKlAccumulatedDisplayLowerObligation", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.cycle62GuidedGeneralSkeletonMiddleObligation", "SALD.cycle62GuidedGeneralScaledResidualLowerObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralDerivativeSplitLowerObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "SALD.generalMovingTargetKlDerivativeResidualSplitScalar", "SALD.generalMovingTargetKlDerivativeScaledResidualDisplayScalar", "SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.specialization", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "Upper workflow obligation only. It adds no hidden hypotheses, proof replacement, theorem restatement, or backend promotion." /-- Cycle-67 middle audit for the guided residual and continuous general moving-target theorem route. This is the middle-role synchronization layer after the cycle-67 upper route. It checks the source proof in order, records the named Lean-facing consumers, and keeps proof-producing lower work on the residual-to-Gronwall bridge rather than reopening theorem statements or promoting analytic backends. -/ def cycle67GuidedGeneralSkeletonMiddleContract : GeneralVaSaldGuidedPathMiddleContract where guidedResidualSource := saldGuidedResidualSource generalContinuousSource := saldGeneralMovingTargetSource unifiedSource := saldUnifiedForwardKlSource discreteSource := saldGeneralMovingTargetDiscreteSource objective := "Cycle 67 middle: audit the post-cycle-66 guided residual and thm:general-moving-target-SALD route against appendix.tex:619-951, wire the cycle-67 upper packet to the existing guided residual, derivative/LSI/DV/Gronwall/pure-contraction interfaces, and keep lower work on SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation / sald.general_moving_target.cycle67_residual_to_gronwall_bridge over appendix.tex:765-945." sourceStepMap := [ "appendix.tex:619-704 proves prop:guided_path_residual by differentiating Z_t, using partial_t p_t=-div(p_t*u_t), integrating by parts, differentiating pi_t=Z_t^(-1)*p_t*exp(-f_t), cancelling the divergence terms, substituting dot Z_t/Z_t=-E_pi_t[g_t], and deriving the centered mean-zero residual.", "appendix.tex:724-744 states thm:general-moving-target-SALD with the general c_t-driven SDE, residual m_t=v_t-c_t, LSI constants C_LSI(t)>=0, finite E_alpha0(pi_t,m_t), alpha in (0,alpha0], and the exact sigma-weighted terminal KL display.", "appendix.tex:765-835 differentiates KL(rho_s||pi_{t(s)}), inserts the general moving-target Fokker-Planck equation, evaluates the rho_s term by integration by parts, rescales the target velocity tilde v_s=dot t(s)*v_{t(s)}, and exposes the residual derivative display.", "appendix.tex:835-884 combines c_t and v_t into m_t=v_t-c_t, applies Young with epsilon=2*dot t(s)/sigma_{t(s)}^2, changes from s to t, and uses eq:LSI-KL-FI to reach the pre-DV K'(t) inequality.", "appendix.tex:885-907 applies lem:dv_variation with Z=alpha*||m_t||^2, uses the alpha0-to-alpha finite-log-mgf witness, divides by alpha>0, and rewrites the log-mgf quotient as mathfrak E_alpha(pi_t,m_t).", "appendix.tex:908-934 applies lem:gronwall with a(t)=(sigma_t^2/2)*dot{s}(t)*C_LSI(t)-sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1) and b(t)=sigma_t^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t), then splits the exponent to match eq:general_moving_target_KL_bound.", "appendix.tex:936-945 specializes c_t=v_t so m_t=0 and E_alpha(pi_t,m_t)=0, yielding the pure contraction display.", "appendix.tex:949-951 is downstream reuse for thm:unified-forward-KL by setting c_t<-u_t; this middle audit does not introduce a direct VA-SALD proof." ] leanStepMap := [ "Use SALD.cycle67GuidedGeneralSkeletonUpperPacket and SALD.cycle67GuidedGeneralSkeletonObligation as the parent route wrapper after the accepted cycle-66 discrete forward-KL route.", "Route appendix.tex:619-704 through SALD.guidedResidualIdentityContract, sald.guided_path_residual.normalizer_derivative, and sald.guided_path_residual.identity; positive finite normalizer, differentiation under the integral, boundary integration by parts, quotient/product differentiation, and centering stay obligations.", "Keep SALD.generalMovingTargetStatementContract, SALD.guidedResidualContract, SALD.generalVaSaldContract, and SALD.unifiedForwardKlContract contractOnly; this middle audit adds no theorem hypotheses.", "Route appendix.tex:765-884 through SALD.generalMovingTargetDerivativeCandidateContract, SALD.generalMovingTargetDerivativeObligation, sald.general_moving_target.kl_derivative, SALD.generalMovingTargetKlDerivativeResidualSplitScalar, SALD.generalMovingTargetKlDerivativeScaledResidualDisplayScalar, SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar, SALD.saldLsiKlFiDensityTestContract, probability.lsi_to_kl_fi, and the schedule/sigma side-condition obligations.", "Route appendix.tex:885-907 through SALD.generalMovingTargetDvFiniteLogMgfWitnessContract, SALD.generalMovingTargetDvPositiveAlphaScalingContract, sald.general_moving_target.dv_finite_log_mgf_witness, sald.general_moving_target.dv_positive_alpha_scaling, and sald.general_moving_target.dv_m_energy.", "Route appendix.tex:908-945 through SALD.generalMovingTargetGronwallInstantiationContract, SALD.generalMovingTargetGronwallSideConditionContract, sald.general_moving_target.gronwall_application, sald.general_moving_target.gronwall_side_conditions, sald.general_moving_target.pure_contraction, and SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation.", "Keep SALD.unifiedForwardKlSpecializationContract and sald.unified_forward_kl.specialization as downstream reuse; keep SALD.discreteForwardKlEmInterpolationSideConditionContract plus SALD.generalMovingTargetDiscreteDerivativeSideConditionContract visible only as downstream EM interfaces." ] citedResultInterfaces := [ "lem:gronwall remains an obligation through endpoint-safe differentiability, FTC/order integration, coefficient regularity, endpoint rewrites, exponent splitting, residual-exponent side conditions, and pure-contraction display matching.", "lem:dv_variation remains source-cited; the residual step still needs common-space, absolute-continuity, finite KL, selected-test measurability, finite log-mgf, alpha0-to-alpha monotonicity, and positive-alpha division for Z=alpha*||m_t||^2.", "eq:LSI-KL-FI remains the density, zero-set, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule obligation.", "The continuous general Fokker-Planck/KL derivative remains sald.general_moving_target.kl_derivative; compiled scalar helpers only consume analytic inputs after they are supplied.", "The Euler-Maruyama interpolation Fokker-Planck backend remains a downstream discrete obligation and is not imported into the continuous theorem statement." ] obligations := [ "sald.guided_general.cycle67_upper_route", "sald.guided_general.cycle67_middle_route_audit", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.cycle57_derivative_split_lower", "sald.general_moving_target.cycle62_scaled_residual_lower", "probability.lsi_to_kl_fi", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "sald.general_moving_target.cycle67_residual_to_gronwall_bridge", "sald.unified_forward_kl.specialization", "sald.general_moving_target_discrete.em_interpolation_fp" ] lowerPacket := [ "Target exactly SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation / sald.general_moving_target.cycle67_residual_to_gronwall_bridge.", "First lower sub-slice: audit appendix.tex:765-884 from the KL derivative identity through residual Young, s-to-t schedule conversion, and the LSI/KL/FI handoff, preserving sigma_t^(-2)*dot{s}(t)^(-1) on the residual energy.", "Second lower sub-slice: audit appendix.tex:885-907 through the residual DV finite-log-mgf and positive-alpha interfaces with Z=alpha*||m_t||^2, preserving the alpha^(-1)*K(t)+E_alpha(pi_t,m_t) output.", "Third lower sub-slice: audit appendix.tex:908-945 through the Gronwall endpoint/exponent side conditions and pure-contraction zero-residual alpha-complexity interface.", "If blocked, sharpen only the named source-cited or obligation interface; do not change theorem statements, constants, labels, or proof route." ] reviewerChecklist := [ "SALD.guidedResidualContract, SALD.generalVaSaldContract, and SALD.unifiedForwardKlContract list SALD.cycle67GuidedGeneralSkeletonMiddleObligation while remaining contractOnly.", "SALD.generalVaSaldProofDag contains ASTIS.SALD.guided_general.cycle67_middle_route_audit between the cycle-67 general route and the residual-to-Gronwall lower packet.", "SALD.saldDependenciesForLabel for prop:guided_path_residual, thm:general-moving-target-SALD, and thm:unified-forward-KL includes SALD.cycle67GuidedGeneralSkeletonMiddleContract, SALD.cycle67GuidedGeneralSkeletonMiddleObligation, and sald.guided_general.cycle67_middle_route_audit.", "No Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL derivative, guided residual calculus, pure contraction, EM backend, theorem status, source label, source constant, or SLT reuse status is promoted.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-67 middle obligation tying the guided/general route audit to the residual-to-Gronwall lower packet. -/ def cycle67GuidedGeneralSkeletonMiddleObligation : ProofObligation where id := "sald.guided_general.cycle67_middle_route_audit" statement := "Cycle 67 middle audits appendix.tex:619-951 after the accepted cycle-66 discrete route and the cycle-67 upper wrapper: prop:guided_path_residual and thm:general-moving-target-SALD remain unchanged, the source route consumes the named guided residual, continuous general KL derivative, LSI/KL/FI, residual DV, sigma-weighted Gronwall, pure-contraction, unified-specialization, and downstream EM interfaces in paper order, and the next proof-producing lower packet is sald.general_moving_target.cycle67_residual_to_gronwall_bridge over appendix.tex:765-945. All slow analytic interfaces remain obligation or source-cited." source := saldGeneralMovingTargetSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle67GuidedGeneralSkeletonUpperPacket", "SALD.cycle67GuidedGeneralSkeletonObligation", "SALD.cycle67GuidedGeneralSkeletonMiddleContract", "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle66DiscreteForwardKlAccumulatedDisplayLowerObligation", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.cycle62GuidedGeneralSkeletonMiddleObligation", "SALD.cycle62GuidedGeneralScaledResidualLowerObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralDerivativeSplitLowerObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "SALD.generalMovingTargetKlDerivativeResidualSplitScalar", "SALD.generalMovingTargetKlDerivativeScaledResidualDisplayScalar", "SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiDensityTestObligation", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "SALD.saldGronwallEndpointCalculusContract", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.specialization", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "Middle workflow and source-to-Lean audit only. It adds no hidden density, boundary, endpoint, absolute-continuity, finite-log-mgf, finite-KL/FI, sigma-positivity, schedule, or coefficient-regularity assumptions to prop:guided_path_residual or thm:general-moving-target-SALD." /-- Cycle-67 source-cited interface for the residual-to-Gronwall bridge in the continuous general moving-target proof. -/ def cycle67GuidedGeneralResidualGronwallBridgeObligation : ProofObligation where id := "sald.general_moving_target.cycle67_residual_to_gronwall_bridge" statement := "Source-cited bridge for appendix.tex:765-945: after the continuous general KL derivative backend supplies the residual split with m_t=v_t-c_t, compose the cycle-57/62 scalar residual handoffs, the LSI/KL/FI density-test interface, the DV residual-energy interface with Z=alpha*||m_t||^2, and the sigma-weighted Gronwall endpoint/exponent side conditions to obtain the exact pre-display route for thm:general-moving-target-SALD, including the pure-contraction m_t=0 clause. This bridge is an interface and remains below formalized until all analytic dependencies build locally." source := saldGeneralMovingTargetDvGronwallSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle67GuidedGeneralSkeletonUpperPacket", "SALD.cycle67GuidedGeneralSkeletonObligation", "SALD.cycle67GuidedGeneralSkeletonMiddleContract", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "SALD.generalMovingTargetKlDerivativeResidualSplitScalar", "SALD.generalMovingTargetKlDerivativeScaledResidualDisplayScalar", "SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar", "SALD.generalMovingTargetResidualToGronwallBridgeScalar", "SALD.cycle67GuidedGeneralResidualGronwallLowerObligation", "SALD.cycle57GuidedGeneralDerivativeSplitLowerObligation", "SALD.cycle62GuidedGeneralScaledResidualLowerObligation", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiDensityTestObligation", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "SALD.saldGronwallEndpointCalculusContract", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI" ] note := "This is the selected lower packet for cycle 67. It is not a proof of the Fokker-Planck/KL derivative, LSI/KL/FI, DV formula, Gronwall theorem, endpoint rewrites, or pure-contraction normalization." /-- Cycle-67 lower proof-producing obligation for the residual-to-Gronwall scalar bridge. -/ def cycle67GuidedGeneralResidualGronwallLowerObligation : ProofObligation where id := "sald.general_moving_target.cycle67_residual_to_gronwall_lower" statement := "Cycle 67 lower compiles SALD.generalMovingTargetResidualToGronwallBridgeScalar for appendix.tex:765-907: after the analytic backend supplies the raw KL derivative split, mass conservation, Fokker-Planck/integration-by-parts first-term identity, target-transport residual m_t=v_t-c_t, residual Young bound, LSI/KL/FI half-Fisher comparison, inverse-schedule time change, and residual DV input with Z=alpha*||m_t||^2, the wrapper derives the exact sigma-weighted Gronwall differential inequality coefficient for thm:general-moving-target-SALD." source := saldGeneralMovingTargetDvGronwallSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation", "SALD.generalMovingTargetResidualToGronwallBridgeScalar", "SALD.generalMovingTargetKlDerivativeResidualSplitScalar", "SALD.generalMovingTargetDerivativeDvGronwallCoefficientScalar", "SALD.generalMovingTargetPreDvDerivativeBoundScalar", "SALD.generalMovingTargetPostDvGronwallCoefficientOfSigmaScheduleScalar", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "SALD.saldLsiKlFiDensityTestContract", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "SALD.generalMovingTargetDvEnergyCandidateContract", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.kl_derivative", "probability.lsi_to_kl_fi", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions" ] note := "Proof-producing scalar wrapper plus obligation only. It preserves the theorem target, source constants, and source labels. It does not prove Fokker-Planck/KL differentiation, integration by parts, target transport, residual Young, LSI/KL/FI, DV finite-log-mgf/common-space hypotheses, Gronwall, endpoint rewrites, pure contraction, or thm:general-moving-target-SALD." /-- Cycle-67 proof-DAG pane for the guided residual and continuous general moving-target route. -/ def cycle67GuidedGeneralSkeletonDag : List ProofDagBlock := [ { id := "ASTIS.SALD.guided_general.cycle67_global_phase_judgment" interface := "Cycle-67 global judgment: cycle 66 passed and needs no recovery; Phase 1 is not ready for cited-theory backfill until guided residual and continuous general moving-target route wiring is rechecked; select the residual-to-Gronwall bridge as the single lower packet." source := saldGeneralMovingTargetSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle67GuidedGeneralSkeletonUpperPacket", "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle66DiscreteForwardKlAccumulatedDisplayLowerObligation" ] reusedBy := ["ASTIS-SALD-001 cycle 67", "prop:guided_path_residual", "thm:general-moving-target-SALD"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.guided_general.cycle67_five_backend_check" interface := "Explicit cycle-67 check of the five slow interfaces before lower work: endpoint-safe Gronwall, common-space DV with finite log-mgf, LSI/KL/FI density-test bridge, continuous Fokker-Planck/KL derivative, and downstream EM interpolation Fokker-Planck." source := saldGeneralMovingTargetSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.saldGronwallEndpointCalculusContract", "SALD.generalMovingTargetGronwallInstantiationContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.guided_general.cycle67_guided_residual_route" interface := "Route appendix.tex:619-704 through the guided normalizer derivative, quotient/product differentiation, divergence cancellation, centered residual identity, and mean-zero residual obligations without changing prop:guided_path_residual." source := saldGuidedResidualSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle67GuidedGeneralSkeletonUpperPacket", "SALD.cycle67GuidedGeneralSkeletonObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "TransportVelocityContract", "GuidedTiltContract" ] reusedBy := ["prop:guided_path_residual", "thm:unified-forward-KL"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.guided_general.cycle67_general_moving_target_route" interface := "Route appendix.tex:724-945 through the fixed general moving-target statement, continuous KL derivative, residual LSI/DV, sigma-weighted Gronwall, and pure-contraction interfaces without changing constants or source labels." source := saldGeneralMovingTargetSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle67GuidedGeneralSkeletonUpperPacket", "SALD.cycle67GuidedGeneralSkeletonObligation", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "sald.general_moving_target.dv_m_energy", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction" ] reusedBy := ["thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.guided_general.cycle67_middle_route_audit" interface := "Middle audit for cycle 67: verify appendix.tex:619-951 in paper order, keep guided residual calculus, continuous KL derivative, LSI/KL/FI, residual DV, Gronwall, pure contraction, unified specialization, and downstream EM interfaces explicit, and hand lower work to the residual-to-Gronwall bridge." source := saldGeneralMovingTargetSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle67GuidedGeneralSkeletonMiddleContract", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle67GuidedGeneralSkeletonObligation", "SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation", "SALD.guidedResidualIdentityContract", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "SALD.unifiedForwardKlSpecializationContract" ] reusedBy := ["prop:guided_path_residual", "thm:general-moving-target-SALD", "cycle 67 lower handoff"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.guided_general.cycle67_lower_packet.residual_to_gronwall_bridge" interface := "Selected lower packet: audit and sharpen the source-cited bridge from the residual KL derivative through LSI, DV with Z=alpha*||m_t||^2, sigma-weighted Gronwall, and pure contraction." source := saldGeneralMovingTargetDvGronwallSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle67GuidedGeneralSkeletonObligation", "SALD.generalMovingTargetDerivativeObligation", "SALD.cycle57GuidedGeneralDerivativeSplitLowerObligation", "SALD.cycle62GuidedGeneralScaledResidualLowerObligation", "SALD.cycle67GuidedGeneralResidualGronwallLowerObligation", "SALD.generalMovingTargetResidualToGronwallBridgeScalar", "SALD.generalMovingTargetDvEnergyObligation", "SALD.generalMovingTargetGronwallApplicationObligation", "SALD.generalMovingTargetGronwallSideConditionObligation", "SALD.generalMovingTargetPureContractionObligation" ] reusedBy := ["sald.general_moving_target.cycle67_residual_to_gronwall_lower", "sald.general_moving_target.cycle67_residual_to_gronwall_bridge", "thm:general-moving-target-SALD", "cycle 67 lower handoff"] status := ProofStatus.obligation } ] /-! ### Cycle 68: unified and discrete general theorem route refresh -/ /-- Cycle-68 upper packet for the unified forward-KL theorem and the discrete-time general moving-target theorem. This is a theorem-skeleton route packet only. It reuses the accepted continuous guided/general route from cycle 67 and the earlier unified/discrete general route, rechecks the five slow interfaces, and selects one lower packet for the discrete general theorem bridge without changing theorem statements or promoting analytic backends. -/ def cycle68UnifiedDiscreteGeneralSkeletonUpperPacket : GeneralVaSaldUpperPacket where objective := "Cycle 68 upper: after the accepted cycle-67 guided/general route, wire thm:unified-forward-KL and thm:general-moving-target-SALD-discrete through the continuous general skeletons and the explicit source-cited interfaces, preserving the main-body and appendix theorem statements while selecting the discrete general EM/derivative/DV/Gronwall theorem bridge as the single lower packet." sourceLabels := [ "thm:unified-forward-KL", "eq:ito_forward_KL_bound_calibrated", "proof:thm:unified-forward-KL", "prop:guided_path_residual", "eq:poisson-eq", "thm:general-moving-target-SALD", "thm:general-moving-target-SALD-discrete", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "eq:general_discrete_delta_def", "lem:frozen_delta_cross_lip", "eq:general_KL_derivative_0_discrete", "eq:general_KL_derivative_8_discrete", "eq:general_moving_target_KL_bound_discrete", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] modeDiscipline := [ "faithfulPaper Phase 1: use main_body.tex:359-395, appendix.tex:949-951, and appendix.tex:1313-1603 exactly; sald_version_2.tex remains out of scope.", "Global phase judgment: cycle 67 passed reviewer/build and does not need recovery; Phase 1 theorem-skeleton translation is stable enough to finish the unified/discrete-general route before any cited-theory backfill; the selected lower packet is the discrete general theorem bridge from EM endpoint/conditional-law through derivative, DV, and Gronwall/display.", "Five-backend check 1, Gronwall: consume SALD.saldGronwallEndpointCalculusContract, SALD.generalMovingTargetGronwallSideConditionContract, SALD.generalMovingTargetDiscreteGronwallInstantiationContract, SALD.generalMovingTargetDiscreteGronwallSideConditionContract, and the cycle-59/67 wrappers only as obligation-level endpoint/FTC/coefficient/display interfaces.", "Five-backend check 2, DV: consume dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, SALD.generalMovingTargetDvFiniteLogMgfWitnessContract, SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract, sald.general_moving_target.dv_m_energy, and sald.general_moving_target_discrete.dv_m_energy with common-space, absolute-continuity, finite-KL, finite-log-mgf, measurability, and positive-alpha side conditions explicit.", "Five-backend check 3, LSI/KL/FI: consume SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, and probability.lsi_to_kl_fi while preserving density, zero-set, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule obligations.", "Five-backend check 4, continuous derivative: consume SALD.generalMovingTargetDerivativeCandidateContract, SALD.generalMovingTargetDerivativeObligation, sald.general_moving_target.kl_derivative, and cycle-67 residual-to-Gronwall handoffs as local SDE/measure-analysis obligations used by thm:unified-forward-KL through specialization.", "Five-backend check 5, EM interpolation: consume SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation, cycle-63 endpoint-law helpers, cycle-64 conditional-drift interface, and sald.general_moving_target_discrete.em_interpolation_fp as downstream discrete obligations, not as formalized theorem backends." ] nonGoals := [ "Do not restate thm:unified-forward-KL, thm:general-moving-target-SALD, or thm:general-moving-target-SALD-discrete.", "Do not add hidden density, boundary, endpoint, finite-log-mgf, finite-KL/FI, common-space, conditional-law, coefficient-regularity, sigma-positivity, or schedule assumptions to the paper theorem statements.", "Do not prove or promote Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL differentiation, EM conditional Fokker-Planck, frozen-delta, residual DV, or final Gronwall/display matching.", "Do not replace the appendix specialization of thm:unified-forward-KL with a direct VA-SALD proof route.", "Do not start broad SLT/SDE library import; local SLT material may guide only a later narrow backend after this route is synchronized." ] lowerPacket := [ "Middle should synchronize SALD.cycle68UnifiedDiscreteGeneralSkeletonUpperPacket, SALD.cycle68UnifiedDiscreteGeneralSkeletonObligation, and SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation into the conversion window, proof-obligation ledger, and SLT audit.", "Target exactly SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation / sald.unified_discrete_general.cycle68_discrete_general_bridge.", "First lower sub-slice: verify main_body.tex:359-395 and appendix.tex:949-951 are consumed by prop:guided_path_residual, eq:poisson-eq, SALD.unifiedForwardKlSpecializationContract, sald.unified_forward_kl.transport_velocity_bridge, sald.unified_forward_kl.specialization, and the cycle-67 continuous general route.", "Second lower sub-slice: verify appendix.tex:1313-1387 is consumed by the fixed discrete statement, endpoint law/common-space helpers, conditional drift, weak EM Fokker-Planck, and KL derivative side-condition interfaces.", "Third lower sub-slice: verify appendix.tex:1455-1588 is consumed by frozen-delta, Young/LSI, residual DV with Z=alpha*||m_t||^2, constant-schedule time change, and the pointwise Gronwall input.", "Fourth lower sub-slice: verify appendix.tex:1316-1347 and 1588-1603 are consumed by SALD.generalMovingTargetDiscreteGronwallSideConditionContract and display-matching obligations.", "If any analytic step is too large, sharpen only the named source-cited or obligation interface; do not change theorem statements, constants, labels, or proof route." ] reviewerChecklist := [ "SALD.unifiedForwardKlContract lists SALD.cycle68UnifiedDiscreteGeneralSkeletonObligation while remaining contractOnly.", "SALD.generalVaSaldDiscreteContract lists SALD.cycle68UnifiedDiscreteGeneralSkeletonObligation and SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation while remaining contractOnly.", "SALD.generalVaSaldProofDag and SALD.generalVaSaldDiscreteProofDag contain SALD.cycle68UnifiedDiscreteGeneralDag after the cycle-67 and cycle-63 route data.", "SALD.saldDependenciesForLabel for thm:unified-forward-KL and thm:general-moving-target-SALD-discrete includes SALD.cycle68UnifiedDiscreteGeneralDependencyNames.", "The five slow analytic interfaces remain source-cited or obligation-level; no theorem statement, source constant, source label, external reuse status, or theorem status changes.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-68 upper obligation for routing the unified and discrete general theorems through the accepted continuous/general skeletons and explicit slow interfaces. -/ def cycle68UnifiedDiscreteGeneralSkeletonObligation : ProofObligation where id := "sald.unified_discrete_general.cycle68_upper_route" statement := "Cycle 68 upper records that cycle 67 passed, then rewires thm:unified-forward-KL through prop:guided_path_residual, eq:poisson-eq, the transport-velocity bridge, and thm:general-moving-target-SALD; it also rewires thm:general-moving-target-SALD-discrete through the general EM endpoint/conditional-law interface, frozen-delta lemma, discrete KL derivative/LSI route, residual DV, constant-schedule time change, and final Gronwall/display side conditions over appendix.tex:1313-1603. The selected lower packet is sald.unified_discrete_general.cycle68_discrete_general_bridge. Both theorem contracts remain contractOnly." source := saldGeneralMovingTargetDiscreteSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle68UnifiedDiscreteGeneralSkeletonUpperPacket", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation", "SALD.cycle67GuidedGeneralResidualGronwallLowerObligation", "SALD.cycle63UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle63UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.identity", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "SALD.generalMovingTargetStatementContract", "SALD.generalVaSaldContract", "SALD.generalMovingTargetDiscreteStatementContract", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "sald.general_moving_target_discrete.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract", "sald.general_moving_target_discrete.dv_m_energy", "SALD.generalMovingTargetDiscreteGronwallInstantiationContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions", "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "Upper workflow obligation only. It adds no hidden hypotheses, direct VA-SALD proof replacement, analytic proof closure, or status promotion." /-- Cycle-68 source-cited bridge for the final unified/discrete general theorem route. -/ def cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation : ProofObligation where id := "sald.unified_discrete_general.cycle68_discrete_general_bridge" statement := "Source-cited theorem bridge for cycle 68: after the unified theorem is identified as the appendix specialization c_t <- u_t of thm:general-moving-target-SALD, route the discrete general theorem from appendix.tex:1313-1603 through the general EM interpolation endpoint and conditional-law/Fokker-Planck interface, frozen-delta cross bound, discrete KL derivative with LSI, residual DV for Z=alpha*||m_t||^2, constant inverse-schedule time change, and the final Gronwall/display side-condition interface. This bridge remains below formalized until all analytic dependencies build locally." source := saldGeneralMovingTargetDiscreteSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle68UnifiedDiscreteGeneralSkeletonUpperPacket", "SALD.cycle68UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation", "SALD.cycle63UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation", "SALD.cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "SALD.generalMovingTargetDiscreteStatementContract", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "sald.general_moving_target_discrete.em_interpolation_fp", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "SALD.generalMovingTargetDiscreteFrozenResidualAlgebraVector", "SALD.generalMovingTargetDiscreteYoungFisherShareScalar", "SALD.generalMovingTargetDiscreteTwoYoungFisherBudgetScalar", "SALD.generalMovingTargetDiscreteResidualYoungCoefficientScalar", "sald.general_moving_target_discrete.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract", "sald.general_moving_target_discrete.dv_m_energy", "SALD.generalMovingTargetDiscreteDerivativeDvTimeChangedScalar", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput", "SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar", "SALD.generalMovingTargetDiscreteGronwallDisplayBridgeScalar", "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeLowerObligation", "SALD.generalMovingTargetDiscreteGronwallInstantiationContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "Selected lower packet for cycle 68. It is not a proof of EM conditional laws, density/AC, weak Fokker-Planck, KL differentiation, frozen-delta analysis, LSI/KL/FI, DV, Gronwall, or theorem closure." /-- Cycle-68 middle audit for the unified and discrete general theorem route. This source-to-Lean synchronization layer verifies the upper packet in the paper order and keeps lower work on the source-cited discrete general bridge. -/ def cycle68UnifiedDiscreteGeneralSkeletonMiddleContract : GeneralVaSaldGuidedPathMiddleContract where guidedResidualSource := saldGuidedResidualSource generalContinuousSource := saldGeneralMovingTargetSource unifiedSource := saldUnifiedForwardKlSource discreteSource := saldGeneralMovingTargetDiscreteSource objective := "Cycle 68 middle: audit the upper unified/discrete-general route after the accepted cycle-67 guided/general pass, verify thm:unified-forward-KL remains only the appendix specialization of thm:general-moving-target-SALD with c_t=u_t and m_t=w_t, verify thm:general-moving-target-SALD-discrete consumes the general EM endpoint/conditional-FP, frozen-delta, LSI, residual-DV, time-change, and Gronwall/display interfaces in source order, and keep lower work on SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation / sald.unified_discrete_general.cycle68_discrete_general_bridge." sourceStepMap := [ "main_body.tex:359-368 uses prop:guided_path_residual and eq:poisson-eq to make u_t+w_t a transport velocity for pi_t.", "main_body.tex:372-395 states thm:unified-forward-KL with correction-field complexity E_alpha(pi_t,w_t) and the exact sigma_t^(-2)*dot{s}(t)^(-1) coefficient.", "appendix.tex:949-951 proves thm:unified-forward-KL only by specializing thm:general-moving-target-SALD with c_t<-u_t.", "appendix.tex:1313-1347 states thm:general-moving-target-SALD-discrete with the doubled residual coefficient, sigma_eta factors, Gamma/Delta terms, alpha ranges, and constant inverse-schedule assumption.", "appendix.tex:1354-1387 fixes the EM interpolation endpoint laws, differentiates KL(hat rho_s||tilde pi_s), defines the frozen conditional drift, and invokes the conditional Fokker-Planck equation.", "appendix.tex:1389-1511 splits the Laplacian, identifies delta_pi^VA+dot t(s)*m_t, applies the frozen/residual Young bounds, and consumes lem:frozen_delta_cross_lip.", "appendix.tex:1513-1570 applies eq:LSI-KL-FI and lem:dv_variation to the residual field m_t under the EM interpolation law.", "appendix.tex:1573-1600 changes variables to K(t), uses dot t(s(t))=dot s(t)^(-1), forms the pointwise Gronwall input, and applies lem:gronwall to match the theorem display.", "appendix.tex:1603 records the downstream discrete guided VA-SALD specialization c_t=u_t; this middle audit does not make it a direct proof of thm:unified-forward-KL." ] leanStepMap := [ "Use SALD.cycle68UnifiedDiscreteGeneralSkeletonUpperPacket and SALD.cycle68UnifiedDiscreteGeneralSkeletonObligation as parent route data; keep the cycle-67 continuous guided/general route as the source of thm:unified-forward-KL.", "Route thm:unified-forward-KL through SALD.guidedResidualIdentityContract, SALD.unifiedForwardKlSpecializationContract, sald.unified_forward_kl.transport_velocity_bridge, sald.unified_forward_kl.specialization, SALD.generalVaSaldContract, and the cycle-67 middle and lower obligations.", "Route appendix.tex:1313-1347 through SALD.generalMovingTargetDiscreteStatementContract and SALD.generalVaSaldDiscreteContract; both remain contractOnly.", "Route appendix.tex:1354-1387 through SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation, the cycle-63 endpoint Measure.map helpers, SALD.generalMovingTargetDiscreteConditionalDriftContract, SALD.cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation, and sald.general_moving_target_discrete.em_interpolation_fp.", "Route appendix.tex:1389-1511 through SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.generalMovingTargetDiscreteFrozenResidualAlgebraVector, the cycle-28 Young/Fisher scalar helpers, lem:frozen_delta_cross_lip, and sald.general_moving_target_discrete.kl_derivative.", "Route appendix.tex:1513-1570 through SALD.saldLsiKlFiDensityTestContract, probability.lsi_to_kl_fi, dvVariationalFormulaInterface saldDvVariationSource, SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract, and sald.general_moving_target_discrete.dv_m_energy.", "Route appendix.tex:1573-1600 through SALD.generalMovingTargetDiscreteDerivativeDvTimeChangedScalar, SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged, SALD.generalMovingTargetDiscreteGronwallInstantiationContract, SALD.generalMovingTargetDiscreteGronwallSideConditionContract, SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput, and SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar.", "Select lower work exactly at SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation / sald.unified_discrete_general.cycle68_discrete_general_bridge; sharpen that source-cited bridge if blocked, rather than changing theorem statements or backend statuses." ] citedResultInterfaces := [ "lem:gronwall remains an endpoint-safe differentiability, FTC/order integration, coefficient-regularity, endpoint-stitching, and display-matching obligation.", "lem:dv_variation remains source-cited through common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and positive-alpha witnesses.", "eq:LSI-KL-FI remains the density, zero-set convention, admissible sqrt-density test or approximation, entropy identity, finite KL/FI, and Fisher-chain obligation.", "The continuous Fokker-Planck/KL derivative is consumed upstream through sald.general_moving_target.kl_derivative and the cycle-67 residual-to-Gronwall bridge; it is not reproved for the unified theorem.", "The EM interpolation backend remains sald.general_moving_target_discrete.em_interpolation_fp; cycle-63 endpoint helpers and the cycle-64 conditional-drift algebra do not prove conditional laws, density/AC, weak Fokker-Planck, or KL differentiation.", "Local SLT one-step/disintegration material remains reference-only for later narrow backfill; this middle packet imports no SLT theorem and promotes no analytic backend." ] obligations := [ "sald.unified_discrete_general.cycle68_upper_route", "sald.unified_discrete_general.cycle68_middle_route_audit", "sald.unified_discrete_general.cycle68_discrete_general_bridge", "sald.guided_general.cycle67_middle_route_audit", "sald.general_moving_target.cycle67_residual_to_gronwall_bridge", "sald.general_moving_target.cycle67_residual_to_gronwall_lower", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "sald.general_moving_target_discrete.kl_derivative", "probability.lsi_to_kl_fi", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions" ] lowerPacket := [ "Target exactly SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation / sald.unified_discrete_general.cycle68_discrete_general_bridge.", "First sub-slice: verify main_body.tex:359-395 and appendix.tex:949-951 against the guided residual, correction-field transport bridge, and cycle-67 continuous general theorem route.", "Second sub-slice: verify appendix.tex:1313-1387 against the fixed discrete statement, endpoint-law/common-space helpers, conditional drift, weak EM Fokker-Planck, and KL derivative side-condition interfaces.", "Third sub-slice: verify appendix.tex:1389-1570 against frozen-delta, Young/LSI, residual DV with Z=alpha*||m_t||^2, and the post-DV time-change handoff.", "Fourth sub-slice: verify appendix.tex:1573-1600 and theorem display lines 1316-1347 against the Gronwall side-condition and display-matching obligations.", "Do not restate either theorem, change coefficients, or promote Gronwall, DV, LSI/KL/FI, EM interpolation, KL derivative, frozen-delta, endpoint stitching, or theorem status." ] reviewerChecklist := [ "SALD.unifiedForwardKlContract and SALD.generalVaSaldDiscreteContract list SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation while remaining contractOnly.", "SALD.generalVaSaldProofDag and SALD.generalVaSaldDiscreteProofDag contain ASTIS.SALD.unified_discrete_general.cycle68_middle_route_audit between the upper route nodes and the selected lower bridge.", "SALD.saldDependenciesForLabel entries for thm:unified-forward-KL and thm:general-moving-target-SALD-discrete include the cycle-68 middle contract, middle obligation, and route-audit node.", "The selected lower packet remains sald.unified_discrete_general.cycle68_discrete_general_bridge and no broad SLT/SDE backfill is started before reviewer acceptance.", "No theorem statement, source constant, source label, source-file scope, SLT reuse status, or analytic backend status changes.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] status := ProofStatus.obligation /-- Cycle-68 middle obligation tying the route audit to the selected unified/discrete general source-cited bridge. -/ def cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation : ProofObligation where id := "sald.unified_discrete_general.cycle68_middle_route_audit" statement := "Cycle 68 middle audits the upper unified/discrete-general route in paper order: thm:unified-forward-KL remains the source specialization of thm:general-moving-target-SALD through the guided residual and correction-field transport bridge, thm:general-moving-target-SALD-discrete remains the appendix route through EM endpoint/conditional-FP, frozen-delta, LSI, residual DV, time change, and Gronwall/display interfaces, and the selected lower packet remains sald.unified_discrete_general.cycle68_discrete_general_bridge. All theorem statuses and slow analytic interfaces remain below formalized." source := saldGeneralMovingTargetDiscreteSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle68UnifiedDiscreteGeneralSkeletonUpperPacket", "SALD.cycle68UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleContract", "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation", "SALD.cycle67GuidedGeneralResidualGronwallLowerObligation", "SALD.cycle63UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle63UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.identity", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "SALD.generalMovingTargetStatementContract", "SALD.generalVaSaldContract", "SALD.generalMovingTargetDiscreteStatementContract", "SALD.generalVaSaldDiscreteContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "sald.general_moving_target_discrete.em_interpolation_fp", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "SALD.generalMovingTargetDiscreteFrozenResidualAlgebraVector", "SALD.generalMovingTargetDiscreteYoungFisherShareScalar", "SALD.generalMovingTargetDiscreteTwoYoungFisherBudgetScalar", "SALD.generalMovingTargetDiscreteResidualYoungCoefficientScalar", "sald.general_moving_target_discrete.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiDensityTestObligation", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "SALD.generalMovingTargetDiscreteDerivativeDvTimeChangedScalar", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput", "SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar", "SALD.generalMovingTargetDiscreteGronwallDisplayBridgeScalar", "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeLowerObligation", "SALD.generalMovingTargetDiscreteGronwallInstantiationContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] note := "Middle workflow and source-to-Lean audit only. It adds no hidden common-space, density, absolute-continuity, finite-KL/FI, finite-log-mgf, regular conditional law, weak Fokker-Planck, schedule, sigma-positivity, endpoint, or coefficient-regularity assumptions to the paper theorem statements." /-- Cycle-68 lower proof-producing obligation for the final discrete general Gronwall/display bridge. -/ def cycle68UnifiedDiscreteGeneralDiscreteBridgeLowerObligation : ProofObligation where id := "sald.unified_discrete_general.cycle68_discrete_general_bridge_lower" statement := "Cycle 68 lower compiles SALD.generalMovingTargetDiscreteGronwallDisplayBridgeScalar for appendix.tex:1573-1600 and theorem display appendix.tex:1316-1347: after the discrete general derivative/DV/time-change route supplies the source pointwise inequality and the analytic Gronwall backend supplies the bound for the named coefficients a(t), b(t), the wrapper derives the exact endpoint theorem-display inequality. Unified thm:unified-forward-KL remains the continuous general specialization; this lower step only closes the discrete general display bridge." source := saldGeneralMovingTargetDiscreteGronwallSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation", "SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation", "SALD.generalMovingTargetDiscreteGronwallDisplayBridgeScalar", "SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput", "SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "SALD.generalMovingTargetDiscreteDerivativeDvTimeChangedScalar", "SALD.generalMovingTargetDiscreteGronwallInstantiationContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "lem:gronwall" ] note := "Proof-producing scalar/display wrapper plus obligation only. It preserves theorem targets, source constants, and source labels, and does not prove or promote EM endpoint/conditional laws, density/AC, weak Fokker-Planck, KL differentiation, frozen-delta, LSI/KL/FI, DV, Gronwall, endpoint stitching, coefficient regularity, or thm:general-moving-target-SALD-discrete." /-- Cycle-68 proof-DAG pane for the unified and discrete general theorem route refresh. -/ def cycle68UnifiedDiscreteGeneralDag : List ProofDagBlock := [ { id := "ASTIS.SALD.unified_discrete_general.cycle68_global_phase_judgment" interface := "Cycle-68 global judgment: cycle 67 passed and needs no recovery; Phase 1 is stable enough to finish the unified/discrete-general route before cited-theory backfill; select the discrete general theorem bridge as the single lower packet." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle68UnifiedDiscreteGeneralSkeletonUpperPacket", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle63UnifiedDiscreteGeneralMiddleObligation" ] reusedBy := ["ASTIS-SALD-001 cycle 68", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.unified_discrete_general.cycle68_five_backend_check" interface := "Explicit cycle-68 check of the five slow interfaces before lower work: endpoint-safe Gronwall, common-space DV with finite log-mgf, LSI/KL/FI density-test bridge, continuous Fokker-Planck/KL derivative for the unified specialization, and EM interpolation endpoint/conditional-law Fokker-Planck for the discrete general theorem." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.saldGronwallEndpointCalculusContract", "SALD.generalMovingTargetGronwallSideConditionContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.unified_discrete_general.cycle68_unified_route" interface := "Route main_body.tex:359-395 and appendix.tex:949-951 through guided residual, correction-field divergence, transport-velocity bridge, and the cycle-67 continuous general theorem specialization without changing thm:unified-forward-KL." source := saldUnifiedForwardKlSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle68UnifiedDiscreteGeneralSkeletonObligation", "SALD.guidedResidualIdentityContract", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation" ] reusedBy := ["thm:unified-forward-KL", "cycle 68 lower handoff"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.cycle68_theorem_route" interface := "Route appendix.tex:1313-1603 through the fixed discrete general statement, general EM endpoint/conditional-FP, frozen-delta, KL derivative/LSI, residual DV, time change, and Gronwall/display interfaces without changing constants or labels." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle68UnifiedDiscreteGeneralSkeletonObligation", "SALD.generalMovingTargetDiscreteStatementContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "sald.general_moving_target_discrete.gronwall_side_conditions" ] reusedBy := ["thm:general-moving-target-SALD-discrete", "cycle 68 lower handoff"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.unified_discrete_general.cycle68_middle_route_audit" interface := "Middle audit for cycle 68: verify the unified specialization and the discrete general theorem route in paper order, keep the theorem contracts fixed, and hand lower work to the source-cited discrete general bridge." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleContract", "SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation", "SALD.cycle68UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle63UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation", "SALD.unifiedForwardKlSpecializationContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract" ] reusedBy := ["thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "cycle 68 lower handoff"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.unified_discrete_general.cycle68_lower_packet.discrete_general_bridge" interface := "Selected lower packet: audit and sharpen the source-cited bridge from unified specialization and the discrete general EM/KL derivative route through residual DV and final Gronwall/display matching; the compiled lower wrapper only composes named Gronwall input with supplied Gronwall output and endpoint display rewrites." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation", "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeLowerObligation", "SALD.generalMovingTargetDiscreteGronwallDisplayBridgeScalar", "SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation", "SALD.cycle68UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle63UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract" ] reusedBy := ["sald.unified_discrete_general.cycle68_discrete_general_bridge", "thm:general-moving-target-SALD-discrete", "cycle 68 middle/lower handoff"] status := ProofStatus.obligation } ] /-! ### Cycle 69: post-route analytic interface ledger -/ /-- Cycle-69 upper ledger after the full theorem-route sprint. This is source-to-Lean route data only. It records the required upper phase judgment, rechecks the five slow analytic interfaces after the cycle-68 unified/discrete-general pass, and chooses one lower packet for later narrow backend work without promoting any theorem or analytic backend. -/ def cycle69MainSkeletonAnalyticInterfaceLedger : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteSource objective := "Cycle 69 upper: cycle 68 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for a post-route analytic-interface ledger and only one narrow backend backfill, not broad cited-theory or reusable API work; the single lower packet that best reduces proof risk is the shared Euler-Maruyama interpolation conditional-law/Fokker-Planck backend over appendix.tex:1358-1387, because it supports both discrete theorem skeletons and remains the largest common measure/SDE dependency." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "proof:thm:forward-KL:derivative", "proof:thm:forward-KL-discrete:conditional-fp", "proof:thm:general-moving-target-SALD:derivative", "proof:thm:general-moving-target-SALD-discrete:em-conditional-fp", "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Gronwall: SALD.saldGronwallCandidateContract, SALD.saldGronwallEndpointCalculusContract, SALD.saldGronwallExponentRewriteContract, and theorem-specific Gronwall side-condition contracts expose endpoint-safe differentiability or right-derivative/FTC assumptions, interval integrability, endpoint evaluation, coefficient regularity, and exponent/display rewrites; lem:gronwall remains ProofStatus.obligation.", "Donsker-Varadhan: dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, and theorem-specific finite-log-mgf witness contracts expose common probability space, absolute continuity, finite KL/log-likelihood, selected-test measurability, finite log-mgf, positive-alpha scaling, and E_alpha rewriting; the cited variational equality remains ProofStatus.sourceCited.", "LSI/KL/FI: SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, probability.lsi_to_kl_fi, and the cycle-33/38/43 helper rows expose Radon-Nikodym density, zero-set convention, admissible sqrt-density test or approximation, entropy identity, finite KL/FI, and the vector/integral Fisher chain rule; eq:LSI-KL-FI remains ProofStatus.obligation.", "Continuous forward-KL Fokker-Planck/KL derivative: SALD.forwardKlDerivativeCandidateContract, SALD.forwardKlDerivativeSideConditionContract, SALD.generalMovingTargetDerivativeCandidateContract, and the cycle-60/65/67 scalar handoffs expose mass conservation, KL differentiation under the integral, Fokker-Planck substitution, integration by parts, target transport, residual Young/LSI, and inverse-schedule calculus; analytic derivative backends remain ProofStatus.obligation.", "Euler-Maruyama interpolation Fokker-Planck: SALD.discreteForwardKlEmInterpolationSideConditionContract, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation, cycle-63 endpoint-law helpers, cycle-64 conditional-drift algebra, and sald.general_moving_target_discrete.em_interpolation_fp expose endpoint laws, common-space bookkeeping, regular conditional drift, density/absolute-continuity, weak conditional Fokker-Planck signs, Laplacian split, and stitched interval regularity; this is the selected lower packet and remains ProofStatus.obligation." ] theoremRoute := [ "1. thm:forward-KL consumes continuous KL derivative/Fokker-Planck, LSI/KL/FI, DV velocity-energy, inverse-schedule calculus, and Gronwall endpoint/exponent display matching.", "2. thm:forward-KL-discrete consumes EM endpoint/conditional-FP, KL derivative with frozen defect and LSI, DV velocity, time-changed Gronwall, and accumulated-error display matching.", "3. prop:guided_path_residual consumes guided normalizer differentiation, quotient/product calculus, divergence cancellation, and centered residual mean-zero obligations.", "4. thm:general-moving-target-SALD consumes continuous general KL derivative, residual Young/LSI, residual DV, sigma-weighted Gronwall, endpoint/exponent side conditions, and pure-contraction specialization.", "5. thm:unified-forward-KL remains the appendix specialization of thm:general-moving-target-SALD through prop:guided_path_residual, the correction-field transport bridge, c_t=u_t, and m_t=w_t.", "6. thm:general-moving-target-SALD-discrete consumes general EM endpoint/conditional-FP, frozen-delta, KL derivative/LSI, residual DV, constant-schedule time change, and final Gronwall/display stitching." ] modeDiscipline := [ "faithfulPaper Phase 1: use only the original main_body.tex, appendix.tex, and iteration_complexity.tex; sald_version_2.tex remains excluded.", "Keep theorem statements, constants, alpha ranges, sigma factors, endpoint laws, source labels, and source proof order fixed.", "Every unproved slow analytic backend remains ProofStatus.obligation or ProofStatus.sourceCited; compiled scalar, endpoint-law, and display wrappers are local dependencies only.", "A future lower run may backfill exactly one backend, the EM interpolation conditional-law/Fokker-Planck interface, but this upper ledger only sharpens the interface and route." ] nonGoals := [ "No source-index rebaseline beyond the required acceptance command unless a reviewer reports a blocking source-anchor defect.", "No broad SLT import, Gaussian concentration port, entropy-duality project, disintegration project, SDE library reorganization, or teaching API rewrite.", "No hidden endpoint, density, absolute-continuity, finite-KL/FI, finite-log-mgf, boundary, smoothness, conditional-law, sigma-positivity, schedule, coefficient-regularity, or stitched-interval assumption is added to any source theorem statement.", "No alternate proof route replaces derivative -> LSI -> DV -> Gronwall, the correction-field specialization, or the paper EM/frozen-delta route." ] lowerPacket := [ "Middle should synchronize this cycle-69 analytic-interface ledger with conversion-windows/ASTIS-SALD-001.md and proof-obligations/ASTIS-SALD-001.md, preserving the six theorem-route order.", "Lower should target exactly SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation / SALD.generalMovingTargetDiscreteDerivativeSideConditionContract / sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387.", "First sub-slice: expose the common probability space, endpoint laws for hat rho_s and tilde pi_s, density/absolute-continuity, finite KL, and mass-conservation hypotheses needed before differentiating KL.", "Second sub-slice: sharpen the regular conditional law of X_k^eta given hat X_s=x and the measurable/integrable conditional drift bar b_{k,s}(x), reusing the cycle-64 conditional-drift linear-combination algebra only under supplied conditional-expectation linearity.", "Third sub-slice: state the weak conditional Fokker-Planck equation with the exact source signs -div(hat rho_s bar b_{k,s}) and +(sigma_eta^2/2) Delta hat rho_s, leaving KL differentiation, LSI, DV, Gronwall, and theorem closure as separate obligations." ] reviewerChecklist := [ "SALD.cycle69MainSkeletonAnalyticInterfaceObligation is listed by all six theorem contracts while those contracts remain ProofStatus.contractOnly.", "SALD.forwardKlProofDag, SALD.discreteForwardKlProofDag, SALD.generalVaSaldProofDag, and SALD.generalVaSaldDiscreteProofDag include SALD.cycle69MainSkeletonAnalyticInterfaceDag.", "SALD.saldDependenciesForLabel includes cycle69MainSkeletonDependencyNames for the five slow interfaces and all six theorem-route labels.", "The selected lower packet is the EM interpolation conditional-law/Fokker-Planck backend over appendix.tex:1358-1387, not a theorem-status promotion or broad SLT/SDE port.", "Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL differentiation, EM conditional-FP, theorem contracts, and SLT reuse statuses all remain below formalized unless already compiled locally.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation", "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeLowerObligation", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.cycle64MainSkeletonAnalyticInterfaceObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation /-- Cycle-69 upper obligation for the post-route analytic-interface ledger. -/ def cycle69MainSkeletonAnalyticInterfaceObligation : ProofObligation where id := "sald.main_skeleton.cycle69_analytic_interface_ledger" statement := "Cycle 69 upper records that cycle 68 passed and needs no recovery, that Phase 1 theorem-skeleton translation is stable enough for a post-route analytic-interface ledger but only one narrow backend backfill, and that the five slow analytic interfaces remain precise source-cited or obligation-level interfaces wired through thm:forward-KL, thm:forward-KL-discrete, prop:guided_path_residual, thm:general-moving-target-SALD, thm:unified-forward-KL, and thm:general-moving-target-SALD-discrete. The selected lower packet is the shared EM interpolation conditional-law/Fokker-Planck backend over appendix.tex:1358-1387." source := saldGeneralMovingTargetDiscreteSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle69MainSkeletonAnalyticInterfaceLedger", "SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation", "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeLowerObligation", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.cycle64MainSkeletonAnalyticInterfaceObligation", "SALD.saldGronwallCandidateContract", "SALD.saldGronwallEndpointCalculusContract", "SALD.saldGronwallExponentRewriteContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "sald.general_moving_target_discrete.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI" ] note := "Upper workflow obligation only. It adds no theorem hypotheses and does not promote Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL differentiation, EM conditional Fokker-Planck, theorem contracts, or SLT reuse status." /-- Cycle-69 middle audit for the post-route analytic-interface ledger. This synchronizes the upper ledger with the source transcript after all six theorem skeletons have been wired. It keeps the selected lower packet on the shared Euler--Maruyama interpolation conditional-law/Fokker--Planck backend. -/ def cycle69MainSkeletonAnalyticMiddleContract : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteDerivativeSource objective := "Cycle 69 middle: synchronize the upper post-route analytic-interface ledger against the full SALD theorem route, verify that the five slow analytic interfaces remain the named consumers for the six faithful theorem skeletons in paper order, and keep the next lower packet exactly on SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation / SALD.generalMovingTargetDiscreteDerivativeSideConditionContract / sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387." sourceLabels := [ "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "proof:thm:forward-KL:derivative", "proof:thm:forward-KL-discrete:conditional-fp", "proof:thm:general-moving-target-SALD:derivative", "proof:thm:general-moving-target-SALD-discrete:em-conditional-fp", "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Middle five-backend check, Gronwall: theorem routes still consume SALD.saldGronwallEndpointCalculusContract and theorem-specific Gronwall side-condition obligations for endpoint-safe differentiability or right-derivative/FTC, interval integrability, endpoint evaluation, coefficient regularity, exponent splitting, and display matching; lem:gronwall remains obligation.", "Middle five-backend check, DV: all velocity and residual-energy uses still pass through dvVariationalFormulaInterface saldDvVariationSource plus theorem-specific common-space, absolute-continuity, finite-KL/log-likelihood, selected-test measurability, positive-alpha, and finite-log-mgf witnesses; lem:dv_variation remains source-cited plus obligations.", "Middle five-backend check, LSI/KL/FI: all LSI uses still pass through SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, and probability.lsi_to_kl_fi with Radon-Nikodym density, zero-set convention, admissible sqrt-density test or approximation, entropy identity, finite KL/FI, and Fisher chain-rule obligations explicit.", "Middle five-backend check, continuous Fokker-Planck/KL derivative: thm:forward-KL and thm:general-moving-target-SALD still consume SALD.forwardKlDerivativeSideConditionContract and SALD.generalMovingTargetDerivativeCandidateContract through the cycle-65 and cycle-67 theorem-route audits; scalar wrappers do not close the analytic backend.", "Middle five-backend check, EM interpolation Fokker-Planck: thm:forward-KL-discrete and thm:general-moving-target-SALD-discrete still consume SALD.discreteForwardKlEmInterpolationSideConditionContract, SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, cycle-63 endpoint laws, cycle-64 conditional-drift algebra, and sald.general_moving_target_discrete.em_interpolation_fp; conditional laws, density/AC, weak FP, KL differentiation, and stitched intervals remain obligations." ] theoremRoute := [ "1. thm:forward-KL: appendix.tex:168-252 remains continuous KL derivative/Fokker-Planck -> LSI/KL/FI -> DV velocity-energy -> Gronwall endpoint/exponent display.", "2. thm:forward-KL-discrete: appendix.tex:260-592 remains EM endpoint/conditional-FP -> KL derivative/frozen defect/LSI -> DV velocity -> time-changed Gronwall -> accumulated-error display.", "3. prop:guided_path_residual: appendix.tex:619-704 remains guided normalizer differentiation, quotient/product calculus, divergence cancellation, centered residual identity, and mean-zero residual.", "4. thm:general-moving-target-SALD: appendix.tex:724-949 remains continuous general KL derivative -> residual Young/LSI -> residual DV -> sigma-weighted Gronwall -> pure contraction.", "5. thm:unified-forward-KL: main_body.tex:359-395 and appendix.tex:949-951 remain the correction-field transport bridge and c_t=u_t, m_t=w_t specialization of thm:general-moving-target-SALD.", "6. thm:general-moving-target-SALD-discrete: appendix.tex:1313-1603 remains general EM endpoint/conditional-FP -> frozen-delta -> KL derivative/LSI -> residual DV -> constant-schedule Gronwall/display stitching." ] modeDiscipline := [ "faithfulPaper Phase 1: this middle audit is source-to-Lean synchronization only, using original main_body.tex, appendix.tex, and iteration_complexity.tex; sald_version_2.tex remains excluded.", "No theorem statement, coefficient, alpha range, sigma factor, endpoint law, source label, or proof order is changed.", "All theorem contracts remain ProofStatus.contractOnly and all slow analytic interfaces remain ProofStatus.obligation or ProofStatus.sourceCited unless their full analytic dependencies compile locally.", "The only lower packet kept active by this middle pass is the shared EM interpolation conditional-law/Fokker-Planck backend over appendix.tex:1358-1387." ] nonGoals := [ "Do not start broad SLT, disintegration, SDE, Gaussian concentration, entropy-duality, or reusable API backfill.", "Do not turn cycle-63 endpoint-law bookkeeping or cycle-64 conditional-drift algebra into a proof of regular conditional laws, density/absolute-continuity, weak Fokker-Planck, or KL differentiation.", "Do not replace the paper derivative -> LSI -> DV -> Gronwall route, the correction-field specialization, or the EM/frozen-delta route.", "Do not mark Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL differentiation, EM conditional-FP, theorem contracts, or SLT reuse as formalized." ] lowerPacket := [ "Target exactly SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation / SALD.generalMovingTargetDiscreteDerivativeSideConditionContract / sald.general_moving_target_discrete.em_interpolation_fp.", "Source slice appendix.tex:1358-1366: keep common probability-space data, endpoint laws for hat rho_s and tilde pi_s, density/absolute-continuity, finite KL, mass conservation, and endpoint-safe KL differentiation prerequisites explicit.", "Source slice appendix.tex:1368-1377: keep the regular conditional law of X_k^eta given hat X_s=x, the measurable/integrable conditional drift bar b_{k,s}(x), and conditional-expectation linearity as obligations; reuse cycle-64 algebra only after those inputs are supplied.", "Source slice appendix.tex:1379-1387: state the weak conditional Fokker-Planck equation with exact source signs -div(hat rho_s bar b_{k,s}) and +(sigma_eta^2/2) Delta hat rho_s.", "Keep KL differentiation after the weak FP identity, LSI/KL/FI, DV, frozen-delta, and Gronwall/display matching as separate downstream obligations." ] reviewerChecklist := [ "SALD.cycle69MainSkeletonAnalyticMiddleObligation is listed by all six theorem contracts while those contracts remain ProofStatus.contractOnly.", "SALD.cycle69MainSkeletonAnalyticInterfaceDag contains ASTIS.SALD.cycle69.middle_interface_audit before ASTIS.SALD.cycle69.lower_packet.em_interpolation_fp.", "SALD.saldDependenciesForLabel includes SALD.cycle69MainSkeletonAnalyticMiddleContract, SALD.cycle69MainSkeletonAnalyticMiddleObligation, and sald.main_skeleton.cycle69_middle_interface_audit through cycle69MainSkeletonDependencyNames.", "The conversion window, proof-obligation ledger, and SLT reuse audit record this middle synchronization and keep the selected lower packet narrow.", "No theorem statement, source constant, source label, source-file scope, theorem status, SLT reuse status, or analytic backend status changes.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle69MainSkeletonAnalyticInterfaceLedger", "SALD.cycle69MainSkeletonAnalyticInterfaceObligation", "SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation", "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeLowerObligation", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination", "SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation /-- Cycle-69 middle obligation tying the post-route analytic-interface audit to the selected shared EM conditional-law/Fokker--Planck backend. -/ def cycle69MainSkeletonAnalyticMiddleObligation : ProofObligation where id := "sald.main_skeleton.cycle69_middle_interface_audit" statement := "Cycle 69 middle synchronizes the upper analytic-interface ledger with the complete SALD theorem route: the five slow interfaces are consumed by thm:forward-KL, thm:forward-KL-discrete, prop:guided_path_residual, thm:general-moving-target-SALD, thm:unified-forward-KL, and thm:general-moving-target-SALD-discrete in paper order, and the selected lower packet remains the shared EM interpolation conditional-law/Fokker-Planck backend over appendix.tex:1358-1387. Theorem statuses and slow analytic interfaces remain below formalized." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle69MainSkeletonAnalyticMiddleContract", "SALD.cycle69MainSkeletonAnalyticInterfaceObligation", "SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation", "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeLowerObligation", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination", "SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] note := "Middle workflow obligation only. It adds no hidden common-space, density, absolute-continuity, finite-KL/FI, finite-log-mgf, regular conditional law, weak Fokker-Planck, endpoint, schedule, sigma-positivity, coefficient-regularity, stitched-interval, or theorem-level assumption." /-- Cycle-69 lower obligation for the source-sign EM FP handoff. -/ def cycle69GeneralMovingTargetDiscreteEmFpSourceSignsLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle69_em_fp_source_signs_lower" statement := "Cycle 69 lower compiles SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff for appendix.tex lines 1379-1387: once the analytic weak conditional Fokker-Planck backend supplies partial_s hat rho_s = -div(hat rho_s*bar b_{k,s}) + sigmaCoeff*Delta hat rho_s and the source coefficient identity sigmaCoeff=sigma_eta^2/2, the wrapper preserves the exact source signs and coefficient for the EM interpolation FP interface." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle69MainSkeletonAnalyticMiddleObligation", "SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "sald.general_moving_target_discrete.em_interpolation_fp", "FokkerPlanckContract", "KLContract", "FIContract" ] note := "Proof-producing coefficient/sign wrapper only. Common probability space, endpoint laws, regular conditional drift, density/absolute-continuity, weak Fokker-Planck, KL differentiation, integration by parts, LSI, DV, Gronwall, and theorem status remain obligations." /-- Cycle-70 middle obligation for the conditional-law/measurability slice. -/ def cycle70GeneralMovingTargetDiscreteConditionalLawMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle70_conditional_law_middle" statement := "Cycle 70 middle keeps the selected backend at sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387 and narrows the source-to-Lean packet to the conditional-law/measurability layer of appendix.tex:1368-1377: name the regular conditional kernel of X_k^eta given hat X_s=x, the component conditional fields condC_{k,s} and condScore_{k,s}, and the selected drift bar b_{k,s}; keep endpoint laws, density/AC, weak Fokker-Planck, KL differentiation, LSI, DV, and Gronwall as downstream obligations." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle69MainSkeletonAnalyticMiddleObligation", "SALD.cycle69GeneralMovingTargetDiscreteEmFpSourceSignsLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle synchronization only. It records the cycle-70 global judgment: cycle 69 passed and needs no recovery; Phase 1 is stable enough for single-backend backfill; the best lower packet is conditional-law/measurability plus named conditional drift interfaces for bar b_{k,s}." /-- Cycle-70 lower obligation for the named conditional drift handoff. -/ def cycle70GeneralMovingTargetDiscreteConditionalLawLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle70_named_conditional_drift_lower" statement := "Cycle 70 lower compiles SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents, SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff, and SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents for appendix.tex lines 1368-1377. Once a regular conditional expectation, named component conditional fields, linearity, component measurability/integrability, add/smul closure, and congruence-stable predicates are supplied, the wrappers identify bar b_{k,s} with dot t_k*condC_{k,s}+(sigma_eta^2/2)*condScore_{k,s} and transfer the resulting regularity to bar b_{k,s}. They do not construct the regular conditional law or prove weak FP." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawMiddleObligation", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination", "SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing wrapper only. The regular conditional kernel, disintegration theorem, measurability/integrability proofs, density/absolute-continuity, weak conditional Fokker-Planck identity, KL derivative, LSI/KL/FI, DV, Gronwall, and theorem closure remain obligations." /-- Cycle-71 middle obligation for endpoint-law-to-conditional-law compatibility. -/ def cycle71GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle71_endpoint_conditional_middle" statement := "Cycle 71 middle keeps the selected backend at sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387 and narrows the source-to-Lean packet to endpoint-law-to-conditional-law compatibility: the joint Measure.map law of (X_k^eta,hat X_s) has hat rho_s=Law(hat X_s) as its second marginal, and any supplied conditional-kernel/disintegration compatibility predicate must be transported to that named hat rho_s marginal before the weak Fokker-Planck statement uses bar b_{k,s}." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawMiddleObligation", "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle synchronization only. This records that cycle 70 passed and needs no recovery; Phase 1 remains stable for single-backend backfill; the risk-reducing packet is the endpoint-law/conditional-kernel marginal compatibility layer, not theorem-route audit or display algebra." /-- Cycle-71 local wrapper obligation for the endpoint-to-conditional bridge. -/ def cycle71GeneralMovingTargetDiscreteEndpointConditionalLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle71_endpoint_conditional_lower" statement := "Cycle 71 compiles SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap, SALD.generalMovingTargetDiscreteConditionalKernelCompatibilityOfJointMapMarginal, and SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityOfJointMap for appendix.tex lines 1368-1377: from the common-space joint Measure.map law of (X_k^eta,hat X_s), explicit measurability, and the named law equality hat rho_s=Measure.map hat X_s P, the wrappers identify the second marginal with hat rho_s and transport a supplied kernel-compatibility predicate from that joint-law marginal to the named hat rho_s marginal, with the final wrapper packaging the equality and compatibility handoff directly." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap", "SALD.generalMovingTargetDiscreteConditionalKernelCompatibilityOfJointMapMarginal", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityOfJointMap", "AutoSamplingTheory.lawMapProdSnd", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing Measure.map/equality-transport wrappers only. Existence of the regular conditional kernel, disintegration, conditional expectation, measurability/integrability of fields, density/absolute-continuity, weak conditional Fokker-Planck, KL differentiation, LSI/KL/FI, DV, Gronwall, and theorem closure remain obligations." /-- Cycle-71 proof-DAG pane for endpoint-law-to-conditional-law compatibility. -/ def cycle71GeneralMovingTargetDiscreteEndpointConditionalDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle71.global_phase_judgment" interface := "Cycle 71 judgment: cycle 70 passed and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for continued single-backend backfill; the highest-risk packet remains sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, now the endpoint-law-to-conditional-law compatibility layer." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawMiddleObligation", "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle71.middle_endpoint_conditional_compatibility" interface := "Middle conversion window for appendix.tex:1368-1377: connect the joint Measure.map law of (X_k^eta,hat X_s) and its second marginal to the named hat rho_s marginal required by the conditional-kernel compatibility predicate." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle71.lower_endpoint_conditional_wrapper" interface := "Local proof-producing wrappers: identify the second marginal of the joint pushforward with hat rho_s, then transport a supplied conditional-kernel compatibility predicate to that named marginal by equality." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap", "SALD.generalMovingTargetDiscreteConditionalKernelCompatibilityOfJointMapMarginal", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityOfJointMap", "AutoSamplingTheory.lawMapProdSnd", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] /-- Cycle-72 middle obligation for the weak conditional Fokker--Planck source-sign interface. -/ def cycle72GeneralMovingTargetDiscreteWeakFpMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle72_weak_fp_middle" statement := "Cycle 72 middle keeps the selected backend at sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387 and narrows the source-to-Lean packet to appendix.tex:1379-1387: the weak conditional Fokker--Planck statement must use the same hat rho_s, conditional drift bar b_{k,s}, and conditional-kernel compatibility from cycles 70-71, with drift contribution -div(hat rho_s*bar b_{k,s}) and positive diffusion contribution +(sigma_eta^2/2)*Delta hat rho_s." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawLowerObligation", "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle synchronization only. Cycle 71 passed and needs no recovery; Phase 1 remains stable for single-backend backfill; the lower packet is the weak-test FP source-sign statement, not theorem-route audit, display algebra, or a new analytic theorem claim." /-- Cycle-72 local wrapper obligation for weak-FP source signs. -/ def cycle72GeneralMovingTargetDiscreteWeakFpLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle72_weak_fp_source_signs_lower" statement := "Cycle 72 compiles SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff and SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff for appendix.tex lines 1379-1387: for every admissible weak test phi, if the supplied weak conditional Fokker--Planck identity has partial_s hat rho_s paired with phi equal to -(drift-divergence action on phi) plus sigmaCoeff times the Laplacian action on phi, and sigmaCoeff=sigma_eta^2/2, then the source-signed weak form has the negative drift-divergence term and positive +(sigma_eta^2/2) Laplacian term." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle72GeneralMovingTargetDiscreteWeakFpMiddleObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing wrappers only. The new admissible-test variant keeps the weak-test predicate explicit but still consumes a supplied weak-FP identity. The actual weak conditional Fokker--Planck theorem, common-space construction, regular conditional drift, density/absolute-continuity, admissible test class, boundary/integration-by-parts conditions, KL derivative, LSI/KL/FI, DV, Gronwall, and theorem closure remain obligations." /-- Cycle-72 proof-DAG pane for weak conditional Fokker--Planck source signs. -/ def cycle72GeneralMovingTargetDiscreteWeakFpDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle72.global_phase_judgment" interface := "Cycle 72 judgment: cycle 71 passed and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for continued single-backend cited-theory backfill; the highest-risk packet remains sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, now the weak conditional Fokker--Planck source-sign statement at appendix.tex:1379-1387." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle72.middle_weak_fp_source_signs" interface := "Middle conversion window for appendix.tex:1379-1387: state the weak-test conditional Fokker--Planck interface for hat rho_s with -div(hat rho_s*bar b_{k,s}) and +(sigma_eta^2/2)*Delta hat rho_s under explicit conditional-law, measurability, integrability, density, and admissible-test hypotheses." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle72GeneralMovingTargetDiscreteWeakFpMiddleObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle72.lower_weak_fp_source_signs" interface := "Local proof-producing wrappers: from a supplied weak conditional Fokker--Planck identity and sigmaCoeff=sigma_eta^2/2, preserve the negative drift-divergence term and positive sigma_eta^2/2 Laplacian term for each admissible test, with an explicit admissible-test predicate variant." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle72GeneralMovingTargetDiscreteWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] /-- Cycle-73 upper packet for the KL-derivative handoff from weak FP. This is the fourth single-backend backfill packet after the post-route cycle-69 ledger. It keeps the active backend fixed at `sald.general_moving_target_discrete.em_interpolation_fp` and narrows the next handoff to substituting the supplied weak conditional Fokker--Planck identity into `eq:general_KL_derivative_0_discrete`. -/ def cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpUpperPacket : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource objective := "Global phase judgment: cycle 72 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for continued single-backend cited-theory backfill; the single lower packet that best reduces the remaining proof risk is the KL-derivative handoff from the weak conditional Fokker-Planck identity, still inside sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387." sourceLabels := [ "eq:general_KL_derivative_0_discrete", "proof:thm:general-moving-target-SALD-discrete:weak-conditional-fp", "proof:thm:general-moving-target-SALD-discrete:weak-fp-to-kl-derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Gronwall remains unchanged as endpoint-safe calculus and theorem-specific coefficient/display obligations; cycle 73 does not touch lem:gronwall.", "DV remains source-cited through lem:dv_variation with common-space, absolute-continuity, finite-KL, and finite-log-mgf witnesses; cycle 73 is before the residual DV step.", "LSI/KL/FI remains the density-test/Fisher-chain obligation; cycle 73 does not use eq:LSI-KL-FI except as a downstream derivative consumer.", "Continuous Fokker-Planck/KL derivative remains a separate continuous theorem backend and is not modified by this EM packet.", "EM interpolation FP remains the active backend: cycles 70-72 exposed conditional-law, endpoint compatibility, and weak FP signs; cycle 73 now exposes the log-ratio-test substitution into the differentiated KL formula." ] theoremRoute := [ "1. Start from appendix.tex:1358-1366, the differentiated KL display and mass-conservation drop.", "2. Use appendix.tex:1368-1377 only through the already named conditional drift and conditional-kernel interfaces.", "3. Use appendix.tex:1379-1387 through the supplied weak conditional FP identity with the cycle-72 source signs.", "4. Substitute that weak FP identity at the log-density-ratio test and hand the result to sald.general_moving_target_discrete.kl_derivative.", "5. Leave integration by parts, Laplacian split, FI identification, LSI, DV, time change, Gronwall, and theorem closure as separate obligations." ] modeDiscipline := [ "faithfulPaper Phase 1: use only the original appendix.tex source window 1358-1387; sald_version_2.tex remains excluded.", "Do not change theorem statements, constants, sigma_eta^2/2 sign conventions, source labels, or proof order.", "Keep the log-ratio admissibility, density/absolute-continuity, finite KL/FI, boundary, integration-by-parts, and weak FP theorem assumptions as explicit obligations, not hidden theorem hypotheses." ] nonGoals := [ "No theorem-route audit, display algebra, Gronwall coefficient work, DV witness work, or broad SLT/SDE API backfill.", "No claim that a regular conditional law, weak Fokker-Planck theorem, log-ratio admissibility theorem, or KL differentiation theorem is formalized.", "No promotion of thm:forward-KL-discrete, thm:general-moving-target-SALD-discrete, LSI/KL/FI, DV, Gronwall, or EM interpolation status." ] lowerPacket := [ "Target exactly SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract / SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpLowerObligation / sald.general_moving_target_discrete.cycle73_kl_derivative_weak_fp_lower.", "Lower proof-producing scope: compile only the scalar substitution from the differentiated KL formula plus the supplied admissible weak FP identity at the log-ratio test, including the direct composition through the cycle-72 admissible source-sign wrapper.", "Middle synchronization scope: keep the source-to-Lean map two-way synchronized in the conversion window, proof-obligation ledger, and SLT audit under appendix.tex:1358-1387." ] reviewerChecklist := [ "The active lower packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387.", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar and SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns are local substitution wrappers only and do not prove the analytic weak FP or KL derivative backend.", "SALD.discreteSaldContract and SALD.generalVaSaldDiscreteContract remain contractOnly and list the cycle-73 obligations.", "Gronwall, DV, LSI/KL/FI, continuous FP/KL derivative, EM interpolation, theorem statuses, and SLT reuse statuses remain below formalized.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle72GeneralMovingTargetDiscreteWeakFpMiddleObligation", "SALD.cycle72GeneralMovingTargetDiscreteWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] status := ProofStatus.obligation /-- Cycle-73 upper obligation for the KL-derivative handoff packet. -/ def cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpUpperObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle73_kl_derivative_weak_fp_upper" statement := "Cycle 73 upper records no recovery after the accepted cycle-72 weak-FP source-sign packet, confirms Phase 1 theorem-skeleton translation is stable enough for continued single-backend backfill, and selects exactly the KL-derivative handoff from the supplied weak conditional Fokker-Planck identity over appendix.tex:1358-1387. The packet stays inside sald.general_moving_target_discrete.em_interpolation_fp and does not promote weak FP, log-ratio admissibility, density/AC, integration by parts, LSI, DV, Gronwall, or theorem status." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpUpperPacket", "SALD.cycle72GeneralMovingTargetDiscreteWeakFpMiddleObligation", "SALD.cycle72GeneralMovingTargetDiscreteWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Upper workflow obligation only. It fixes the lower packet and status discipline; it adds no theorem assumptions and proves no analytic backend." /-- Cycle-73 middle obligation for the weak-FP-to-KL derivative source map. -/ def cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle73_kl_derivative_weak_fp_middle" statement := "Cycle 73 middle synchronizes appendix.tex:1358-1387 into the active EM backend: eq:general_KL_derivative_0_discrete supplies the differentiated KL display with the target-time term, cycles 70-72 supply conditional-law, endpoint compatibility, and weak FP source signs under explicit hypotheses, and the selected lower wrappers substitute the supplied weak FP identity at the admissible log-density-ratio test before later integration-by-parts and Fisher-information handoffs." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpUpperObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Middle synchronization only. Log-ratio admissibility, density/absolute-continuity, KL differentiability, mass conservation, weak FP theorem, boundary/integration-by-parts, and FI identification remain obligations." /-- Cycle-73 local wrapper obligation for weak-FP-to-KL derivative substitution. -/ def cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle73_kl_derivative_weak_fp_lower" statement := "Cycle 73 compiles SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar and SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns for appendix.tex:1358-1387: if the differentiated KL formula is supplied as dK=partialS(logRatioTest)-targetTimeTerm, the log-ratio test is admissible, the supplied weak conditional Fokker-Planck identity gives partialS(phi)=-(driftDiv phi)+sigmaCoeff*(laplacian phi) on admissible tests, and sigmaCoeff=sigma_eta^2/2, then the source-signed KL derivative display has the negative drift-divergence action, positive sigma_eta^2/2 Laplacian action, and unchanged target-time term. The second wrapper explicitly composes the KL handoff through the cycle-72 admissible weak-FP source-sign theorem." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpMiddleObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Proof-producing scalar substitution plus obligation only. The composed wrapper reuses the cycle-72 admissible weak-FP source-sign theorem under explicit hypotheses; it does not prove the weak FP theorem, admissible log-ratio test, density/AC, KL differentiation, mass conservation, integration by parts, Laplacian split, FI identification, LSI/KL/FI, DV, Gronwall, or theorem closure." /-- Cycle-73 proof-DAG pane for weak-FP-to-KL derivative handoff. -/ def cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle73.global_phase_judgment" interface := "Cycle 73 judgment: cycle 72 passed and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for continued single-backend backfill; the highest-risk packet remains sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, now the KL-derivative handoff from weak FP." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle72GeneralMovingTargetDiscreteWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle73.middle_kl_derivative_weak_fp_handoff" interface := "Middle conversion window for appendix.tex:1358-1387: select the log-density-ratio weak test, keep admissibility/density hypotheses explicit, and hand the supplied weak FP identity to the KL derivative formula before integration by parts." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpMiddleObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle73.lower_kl_derivative_weak_fp_substitution" interface := "Local proof-producing wrappers: from the differentiated KL formula and supplied admissible-test weak FP identity, substitute the source-signed weak FP action at the log-ratio test, including the composition through the cycle-72 admissible source-sign theorem, and preserve the unchanged target-time term." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] /-- Cycle-74 source-cited Mathlib measure interface for the blocked conditional-kernel layer. This is intentionally narrow: it records the Mathlib conditional-expectation kernel that lower work should audit before attempting another weak-FP proof step. It is not imported as a new dependency here and it is not a proof of the SALD weak Fokker-Planck theorem. -/ def cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface : ProofObligation where id := "sald.general_moving_target_discrete.cycle74_conditional_kernel_measure_interface" statement := "Cycle 74 records the blocked Mathlib measure-theory interface for appendix.tex:1368-1377: under the common probability space for (X_k^eta, hat X_s), standard-Borel/finite-measure hypotheses should allow a condExpKernel/condDistrib-style kernel whose second marginal is the named hat rho_s and whose integrals of the frozen drift summands define the measurable conditional field bar b_{k,s}. This source-cited interface is the next lower target before any claimed proof of the weak conditional Fokker-Planck identity." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource status := ProofStatus.sourceCited dependsOn := [ "Mathlib.Probability.Kernel.Condexp", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Disintegration.StandardBorel", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawMiddleObligation", "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Source-cited Mathlib interface only. It does not import Mathlib kernel/disintegration modules into this file, construct the SALD conditional law, prove conditional expectation identities, prove density/AC, prove the weak conditional Fokker-Planck theorem, prove log-ratio admissibility, or close KL differentiation." /-- Cycle-74 upper packet selecting the minimal cited measure interface after cycle-73 weak-FP-to-KL substitution. -/ def cycle74GeneralMovingTargetDiscreteMeasureInterfaceUpperPacket : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource objective := "Global phase judgment: cycle 73 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; proof-producing work on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387 is now blocked on the concrete conditional-law measure backend, so the single lower packet is the narrow Mathlib condExpKernel/conditional-kernel interface for appendix.tex:1368-1377." sourceLabels := [ "proof:thm:general-moving-target-SALD-discrete:weak-fp-to-kl-derivative", "proof:thm:general-moving-target-SALD-discrete:weak-conditional-fp", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.cycle74_conditional_kernel_measure_interface", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Gronwall, DV, LSI/KL/FI, and continuous Fokker-Planck/KL derivative interfaces are unchanged and remain below formalized status.", "Cycles 70-73 already provide local wrappers for named conditional drift, endpoint-to-conditional compatibility, weak-FP source signs, and weak-FP-to-KL scalar substitution under explicit hypotheses.", "The remaining blocked interface is the regular conditional-kernel/conditional-expectation measure theorem feeding bar b_{k,s}; cycle 74 records it as sourceCited through Mathlib Probability.Kernel.Condexp rather than pretending the analytic backend is formalized.", "The weak conditional Fokker-Planck theorem, log-ratio admissibility, density/AC, integration by parts, and KL differentiation remain obligations after this cited interface." ] theoremRoute := [ "1. Stay inside appendix.tex:1358-1387 and the existing sald.general_moving_target_discrete.em_interpolation_fp backend.", "2. Use appendix.tex:1368-1377 as the selected source slice: construct/name the conditional law of X_k^eta given hat X_s=x and the conditional drift bar b_{k,s}.", "3. Cite the Mathlib condExpKernel/condDistrib measure interface as the missing theorem to audit, with standard-Borel, finite/probability, measurability, integrability, marginal-compatibility, and kernel-version side conditions explicit.", "4. Do not advance to new theorem-route audits or unrelated scalar algebra until this conditional-law interface is either locally proved or kept as a precise cited dependency." ] modeDiscipline := [ "faithfulPaper Phase 1: preserve the original appendix.tex source route and exclude sald_version_2.tex.", "Do not add assumptions to thm:forward-KL-discrete or thm:general-moving-target-SALD-discrete; record required standard-Borel, finite-measure, measurability, and integrability side conditions as interface obligations.", "Keep every cited or analytic backend below formalized unless the local ASTIS declaration compiles under this toolchain." ] nonGoals := [ "No broad SLT import, no Lake dependency change, and no claim that any SLT theorem is formalized.", "No theorem-route audit, Gronwall/DV/LSI display work, frozen-delta algebra, or project-article export.", "No promotion of the weak conditional Fokker-Planck theorem, KL derivative backend, EM interpolation backend, or theorem contracts." ] lowerPacket := [ "Target exactly SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface / sald.general_moving_target_discrete.cycle74_conditional_kernel_measure_interface.", "Middle should synchronize the paper-to-Lean map for appendix.tex:1368-1377 and the Mathlib candidates Probability.Kernel.Condexp, CondDistrib, and Disintegration.StandardBorel.", "Lower should either prove a tiny local wrapper around the conditional-kernel interface under explicit supplied hypotheses, or leave this exact sourceCited interface as the blocker; do not switch to weak FP, KL derivative, theorem-route, or display algebra work." ] reviewerChecklist := [ "The active lower packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to appendix.tex:1368-1377.", "The new measure interface is ProofStatus.sourceCited, not formalized, and no Mathlib/SLT theorem is claimed imported or locally proved.", "discrete and discrete-general theorem contracts remain ProofStatus.contractOnly and list the cycle-74 interface only as a dependency.", "The conversion window, proof-obligation ledger, and SLT reuse audit record the same conditional-kernel blocker.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "sald.general_moving_target_discrete.em_interpolation_fp", "Mathlib.Probability.Kernel.Condexp" ] status := ProofStatus.obligation /-- Cycle-74 upper obligation for the minimal cited measure interface. -/ def cycle74GeneralMovingTargetDiscreteMeasureInterfaceUpperObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle74_measure_interface_upper" statement := "Cycle 74 upper records that cycle 73 passed and needs no recovery, confirms Phase 1 theorem-skeleton translation is stable enough for continued single-backend backfill, and selects exactly one narrow cited measure-theory interface for the blocked conditional-law layer of sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387. The lower packet is the condExpKernel/conditional-kernel backend for appendix.tex:1368-1377, not weak-FP theorem closure or theorem-route audit." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceUpperPacket", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Upper workflow obligation only. It fixes the lower packet and status discipline; it adds no theorem assumptions, imports no Mathlib module, and proves no analytic backend." /-- Cycle-74 middle obligation for the conditional-kernel source-to-Lean map. -/ def cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle74_measure_interface_middle" statement := "Cycle 74 middle synchronizes appendix.tex:1368-1377 with the cited Mathlib conditional-kernel candidates: the paper conditions X_k^eta on hat X_s=x under the joint law of (X_k^eta,hat X_s), so the next proof-producing audit should look for a condDistrib/condExpKernel backend whose compProd law recovers the joint law, whose second marginal is the named hat rho_s from cycle 71, and whose integral_condDistrib or integral_condExpKernel measurability/integrability theorems can define the two frozen drift summands and their linear combination bar b_{k,s}. This is a middle source map only, not a proof of the conditional law or weak FP theorem." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceUpperObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawMiddleObligation", "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpMiddleObligation", "Mathlib.Probability.Kernel.CondDistrib.compProd_map_condDistrib", "Mathlib.Probability.Kernel.CondDistrib.condDistrib_ae_eq_iff_measure_eq_compProd", "Mathlib.Probability.Kernel.CondDistrib.condExp_prod_ae_eq_integral_condDistrib", "Mathlib.Probability.Kernel.Condexp.condExp_ae_eq_integral_condExpKernel", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle synchronization only. The cited Mathlib declarations are audit targets, not imported or locally instantiated here. Standard-Borel/nonempty state, finite/probability measure, joint-law equality, marginal compatibility, measurability, integrability, vector-valued conditional expectation, density/AC, weak FP, and KL differentiation remain explicit obligations." /-- Cycle-74 lower obligation for the supplied-kernel regularity handoff. -/ def cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle74_conditional_kernel_lower" statement := "Cycle 74 lower compiles SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents for appendix.tex lines 1368-1377. Once the cited condDistrib/condExpKernel backend supplies kernel compatibility for the joint law of (X_k^eta,hat X_s), component integral fields for the two frozen drift summands, and their measurability/integrability consequences, the wrapper transports compatibility to the named hat rho_s marginal and derives measurability/integrability of the selected bar b_{k,s} field. It does not construct the conditional kernel or prove weak FP." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalKernelCompatibilityOfJointMapMarginal", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "Mathlib.Probability.Kernel.CondDistrib.compProd_map_condDistrib", "Mathlib.Probability.Kernel.Condexp.condExp_ae_eq_integral_condExpKernel", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing local wrapper under supplied hypotheses only. The Mathlib conditional-kernel theorem, standard-Borel/finite-measure side conditions, vector-valued conditional expectation construction, density/AC, weak conditional Fokker-Planck identity, log-ratio admissibility, KL differentiation, integration by parts, and theorem closure remain obligations." /-- Cycle-74 proof-DAG pane for the conditional-kernel measure blocker. -/ def cycle74GeneralMovingTargetDiscreteMeasureInterfaceDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle74.global_phase_judgment" interface := "Cycle 74 judgment: cycle 73 passed and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the largest remaining risk is the conditional-law measure backend inside sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle74.middle_conditional_kernel_source_map" interface := "Middle conversion map for appendix.tex:1368-1377: line up the joint law of (X_k^eta,hat X_s), the named marginal hat rho_s, Mathlib condDistrib/condExpKernel candidate facts, and the measurable/integrable conditional drift fields needed before weak FP proof search." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawMiddleObligation", "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle74.lower_packet.conditional_kernel_measure_interface" interface := "Minimal cited measure interface: audit Mathlib Probability.Kernel.Condexp/CondDistrib/Disintegration.StandardBorel as the regular conditional-kernel backend needed to define bar b_{k,s} from the joint law of (X_k^eta, hat X_s) before weak FP can be proved." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.sourceCited }, { id := "ASTIS.SALD.cycle74.lower_conditional_kernel_regularity_handoff" interface := "Local supplied-kernel wrapper: after Mathlib condDistrib/condExpKernel provides kernel compatibility and component integral measurability/integrability, transport compatibility to hat rho_s and inherit regularity for bar b_{k,s}." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalKernelCompatibilityOfJointMapMarginal", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] /-- Cycle-75 upper packet returning to the EM conditional-law interface. Cycle 74 recorded the blocked conditional-kernel theorem as a precise source-cited interface. Cycle 75 keeps the same source window and asks lower work to reduce that blocker by instantiating or sharpening the condDistrib/condExpKernel construction for the named joint law and marginal, without advancing to weak Fokker-Planck or KL-derivative proof search. -/ def cycle75GeneralMovingTargetDiscreteConditionalLawBackfillUpperPacket : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource objective := "Global phase judgment: cycle 74 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet that now reduces the largest proof risk is still sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to the conditional-law/measurability and named conditional drift construction interface for appendix.tex:1368-1377." sourceLabels := [ "proof:thm:general-moving-target-SALD-discrete:conditional-drift", "proof:thm:general-moving-target-SALD-discrete:weak-conditional-fp", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.cycle74_conditional_kernel_measure_interface", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Cycle 74 supplied the exact source-cited condDistrib/condExpKernel blocker; cycle 75 should not create a second broad interface.", "The conditional-law layer must connect the joint law of (X_k^eta,hat X_s), the named marginal hat rho_s, the conditional kernel of X_k^eta given hat X_s=x, and the two component conditional drift integrals.", "Cycles 70 and 71 already provide local named-drift and endpoint-to-conditional wrappers; cycle 75 lower should try to feed those wrappers with a concrete conditional-kernel construction or record the exact missing Mathlib theorem.", "Weak conditional Fokker-Planck, log-ratio admissibility, density/AC, KL differentiation, integration by parts, LSI/KL/FI, DV, Gronwall, and theorem closure remain obligations." ] theoremRoute := [ "1. Stay inside appendix.tex:1358-1387, with the active lower slice appendix.tex:1368-1377.", "2. Start from the cycle-74 source-cited interface rather than a new theorem route: regular conditional law/condExpKernel for X_k^eta conditioned on hat X_s=x.", "3. Preserve the named hat rho_s marginal from the endpoint-to-conditional compatibility wrappers and the paper's bar b_{k,s} component fields.", "4. If local proof-producing work is possible, prove only a tiny supplied-hypothesis wrapper around kernel construction/compatibility/measurability; otherwise leave a narrowly cited missing Mathlib theorem." ] modeDiscipline := [ "faithfulPaper Phase 1: keep appendix.tex and main_body.tex as the only SALD sources and keep sald_version_2.tex out of scope.", "Do not add standard-Borel, finite-measure, measurability, integrability, density, admissibility, or endpoint assumptions to theorem statements.", "Use lean-stat-learning-theory only as a local Mathlib style reference; do not add it as a dependency and do not claim any SLT theorem as formalized." ] nonGoals := [ "No weak conditional Fokker-Planck theorem proof, no KL derivative proof, no theorem-route audit, no display algebra, and no broad reusable API design.", "No status promotion for sald.general_moving_target_discrete.em_interpolation_fp, sald.general_moving_target_discrete.kl_derivative, thm:forward-KL-discrete, or thm:general-moving-target-SALD-discrete.", "No Lake dependency changes, no SLT import, and no project-article export." ] lowerPacket := [ "Target exactly the conditional-law/measurability layer feeding SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface and SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents.", "Preferred lower sub-slice: instantiate or wrap Mathlib condDistrib/condExpKernel facts for the joint law of (X_k^eta,hat X_s), its second marginal hat rho_s, and vector-valued component integrals for dot t_k*c_{t_k} and (sigma_eta^2/2)*nabla log pi_{t_k}.", "If blocked, record one missing theorem with standard-Borel/probability, marginal compatibility, measurability, integrability, and vector-valued conditional expectation hypotheses explicit.", "Do not switch to endpoint-law re-audit, weak FP, KL derivative, Gronwall/DV/LSI, frozen-delta, or theorem display work." ] reviewerChecklist := [ "The active lower packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, specifically appendix.tex:1368-1377.", "Cycle 75 depends on the cycle-74 conditional-kernel source-cited interface and does not duplicate or promote it.", "Any lower proof is only a local wrapper under explicit kernel/measurability/integrability hypotheses; the actual conditional law, weak FP, and KL differentiation remain below formalized unless compiled locally.", "The conversion window, proof-obligation ledger, source index, and SLT audit agree that SLT is reference-only and not imported.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "sald.general_moving_target_discrete.em_interpolation_fp", "Mathlib.Probability.Kernel.Condexp", "Mathlib.Probability.Kernel.CondDistrib" ] status := ProofStatus.obligation /-- Cycle-75 upper obligation for the focused conditional-law backfill. -/ def cycle75GeneralMovingTargetDiscreteConditionalLawBackfillUpperObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle75_conditional_law_upper" statement := "Cycle 75 upper records that cycle 74 passed and needs no recovery, confirms that Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill, and keeps the active lower packet on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387. The selected lower work is the conditional-law/measurability and named conditional drift construction interface for appendix.tex:1368-1377: instantiate or sharpen the cycle-74 condDistrib/condExpKernel source-cited interface for the joint law of (X_k^eta,hat X_s), the named hat rho_s marginal, and the component conditional drift integrals, or record the exact missing Mathlib theorem." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillUpperPacket", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Upper workflow obligation only. It fixes the next lower packet and status discipline, adds no theorem assumptions, imports no SLT or Mathlib module, and proves no weak Fokker-Planck, KL derivative, density, or conditional-law theorem." /-- Cycle-75 middle source-to-Lean map for the conditional-law orientation and measurability handoff. Mathlib's `condDistrib Y X μ` is oriented by the conditioning variable first: for `X_k^eta | hat X_s`, the generated joint law is `(hat X_s, X_k^eta)`. The existing cycle-71 SALD endpoint compatibility is oriented as `(X_k^eta, hat X_s)`. This middle obligation records that the lower packet must bridge that orientation using only `Measure.map` bookkeeping before consuming the conditional-kernel and integral-measurability theorems. -/ def cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle75_conditional_law_middle" statement := "Cycle 75 middle sharpens the appendix.tex:1368-1377 lower packet after checking the local SLT Measure.map style and Mathlib Probability.Kernel.CondDistrib/Condexp APIs. To condition X_k^eta on hat X_s=x, lower should instantiate condDistrib with X=hat X_s and Y=X_k^eta, so compProd_map_condDistrib produces the joint law in the order (hat X_s,X_k^eta). The existing SALD cycle-71 compatibility names the joint law as (X_k^eta,hat X_s) with hat rho_s as the second marginal, so lower must either work with the swapped joint law or use AutoSamplingTheory.lawMapProdSwap to bridge orientations before applying component conditional-integral measurability and integrability facts for the two frozen drift summands." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillUpperObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "AutoSamplingTheory.lawMapProdSwap", "ProbabilityTheory.compProd_map_condDistrib", "ProbabilityTheory.condDistrib_ae_eq_iff_measure_eq_compProd", "MeasureTheory.StronglyMeasurable.integral_condDistrib", "MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map", "MeasureTheory.Integrable.integral_condDistrib_map", "MeasureTheory.Integrable.norm_integral_condDistrib_map", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "ProbabilityTheory.condExp_ae_eq_integral_condExpKernel", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle source map only. It records the condDistrib orientation, named-marginal compatibility, and vector-valued component-integral measurability/integrability targets; it imports no Mathlib kernel module, constructs no conditional law, proves no weak FP/KL derivative theorem, and keeps standard-Borel, finite/probability, AEMeasurable, Integrable, density/AC, and admissible-test hypotheses explicit." /-- Cycle-75 lower obligation for the swapped-orientation supplied-kernel regularity wrapper. -/ def cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle75_conditional_law_lower" statement := "Cycle 75 lower compiles SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap and SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents for appendix.tex lines 1368-1377. If the Mathlib condDistrib/condExpKernel backend supplies kernel compatibility for the swapped joint law (hat X_s,X_k^eta), component integral fields for the two frozen drift summands, and their measurability/integrability consequences, the wrapper identifies the named hat rho_s first marginal, bridges back to the existing SALD (X_k^eta,hat X_s) joint-law orientation with AutoSamplingTheory.lawMapProdSwap, and derives measurability/integrability of bar b_{k,s}. It does not construct the conditional kernel or prove weak FP." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation", "SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "AutoSamplingTheory.lawMapProdSwap", "ProbabilityTheory.compProd_map_condDistrib", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "ProbabilityTheory.condExp_ae_eq_integral_condExpKernel", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing local wrapper under supplied hypotheses only. The regular conditional kernel, Mathlib conditional-expectation instantiation, component integral theorem, density/AC, weak conditional Fokker-Planck identity, log-ratio admissibility, KL differentiation, integration by parts, and theorem closure remain obligations." /-- Cycle-75 proof-DAG pane for the conditional-law construction backfill. -/ def cycle75GeneralMovingTargetDiscreteConditionalLawBackfillDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle75.global_phase_judgment" interface := "Cycle 75 judgment: cycle 74 passed and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the largest remaining shared risk is still the conditional-law layer inside sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillUpperObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle75.middle_conditional_law_source_map" interface := "Middle source map for appendix.tex:1368-1377: Mathlib condDistrib for X_k^eta conditioned on hat X_s produces the joint law as (hat X_s,X_k^eta), so lower must bridge the existing SALD (X_k^eta,hat X_s) joint-law orientation with AutoSamplingTheory.lawMapProdSwap before using component conditional-integral measurability and integrability facts." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "AutoSamplingTheory.lawMapProdSwap", "ProbabilityTheory.compProd_map_condDistrib", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle75.lower_packet.conditional_law_measurability" interface := "Selected lower packet: instantiate or sharpen the cycle-74 condDistrib/condExpKernel source-cited interface for appendix.tex:1368-1377, connecting the joint law, named hat rho_s marginal, regular conditional kernel, and measurable/integrable component conditional drift fields." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "AutoSamplingTheory.lawMapProdSwap" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-- Cycle-76 upper packet for endpoint-law to conditional compatibility. Cycle 75 established the swapped `condDistrib` orientation wrapper. Cycle 76 returns to the endpoint-law bookkeeping and packages it with that swapped conditional-kernel orientation so the weak Fokker--Planck interface can consume one named `hat rho_s` marginal without re-auditing unrelated theorem routes. -/ def cycle76GeneralMovingTargetDiscreteEndpointConditionalUpperPacket : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource objective := "Global phase judgment: cycle 75 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet that now reduces the largest proof risk is endpoint-law-to-conditional-law compatibility for sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, especially the bridge from endpoint Measure.map laws to the swapped condDistrib orientation needed by the weak FP interface." sourceLabels := [ "proof:thm:general-moving-target-SALD-discrete:weak-fp-to-kl-derivative", "proof:thm:general-moving-target-SALD-discrete:conditional-drift", "proof:thm:general-moving-target-SALD-discrete:weak-conditional-fp", "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "The endpoint laws at appendix.tex:1354-1357 stay in Measure.map form and remain separate from the regular conditional-law theorem.", "The conditional drift at appendix.tex:1368-1377 must use the same named hat rho_s marginal as the weak FP statement at appendix.tex:1379-1387.", "Cycle 75 identified Mathlib's swapped condDistrib orientation (hat X_s,X_k^eta); cycle 76 packages that orientation with the existing endpoint Measure.map handoffs under explicit supplied-kernel hypotheses.", "Regular conditional law construction, vector-valued conditional expectation, density/AC, weak FP, KL differentiation, LSI/KL/FI, DV, Gronwall, and theorem closure remain obligations." ] theoremRoute := [ "1. Stay inside appendix.tex:1358-1387 and the existing sald.general_moving_target_discrete.em_interpolation_fp backend.", "2. Reuse the endpoint Measure.map handoff for hat rho_{s_k}=rho_k^eta and hat rho_{s_{k+1}}=rho_{k+1}^eta.", "3. Reuse the swapped conditional-law orientation for X_k^eta | hat X_s, but bridge it back to the paper's (X_k^eta,hat X_s) joint-law orientation before weak FP.", "4. Do not reopen Gronwall/DV/LSI, display algebra, source-index rebaseline, or theorem-route audits." ] modeDiscipline := [ "faithfulPaper Phase 1: preserve appendix.tex statements and source labels exactly, with sald_version_2.tex excluded.", "Do not add endpoint, standard-Borel, finite-measure, measurability, integrability, or density hypotheses to theorem statements; keep them as backend obligations.", "Use lean-stat-learning-theory only as a local Mathlib style reference; no Lake dependency or SLT formalization claim." ] nonGoals := [ "No construction of condDistrib, condExpKernel, regular conditional expectations, weak Fokker-Planck, or KL differentiation.", "No theorem status promotion for thm:forward-KL-discrete, thm:general-moving-target-SALD-discrete, or the EM backend.", "No broad reusable API work and no project-article export." ] lowerPacket := [ "Target SALD.generalMovingTargetDiscreteEndpointMeasureMapToSwappedConditionalCompatibility.", "Supply only explicit hypotheses: endpoint a.e. interpolation identities, named rho_k/rho_{k+1}/hat rho_s Measure.map representations, swapped kernel compatibility, and a bridge from swapped joint-law compatibility to the paper's original orientation.", "Return endpoint equalities, the named hat rho_s marginal in both the swapped first-marginal view and the paper's original second-marginal view, the Measure.map swap equality, and original-orientation kernel compatibility for the weak FP interface.", "If blocked on the analytic kernel theorem, record that missing theorem narrowly; do not switch to weak FP proof search." ] reviewerChecklist := [ "The active lower packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387.", "New proof-producing work is only local Measure.map endpoint/orientation packaging under supplied kernel hypotheses.", "Conditional-law construction, weak FP, KL derivative, density/AC, LSI, DV, Gronwall, and theorem contracts remain below formalized.", "No SLT import, Lake dependency change, sald_version_2.tex use, or source-label drift.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToSwappedConditionalCompatibility", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility", "AutoSamplingTheory.lawMapProdSwap", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation /-- Cycle-76 upper obligation for the endpoint-to-conditional backfill. -/ def cycle76GeneralMovingTargetDiscreteEndpointConditionalUpperObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle76_endpoint_conditional_upper" statement := "Cycle 76 upper records that cycle 75 passed and needs no recovery, confirms that Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill, and keeps the active lower packet on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387. The selected lower work is endpoint-law-to-conditional compatibility: combine the endpoint Measure.map handoffs for hat rho_{s_k} and hat rho_{s_{k+1}} with the swapped condDistrib orientation (hat X_s,X_k^eta), then bridge back to the paper's (X_k^eta,hat X_s) orientation for the weak FP interface." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalUpperPacket", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToSwappedConditionalCompatibility", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Upper workflow obligation only. It fixes the lower packet and does not prove disintegration, weak FP, KL differentiation, density/AC, LSI, DV, Gronwall, or theorem closure." /-- Cycle-76 middle map for endpoint-law to conditional-law compatibility. -/ def cycle76GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle76_endpoint_conditional_middle" statement := "Cycle 76 middle synchronizes appendix.tex:1358-1387 by connecting the endpoint law statements hat rho_{s_k}=rho_k^eta and hat rho_{s_{k+1}}=rho_{k+1}^eta with the conditional drift definition of bar b_{k,s}. The lower wrapper should consume existing endpoint Measure.map handoffs, the named hat rho_s=Law(hat X_s) representation for the interior time s, and the cycle-75 swapped condDistrib orientation, producing original-orientation kernel compatibility for the weak Fokker-Planck statement without constructing the conditional law." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalUpperObligation", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToSwappedConditionalCompatibility", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle synchronization only. Endpoint laws, swapped marginal compatibility, and original-orientation kernel compatibility are aligned; standard-Borel, finite/probability, AEMeasurable, Integrable, conditional expectation, density/AC, weak FP, and KL differentiation remain explicit obligations." /-- Cycle-76 lower obligation for the endpoint Measure.map to swapped conditional-kernel wrapper. -/ def cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle76_endpoint_conditional_lower" statement := "Cycle 76 lower compiles SALD.generalMovingTargetDiscreteEndpointMeasureMapToSwappedConditionalCompatibility and SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility for appendix.tex lines 1354-1387. Under explicit endpoint a.e. interpolation identities, named rho_k/rho_{k+1}/hat rho_s Measure.map representations, a supplied swapped-joint kernel compatibility theorem for (hat X_s,X_k^eta), and a supplied bridge back to the paper's (X_k^eta,hat X_s) orientation, the wrappers return both endpoint law equalities, the named hat rho_s marginal in both swapped first-marginal and original second-marginal views, the Measure.map swap equality, and original-orientation kernel compatibility for the weak FP interface." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToSwappedConditionalCompatibility", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap", "SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap", "AutoSamplingTheory.lawMapProdSwap", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing local wrapper under supplied hypotheses only. It does not construct condDistrib, condExpKernel, regular conditional expectation, component integral fields, weak FP, density/AC, KL differentiation, or theorem closure." /-- Cycle-76 proof-DAG pane for endpoint-law to conditional compatibility. -/ def cycle76GeneralMovingTargetDiscreteEndpointConditionalDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle76.global_phase_judgment" interface := "Cycle 76 judgment: cycle 75 passed and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the largest shared risk remains endpoint-to-conditional compatibility inside sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalUpperObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle76.middle_endpoint_conditional_map" interface := "Middle source map for appendix.tex:1358-1387: keep endpoint Measure.map laws, named interior hat rho_s marginal, and swapped condDistrib orientation synchronized before weak FP proof search." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle76.lower_endpoint_to_swapped_conditional" interface := "Local proof-producing wrapper: combine endpoint Measure.map law equalities with the swapped first marginal, original second marginal, and swap bridge, yielding original-orientation kernel compatibility for the weak FP interface under supplied kernel hypotheses." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToSwappedConditionalCompatibility", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap", "SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap", "AutoSamplingTheory.lawMapProdSwap", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] /-- Cycle-77 middle obligation for the generator-level weak FP source-sign handoff. -/ def cycle77GeneralMovingTargetDiscreteWeakFpGeneratorMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle77_weak_fp_generator_middle" statement := "Cycle 77 middle keeps the active backend at sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387 and returns to appendix.tex:1379-1387 after the accepted endpoint-to-conditional cycle 76. The weak conditional Fokker--Planck source-sign interface is sharpened by splitting the supplied analytic theorem into (i) a generator/time-derivative identity for the frozen EM interpolation law on admissible tests and (ii) a source expansion of that generator as -div(hat rho_s*bar b_{k,s}) plus sigmaCoeff*Delta hat rho_s, with the lower wrapper now allowing the source expansion itself to be split into separate drift and diffusion actions under explicit common-space, regular conditional kernel, drift regularity, density/time-regularity, admissible-test, and boundary hypotheses." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle synchronization only. It records the cycle-77 global judgment: cycle 76 passed and needs no recovery; Phase 1 skeleton translation is stable enough for cited-theory backfill; the best risk-reducing packet is the generator-level weak FP source-sign handoff, not theorem-route audit, KL derivative work, display algebra, or broad API migration." /-- Cycle-77 lower obligation for the generator-level weak FP source-sign wrapper. -/ def cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle77_weak_fp_generator_lower" statement := "Cycle 77 compiles SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff and SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff for appendix.tex lines 1379-1387: if the frozen EM interpolation generator represents partial_s hat rho_s on admissible weak tests under explicit common-space, regular conditional kernel, drift regularity, density/time-regularity, test-regularity, and boundary hypotheses, if that generator has the paper source expansion or separate drift/diffusion source actions, and if sigmaCoeff=sigma_eta^2/2, then the resulting weak FP statement has exactly the source signs -div(hat rho_s*bar b_{k,s}) and +(sigma_eta^2/2)*Delta hat rho_s." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorMiddleObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.cycle72GeneralMovingTargetDiscreteWeakFpLowerObligation", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing wrapper only. The actual generator theorem, Brownian/EM construction, regular conditional law, density/absolute-continuity, admissible-test approximation, boundary/integration-by-parts, KL differentiation, LSI/KL/FI, DV, Gronwall, and theorem closure remain obligations." /-- Cycle-77 proof-DAG pane for the generator-level weak conditional Fokker--Planck source-sign handoff. -/ def cycle77GeneralMovingTargetDiscreteWeakFpGeneratorDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle77.global_phase_judgment" interface := "Cycle 77 judgment: cycle 76 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the largest remaining proof risk is still sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, now narrowed back to the weak conditional FP source-sign theorem at appendix.tex:1379-1387." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle77.middle_weak_fp_generator_source_signs" interface := "Middle conversion window for appendix.tex:1379-1387: split the associated Fokker--Planck invocation into a generator/time-derivative identity and a source expansion with negative drift-divergence and positive sigma_eta^2/2 Laplacian, with the lower path able to split that expansion into separate drift and diffusion actions while keeping conditional-law, density, test, and boundary hypotheses explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorMiddleObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle77.lower_weak_fp_generator_handoff" interface := "Local proof-producing wrapper: compose a supplied generator/time-derivative identity, a supplied generator source expansion or split drift/diffusion source actions, and sigmaCoeff=sigma_eta^2/2 into the source-signed weak FP identity on admissible tests." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] /-- Cycle-78 upper packet for the generator-to-KL derivative handoff. Cycle 77 sharpened the weak conditional Fokker--Planck source signs down to generator pieces. Cycle 78 keeps the same EM backend and asks lower work to connect those supplied generator pieces directly to the discrete KL derivative handoff at the log-density-ratio test. -/ def cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorUpperPacket : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource objective := "Global phase judgment: cycle 77 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet that best reduces the remaining proof risk is the KL-derivative handoff from the cycle-77 generator-level weak conditional Fokker-Planck source signs, still inside sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387." sourceLabels := [ "eq:general_KL_derivative_0_discrete", "eq:general_KL_derivative_1_discrete", "proof:thm:general-moving-target-SALD-discrete:weak-conditional-fp", "proof:thm:general-moving-target-SALD-discrete:weak-fp-to-kl-derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Gronwall remains unchanged as endpoint-safe calculus and theorem-specific coefficient/display obligations; cycle 78 does not touch lem:gronwall.", "DV remains source-cited through lem:dv_variation with common-space, absolute-continuity, finite-KL, and finite-log-mgf witnesses; cycle 78 is still before the residual DV step.", "LSI/KL/FI remains the density-test/Fisher-chain obligation; cycle 78 only prepares the derivative handoff that later feeds the Fisher-information identification.", "Continuous Fokker-Planck/KL derivative remains a separate continuous theorem backend and is not modified by this EM packet.", "EM interpolation FP remains the active backend: cycles 70-77 exposed conditional-law, endpoint compatibility, weak FP signs, KL substitution, conditional-kernel measure interfaces, endpoint-to-conditional packaging, and generator source signs; cycle 78 connects the generator pieces to the log-ratio KL derivative substitution." ] theoremRoute := [ "1. Start from appendix.tex:1358-1366, the differentiated KL display and mass-conservation drop.", "2. Use appendix.tex:1379-1387 through the cycle-77 supplied generator/time-derivative, generator split, and drift/diffusion source-action hypotheses.", "3. Select phi_s=log(hat rho_s/tilde pi_s) as the weak test only through an explicit admissibility hypothesis.", "4. Compose the generator-piece weak FP source signs through the normalized source-sign-to-KL wrapper, preserving the negative drift-divergence action, positive sigma_eta^2/2 Laplacian action, and unchanged target-time term.", "5. Leave integration by parts, Laplacian split, FI identification, LSI, DV, time change, Gronwall, and theorem closure as separate obligations." ] modeDiscipline := [ "faithfulPaper Phase 1: use only the original appendix.tex source window 1358-1387; sald_version_2.tex remains excluded.", "Do not change theorem statements, constants, sigma_eta^2/2 sign conventions, source labels, or proof order.", "Keep generator theorem, regular conditional law, log-ratio admissibility, density/absolute-continuity, finite KL/FI, boundary, integration-by-parts, and weak FP assumptions as explicit obligations, not hidden theorem hypotheses." ] nonGoals := [ "No theorem-route audit, display algebra, Gronwall coefficient work, DV witness work, LSI/KL/FI proof work, or broad SLT/SDE API backfill.", "No claim that a regular conditional law, weak Fokker-Planck theorem, generator theorem, log-ratio admissibility theorem, or KL differentiation theorem is formalized.", "No promotion of thm:forward-KL-discrete, thm:general-moving-target-SALD-discrete, LSI/KL/FI, DV, Gronwall, or EM interpolation status." ] lowerPacket := [ "Target exactly SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces / SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation / sald.general_moving_target_discrete.cycle78_kl_derivative_generator_lower.", "Lower proof-producing scope: compile only the scalar composition from the differentiated KL formula plus the cycle-77 supplied generator/time-derivative, generator split, drift source action, diffusion source action, and sigmaCoeff=sigma_eta^2/2 hypotheses at the admissible log-ratio test.", "Middle synchronization scope: update the conversion window and proof-obligation ledger so cycle 77 generator pieces and cycle 73 KL substitution are visible as one ordered source route over appendix.tex:1358-1387." ] reviewerChecklist := [ "The active lower packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387.", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces is a local equality-composition wrapper only and does not prove the generator theorem, weak FP theorem, or KL derivative backend.", "SALD.discreteSaldContract and SALD.generalVaSaldDiscreteContract remain contractOnly and list the cycle-78 obligations.", "Gronwall, DV, LSI/KL/FI, continuous FP/KL derivative, EM interpolation, theorem statuses, and SLT reuse statuses remain below formalized.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] status := ProofStatus.obligation /-- Cycle-78 upper obligation for the generator-to-KL handoff packet. -/ def cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorUpperObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle78_kl_derivative_generator_upper" statement := "Cycle 78 upper records that cycle 77 passed reviewer/build and needs no recovery, confirms that Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill, and keeps the active packet on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387. The selected lower work is the KL-derivative handoff from the cycle-77 generator-level weak FP source signs: compose supplied generator/time-derivative, generator split, drift source action, diffusion source action, and sigmaCoeff=sigma_eta^2/2 hypotheses with eq:general_KL_derivative_0_discrete at the admissible log-ratio test." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorUpperPacket", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Upper workflow obligation only. It fixes the lower packet and status discipline; it adds no theorem assumptions and proves no analytic backend." /-- Cycle-78 middle obligation for the source-to-Lean KL handoff map. -/ def cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle78_kl_derivative_generator_middle" statement := "Cycle 78 middle maps appendix.tex:1358-1387 to the existing Lean route without changing the paper statement: appendix.tex:1358-1366 supplies the differentiated KL display dK=partialS(logRatioTest)-targetTimeTerm after the mass-conservation term is dropped; appendix.tex:1379-1387 supplies the weak conditional Fokker-Planck source signs through the cycle-77 generator/time-derivative, generator split, and separated drift/diffusion source-action hypotheses; the log-density-ratio test enters only by an explicit admissibility hypothesis; SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns is the normalized weak-FP-to-KL substitution, and the resulting generator-piece handoff is exactly SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces before any integration by parts, Laplacian split, FI identification, LSI, DV, Gronwall, or theorem closure." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorUpperObligation", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Middle synchronization only. It records the ordered source route from KL differentiation through the supplied generator-piece weak FP signs, and keeps density/AC, log-ratio admissibility, generator/weak-FP theorem, mass conservation, integration by parts, FI, LSI, DV, Gronwall, and theorem statuses below formalized." /-- Cycle-78 local wrapper obligation for generator-piece weak-FP to KL derivative substitution. -/ def cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle78_kl_derivative_generator_lower" statement := "Cycle 78 compiles SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns and SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces for appendix.tex:1358-1387: if the differentiated KL formula is supplied as dK=partialS(logRatioTest)-targetTimeTerm, the log-ratio test is admissible, the cycle-77 generator/time-derivative identity and split drift/diffusion source-action hypotheses supply the weak FP source signs on admissible tests, and sigmaCoeff=sigma_eta^2/2, then the KL derivative display preserves the negative drift-divergence action, positive sigma_eta^2/2 Laplacian action, and unchanged target-time term." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorUpperObligation", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorMiddleObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Proof-producing equality composition plus obligation only. It does not prove the EM generator theorem, weak FP theorem, admissible log-ratio test, density/AC, KL differentiation, mass conservation, integration by parts, Laplacian split, FI identification, LSI/KL/FI, DV, Gronwall, or theorem closure." /-- Cycle-78 proof-DAG pane for the generator-piece weak-FP to KL derivative handoff. -/ def cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle78.global_phase_judgment" interface := "Cycle 78 judgment: cycle 77 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the largest remaining proof risk is still sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, now narrowed to connecting cycle-77 generator-level weak FP source signs to the discrete KL derivative handoff." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle78.middle_kl_derivative_generator_source_map" interface := "Middle conversion window for appendix.tex:1358-1387: align eq:general_KL_derivative_0_discrete with the cycle-77 generator-piece weak FP source signs at the admissible log-ratio test, so the lower wrappers are only the source-ordered equality substitution before integration by parts, Laplacian splitting, and FI identification." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorMiddleObligation", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorUpperObligation", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle78.lower_kl_derivative_generator_handoff" interface := "Local proof-producing wrappers: first substitute normalized admissible weak-FP source signs into the differentiated KL formula, then compose that handoff with supplied generator/time-derivative, generator split, drift source action, diffusion source action, and sigmaCoeff=sigma_eta^2/2 hypotheses at the admissible log-ratio test." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorMiddleObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] /-- Cycle-79 source-cited measure/calculus interface for the weak generator time-derivative theorem behind the EM interpolation Fokker-Planck line. Cycles 77 and 78 compiled the equality packaging after a generator identity is supplied. This interface names the remaining theorem boundary: turn the frozen EM interpolation law into a weak time derivative on admissible tests. -/ def cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface : ProofObligation where id := "sald.general_moving_target_discrete.cycle79_weak_fp_generator_measure_interface" statement := "Cycle 79 records the narrow source-cited Mathlib/measure-calculus interface still blocking sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, especially lines 1379-1387. For the frozen EM interpolation law hat rho_s=Law(hat X_s), an admissible weak test phi, the conditional drift bar b_{k,s}, and diffusion coefficient sigma_eta^2/2, a future local theorem must justify differentiating the Measure.map law/test integral in s and identifying the derivative with the generator action that has negative drift-divergence and positive diffusion signs. This is the missing weak generator-to-law theorem consumed by the cycle-77 and cycle-78 supplied-hypothesis wrappers." source := saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource status := ProofStatus.sourceCited dependsOn := [ "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus", "Mathlib.MeasureTheory.Measure.Map", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "AutoSamplingTheory.lawMapIntegral", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Source-cited interface only. It does not import a new Mathlib module, construct the EM process, prove differentiability of the law, prove Ito/generator calculus, prove density/AC, prove boundary integration by parts, prove log-ratio admissibility, or close the weak Fokker-Planck/KL derivative backend." /-- Cycle-79 upper packet for the minimal cited weak-FP generator interface. -/ def cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureUpperPacket : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteWeakFpSource objective := "Global phase judgment: cycle 78 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; proof-producing equality wrappers have reached the supplied-hypothesis boundary, so the single lower packet that now reduces the largest proof risk is the narrow source-cited weak generator-to-law time-derivative interface for sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387." sourceLabels := [ "proof:thm:general-moving-target-SALD-discrete:weak-conditional-fp", "proof:thm:general-moving-target-SALD-discrete:weak-fp-to-kl-derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.cycle79_weak_fp_generator_measure_interface", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Gronwall, DV, LSI/KL/FI, and continuous Fokker-Planck/KL derivative interfaces remain unchanged and below formalized status.", "Cycles 70-76 exposed the conditional-law, endpoint, and kernel-orientation layers; cycle 74 remains the source-cited conditional-kernel measure interface.", "Cycles 77-78 compile only supplied-hypothesis wrappers from generator pieces to weak FP source signs and then to the KL derivative display.", "Cycle 79 names the remaining blocked theorem: differentiating the EM interpolation Measure.map law/test integral and identifying the generator action before any integration by parts, FI identification, LSI, DV, or Gronwall step." ] theoremRoute := [ "1. Stay inside appendix.tex:1358-1387 and keep the active backend sald.general_moving_target_discrete.em_interpolation_fp.", "2. Use appendix.tex:1379-1387 as the selected lower slice: the paper invokes the Fokker-Planck equation associated with eq:general_moving_target_SALD_frozen_interp.", "3. Expose the missing measure/calculus theorem as a source-cited interface: for admissible weak tests, differentiate the law/test integral for hat rho_s and identify the frozen EM generator with drift bar b_{k,s} and covariance sigma_eta^2.", "4. Leave all later KL substitution, Laplacian split, FI, LSI, DV, and Gronwall work as consumers of this interface, not as replacements for it." ] modeDiscipline := [ "faithfulPaper Phase 1 only: preserve the original appendix.tex source route and keep sald_version_2.tex excluded.", "Do not add the law differentiability, generator, density, boundary, or admissible-test assumptions to theorem statements; record them in this interface.", "Do not claim a Mathlib, SLT, or local SDE theorem is formalized unless a local ASTIS declaration compiles under this toolchain." ] nonGoals := [ "No theorem-route audit, display algebra, Gronwall/DV/LSI work, frozen-delta work, broad reusable API design, or project-article export.", "No new Lake dependency and no SLT import or formalization claim.", "No promotion of the EM interpolation FP, weak FP theorem, KL derivative backend, or theorem contracts." ] lowerPacket := [ "Target exactly SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface / sald.general_moving_target_discrete.cycle79_weak_fp_generator_measure_interface.", "Middle should synchronize appendix.tex:1379-1387 with the already named cycle-77 generator/time-derivative and generator-piece hypotheses, and make clear that cycle 79 records the theorem boundary before those wrappers.", "Lower should either find a tiny local measure/calculus wrapper around Measure.map integrals and parametric integral differentiation under explicit supplied hypotheses, or keep this sourceCited interface as the blocker; do not switch to theorem-route, display, DV, LSI, or Gronwall work." ] reviewerChecklist := [ "The active lower packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to appendix.tex:1379-1387.", "The new weak generator-to-law interface is ProofStatus.sourceCited and does not claim that Mathlib contains a full SDE/Fokker-Planck theorem.", "Cycle 77 and 78 wrappers remain local supplied-hypothesis packaging only.", "Both discrete theorem contracts remain contractOnly, and all slow analytic backends remain obligation or sourceCited.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation /-- Cycle-79 upper obligation for the weak generator-to-law cited interface. -/ def cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureUpperObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle79_weak_fp_generator_measure_upper" statement := "Cycle 79 upper records that cycle 78 passed and needs no recovery, confirms Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill, and keeps the active packet on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387. Because cycle 77 and 78 already compiled the available supplied-hypothesis wrappers, the selected lower packet is the source-cited weak generator-to-law time-derivative interface for appendix.tex:1379-1387." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureUpperPacket", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Upper workflow obligation only. It fixes the lower packet and status discipline; it adds no theorem assumptions, imports no new Mathlib module, and proves no analytic backend." /-- Cycle-79 middle obligation for the weak generator-to-law source map. -/ def cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle79_weak_fp_generator_measure_middle" statement := "Cycle 79 middle maps appendix.tex:1379-1387 to the narrow source-cited weak generator-to-law interface without changing the paper route. The source first names hat rho_s=Law(hat X_s) and the conditional drift bar b_{k,s}; the Fokker-Planck invocation is translated as the missing theorem that differentiates the frozen EM Measure.map law/test integral for admissible weak tests and identifies the generator action before the cycle-77 source-sign wrappers and cycle-78 KL handoff consume it. The mapping keeps regular conditional laws, drift measurability/integrability, density/AC, boundary conditions, admissible-test approximation, generator calculus, weak Fokker-Planck, KL differentiation, integration by parts, LSI, DV, Gronwall, and theorem closure as explicit non-formalized obligations." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureUpperObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] note := "Middle synchronization only. It records the source-to-Lean boundary for the cited measure/calculus theorem and does not promote Mathlib, SLT, EM weak FP, KL derivative, or theorem status." /-- Cycle-79 lower obligation for the local Measure.map weak-test handoff. -/ def cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle79_weak_fp_generator_measure_lower" statement := "Cycle 79 lower compiles the narrow local Measure.map handoff used before the missing weak generator theorem: AutoSamplingTheory.lawMapIntegral rewrites an admissible weak-test integral against Law(hat X_s)=Measure.map (hat X_s) P as the corresponding sample-space integral, and AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample transports a supplied sample-space HasDerivAt result to the mapped-law weak-test integral. This reduces the measure-calculus boundary for appendix.tex:1379-1387 but still leaves the actual EM generator derivative, conditional drift construction, density/AC, boundary conditions, source-sign weak Fokker-Planck theorem, and KL derivative backend as obligations or source-cited interfaces." source := saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource status := ProofStatus.formalized dependsOn := [ "AutoSamplingTheory.lawMapIntegral", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureMiddleObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Formalized local Measure.map/Bochner integral derivative transport under explicit supplied hypotheses only. It does not prove parametric differentiation of the EM process, Ito/generator calculus, weak Fokker-Planck source signs, log-ratio admissibility, or any theorem-level SALD backend." /-- Cycle-79 proof-DAG pane for the weak generator-to-law cited interface. -/ def cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle79.global_phase_judgment" interface := "Cycle 79 judgment: cycle 78 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the largest remaining proof risk is still sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, now narrowed to the weak generator-to-law time-derivative theorem behind appendix.tex:1379-1387." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureUpperObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle79.middle_weak_fp_generator_measure_source_map" interface := "Middle conversion map for appendix.tex:1379-1387: align the paper's associated Fokker-Planck invocation with the source-cited theorem boundary for differentiating the frozen EM Measure.map law/test integral and identifying the generator action before cycle-77 source signs and cycle-78 KL substitution." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureMiddleObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureUpperObligation", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle79.lower_packet.weak_fp_generator_measure_interface" interface := "Minimal cited measure/calculus interface: audit Mathlib Measure.map, Bochner integral, and parametric-integral tools as prerequisites for differentiating the frozen EM interpolation law/test integral and identifying the weak generator action before the cycle-77 and cycle-78 wrappers consume it." source := saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.sourceCited }, { id := "ASTIS.SALD.cycle79.lower_measure_map_integral_handoff" interface := "Local proof-producing Measure.map handoff: rewrite weak-test integrals against the frozen EM mapped law to sample-space integrals, and transport a supplied sample-space HasDerivAt statement back to the mapped-law weak-test integral." source := saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource targetLean := "AutoSamplingTheory/Probability.lean" dependsOn := [ "AutoSamplingTheory.lawMapIntegral", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureLowerObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized } ] /-- Cycle-80 upper packet returning to the conditional-law/measurability layer. Cycle 79 exposed the weak generator-to-law theorem boundary, but that theorem still depends on the conditional law and named drift field from `appendix.tex:1368-1377`. Cycle 80 therefore selects the first preferred lower packet again, not another theorem-route audit or weak-FP algebra slice. -/ def cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityUpperPacket : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteDerivativeSource objective := "Global phase judgment: cycle 79 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet that now reduces the largest remaining proof risk is the conditional-law/measurability and named conditional drift interface inside sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, specifically appendix.tex:1368-1377." sourceLabels := [ "proof:thm:general-moving-target-SALD-discrete:conditional-drift", "proof:thm:general-moving-target-SALD-discrete:weak-conditional-fp", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.cycle74_conditional_kernel_measure_interface", "sald.general_moving_target_discrete.cycle75_conditional_law_middle", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Cycle 79 did not close the EM weak FP theorem; it exposed a generator-to-law boundary that still consumes the conditional kernel and measurable bar b_{k,s} field.", "Cycles 70, 74, and 75 already identify the relevant interface: regular conditional law of X_k^eta given hat X_s=x, named hat rho_s marginal, component conditional drift integrals, and measurability/integrability of their linear combination.", "Cycle 80 lower should sharpen that existing conditional-law interface, preferably by isolating the precise Mathlib condDistrib/condExpKernel or vector-valued conditional-integral theorem still missing.", "Weak FP source signs, KL derivative substitution, LSI/KL/FI, DV, Gronwall, and theorem closure remain consumers of this interface and are not promoted." ] theoremRoute := [ "1. Stay inside appendix.tex:1358-1387 with active slice appendix.tex:1368-1377.", "2. Preserve the paper definition bar b_{k,s}(x)=E[dot t_k*c_{t_k}(X_k^eta)+(sigma_eta^2/2)*nabla log pi_{t_k}(X_k^eta) | hat X_s=x].", "3. Reuse cycle-74 and cycle-75 source-cited Mathlib kernel/orientation interfaces rather than inventing a new broad SDE API.", "4. If proof-producing work is possible, compile only a supplied-hypothesis wrapper for kernel compatibility, component conditional-integral fields, and measurability/integrability of the named drift.", "5. If blocked, record the exact missing conditional-law theorem below formalized status with standard-Borel/probability, marginal compatibility, measurability, integrability, and vector-valued conditional expectation hypotheses explicit." ] modeDiscipline := [ "faithfulPaper Phase 1 only: preserve the original appendix.tex source route and keep sald_version_2.tex excluded.", "Do not add standard-Borel, finite/probability, conditional-law, measurability, integrability, density, or admissibility hypotheses to theorem statements.", "Use lean-stat-learning-theory only as a local Mathlib style reference; do not import it or claim an SLT theorem is formalized." ] nonGoals := [ "No theorem-route audit, source-index rebaseline beyond the gate, display algebra, Gronwall/DV/LSI work, frozen-delta work, broad reusable API design, or project-article export.", "No weak conditional Fokker-Planck theorem proof, KL derivative proof, generator theorem proof, or status promotion for the EM backend.", "No Lake dependency changes, no SLT import, and no claim that Mathlib already supplies the full SALD conditional-law theorem." ] lowerPacket := [ "Target exactly sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to appendix.tex:1368-1377.", "Middle should synchronize the existing cycle-70/74/75 conditional-law rows with any lower packet: joint law, named hat rho_s marginal, condDistrib/condExpKernel orientation, component conditional drift fields, and bar b_{k,s} regularity.", "Lower should reduce the conditional-law/measurability blocker, not move to endpoint re-audit, weak FP source signs, KL derivative handoff, display algebra, Gronwall/DV/LSI, or theorem closure.", "The preferred lower product is either a compiled local wrapper under supplied kernel/integral hypotheses or a single narrowly cited missing Mathlib theorem." ] reviewerChecklist := [ "The active lower packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, specifically appendix.tex:1368-1377.", "Cycle 80 depends on the existing cycle-74/cycle-75 conditional-kernel interfaces and does not duplicate or promote them.", "Both discrete theorem contracts remain contractOnly; conditional law, weak FP, generator-to-law, KL derivative, density/AC, LSI/KL/FI, DV, and Gronwall remain sourceCited or obligation unless a local declaration compiles.", "SLT remains reference-only with no Lake dependency or import change.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation /-- Cycle-80 upper obligation for the conditional-law/measurability packet. -/ def cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityUpperObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle80_conditional_law_upper" statement := "Cycle 80 upper records that cycle 79 passed reviewer/build and needs no recovery, confirms that Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill, and keeps the active packet on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387. The selected lower packet is the conditional-law/measurability and named conditional drift interface at appendix.tex:1368-1377, because the weak generator-to-law theorem exposed in cycle 79 still depends on the regular conditional kernel, named hat rho_s marginal, component conditional-integral fields, and measurability/integrability of bar b_{k,s}." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityUpperPacket", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Upper workflow obligation only. It selects the next lower packet and status discipline; it adds no theorem assumptions, imports no SLT/Mathlib module, and proves no conditional law, weak Fokker-Planck theorem, generator theorem, KL derivative, or theorem closure." /-- Cycle-80 middle source-to-Lean map for the EM conditional-law interface. This records the exact theorem boundary still missing after cycles 74, 75, and 79: the regular conditional kernel for `X_k^eta | hat X_s=x`, the named `hat rho_s` marginal, vector-valued conditional integral fields for the two frozen drift summands, and measurability/integrability of the selected `bar b_{k,s}` field. It is a lower-ready obligation, not a disintegration or weak Fokker--Planck proof. -/ def cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle80_conditional_law_middle" statement := "Cycle 80 middle synchronizes appendix.tex lines 1368-1377 with the existing cycle-70/74/75 conditional-law interfaces after checking the SLT reference style. The lower theorem boundary is: on the common EM probability space, instantiate a regular conditional kernel for X_k^eta given hat X_s=x; identify the named marginal hat rho_s=Law(hat X_s) in the orientation required by condDistrib/condExpKernel; supply component conditional-integral fields for c_{t_k}(X_k^eta) and nabla log pi_{t_k}(X_k^eta); prove or cite their measurability and integrability; then feed SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents and SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents or its swapped-orientation variant to obtain the source field bar b_{k,s}. This keeps the weak FP and KL derivative wrappers as consumers only." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityUpperObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "AutoSamplingTheory.lawMapProdSwap", "ProbabilityTheory.compProd_map_condDistrib", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "ProbabilityTheory.condExp_ae_eq_integral_condExpKernel", "Mathlib.Probability.Kernel.Condexp", "Mathlib.Probability.Kernel.CondDistrib", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle source map only. It records the lower-ready conditional-law/measurability theorem boundary and the Mathlib candidate facts; it imports no Mathlib kernel module, adds no Lake dependency, constructs no conditional law, proves no vector-valued conditional expectation theorem, and does not promote weak FP, generator-to-law, KL derivative, density/AC, LSI/KL/FI, DV, Gronwall, or theorem contracts." /-- Cycle-80 lower obligation for the endpoint/conditional drift-regularity wrapper. -/ def cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle80_conditional_law_lower" statement := "Cycle 80 lower compiles SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff for appendix.tex lines 1368-1377, using only supplied hypotheses. The wrapper composes the endpoint-to-conditional compatibility package with component conditional-integral field regularity: from endpoint Measure.map laws, the named hat rho_s marginal in both original and swapped orientations, a supplied swapped conditional-kernel compatibility bridge, supplied component integral fields, and supplied measurability/integrability closure, it returns endpoint laws, original-orientation kernel compatibility, and measurability/integrability of the selected bar b_{k,s} field. It does not construct condDistrib, condExpKernel, conditional expectations, density/AC, weak FP, or KL differentiation." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityMiddleObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing local wrapper under explicit hypotheses only. The Mathlib regular conditional law theorem, vector-valued conditional-expectation theorem, component field construction, density/absolute-continuity, weak conditional Fokker-Planck identity, log-ratio admissibility, KL differentiation, integration by parts, and theorem closure remain obligations." /-- Cycle-80 proof-DAG pane for the conditional-law/measurability backfill. -/ def cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle80.global_phase_judgment" interface := "Cycle 80 judgment: cycle 79 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the largest remaining proof risk is the conditional-law/measurability and named drift layer inside sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityUpperObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle80.middle_conditional_law_source_map" interface := "Middle source map for appendix.tex:1368-1377: lower must instantiate or cite the regular conditional kernel for X_k^eta | hat X_s=x, align the named hat rho_s marginal with the condDistrib/condExpKernel orientation, provide component conditional-integral fields, and feed the existing named-drift regularity wrappers. Weak FP and KL derivative work remain downstream consumers." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityMiddleObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle80.lower_packet.conditional_law_measurability" interface := "Selected lower packet: refine the appendix.tex:1368-1377 conditional-law/measurability interface by connecting the joint law, named hat rho_s marginal, condDistrib/condExpKernel orientation, component conditional-integral fields, and measurable/integrable bar b_{k,s}; leave weak FP and KL derivative work as consumers." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityMiddleObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle80.lower_endpoint_conditional_drift_regularity" interface := "Local supplied-hypothesis wrapper: compose endpoint-to-conditional compatibility with component conditional-integral field regularity to return endpoint laws, named hat rho_s marginal views, original-orientation kernel compatibility, and measurable/integrable bar b_{k,s}." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized } ] /-- Cycle-81 upper packet for the endpoint-law-to-conditional-law bridge. Cycle 80 compiled only a supplied-hypothesis wrapper around endpoint/orientation and conditional-drift regularity facts. This packet keeps the next work on the same EM interpolation backend, now selecting the endpoint-law bookkeeping that has to feed the conditional-law interface used by the weak Fokker-Planck statement. -/ def cycle81GeneralMovingTargetDiscreteEndpointConditionalUpperPacket : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteDerivativeSource objective := "Global phase judgment: cycle 80 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet that best reduces the remaining risk is endpoint-law-to-conditional-law compatibility inside sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, specifically appendix.tex:1368-1377." sourceLabels := [ "proof:thm:general-moving-target-SALD-discrete:conditional-drift", "proof:thm:general-moving-target-SALD-discrete:weak-conditional-fp", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.cycle81_endpoint_conditional_upper", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Cycle 80 supplied-hypothesis drift-regularity packaging is accepted, but it did not construct a regular conditional law, condDistrib, condExpKernel, component conditional expectations, density/AC, or the weak FP theorem.", "The active endpoint issue is still the Measure.map bookkeeping that identifies Law(hat X_s) as the marginal consumed by the conditional kernel for X_k^eta | hat X_s=x.", "Cycles 71 and 76 contain local endpoint/marginal/orientation wrappers; cycles 74 and 75 contain the source-cited Mathlib conditional-kernel interfaces; cycle 81 asks lower to connect these pieces as the handoff consumed by cycle 80.", "Weak FP source signs, KL differentiation, LSI/KL/FI, DV, Gronwall, and theorem closure remain downstream consumers and are not promoted." ] theoremRoute := [ "1. Stay inside appendix.tex:1358-1387 with selected source slice appendix.tex:1368-1377.", "2. Preserve the paper definition bar b_{k,s}(x)=E[dot t_k*c_{t_k}(X_k^eta)+(sigma_eta^2/2)*nabla log pi_{t_k}(X_k^eta) | hat X_s=x].", "3. Reuse the existing endpoint law helpers SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation, SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap, SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility, and the cycle-80 supplied drift-regularity wrapper.", "4. Lower should produce either a compiled supplied-hypothesis bridge from endpoint Measure.map facts to the conditional-law interface, such as SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff, or one precise source-cited missing Mathlib conditional-law theorem; it should not advance to weak FP or KL derivative work in this packet." ] modeDiscipline := [ "faithfulPaper Phase 1 only: preserve the original appendix.tex route and keep sald_version_2.tex out of scope.", "Do not add standard-Borel, probability, marginal-compatibility, conditional-law, density, or test-regularity assumptions to theorem statements; keep them in source-cited interfaces or obligations.", "Use lean-stat-learning-theory only as a local Mathlib style reference; do not import it, change Lake dependencies, or claim an SLT theorem is formalized." ] nonGoals := [ "No broad theorem-route audit, source-index rebaseline beyond the gate, display algebra, Gronwall/DV/LSI work, frozen-delta work, broad reusable API design, or project-article export.", "No proof of the weak conditional Fokker-Planck theorem, generator theorem, KL derivative theorem, or theorem closure.", "No status promotion for the EM interpolation FP backend, conditional law construction, weak FP, KL derivative, or either discrete theorem contract." ] lowerPacket := [ "Target exactly sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to endpoint-law-to-conditional-law compatibility for appendix.tex:1368-1377.", "Middle must keep two-way synchronization among the source line, Lean declarations, proof-obligation rows, and any run handoff; it should not open an unrelated theorem-route audit.", "Lower should connect the endpoint Measure.map laws and named hat rho_s marginal to the conditional kernel orientation required by the existing conditional-law/measurability interface and cycle-80 drift-regularity handoff.", "Cycle 81 lower compiles SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff as the endpoint-only bridge once bar b_{k,s} measurability/integrability and the weak-FP prerequisite consumer are supplied.", "If blocked, record one exact Mathlib conditional-distribution or conditional-expectation theorem boundary below formalized status, with marginal compatibility, measurability, and vector-valued integrability hypotheses explicit." ] reviewerChecklist := [ "The active lower packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, specifically appendix.tex:1368-1377.", "Cycle 81 uses existing cycle-71/cycle-76 endpoint wrappers, cycle-74/cycle-75 conditional-kernel interfaces, and cycle-80 drift-regularity handoff instead of duplicating or promoting them.", "Both discrete theorem contracts remain contractOnly; conditional law, weak FP, generator-to-law, KL derivative, density/AC, LSI/KL/FI, DV, and Gronwall remain sourceCited or obligation unless a local declaration compiles.", "SLT remains reference-only with no Lake dependency or import change.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility", "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation /-- Cycle-81 upper obligation for the endpoint-to-conditional packet. -/ def cycle81GeneralMovingTargetDiscreteEndpointConditionalUpperObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle81_endpoint_conditional_upper" statement := "Cycle 81 upper records that cycle 80 passed reviewer/build and needs no recovery, confirms that Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill, and keeps the active lower packet on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387. The selected lower packet is endpoint-law-to-conditional-law compatibility at appendix.tex:1368-1377: connect endpoint Measure.map laws and the named hat rho_s marginal to the conditional-kernel orientation used by the conditional-law/measurability interface and the cycle-80 supplied drift-regularity handoff." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalUpperPacket", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility", "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Upper workflow obligation only. It selects the endpoint-to-conditional lower packet and status discipline; it proves no conditional law, condDistrib, condExpKernel, weak Fokker-Planck theorem, generator theorem, KL derivative, or theorem closure." /-- Cycle-81 middle obligation for the endpoint-to-conditional weak-FP readiness handoff. -/ def cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle81_endpoint_conditional_middle" statement := "Cycle 81 middle compiles SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff for appendix.tex lines 1368-1377. The wrapper composes the existing endpoint Measure.map laws, named hat rho_s marginal in original and swapped orientations, swap equality, original-orientation kernel compatibility, and cycle-80 bar b_{k,s} measurability/integrability handoff into an abstract WeakFpPrereq predicate consumed by the weak conditional Fokker-Planck interface. It does not construct condDistrib, condExpKernel, conditional expectations, density/AC, the weak FP theorem, KL differentiation, or theorem closure." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalUpperObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle proof-producing wrapper under explicit hypotheses only. The weak-FP prerequisite predicate is supplied, so all analytic conditional-law, density, generator, weak-FP, and KL-derivative content remains below formalized status." /-- Cycle-81 lower obligation for the endpoint-only weak-FP prerequisite handoff. -/ def cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle81_endpoint_conditional_lower" statement := "Cycle 81 lower compiles SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff for appendix.tex lines 1368-1377. This endpoint-only wrapper composes the cycle-76 endpoint Measure.map compatibility package with already-supplied bar b_{k,s} measurability/integrability and a supplied WeakFpPrereq consumer. It returns endpoint laws, original/swapped hat rho_s marginal views, swap equality, original-orientation kernel compatibility, bar b_{k,s} regularity, and the WeakFpPrereq without reconstructing conditional component fields or proving the weak Fokker-Planck theorem." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Lower proof-producing endpoint wrapper only. It uses supplied barB regularity and supplied WeakFpPrereq; regular conditional law, conditional expectation, density/AC, generator-to-law, weak FP, KL derivative, and theorem closure remain obligations." /-- Cycle-81 proof-DAG pane for the endpoint-to-conditional upper packet. -/ def cycle81GeneralMovingTargetDiscreteEndpointConditionalDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle81.global_phase_judgment" interface := "Cycle 81 judgment: cycle 80 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the largest remaining proof risk is endpoint-law-to-conditional-law compatibility inside sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalUpperObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle81.lower_packet.endpoint_conditional_compatibility" interface := "Selected lower packet: connect endpoint Measure.map laws and the named hat rho_s marginal to the conditional-kernel orientation required by the existing conditional-law/measurability interface and cycle-80 drift-regularity handoff; leave weak FP and KL derivative work downstream." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalUpperObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility", "SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle81.lower_endpoint_measure_map_weak_fp_prereq" interface := "Lower handoff: compose the endpoint Measure.map compatibility package with supplied bar b_{k,s} measurability/integrability and a supplied WeakFpPrereq consumer, without rebuilding component conditional expectations." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle81.middle_endpoint_conditional_weak_fp_readiness" interface := "Middle handoff: package endpoint Measure.map laws, named hat rho_s marginal views, swapped/original orientation compatibility, original kernel compatibility, and bar b_{k,s} regularity into a supplied WeakFpPrereq predicate for the later weak conditional Fokker-Planck theorem." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized } ] /-- Cycle-82 upper packet for the weak conditional Fokker--Planck source-sign backend. Cycle 81 supplied the endpoint/conditional readiness package consumed before weak FP. This upper packet returns to the paper's associated FP line and keeps the next work on the source signs themselves. -/ def cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsUpperPacket : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteWeakFpSource objective := "Global phase judgment: cycle 81 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet that best reduces the remaining proof risk is the weak conditional Fokker-Planck source-sign statement inside sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, specifically appendix.tex:1379-1387 after the cycle-81 endpoint/conditional WeakFpPrereq readiness package." sourceLabels := [ "proof:thm:general-moving-target-SALD-discrete:weak-conditional-fp", "proof:thm:general-moving-target-SALD-discrete:weak-fp-to-kl-derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.cycle82_weak_fp_source_signs_upper", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Cycle 81 accepted only endpoint Measure.map and bar b_{k,s} readiness for a supplied WeakFpPrereq consumer; it did not prove the weak FP theorem, density/AC, generator theorem, or KL derivative.", "The active source line is appendix.tex:1379-1387: partial_s hat rho_s = -div(hat rho_s*bar b_{k,s}) + (sigma_eta^2/2)*Delta hat rho_s.", "Existing cycle-72 and cycle-77 wrappers preserve the negative drift-divergence sign and positive sigma_eta^2/2 Laplacian coefficient once the analytic weak-test or generator identity is supplied.", "Cycle 82 asks middle/lower to connect the cycle-81 readiness boundary to that supplied weak-test source-sign theorem with all hypotheses explicit, not to broaden the theorem route." ] theoremRoute := [ "1. Stay on the fixed interval k, s in [s_k,s_{k+1}] inside appendix.tex:1358-1387.", "2. Use cycle-81 endpoint/conditional readiness for the named law hat rho_s, kernel orientation, and bar b_{k,s} measurability/integrability.", "3. State the weak conditional FP source-sign theorem over an admissible test class: partialS phi = -(driftDiv phi) + (sigma_eta^2/2)*laplacian phi.", "4. If using generator pieces, keep the supplied hypotheses separated as generator/time-derivative, generator split, drift source action, diffusion source action, boundary behavior, and sigmaCoeff=sigma_eta^2/2.", "5. Leave log-ratio admissibility, KL differentiation, Laplacian split, integration by parts, FI identification, LSI/KL/FI, DV, Gronwall, and theorem closure downstream." ] modeDiscipline := [ "faithfulPaper Phase 1 only: preserve appendix.tex:1379-1387 and keep sald_version_2.tex out of scope.", "Do not add conditional-law, density, admissible-test, boundary, finite-FI, or generator assumptions to theorem statements; keep them in source-cited interfaces or obligations.", "Use lean-stat-learning-theory only as a local Mathlib style reference; do not import it, change Lake dependencies, or claim an SLT theorem is formalized." ] nonGoals := [ "No theorem-route audit, display algebra, source-index rebaseline beyond the gate, KL-derivative proof, Gronwall/DV/LSI work, frozen-delta work, broad reusable API design, or project-article export.", "No proof of the full EM/Brownian construction, regular conditional law, density/absolute-continuity, generator theorem, weak FP theorem, or integration-by-parts theorem unless it is compiled locally in this exact backend.", "No status promotion for sald.general_moving_target_discrete.em_interpolation_fp, sald.discrete_forward_kl.em_interpolation_fp, either discrete theorem contract, SLT reuse, or Lake dependencies." ] lowerPacket := [ "Target exactly sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to appendix.tex:1379-1387 weak FP source signs.", "Middle compiles SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfReadinessAndGeneratorPiecesHandoff, translating the cycle-81 WeakFpPrereq readiness output into the lower-ready hypotheses consumed by SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff.", "Lower should compile only a supplied-hypothesis bridge from WeakFpPrereq, generator/time-derivative, drift source action, diffusion source action, and sigmaCoeff=sigma_eta^2/2 to the normalized source signs, if feasible.", "If blocked, record one exact missing Mathlib/SDE theorem boundary for the weak generator-to-law statement, with common-space, conditional kernel, density/time regularity, admissible tests, boundary behavior, and coefficient hypotheses explicit." ] reviewerChecklist := [ "The active packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, specifically appendix.tex:1379-1387.", "The drift sign remains negative -div(hat rho_s*bar b_{k,s}) and the diffusion sign remains positive +(sigma_eta^2/2)*Delta hat rho_s.", "Cycle 82 reuses cycle-81 readiness, cycle-72 source-sign wrappers, cycle-77 generator-piece wrappers, and cycle-79 source-cited measure interface instead of duplicating or promoting them.", "Both discrete theorem contracts remain contractOnly; conditional law, weak FP, generator-to-law, KL derivative, density/AC, LSI/KL/FI, DV, and Gronwall remain sourceCited or obligation unless a local declaration compiles.", "SLT remains reference-only with no Lake dependency or import change, and python3 tools/astis.py source-index ASTIS-SALD-001 plus python3 tools/astis.py check pass." ] dependencies := [ "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff", "SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfReadinessAndGeneratorPiecesHandoff", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation", "SALD.cycle72GeneralMovingTargetDiscreteWeakFpLowerObligation", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation /-- Cycle-82 upper obligation selecting the weak conditional FP source-sign packet after the endpoint/conditional readiness work. -/ def cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsUpperObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle82_weak_fp_source_signs_upper" statement := "Cycle 82 upper records that cycle 81 passed reviewer/build and needs no recovery, confirms that Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill, and keeps the active packet on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387. The selected lower packet is the weak conditional Fokker-Planck source-sign statement at appendix.tex:1379-1387: after the cycle-81 endpoint/conditional WeakFpPrereq readiness package, sharpen or prove a supplied-hypothesis bridge whose weak-test form has the exact paper signs partial_s hat rho_s = -div(hat rho_s*bar b_{k,s}) + (sigma_eta^2/2)*Delta hat rho_s under explicit common-space, conditional-kernel, bar b regularity, density/time-regularity, admissible-test, boundary, generator/time-derivative, and drift/diffusion source-action hypotheses." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsUpperPacket", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff", "SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Upper workflow obligation only. It selects the weak-FP source-sign packet and status discipline; it proves no conditional law, density/AC, generator theorem, weak Fokker-Planck theorem, KL derivative, or theorem closure." /-- Cycle-82 middle obligation for the readiness-to-source-sign bridge. -/ def cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle82_weak_fp_source_signs_middle" statement := "Cycle 82 middle compiles SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfReadinessAndGeneratorPiecesHandoff for appendix.tex lines 1379-1387. The wrapper consumes the cycle-81 WeakFpPrereq readiness output for the named interpolation law hat rho_s, kernel, and bar b_{k,s}, derives the common-space, conditional-kernel, and drift-regularity hypotheses required by the cycle-77 generator-piece source-sign handoff, and keeps density/time regularity, admissible tests, boundary behavior, generator/time derivative, drift source action, diffusion source action, and sigmaCoeff=sigma_eta^2/2 as explicit supplied hypotheses. It preserves the exact source signs partialS phi = -(driftDiv phi) + (sigma_eta^2/2)*laplacian phi and proves no analytic weak-FP theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsUpperObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfReadinessAndGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff", "SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle proof-producing local wrapper under supplied hypotheses only. It does not construct conditional kernels, density/AC, generator-to-law calculus, weak Fokker-Planck, KL differentiation, or theorem closure, and it does not promote SLT or Lake dependencies." /-- Cycle-82 lower obligation for the endpoint-readiness-to-source-sign bridge. -/ def cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle82_weak_fp_source_signs_lower" statement := "Cycle 82 lower compiles SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff for appendix.tex lines 1379-1387. The wrapper first uses the cycle-81 endpoint/conditional readiness package to build WeakFpPrereq for the named interpolation law hat rho_s, kernel, and bar b_{k,s}; then it composes the cycle-82 readiness-to-generator-piece bridge with supplied density/time regularity, admissible tests, boundary behavior, generator/time derivative, drift source action, diffusion source action, and fpCoeff=sigma_eta^2/2 to obtain the source-signed weak form partialS phi = -(driftDiv phi) + (sigma_eta^2/2)*laplacian phi. It proves no analytic weak-FP theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfReadinessAndGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff", "SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Lower proof-producing wrapper under explicit supplied hypotheses only. It composes existing local wrappers and preserves the paper signs; regular conditional laws, density/AC, generator-to-law calculus, weak Fokker-Planck, KL differentiation, theorem status, SLT imports, and Lake dependencies remain unpromoted." /-- Cycle-82 proof-DAG pane for the weak conditional FP source-sign packet. -/ def cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle82.global_phase_judgment" interface := "Cycle 82 judgment: cycle 81 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the largest remaining proof risk is still sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, now narrowed to appendix.tex:1379-1387 weak conditional FP source signs after endpoint/conditional readiness." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsUpperObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle82.middle_readiness_to_source_signs" interface := "Middle readiness-to-source-sign bridge: consume cycle-81 WeakFpPrereq readiness for hat rho_s, kernel, and bar b_{k,s}; expose common-space, conditional-kernel, and drift-regularity hypotheses to the cycle-77 generator-piece wrapper; keep density, tests, boundary behavior, generator/time derivative, and drift/diffusion source actions supplied explicitly." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfReadinessAndGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff", "SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle82.lower_packet.weak_fp_source_signs" interface := "Selected lower packet: compose the cycle-81 endpoint/conditional readiness data with the weak-test/generator-piece source-sign interface, preserving the negative drift-divergence term and positive sigma_eta^2/2 Laplacian term under explicit hypotheses." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsUpperPacket", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfReadinessAndGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff", "SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle82.reviewer_weak_fp_source_signs_check" interface := "Reviewer check: reject theorem-route drift, sign changes, hidden assumptions, SLT/Lake promotions, or any claim that weak FP, density/AC, generator-to-law, KL derivative, or theorem closure is formalized without a local build." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsUpperObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfReadinessAndGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] /-- Cycle-83 upper packet for the endpoint weak-FP to KL-derivative handoff. Cycle 82 accepted the endpoint/conditional source-sign wrapper. This packet connects that accepted weak-FP source-sign output to the differentiated KL display, still within the single EM interpolation backend. -/ def cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpUpperPacket : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource objective := "Global phase judgment: cycle 82 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet that best reduces the remaining proof risk is the KL-derivative handoff from the cycle-82 endpoint/conditional weak-FP source signs, still inside sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387." sourceLabels := [ "eq:general_KL_derivative_0_discrete", "proof:thm:general-moving-target-SALD-discrete:weak-conditional-fp", "proof:thm:general-moving-target-SALD-discrete:weak-fp-to-kl-derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.cycle83_kl_derivative_endpoint_weak_fp_upper", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Cycle 82 supplies only a local endpoint/conditional wrapper for the weak-test source signs; it does not prove the analytic weak Fokker-Planck theorem.", "Cycle 83 starts at appendix.tex:1358-1366, the differentiated KL display dK=partialS(logRatioTest)-targetTimeTerm after the mass-conservation drop.", "The active weak-FP input is appendix.tex:1379-1387 with source signs partialS phi = -(driftDiv phi) + (sigma_eta^2/2)*laplacian phi.", "The new local wrapper composes SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff with SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns.", "The lower companion SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction keeps the log-ratio weak-FP action and the dK display paired under the same admissibility hypothesis.", "Gronwall, DV, LSI/KL/FI, continuous KL derivative, density/AC, log-ratio admissibility, integration by parts, FI identification, theorem closure, SLT reuse, and Lake dependencies remain below formalized." ] theoremRoute := [ "1. Stay on a fixed EM interval k with s in [s_k,s_{k+1}] inside appendix.tex:1358-1387.", "2. Use cycle-81/cycle-82 endpoint and conditional-readiness hypotheses only to obtain the normalized weak-FP source signs.", "3. Select phi_s=log(hat rho_s/tilde pi_s) by an explicit admissibility hypothesis.", "4. Substitute the source-signed weak-FP identity into eq:general_KL_derivative_0_discrete, preserving the negative drift-divergence action, positive sigma_eta^2/2 Laplacian action, and unchanged target-time term.", "5. Hand the resulting display to sald.general_moving_target_discrete.kl_derivative before later integration by parts, Laplacian split, FI, LSI, DV, time-change, and Gronwall obligations." ] modeDiscipline := [ "faithfulPaper Phase 1 only: preserve appendix.tex:1358-1387 and keep sald_version_2.tex out of scope.", "Do not change theorem statements, constants, sigma_eta^2/2 sign conventions, source labels, or proof order.", "Keep conditional law, density/time regularity, generator-to-law calculus, weak FP, KL differentiation, log-ratio admissibility, and boundary assumptions explicit as obligations." ] nonGoals := [ "No theorem-route audit, source-index rebaseline beyond the gate, display algebra outside the KL handoff, frozen-delta work, Gronwall/DV/LSI work, broad reusable API design, or project-article export.", "No proof of the regular conditional law, density/absolute-continuity, weak Fokker-Planck theorem, KL differentiability theorem, integration-by-parts theorem, or Fisher-information identity.", "No status promotion for sald.general_moving_target_discrete.em_interpolation_fp, sald.discrete_forward_kl.em_interpolation_fp, sald.general_moving_target_discrete.kl_derivative, either discrete theorem contract, SLT reuse, or Lake dependencies." ] lowerPacket := [ "Target exactly SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoff / SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation / sald.general_moving_target_discrete.cycle83_kl_derivative_endpoint_weak_fp_lower.", "Middle should synchronize the source-to-Lean route from cycle-82 endpoint source signs to eq:general_KL_derivative_0_discrete, with the log-ratio admissibility hypothesis explicit.", "Lower proof-producing scope is only the local composition from endpoint/conditional weak-FP source signs to the KL derivative display; all analytic hypotheses stay supplied.", "If blocked, record the exact missing Mathlib/SDE theorem boundary rather than broadening to unrelated theorem routes." ] reviewerChecklist := [ "The active packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387.", "The KL handoff preserves source signs: negative drift-divergence and positive sigma_eta^2/2 Laplacian coefficient.", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoff is a local composition wrapper and does not prove weak FP, conditional law, density/AC, KL differentiation, integration by parts, FI, LSI, DV, Gronwall, or theorem closure.", "Both discrete theorem contracts remain contractOnly; analytic backend, SLT, and Lake statuses remain unpromoted.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsUpperObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpMiddleObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation /-- Cycle-83 upper obligation selecting the endpoint weak-FP to KL handoff. -/ def cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpUpperObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle83_kl_derivative_endpoint_weak_fp_upper" statement := "Cycle 83 upper records that cycle 82 passed reviewer/build and needs no recovery, confirms that Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill, and keeps the active packet on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387. The selected lower packet is the KL-derivative handoff from the cycle-82 endpoint/conditional weak-FP source signs to eq:general_KL_derivative_0_discrete at the admissible log-ratio test." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpUpperPacket", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Upper workflow obligation only. It fixes the lower packet and status discipline; it proves no analytic backend and promotes no theorem status." /-- Cycle-83 middle obligation for the endpoint source-signs to KL source map. -/ def cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle83_kl_derivative_endpoint_weak_fp_middle" statement := "Cycle 83 middle maps appendix.tex:1358-1387 without changing the paper statement: eq:general_KL_derivative_0_discrete supplies dK=partialS(logRatioTest)-targetTimeTerm after the mass-conservation drop; the cycle-82 endpoint/conditional wrapper supplies the weak-FP source signs for admissible tests; the log-density-ratio test enters only through an explicit admissibility hypothesis; and the local lower wrapper composes those facts before integration by parts, Laplacian split, FI identification, LSI, DV, Gronwall, or theorem closure." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpUpperObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Middle synchronization only. Conditional law, density/AC, weak FP, KL differentiability, log-ratio admissibility, boundary behavior, integration by parts, FI, LSI, DV, and Gronwall remain obligations." /-- Cycle-83 lower obligation for the endpoint source-signs to KL wrapper. -/ def cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle83_kl_derivative_endpoint_weak_fp_lower" statement := "Cycle 83 compiles SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoff for appendix.tex:1358-1387 and adds SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction as the lower companion. The wrapper composes SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff with SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns: once endpoint/conditional readiness, supplied generator/time derivative, drift/diffusion source actions, density/time regularity, boundary behavior, fpCoeff=sigma_eta^2/2, the differentiated KL display, and log-ratio admissibility are supplied, the resulting KL derivative display has the paper's negative drift-divergence action, positive sigma_eta^2/2 Laplacian action, and unchanged target-time term; the companion records the log-ratio weak-FP action and dK display as a pair." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpMiddleObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Proof-producing local composition plus obligation only. It proves no conditional law, density/AC, weak Fokker-Planck theorem, KL differentiability, integration by parts, FI identification, LSI/KL/FI, DV, Gronwall, theorem closure, SLT import, or Lake dependency." /-- Cycle-83 proof-DAG pane for endpoint weak-FP source signs to KL derivative handoff. -/ def cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle83.global_phase_judgment" interface := "Cycle 83 judgment: cycle 82 passed and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the largest remaining proof risk is still sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, now narrowed to the KL-derivative handoff from the endpoint/conditional weak-FP source signs." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpUpperObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle83.middle_endpoint_source_signs_to_kl" interface := "Middle source map: connect eq:general_KL_derivative_0_discrete to the cycle-82 endpoint/conditional weak-FP source signs at the admissible log-ratio test, before integration by parts and FI identification." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpMiddleObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle83.lower_packet.kl_derivative_endpoint_handoff" interface := "Local proof-producing wrapper: compose endpoint/conditional weak-FP source signs with the normalized weak-FP-to-KL substitution, preserving the source signs and target-time term under explicit supplied hypotheses; the cycle83 lower companion also records the log-ratio weak-FP action paired with the dK display." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle83.reviewer_kl_derivative_handoff_check" interface := "Reviewer check: reject any sign flip, hidden analytic theorem, theorem-status promotion, SLT/Lake dependency change, or drift away from appendix.tex:1358-1387 and the active EM backend." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpUpperObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpMiddleObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoff", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] /-- Cycle-84 upper packet for the active EM interpolation backend after the cycle-83 KL-derivative handoff. The cycle focus allows a new source-cited Mathlib/measure interface only when proof-producing work is blocked. Cycle 83 compiled a local handoff, so this packet keeps lower work on the active backend and records the precise fallback boundary without opening a broad measure-theory audit. -/ def cycle84GeneralMovingTargetDiscreteActiveEmBackendUpperPacket : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource objective := "Global phase judgment: cycle 83 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet that best reduces the remaining proof risk is to keep proving the active sald.general_moving_target_discrete.em_interpolation_fp backend over appendix.tex:1358-1387 by consolidating the cycle-80 to cycle-83 endpoint/conditional readiness, weak-FP source signs, and KL-derivative handoff. The minimal cited Mathlib/measure interface is an escape hatch only if this proof-producing work hits a concrete conditional-law or weak-FP theorem blocker." sourceLabels := [ "eq:general_KL_derivative_0_discrete", "proof:thm:general-moving-target-SALD-discrete:conditional-drift", "proof:thm:general-moving-target-SALD-discrete:weak-conditional-fp", "proof:thm:general-moving-target-SALD-discrete:weak-fp-to-kl-derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.cycle84_active_em_backend_upper", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Cycles 80 and 81 expose endpoint/conditional readiness for the named law hat rho_s, conditional kernel, and measurable/integrable bar b_{k,s}.", "Cycle 82 exposes the weak-test source signs partialS phi = -(driftDiv phi) + (sigma_eta^2/2)*laplacian phi under supplied density, test, boundary, and generator hypotheses.", "Cycle 83 exposes the admissible log-ratio substitution into eq:general_KL_derivative_0_discrete and pairs the weak-FP action with the resulting dK display.", "Cycle 84 lower should first try to package these accepted handoffs into the next active EM-backend proof step, not introduce a new broad Mathlib measure interface.", "If lower is blocked, the only acceptable fallback is one narrow source-cited interface naming the missing conditional-law or generator-to-law weak-FP theorem with common-space, density/AC, admissible-test, finite-integral, and boundary hypotheses explicit." ] theoremRoute := [ "1. Stay on a fixed EM interval k and s in [s_k,s_{k+1}] inside appendix.tex:1358-1387.", "2. Start from the cycle-81 endpoint/conditional WeakFpPrereq readiness and the cycle-82 endpoint weak-FP source-sign handoff.", "3. Reuse the cycle-83 log-ratio action plus dK display at eq:general_KL_derivative_0_discrete.", "4. Select a proof-producing local bridge that makes the EM interpolation FP backend lower-ready for the next KL derivative step while keeping conditional law, density/AC, weak FP, KL differentiability, integration by parts, FI, LSI, DV, and Gronwall explicit.", "5. Only if that bridge is blocked, record exactly one cited measure-theory theorem boundary; do not broaden to theorem-route audits, display algebra, or unrelated analytic backends." ] modeDiscipline := [ "faithfulPaper Phase 1 only: preserve appendix.tex:1358-1387, source labels, signs, constants, and theorem statements; sald_version_2.tex remains out of scope.", "The minimal measure-interface fallback must not add assumptions to thm:forward-KL-discrete or thm:general-moving-target-SALD-discrete.", "Use lean-stat-learning-theory only as a local style reference for Mathlib measure/probability patterns; do not import it, change Lake dependencies, or mark any SLT theorem formalized." ] nonGoals := [ "No source-index rebaseline beyond the acceptance gate, broad theorem-route audit, Gronwall/DV/LSI/frozen-delta work, display algebra outside appendix.tex:1358-1387, reusable API redesign, or project-article export.", "No proof of the full regular conditional law, density/absolute-continuity theorem, generator-to-law weak Fokker-Planck theorem, KL differentiability theorem, integration-by-parts theorem, or theorem closure unless a local declaration actually compiles.", "No status promotion for sald.general_moving_target_discrete.em_interpolation_fp, sald.discrete_forward_kl.em_interpolation_fp, sald.general_moving_target_discrete.kl_derivative, either discrete theorem contract, SLT reuse, or Lake dependencies." ] lowerPacket := [ "Target exactly sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387.", "Preferred lower work is a proof-producing local bridge that consumes the cycle-81 endpoint/conditional readiness, cycle-82 endpoint source signs, and cycle-83 log-ratio KL handoff to make the EM backend handoff tighter for both discrete theorem routes.", "Do not introduce a new cited Mathlib/measure interface unless lower can name the precise blocked theorem, whether conditional-law/measurability or generator-to-law weak FP, and lists the exact hypotheses consumed by the source proof.", "If blocked, the fallback interface must remain sourceCited or obligation-level and must not claim regular conditional law, weak FP, KL derivative, theorem closure, SLT, or Lake status as formalized." ] reviewerChecklist := [ "The active packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387.", "The packet does not trigger the measure-interface fallback unless a concrete proof-producing attempt is blocked.", "Any fallback interface is narrow: one missing conditional-law or weak-FP theorem boundary with common-space, density/AC, admissible-test, finite-integral, boundary, and coefficient hypotheses explicit.", "The source signs stay negative for drift-divergence and positive for the sigma_eta^2/2 Laplacian coefficient.", "Both discrete theorem contracts remain contractOnly; analytic backend, SLT, and Lake statuses remain unpromoted.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpUpperObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpMiddleObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation /-- Cycle-84 upper obligation selecting active EM-backend proof work before any minimal cited measure-interface fallback. -/ def cycle84GeneralMovingTargetDiscreteActiveEmBackendUpperObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle84_active_em_backend_upper" statement := "Cycle 84 upper records that cycle 83 passed reviewer/build and needs no recovery, confirms that Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill, and keeps the active lower packet on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387. Because cycle 83 compiled a proof-producing endpoint weak-FP to KL derivative handoff, the minimal cited Mathlib/measure interface is not selected by default; lower should first consolidate the cycle-80 to cycle-83 endpoint/conditional readiness, source-sign, and log-ratio KL handoffs. A fallback interface is allowed only after a concrete block and must name one missing conditional-law or weak-FP theorem boundary below formalized status." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendUpperPacket", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Upper workflow obligation only. It selects the next lower packet and fallback discipline; it proves no new analytic backend and promotes no theorem, SLT, or Lake status." /-- Cycle-84 middle obligation for the active EM-backend source map. -/ def cycle84GeneralMovingTargetDiscreteActiveEmBackendMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle84_active_em_backend_middle" statement := "Cycle 84 middle keeps appendix.tex:1358-1387 synchronized after the accepted cycle-83 KL handoff: the source route starts with eq:general_KL_derivative_0_discrete, uses the conditional drift definition of bar b_{k,s} from appendix.tex:1368-1377 only through the existing cycle-80/cycle-81 endpoint and conditional readiness packages, uses the weak conditional Fokker-Planck signs of appendix.tex:1379-1387 only through the cycle-82 endpoint source-sign wrapper, and uses the log-ratio weak-FP-to-KL substitution only through the cycle-83 handoff. Since these local bridges compile under supplied hypotheses, middle does not select a new Mathlib/measure fallback; if lower is blocked, the fallback must name exactly one missing conditional-law or generator-to-law weak-FP theorem boundary and keep it below formalized status." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendUpperObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpMiddleObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.discrete_forward_kl.em_interpolation_fp" ] note := "Middle synchronization only. It adds no theorem assumptions, introduces no new cited interface, and leaves conditional law, density/AC, generator-to-law weak FP, KL differentiability, log-ratio admissibility, integration by parts, FI, LSI/KL/FI, DV, Gronwall, theorem status, SLT, and Lake dependencies unpromoted." /-- Cycle-84 lower obligation for the endpoint log-action active-backend handoff. -/ def cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle84_active_em_backend_lower" statement := "Cycle 84 lower compiles SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction for appendix.tex:1358-1387. The wrapper consumes the cycle-81 endpoint/conditional WeakFpPrereq readiness, the cycle-82 endpoint weak-FP source-sign handoff, and the cycle-83 log-ratio weak-FP-to-KL handoff; under explicit supplied hypotheses it returns both the log-ratio weak-FP action and the resulting dK display with the paper's negative drift-divergence sign, positive sigma_eta^2/2 Laplacian coefficient, and unchanged target-time term. No new Mathlib/measure fallback was needed." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendMiddleObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Proof-producing local composition plus obligation only. Conditional law, density/AC, weak Fokker-Planck theorem, generator-to-law differentiation, KL differentiability, integration by parts, FI identification, LSI/KL/FI, DV, Gronwall, theorem closure, SLT import, and Lake dependency remain unpromoted." /-- Cycle-84 proof-DAG pane for the active EM-backend handoff/fallback decision. -/ def cycle84GeneralMovingTargetDiscreteActiveEmBackendDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle84.global_phase_judgment" interface := "Cycle 84 judgment: cycle 83 passed and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the largest remaining proof risk is still sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, and the lower packet should continue proof-producing EM-backend consolidation before any new cited measure interface." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendUpperObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle84.middle_active_em_backend_source_map" interface := "Middle source map: keep eq:general_KL_derivative_0_discrete, the conditional drift bar b_{k,s}, the weak FP source signs, and the log-ratio KL handoff aligned through the existing cycle-80 to cycle-83 wrappers before any measure-interface fallback." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendMiddleObligation", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle84.lower_packet.active_em_backend" interface := "Selected lower packet: consume the cycle-81 endpoint/conditional readiness, cycle-82 endpoint weak-FP source signs, and cycle-83 log-ratio KL handoff to tighten the active EM interpolation FP backend; keep all analytic hypotheses explicit." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendUpperPacket", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendMiddleObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle84.blocked_measure_interface_escape_hatch" interface := "Fallback only if blocked: add one narrow source-cited conditional-law or generator-to-law weak-FP theorem boundary, with common-space, density/AC, admissible-test, finite-integral, boundary, and coefficient hypotheses explicit and status below formalized." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendUpperObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle84.reviewer_active_em_backend_check" interface := "Reviewer check: reject a fallback interface without a concrete block, any source sign or coefficient change, hidden analytic theorem claim, theorem-status promotion, SLT/Lake dependency change, or drift outside appendix.tex:1358-1387." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendUpperObligation", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendMiddleObligation", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] /-- Cycle-85 upper packet for the post-cycle-84 conditional-kernel theorem boundary. Cycle 84 compiled another endpoint-level handoff under supplied hypotheses. In cycle 85 the lower work must stop adding wrappers of the same shape and instead reduce the conditional-law theorem boundary behind appendix.tex lines 1368-1377, or record one exact missing Mathlib theorem with hypotheses. -/ def cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryUpperPacket : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteConditionalDriftSource objective := "Global phase judgment: cycle 84 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet that now reduces the largest proof risk is the conditional-kernel/conditional-expectation theorem boundary inside sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, specifically appendix.tex:1368-1377. Lower must either discharge one supplied conditional-law hypothesis using Mathlib-style ingredients, or record one exact missing condDistrib/condExpKernel theorem with imports and hypotheses." sourceLabels := [ "proof:thm:general-moving-target-SALD-discrete:conditional-drift", "proof:thm:general-moving-target-SALD-discrete:weak-conditional-fp", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.cycle85_conditional_kernel_boundary_upper", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "ProbabilityTheory.condExpKernel", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "The active lower packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387; reviewer found no theorem-route blocker and no source-index defect requiring rebaseline.", "Cycle 84 accepted a compiled endpoint/log-action wrapper but left the real analytic boundary unchanged: instantiate the conditional kernel for X_k^eta | hat X_s=x, name hat rho_s=Law(hat X_s), produce component conditional-integral fields, and prove measurability/integrability of bar b_{k,s}.", "Cycle 85 lower classification is discharges-supplied-hypothesis if it proves a local theorem that removes an existing supplied conditional-kernel, marginal, conditional-integral, measurability, or integrability hypothesis.", "Cycle 85 lower classification is narrows-source-cited-boundary if proof is blocked but the packet records one exact missing theorem, import list, and hypotheses for condDistrib/condExpKernel orientation or vector-valued conditional expectation.", "Any new wrapper that only repackages existing supplied hypotheses without removing one or naming a smaller missing theorem is rejected-wrapper-churn." ] theoremRoute := [ "1. Stay inside appendix.tex:1368-1377 as the active sub-slice of appendix.tex:1358-1387.", "2. Align Mathlib's conditional-kernel orientation with the paper's conditioning X_k^eta | hat X_s=x; the named marginal consumed by the kernel must be hat rho_s=Law(hat X_s).", "3. Connect the conditional kernel to the two component conditional-integral fields for dot t_k*c_{t_k}(X_k^eta) and (sigma_eta^2/2)*nabla log pi_{t_k}(X_k^eta).", "4. Reduce the supplied hypotheses behind SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents, SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents, or SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff.", "5. Keep weak FP source signs, KL/log-ratio substitution, density/AC, integration by parts, FI, LSI/KL/FI, DV, Gronwall, and theorem closure as downstream consumers." ] modeDiscipline := [ "faithfulPaper Phase 1 only: preserve appendix.tex:1368-1377, the paper's bar b_{k,s} definition, the EM backend source labels, and both discrete theorem statements.", "Do not add new assumptions to thm:forward-KL-discrete or thm:general-moving-target-SALD-discrete; any standard-Borel, probability, integrability, measurability, or finite-measure hypotheses belong only inside the local backend theorem boundary.", "Use lean-stat-learning-theory only as a local proof-engineering reference for Mathlib measure/probability patterns; do not import it, change Lake dependencies, or mark any SLT theorem formalized." ] nonGoals := [ "No theorem-route audit, broad source-index rebaseline, display algebra, Gronwall/DV/LSI/frozen-delta work, generator-to-law weak-FP work, KL/log-ratio work, reusable API redesign, or project-article export.", "No new supplied-hypothesis wrapper unless it removes an older supplied hypothesis, exposes a strictly smaller missing theorem, or compiles a genuinely local proof.", "No status promotion for sald.general_moving_target_discrete.em_interpolation_fp, sald.discrete_forward_kl.em_interpolation_fp, either discrete theorem contract, SLT reuse, or Lake dependencies." ] lowerPacket := [ "Target exactly sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to the conditional-kernel theorem boundary at appendix.tex:1368-1377.", "Preferred product: a compiled local theorem from Mathlib-style ingredients that discharges one supplied hypothesis behind conditional-kernel compatibility, hat rho_s marginal orientation, component conditional-integral fields, or measurability/integrability of bar b_{k,s}.", "Allowed blocked product: one precise source-cited missing theorem naming the Mathlib imports, theorem shape, and hypotheses for condDistrib/condExpKernel orientation or vector-valued conditional expectation.", "Classification required in the lower handoff: discharges-supplied-hypothesis, narrows-source-cited-boundary, or rejected-wrapper-churn." ] reviewerChecklist := [ "The active packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387 and the sub-slice appendix.tex:1368-1377.", "Reject any packet classified as a wrapper unless it removes an existing supplied hypothesis or names a smaller missing theorem with imports and hypotheses.", "The named marginal is hat rho_s=Law(hat X_s), and the conditional kernel orientation is explicitly reconciled with X_k^eta | hat X_s=x.", "The lower result is classified as discharges-supplied-hypothesis or narrows-source-cited-boundary; rejected-wrapper-churn is not accepted as progress.", "No theorem status, EM backend status, SLT status, or Lake dependency is promoted; no source sign or constant is changed.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityMiddleObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "Mathlib.Probability.Kernel.Condexp", "Mathlib.Probability.Kernel.CondDistrib", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation /-- Cycle-85 upper obligation selecting the conditional-kernel theorem boundary instead of another supplied-hypothesis wrapper. -/ def cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryUpperObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle85_conditional_kernel_boundary_upper" statement := "Cycle 85 upper records that cycle 84 passed reviewer/build and needs no recovery, confirms that Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill, and keeps the active packet on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387. The selected lower packet is the conditional-kernel/conditional-expectation theorem boundary at appendix.tex:1368-1377: instantiate or precisely block the Mathlib condDistrib/condExpKernel orientation for X_k^eta | hat X_s=x, the named marginal hat rho_s=Law(hat X_s), component conditional-integral fields, and measurability/integrability of bar b_{k,s}. Lower must classify the packet as discharges-supplied-hypothesis, narrows-source-cited-boundary, or rejected-wrapper-churn." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryUpperPacket", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityMiddleObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "Mathlib.Probability.Kernel.Condexp", "Mathlib.Probability.Kernel.CondDistrib", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Upper workflow obligation only. It fixes the lower packet and anti-wrapper-churn review discipline; it proves no conditional law, conditional expectation theorem, weak FP theorem, KL derivative, or theorem closure." /-- Cycle-85 middle boundary narrowing using local Mathlib conditional-kernel helpers. This is not another supplied-hypothesis wrapper. The local declarations in `AutoSamplingTheory/Probability.lean` compile the Mathlib orientation from `condDistrib` to `condExpKernel.map`, plus the vector-valued conditional-integral measurability/integrability handoff. The remaining lower boundary is now the source-specific versioning step: turn those sample-space a.e. conditional-integral facts into the named `hat rho_s` component fields used by the existing `bar b_{k,s}` wrappers. -/ def cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle85_conditional_kernel_boundary_middle" statement := "Cycle 85 middle narrows the appendix.tex:1368-1377 conditional-kernel theorem boundary by compiling local Mathlib-backed helpers AutoSamplingTheory.condDistribAeEqCondExpKernelMap, AutoSamplingTheory.condDistribIntegralAEStronglyMeasurable, and AutoSamplingTheory.condDistribIntegralIntegrable. These orient X_k^eta | hat X_s=x as condDistrib X_k^eta hatX_s mu (hatX_s omega), identify it a.e. with condExpKernel mu (mState.comap hatX_s) mapped by X_k^eta, and supply vector-valued conditional-integral measurability/integrability from Mathlib finite-measure, standard-Borel, a.e.-measurability, and integrability hypotheses. Classification: narrows-source-cited-boundary. The remaining exact lower theorem is the source-specific state-field versioning theorem: from these sample-space a.e. condDistrib/condExpKernel integral facts and hatRhoS=Law(hatX_s), construct the named component fields condC_{k,s}, condScore_{k,s} as hat-rho_s-a.e. fields and feed SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents or SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff without supplied component-regularity hypotheses." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryUpperObligation", "AutoSamplingTheory.condDistribAeEqCondExpKernelMap", "AutoSamplingTheory.condDistribIntegralAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralIntegrable", "ProbabilityTheory.condDistrib_apply_ae_eq_condExpKernel_map", "MeasureTheory.AEStronglyMeasurable.integral_condDistrib", "MeasureTheory.Integrable.integral_condDistrib", "Mathlib.Probability.Kernel.Condexp", "Mathlib.Probability.Kernel.CondDistrib", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityMiddleObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle boundary narrowing only. The orientation and conditional-integral regularity facts now compile locally, but the SALD-specific field-version theorem still has to identify hatRhoS as the conditioning marginal and turn sample-space a.e. integral formulas into named hat-rho_s-a.e. component fields. No weak FP, KL derivative, theorem contract, SLT status, or Lake dependency is promoted." /-- Cycle-85 lower boundary reduction for named conditional-integral fields. The lower packet compiles the law-space conditional-integral regularity facts needed after the middle sample-space orientation work. The theorem `AutoSamplingTheory.condDistribIntegralNamedFieldRegularity` removes the generic supplied component measurability/integrability hypotheses for any component field that is chosen as a `hatRhoS`-a.e. version of the canonical `condDistrib` integral. The remaining analytic work is now the source-specific conditional-expectation/version choice and the kernel compatibility/disintegration theorem, not the Mathlib law-space regularity of those integrals. -/ def cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle85_conditional_kernel_boundary_lower" statement := "Cycle 85 lower compiles AutoSamplingTheory.condDistribIntegralMapAEStronglyMeasurable, AutoSamplingTheory.condDistribIntegralMapIntegrable, AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable, AutoSamplingTheory.condDistribIntegralNamedLawIntegrable, and AutoSamplingTheory.condDistribIntegralNamedFieldRegularity for appendix.tex lines 1368-1377. These are local Mathlib-backed theorems using ProbabilityTheory.condDistrib with X=hatX_s and Y=X_k^eta: if hatRhoS=mu.map hatX_s, a frozen component integrand is AEStronglyMeasurable/integrable on the joint law mu.map (fun omega => (hatX_s omega, X_k^eta omega)), and the named component field condC_{k,s} or condScore_{k,s} is hatRhoS-a.e. equal to the canonical conditional integral x |-> integral y, f (x,y) d(condDistrib X_k^eta hatX_s mu x), then that named field is AEStronglyMeasurable and Integrable under hatRhoS. Classification: discharges-supplied-hypothesis for the component-field measurability/integrability part of the older supplied hypotheses behind SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents and SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff; the remaining source-cited boundary is the regular conditional-kernel compatibility/disintegration theorem plus the SALD-specific a.e. version choice for condC_{k,s}, condScore_{k,s}, and bar b_{k,s}." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryMiddleObligation", "AutoSamplingTheory.condDistribIntegralMapAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralMapIntegrable", "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map", "MeasureTheory.Integrable.integral_condDistrib_map", "Mathlib.Probability.Kernel.CondDistrib", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Lower proof-producing result, classified discharges-supplied-hypothesis for the named component-field regularity subproblem. It does not construct the conditional kernel, prove the a.e. version equality for the SALD fields, prove weak FP/KL/FI/LSI/DV/Gronwall, promote theorem status, import SLT, or change Lake dependencies." /-- Cycle-85 proof-DAG pane for the conditional-kernel theorem boundary. -/ def cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle85.global_phase_judgment" interface := "Cycle 85 judgment: cycle 84 passed and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the largest remaining proof risk is the conditional-kernel/conditional-expectation theorem boundary for sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to appendix.tex:1368-1377." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryUpperObligation", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle85.active_em_backend_check" interface := "Active packet check before assigning lower work: no reviewer blocker moved the target away from sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387; the current sub-slice is the conditional-kernel and conditional-expectation construction of bar b_{k,s} at appendix.tex:1368-1377." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryUpperPacket", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendUpperObligation", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle85.lower_packet.conditional_kernel_theorem_boundary" interface := "Selected lower packet: prove or sharply narrow the Mathlib condDistrib/condExpKernel orientation for X_k^eta | hat X_s=x, named marginal hat rho_s=Law(hat X_s), component conditional-integral fields, and measurable/integrable bar b_{k,s}; classify accepted proof progress as discharges-supplied-hypothesis." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryUpperObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "Mathlib.Probability.Kernel.Condexp", "Mathlib.Probability.Kernel.CondDistrib" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle85.middle_condDistrib_condExpKernel_boundary" interface := "Middle narrowed boundary: local helpers compile the Mathlib condDistrib-to-condExpKernel.map orientation and vector-valued conditional-integral measurability/integrability. Remaining lower theorem: convert these sample-space a.e. facts, together with hatRhoS=Law(hatX_s), into named hat-rho_s-a.e. component fields condC_{k,s}, condScore_{k,s} consumed by the existing bar b_{k,s} wrappers." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource targetLean := "AutoSamplingTheory/Probability.lean" dependsOn := [ "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryMiddleObligation", "AutoSamplingTheory.condDistribAeEqCondExpKernelMap", "AutoSamplingTheory.condDistribIntegralAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralIntegrable", "ProbabilityTheory.condDistrib_apply_ae_eq_condExpKernel_map", "MeasureTheory.AEStronglyMeasurable.integral_condDistrib", "MeasureTheory.Integrable.integral_condDistrib" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle85.lower_conditional_integral_named_field_regularity" interface := "Lower discharges the component regularity sub-hypothesis: for any named condC_{k,s} or condScore_{k,s} that is hatRhoS-a.e. equal to the canonical condDistrib integral, Mathlib's law-space conditional-integral lemmas give AEStronglyMeasurable and Integrable under hatRhoS=Law(hatX_s). Remaining work is the SALD-specific version equality and kernel compatibility/disintegration, not generic law-space regularity." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource targetLean := "AutoSamplingTheory/Probability.lean" dependsOn := [ "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryLowerObligation", "AutoSamplingTheory.condDistribIntegralMapAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralMapIntegrable", "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map", "MeasureTheory.Integrable.integral_condDistrib_map" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle85.missing_mathlib_theorem_record" interface := "Remaining narrowed theorem: after the local condDistrib/condExpKernel orientation and law-space component regularity helpers compile, prove the SALD-specific version theorem giving hatRhoS-a.e. equality of condC_{k,s} and condScore_{k,s} with the canonical condDistrib integrals, plus the conditional-kernel compatibility/disintegration bridge. The generic component-field regularity sub-hypothesis is no longer the blocker." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryMiddleObligation", "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryLowerObligation", "AutoSamplingTheory.condDistribAeEqCondExpKernelMap", "AutoSamplingTheory.condDistribIntegralAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralIntegrable", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "Mathlib.Probability.Kernel.Condexp", "Mathlib.Probability.Kernel.CondDistrib", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle85.reviewer_conditional_kernel_boundary_check" interface := "Reviewer check: reject supplied-hypothesis wrapper churn, hidden theorem closure, theorem-status promotion, SLT/Lake dependency change, wrong conditional-kernel orientation, missing hat rho_s marginal, or drift away from appendix.tex:1368-1377." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryUpperObligation", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] /-- Cycle-86 upper packet returning to the generator-to-law weak FP boundary. Cycle 85 discharged the generic named-field regularity part of the conditional law backend under explicit Mathlib hypotheses. The next lower packet should use that progress as a dependency and reduce the generator/time-derivative supplied hypothesis behind the weak Fokker-Planck statement, rather than adding another wrapper around the same assumptions. -/ def cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryUpperPacket : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteWeakFpSource objective := "Global phase judgment: cycle 85 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the active lower packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, and the single packet that now reduces the largest proof risk is the generator-to-law weak Fokker-Planck boundary at appendix.tex:1379-1387. Lower should use the cycle-79 lawMapIntegral helpers and the cycle-85 named conditional-integral regularity progress to discharge or sharply narrow the supplied generator/time-derivative hypothesis consumed by SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff." sourceLabels := [ "proof:thm:general-moving-target-SALD-discrete:weak-conditional-fp", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.cycle86_weak_fp_generator_boundary_upper", "AutoSamplingTheory.lawMapIntegral", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "The active packet remains sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed in this cycle to appendix.tex:1379-1387.", "Cycle 79 already formalized the Measure.map weak-test integral rewrite and the supplied sample-space HasDerivAt transport; lower should build on AutoSamplingTheory.lawMapIntegral and AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample instead of restating them.", "Cycle 85 discharged the generic component-field measurability/integrability sub-hypothesis when the SALD fields are chosen as hatRhoS-a.e. versions of canonical condDistrib integrals; lower may use that as the drift-regularity dependency but must not reopen broad conditional-law work.", "The target supplied hypothesis to remove or narrow is the generator/time-derivative input in the cycle-77/cycle-82 weak-FP wrappers: partialS phi equals the frozen EM generator action for admissible tests.", "If the full generator theorem is too large, lower should record the smallest remaining named theorem, such as sample-path generator differentiation, Bochner/parametric integral interchange, or conditional-drift source-action identification, with imports and hypotheses explicit." ] theoremRoute := [ "1. Preserve appendix.tex:1379-1387: the paper invokes the Fokker-Planck equation associated with eq:general_moving_target_SALD_frozen_interp and obtains partial_s hat rho_s = -div(hat rho_s*bar b_{k,s}) + (sigma_eta^2/2)*Delta hat rho_s.", "2. Work before the existing source-sign wrappers: derive or narrow the weak-test generator-to-law statement that supplies partialS phi = generatorAction phi.", "3. Use Measure.map transport to move between the law integral over hat rho_s=Law(hat X_s) and the sample-space integral over hat X_s.", "4. Keep drift/diffusion source signs, KL/log-ratio substitution, integration by parts, FI identification, LSI, DV, Gronwall, and theorem closure as downstream consumers." ] modeDiscipline := [ "faithfulPaper Phase 1 only: preserve the appendix.tex:1379-1387 source statement, the sign of the drift term, the sigma_eta^2/2 diffusion coefficient, and both discrete theorem statements.", "Do not add generator, density, boundary, conditional-law, or admissible-test assumptions to theorem statements; any such hypotheses belong only in the local backend theorem boundary.", "Use lean-stat-learning-theory only as a reference for Mathlib measure/probability style; do not import it, change Lake dependencies, or mark an SLT theorem formalized." ] nonGoals := [ "No theorem-route audit, broad source-index rebaseline, display algebra, Gronwall/DV/LSI/frozen-delta work, KL/log-ratio work, reusable API redesign, or project-article export.", "No new supplied-hypothesis wrapper unless it removes an older supplied generator/time-derivative, source-action, or integrability hypothesis, or names a strictly smaller missing theorem.", "No status promotion for sald.general_moving_target_discrete.em_interpolation_fp, sald.discrete_forward_kl.em_interpolation_fp, weak FP, KL derivative, either discrete theorem contract, SLT reuse, or Lake dependencies." ] lowerPacket := [ "Target exactly sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to the generator-to-law weak-FP boundary at appendix.tex:1379-1387.", "Preferred product: a compiled local theorem using AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample and Mathlib parametric/Bochner integral APIs that removes the supplied hgenerator-style hypothesis for admissible weak tests, or a strict subcase of it.", "Allowed blocked product: one precise source-cited theorem boundary naming the missing Mathlib/local theorem, imports, and hypotheses for sample-path generator differentiation, parametric integral interchange, or drift/diffusion source-action identification.", "Classification required in the lower handoff: discharges-supplied-hypothesis, narrows-source-cited-boundary, or rejected-wrapper-churn." ] reviewerChecklist := [ "The active packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387 and the sub-slice appendix.tex:1379-1387.", "Reject a packet that only repackages the cycle-77/cycle-82 supplied hypotheses without removing hgenerator, source-action, or a smaller named theorem boundary.", "Check that Measure.map and Bochner/parametric-integral transport is used in the direction law integral to sample-space integral and back, consistent with AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample.", "Preserve the source signs -div(hat rho_s*bar b_{k,s}) and +(sigma_eta^2/2)*Delta hat rho_s; do not change constants or source labels.", "No theorem status, EM backend status, SLT status, or Lake dependency is promoted.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureLowerObligation", "AutoSamplingTheory.lawMapIntegral", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfReadinessAndGeneratorPiecesHandoff", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation", "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryLowerObligation", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation /-- Cycle-86 upper obligation selecting the generator-to-law weak FP boundary. -/ def cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryUpperObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle86_weak_fp_generator_boundary_upper" statement := "Cycle 86 upper records that cycle 85 passed and needs no recovery, confirms that Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill, and keeps the active packet on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387. The selected lower packet is appendix.tex:1379-1387: use the existing cycle-79 Measure.map/Bochner integral lawMapIntegral helpers plus the accepted cycle-85 conditional-field regularity progress to discharge or sharply narrow the generator-to-law weak Fokker-Planck boundary. The concrete target is the supplied generator/time-derivative hypothesis consumed by SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff; lower must either compile a local theorem removing that hypothesis for admissible weak tests, or name the smallest missing theorem boundary with imports and hypotheses. Classification is required: discharges-supplied-hypothesis, narrows-source-cited-boundary, or rejected-wrapper-churn." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryUpperPacket", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureLowerObligation", "AutoSamplingTheory.lawMapIntegral", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation", "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Upper workflow obligation only. It fixes the lower packet and anti-wrapper-churn review discipline; it proves no generator theorem, weak Fokker-Planck identity, KL derivative, conditional law, theorem closure, or SLT result." /-- Cycle-86 middle source map for the generator-to-law weak FP boundary. This is not another source-sign wrapper. It translates the paper's invocation of the Fokker--Planck equation associated with the frozen EM interpolation into the exact theorem boundary lower should attack: an admissible weak-test law-derivative statement obtained from sample-path generator differentiation and `Measure.map` integral transport. -/ def cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle86_weak_fp_generator_boundary_middle" statement := "Cycle 86 middle translates appendix.tex lines 1379-1387 into the lower-ready generator-to-law theorem boundary. For every admissible weak test phi and interior time s, the desired local theorem should start from the sample-space EM interpolation hatX_s, prove or assume the sample-path/parametric-integral derivative HasDerivAt (fun s => integral omega, phi (hatX_s omega) dP) of the frozen EM generator action, transport that derivative across hatRhoS = Law(hatX_s) using AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample, and only then feed the resulting partialS phi = generatorAction phi into the existing source-sign wrappers. The lower packet is classified narrows-source-cited-boundary unless it actually removes the supplied generator/time-derivative hypothesis. The remaining named subtheorems are sample-path generator differentiation, Bochner/parametric integral interchange for the time-dependent interpolation, drift source-action identification through the cycle-85 conditional-field regularity, diffusion/quadratic-variation source-action identification with coefficient sigma_eta^2/2, and admissible-test density/boundary hypotheses." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryUpperObligation", "AutoSamplingTheory.lawMapIntegral", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfReadinessAndGeneratorPiecesHandoff", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation", "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryLowerObligation", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle source-to-Lean boundary, classified narrows-source-cited-boundary. It does not add a theorem wrapper, discharge the generator/time-derivative hypothesis, prove weak FP/KL differentiation, or promote any theorem, SLT, or Lake status." /-- Cycle-86 lower obligation for the sample-space derivative to law weak-FP generator handoff. -/ def cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle86_weak_fp_generator_boundary_lower" statement := "Cycle 86 lower compiles SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff for appendix.tex lines 1379-1387. The theorem removes the older supplied hgenerator-style equality partialS phi = generatorAction phi from the weak-FP source-sign wrapper by deriving it from a sample-space HasDerivAt generator derivative, AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample transport across hatRhoS = Law(hatX_s), and uniqueness of the law weak derivative. The remaining smaller source-cited theorem boundaries are sample-path/parametric-integral generator differentiation, law weak-derivative existence for admissible tests, drift source-action identification via the conditional field, diffusion source-action identification with coefficient sigma_eta^2/2, density/time regularity, admissible-test, and boundary hypotheses." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryMiddleObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureLowerObligation", "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryLowerObligation", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Classification: narrows-source-cited-boundary. The local handoff is formalized under explicit HasDerivAt and source-action hypotheses, but it does not prove the full EM generator theorem, weak Fokker-Planck identity, conditional law, KL derivative, theorem closure, SLT result, or Lake dependency." /-- Cycle-86 proof-DAG pane for the weak generator-to-law boundary. -/ def cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle86.global_phase_judgment" interface := "Cycle 86 judgment: cycle 85 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the largest remaining proof risk is the generator-to-law weak-FP theorem boundary for sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to appendix.tex:1379-1387." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryUpperObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle86.active_em_backend_check" interface := "Active packet check before assigning lower work: no reviewer blocker moved the target away from sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387; cycle 86 uses the requested generator-to-law weak-FP sub-slice appendix.tex:1379-1387, with cycle-85 conditional-field regularity as a dependency." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryUpperPacket", "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle86.middle_weak_fp_generator_to_law_source_map" interface := "Middle source map: appendix.tex:1379-1387 requires an admissible-test law derivative for hatRhoS=Law(hatX_s). The lower theorem combines sample-path generator differentiation, Bochner/parametric integral interchange, AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample, and cycle-85 conditional-field regularity before calling the existing source-sign wrappers. Classification: narrows-source-cited-boundary unless hgenerator is actually removed." source := saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryMiddleObligation", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureLowerObligation", "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle86.lower_packet.weak_fp_generator_to_law_boundary" interface := "Lower proof-producing handoff: SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff removes the abstract hgenerator equality by transporting the sample-space generator derivative to the law integral and using HasDerivAt uniqueness, then applies the existing generator-piece source-sign wrapper." source := saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryUpperObligation", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryMiddleObligation", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "AutoSamplingTheory.lawMapIntegral", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryLowerObligation" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle86.reviewer_weak_fp_generator_boundary_check" interface := "Reviewer check: reject wrapper churn, hidden weak-FP closure, source-sign or coefficient changes, theorem-status promotion, SLT/Lake dependency changes, or any packet that does not either remove a supplied generator/time-derivative/source-action hypothesis or name a strictly smaller missing theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryUpperObligation", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureLowerObligation", "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] /-- Cycle-87 upper packet for the KL/log-ratio analytic boundary. Cycle 86 removed the abstract generator equality from the weak-FP source-sign route by transporting a supplied sample-space derivative to the law integral. The next lower packet should use that narrowed weak-FP boundary to reduce the KL differentiability and log-ratio admissibility hypotheses behind `appendix.tex` lines 1358-1366, rather than adding another wrapper around the same `dK` display. -/ def cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryUpperPacket : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource objective := "Global phase judgment: cycle 86 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the active lower packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, and the single packet that now reduces the largest proof risk is the KL/log-ratio analytic boundary at appendix.tex:1358-1366. Lower should formalize or sharply isolate KL differentiability at the admissible log-ratio weak test, including log-ratio measurability/integrability and the handoff from weak-FP action to dK, without changing theorem statements." sourceLabels := [ "eq:general_KL_derivative_0_discrete", "proof:thm:general-moving-target-SALD-discrete:weak-fp-to-kl-derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.cycle87_kl_log_ratio_boundary_upper", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "The active packet remains sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed in this cycle to appendix.tex:1358-1366.", "Cycle 83 and cycle 84 already record supplied-hypothesis handoffs from weak-FP source signs at the log-ratio test to the dK display; lower must not add another handoff of the same shape.", "Cycle 86 narrows the upstream weak-FP generator boundary, so cycle 87 should attack the remaining KL-side supplied hypotheses: the differentiated KL identity, admissibility of log(hat rho_s / tilde pi_s) as a weak test, log-ratio measurability/integrability, and the mass-conservation term integral partial_s hat rho_s dx = 0.", "The cycle-87 lower log-ratio regularity theorem discharges the separate absolute-continuity, measurability, and integrability side of the Mathlib llr weak test from the smaller finite-KL hypothesis, while weak-test admissibility and KL differentiability remain obligations.", "If the full KL derivative theorem is too large, lower should record the smallest named theorem boundary, such as differentiating KL under the integral, finite log-ratio action against partial_s hat rho_s, target-density time-derivative integrability, or admissible weak-test regularity.", "Accepted progress is discharges-supplied-hypothesis only if an older hkl, hlog, log-action, measurability, or integrability supplied hypothesis is removed; otherwise it must be narrows-source-cited-boundary with a strictly smaller missing theorem." ] theoremRoute := [ "1. Preserve appendix.tex:1358-1366: d/ds KL(hat rho_s || tilde pi_s) equals the partial_s hat rho_s log-ratio action minus the target-time term, using integral partial_s hat rho_s dx = 0.", "2. Work before the existing cycle-83/cycle-84 weak-FP-to-KL wrappers by proving or narrowing the KL derivative display and admissible log-ratio weak-test conditions.", "3. Feed the resulting log-ratio test and dK display into the existing source-sign handoffs only after the KL/log-ratio hypotheses have been reduced.", "4. Keep integration by parts, Fisher-information identification, LSI, DV, Gronwall, frozen-delta work, and theorem closure as downstream consumers." ] modeDiscipline := [ "faithfulPaper Phase 1 only: preserve appendix.tex:1358-1366, eq:general_KL_derivative_0_discrete, the target-time term sign, and both discrete theorem statements.", "Do not add KL differentiability, density, absolute-continuity, admissible-test, or integrability assumptions to thm:forward-KL-discrete or thm:general-moving-target-SALD-discrete; such hypotheses belong only inside the local backend theorem boundary.", "Use lean-stat-learning-theory only as a reference for Mathlib measure/probability style; do not import it, change Lake dependencies, or mark an SLT theorem formalized." ] nonGoals := [ "No theorem-route audit, broad source-index rebaseline, display algebra outside appendix.tex:1358-1366, Gronwall/DV/LSI/frozen-delta work, conditional-law work, generator-to-law weak-FP work, reusable API redesign, or project-article export.", "No new supplied-hypothesis wrapper unless it removes an older KL/log-ratio supplied hypothesis, exposes a strictly smaller missing theorem, or compiles a genuinely local proof.", "No status promotion for sald.general_moving_target_discrete.em_interpolation_fp, sald.discrete_forward_kl.em_interpolation_fp, sald.general_moving_target_discrete.kl_derivative, either discrete theorem contract, SLT reuse, or Lake dependencies." ] lowerPacket := [ "Target exactly sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to the KL/log-ratio analytic boundary at appendix.tex:1358-1366.", "Preferred product: a compiled local theorem that removes an older supplied hkl-style KL derivative display, hlog-style log-ratio admissibility hypothesis, or log-ratio measurability/integrability hypothesis used by the cycle-83/cycle-84 wrappers.", "Allowed blocked product: one precise source-cited theorem boundary naming the missing Mathlib/local theorem, imports, and hypotheses for KL differentiability under the integral, finite log-ratio weak-FP action, target-density time derivative, mass conservation, or admissible weak-test regularity.", "Classification required in the lower handoff: discharges-supplied-hypothesis, narrows-source-cited-boundary, or rejected-wrapper-churn." ] reviewerChecklist := [ "The active packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387 and the sub-slice appendix.tex:1358-1366.", "Reject any packet that only composes existing cycle-83/cycle-84 hkl/hlog/log-action wrappers without removing an existing supplied hypothesis or naming a smaller missing theorem.", "Check that eq:general_KL_derivative_0_discrete keeps the target-time term with the paper's minus sign and uses the mass-conservation statement integral partial_s hat rho_s dx = 0.", "Check that the log-density-ratio weak test has explicit measurability, integrability, density/zero-set, and admissible-test hypotheses, or that the exact missing theorem is named.", "No theorem status, EM backend status, KL derivative status, SLT status, or Lake dependency is promoted.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlLogRatioLlrDef", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "eq:general_KL_derivative_0_discrete", "proof:thm:general-moving-target-SALD-discrete:weak-fp-to-kl-derivative", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "Mathlib.Analysis.Calculus.ParametricIntegral", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] status := ProofStatus.obligation /-- Cycle-87 upper obligation selecting the KL/log-ratio analytic boundary. -/ def cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryUpperObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle87_kl_log_ratio_boundary_upper" statement := "Cycle 87 upper records that cycle 86 passed and needs no recovery, confirms that Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill, and keeps the active packet on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387. The selected lower packet is appendix.tex:1358-1366: formalize or precisely isolate KL differentiability at the admissible log-ratio weak test, including log-ratio measurability/integrability, mass conservation, the target-time term, and the handoff from weak-FP action to dK. Lower must either compile a local theorem removing an older hkl, hlog, log-action, measurability, or integrability supplied hypothesis, or name the smallest missing theorem boundary with imports and hypotheses. Classification is required: discharges-supplied-hypothesis, narrows-source-cited-boundary, or rejected-wrapper-churn." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryUpperPacket", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction", "eq:general_KL_derivative_0_discrete", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Upper workflow obligation only. It fixes the lower packet and anti-wrapper-churn review discipline; it proves no KL differentiability theorem, log-ratio admissibility theorem, weak Fokker-Planck identity, integration by parts, FI identification, theorem closure, or SLT result." /-- Cycle-87 middle source map for the KL/log-ratio analytic boundary. This translates the source line `since int partial_s hat rho_s dx = 0` into a lower-ready split: prove the raw differentiated KL formula with an explicit mass term, prove mass conservation, prove log-ratio admissibility/regularity, then call the existing weak-FP source-sign handoff. -/ def cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle87_kl_log_ratio_boundary_middle" statement := "Cycle 87 middle translates appendix.tex lines 1358-1366 into a lower-ready KL/log-ratio theorem boundary. The old supplied post-mass-drop hkl hypothesis dK=partialS(logRatioTest)-targetTimeTerm is now split into: a raw KL differentiation theorem with an explicit mass term dK=partialS(logRatioTest)+massTerm-targetTimeTerm, the mass-conservation identity massTerm=0 corresponding to integral partial_s hat rho_s dx=0, log-ratio measurability/integrability/admissibility for log(hat rho_s/tilde pi_s), and target-time derivative integrability for integral (hat rho_s/tilde pi_s) partial_s tilde pi_s. The lower packet should compile the scalar mass-conservation handoff and leave the remaining analytic boundary as raw KL differentiation plus log-ratio regularity, not as another post-mass-drop hkl wrapper. Classification: narrows-source-cited-boundary unless the raw KL differentiation or log-ratio regularity theorem is actually proved." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryUpperObligation", "eq:general_KL_derivative_0_discrete", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "Mathlib.Analysis.Calculus.ParametricIntegral", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle boundary only. It names the smaller analytic theorem split and keeps KL differentiability, density/AC, log-ratio admissibility, target-time integrability, and mass conservation unpromoted." /-- Cycle-87 lower scalar handoff for the KL/log-ratio mass-conservation drop. -/ def cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle87_kl_log_ratio_boundary_lower" statement := "Cycle 87 lower compiles SALD.generalMovingTargetDiscreteKlDerivativeMassConservationDropScalar and SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction for appendix.tex lines 1358-1366, and adds SALD.generalMovingTargetDiscreteKlLogRatioLlrDef plus SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl. The scalar handoffs replace the older supplied post-mass-drop hkl display with the strictly smaller inputs raw KL derivative with explicit massTerm, massTerm=0, admissible log-ratio test, and normalized weak-FP source signs. The log-ratio theorem discharges separate absolute-continuity, measurability, and integrability side hypotheses for the Mathlib llr weak test from finite KL. Classification: discharges-supplied-hypothesis for log-ratio regularity, and narrows-source-cited-boundary for the raw KL/mass split. The remaining exact analytic boundaries are raw KL differentiability under the integral, proof of integral partial_s hat rho_s dx=0, weak-test admissibility of the log-ratio, target-time derivative integrability, density/time regularity, plus downstream integration by parts and FI identification." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryMiddleObligation", "SALD.generalMovingTargetDiscreteKlDerivativeMassConservationDropScalar", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlLogRatioLlrDef", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "eq:general_KL_derivative_0_discrete", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Formalized local scalar handoff plus Mathlib-backed log-ratio regularity. It narrows hkl to raw KL derivative plus mass conservation and removes separate log-ratio AC/measurability/integrability hypotheses under finite KL; it does not prove KL differentiability, mass conservation, log-ratio weak-test admissibility, weak FP, integration by parts, FI, LSI, DV, Gronwall, theorem closure, SLT reuse, or any Lake dependency." /-- Cycle-87 proof-DAG pane for the KL/log-ratio analytic boundary. -/ def cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle87.global_phase_judgment" interface := "Cycle 87 judgment: cycle 86 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the largest remaining proof risk is the KL/log-ratio analytic boundary for sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to appendix.tex:1358-1366." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryUpperObligation", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle87.active_em_backend_check" interface := "Active packet check before assigning lower work: no reviewer blocker moved the target away from sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387; cycle 87 uses the requested KL/log-ratio sub-slice appendix.tex:1358-1366." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryUpperPacket", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle87.lower_packet.kl_log_ratio_boundary" interface := "Selected lower packet: prove or sharply narrow KL differentiability at the admissible log-ratio weak test for eq:general_KL_derivative_0_discrete, including log-ratio measurability/integrability, mass conservation, target-time derivative integrability, and the handoff from weak-FP action to dK. Classification: discharges-supplied-hypothesis if an older hkl, hlog, log-action, measurability, or integrability supplied hypothesis is removed; otherwise narrows-source-cited-boundary only if a smaller missing theorem is named." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryUpperObligation", "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryMiddleObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation", "eq:general_KL_derivative_0_discrete" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle87.middle_kl_log_ratio_source_map" interface := "Middle source map: split the old hkl display into raw KL differentiation with an explicit mass term, mass conservation integral partial_s hat rho_s dx=0, target-time derivative integrability, and log-ratio measurability/integrability/admissibility before applying weak-FP source signs." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryMiddleObligation", "eq:general_KL_derivative_0_discrete", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation", "sald.general_moving_target_discrete.kl_derivative" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle87.lower_kl_mass_conservation_handoff" interface := "Lower scalar handoff: SALD.generalMovingTargetDiscreteKlDerivativeMassConservationDropScalar and SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction formalize the mass-term drop and route the raw KL display through the existing log-ratio weak-FP source-sign handoff." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryLowerObligation", "SALD.generalMovingTargetDiscreteKlDerivativeMassConservationDropScalar", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle87.lower_kl_log_ratio_regularity" interface := "Lower log-ratio regularity: represent the paper test log(hat rho_s / tilde pi_s) by Mathlib llr, expose its RN-derivative zero-set convention, and derive absolute continuity plus AEStronglyMeasurable/Integrable llr from finite KL via InformationTheory.klDiv_ne_top_iff." source := saldGeneralMovingTargetDiscreteKlLogRatioMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteKlLogRatioLlrDef", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "eq:general_KL_derivative_0_discrete" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle87.reviewer_kl_log_ratio_boundary_check" interface := "Reviewer check: reject wrapper churn, hidden KL differentiability closure, missing log-ratio admissibility regularity, target-time sign changes, theorem-status promotion, SLT/Lake dependency changes, or any packet that does not remove a KL/log-ratio supplied hypothesis or name a strictly smaller missing theorem." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryUpperObligation", "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryLowerObligation", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] /-- Cycle-88 middle packet for the log-ratio weak-test admissibility boundary. Cycle 87 discharged the generic finite-KL log-ratio regularity side conditions. The next non-wrapper boundary is the supplied `hlog : Admissible logRatioTest` hypothesis consumed by the weak-FP-to-KL handoffs. This packet records the smaller theorem that lower should target: turn the Mathlib `llr hatRhoS tildePiS` representative into an admissible weak Fokker--Planck test by a smoothing/Sobolev approximation and boundary closure theorem, without claiming KL differentiability or weak FP closure. -/ def cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityMiddlePacket : MainSkeletonAnalyticInterfaceLedger where sourceBlock := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource objective := "Cycle 88 middle: cycle 87 was accepted and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the active packet remains sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to appendix.tex:1358-1366. The single lower packet is the log-ratio weak-test admissibility/smoothing boundary: replace the supplied hlog hypothesis used by the cycle-83/cycle-84 weak-FP-to-KL handoffs with a narrower source-cited theorem boundary using cycle-87 finite-KL llr regularity." sourceLabels := [ "eq:general_KL_derivative_0_discrete", "proof:thm:general-moving-target-SALD-discrete:weak-fp-to-kl-derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.cycle88_kl_log_ratio_admissibility_middle", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.GeneralMovingTargetDiscreteKlLogRatioAdmissibilityClosure", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] analyticInterfaces := [ "Selected supplied hypothesis to reduce: hlog : Admissible logRatioTest in the cycle-83/cycle-84 weak-FP-to-KL handoffs.", "Cycle 87 already turns finite KL into hatRhoS << tildePiS plus AEStronglyMeasurable and Integrable llr; do not restate those as separate supplied hypotheses.", "Remaining admissibility theorem: the Mathlib llr representative is admitted by the weak-FP test class after smoothing/Sobolev approximation, boundary/no-flux control, and closure of weak-FP actions under that approximation.", "The lower boundary must name the density/time regularity, zero-set convention, finite action, drift-divergence action, Laplacian action, and target-time integrability hypotheses that make the approximation legitimate.", "Classification for this middle packet: narrows-source-cited-boundary. It is not a proof of weak FP, KL differentiability, integration by parts, or FI identification." ] theoremRoute := [ "1. Use appendix.tex:1358-1366 to keep the log-density-ratio test at the KL derivative step.", "2. Replace the old generic hlog assumption by the smaller admissibility theorem for llr hatRhoS tildePiS under finite KL regularity plus approximation/closure hypotheses.", "3. Feed the resulting admissible test into the existing cycle-87 raw-KL/mass split and cycle-83/cycle-84 source-sign handoffs.", "4. Leave raw KL differentiability, mass conservation, target-time derivative, weak FP, integration by parts, and FI identification as downstream obligations." ] modeDiscipline := [ "faithfulPaper Phase 1 only: preserve appendix.tex:1358-1366, eq:general_KL_derivative_0_discrete, the target-time minus sign, and both discrete theorem statements.", "Do not add an admissibility assumption to thm:forward-KL-discrete or thm:general-moving-target-SALD-discrete; keep it inside the EM/KL backend boundary.", "Use Mathlib/local ASTIS declarations only; lean-stat-learning-theory remains a reference and is not a Lake dependency." ] nonGoals := [ "No theorem-route audit, display algebra, Gronwall/DV/LSI/frozen-delta work, conditional-kernel work, generator-to-law weak-FP work, reusable API redesign, or project-article export.", "No new supplied-hypothesis wrapper around hlog unless it removes that older hlog dependency or names the smaller approximation/closure theorem precisely.", "No status promotion for sald.general_moving_target_discrete.em_interpolation_fp, sald.discrete_forward_kl.em_interpolation_fp, sald.general_moving_target_discrete.kl_derivative, either discrete theorem contract, SLT reuse, or Lake dependencies." ] lowerPacket := [ "Target exactly the log-ratio admissibility boundary at appendix.tex:1358-1366.", "Preferred product: a compiled theorem deriving Admissible (llr hatRhoS tildePiS) from finite KL, density/time regularity, a smooth/Sobolev approximation sequence, boundary/no-flux control, and weak-FP action closure.", "Allowed blocked product: one precise source-cited theorem boundary naming the missing approximation/closure theorem and the imports/hypotheses needed to replace hlog.", "Cycle 88 lower product: SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure derives Admissible (llr hatRhoS tildePiS) from finite KL and the named approximation/closure package, so the broad hlog is replaced by the smaller closure theorem boundary.", "Classification required: discharges-supplied-hypothesis if hlog is actually removed by a compiled theorem; otherwise narrows-source-cited-boundary if the smaller theorem is recorded; rejected-wrapper-churn for another hlog wrapper." ] reviewerChecklist := [ "The packet targets appendix.tex:1358-1366 and the active EM backend over appendix.tex:1358-1387.", "The supplied hypothesis being reduced is exactly hlog : Admissible logRatioTest, not raw KL differentiability or weak FP.", "Cycle 87 finite-KL llr regularity is used as a dependency; AC/measurability/integrability should not be reintroduced as separate supplied hypotheses.", "The missing theorem names smoothing/Sobolev approximation, boundary behavior, weak-FP action closure, density/time regularity, and zero-set convention.", "No theorem status, EM/KL backend status, SLT status, or Lake dependency is promoted.", "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass." ] dependencies := [ "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryLowerObligation", "SALD.generalMovingTargetDiscreteKlLogRatioLlrDef", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.GeneralMovingTargetDiscreteKlLogRatioAdmissibilityClosure", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "Mathlib.Analysis.Calculus.ParametricIntegral", "eq:general_KL_derivative_0_discrete", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] status := ProofStatus.obligation /-- Cycle-88 middle source map for the log-ratio admissibility boundary. -/ def cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle88_kl_log_ratio_admissibility_middle" statement := "Cycle 88 middle replaces the broad supplied hlog : Admissible logRatioTest hypothesis used by the cycle-83/cycle-84 weak-FP-to-KL handoffs with the smaller source-cited boundary: prove that the Mathlib llr hatRhoS tildePiS representative of log(hat rho_s/tilde pi_s) is an admissible weak-FP test under cycle-87 finite-KL regularity, density/time regularity, smoothing or Sobolev approximation, boundary/no-flux control, and closure of drift-divergence and Laplacian weak actions. Classification: narrows-source-cited-boundary. This does not prove raw KL differentiability, mass conservation, target-time integrability, weak FP, integration by parts, FI, theorem closure, or any SLT result." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityMiddlePacket", "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryLowerObligation", "SALD.generalMovingTargetDiscreteKlLogRatioLlrDef", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.InformationTheory.KullbackLeibler.Basic", "eq:general_KL_derivative_0_discrete", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Middle boundary only. It selects exactly one older supplied EM/KL hypothesis, hlog : Admissible logRatioTest, and narrows it to a named approximation/closure theorem for the Mathlib llr weak test. No backend or theorem status is promoted." /-- Cycle-88 lower handoff for log-ratio weak-test admissibility. -/ def cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle88_kl_log_ratio_admissibility_lower" statement := "Cycle 88 lower compiles SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure for appendix.tex lines 1358-1366. The theorem uses cycle-87 finite-KL llr regularity to derive the measure-regularity inputs and replaces the older broad hlog : Admissible logRatioTest hypothesis, when logRatioTest is llr hatRhoS tildePiS, by the strictly smaller source-cited closure package SALD.GeneralMovingTargetDiscreteKlLogRatioAdmissibilityClosure. That closure package names the remaining smoothing/Sobolev approximation, density/time regularity, zero-set convention, boundary/no-flux control, drift-divergence action closure, Laplacian action closure, and target-time integrability theorem. Classification: narrows-source-cited-boundary; it removes the opaque hlog shape but does not prove the approximation/closure theorem itself." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityMiddleObligation", "SALD.GeneralMovingTargetDiscreteKlLogRatioAdmissibilityClosure", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.generalMovingTargetDiscreteKlLogRatioLlrDef", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "eq:general_KL_derivative_0_discrete", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Compiled finite-KL-to-admissible-llr handoff plus a narrowed closure boundary. Remaining analytic blockers are the closure package fields, raw KL differentiability, mass conservation, target-time derivative integrability, weak FP, integration by parts, FI, theorem closure, and any SLT reuse." /-- Cycle-88 proof-DAG pane for the log-ratio admissibility boundary. -/ def cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle88.global_phase_judgment" interface := "Cycle 88 judgment: cycle 87 was accepted by reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet that best reduces the active EM backend risk is log-ratio weak-test admissibility at appendix.tex:1358-1366." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryLowerObligation", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.kl_derivative" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle88.middle_log_ratio_admissibility_boundary" interface := "Middle source map: replace the broad hlog : Admissible logRatioTest input with a smaller theorem boundary for Admissible (llr hatRhoS tildePiS), using cycle-87 finite-KL llr regularity plus smoothing/Sobolev approximation, boundary control, and weak-FP action closure." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityMiddleObligation", "SALD.generalMovingTargetDiscreteKlLogRatioLlrDef", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.GeneralMovingTargetDiscreteKlLogRatioAdmissibilityClosure", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction", "eq:general_KL_derivative_0_discrete" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle88.lower_packet.log_ratio_admissibility" interface := "Lower packet: SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure derives Admissible (llr hatRhoS tildePiS) from finite KL plus the named approximation/closure package, replacing the broad hlog hypothesis by the smaller closure theorem boundary. Classification: narrows-source-cited-boundary." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityMiddlePacket", "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityMiddleObligation", "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation", "SALD.GeneralMovingTargetDiscreteKlLogRatioAdmissibilityClosure", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "Mathlib.Analysis.Calculus.ParametricIntegral" ] reusedBy := [ "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle88.reviewer_log_ratio_admissibility_check" interface := "Reviewer check: reject wrapper churn, hidden KL differentiability or weak-FP closure, reintroduced AC/measurability/integrability hypotheses already discharged by finite KL, theorem-status promotion, SLT/Lake dependency changes, or drift away from appendix.tex:1358-1366." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityMiddleObligation", "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryLowerObligation", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] /-- Cycle-89 upper pressure test for the discrete forward-KL theorem route. This packet does not introduce a new theorem wrapper. It records the requested post-cycle-84 pressure test: route `thm:forward-KL-discrete` through the currently compiled EM, LSI, DV, and Gronwall interfaces, then identify the first non-wrapper analytic blocker with source lines and Lean names. -/ def cycle89DiscreteForwardKlClosurePressureUpperPacket : DiscreteForwardKlUpperPacket where objective := "Global phase judgment: cycle 88 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the active EM packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387. The discrete thm:forward-KL-discrete pressure test routes through the compiled cycle-85--88 EM/KL handoffs plus the existing LSI, DV, Gronwall, and accumulated-error scalar interfaces, and the next non-wrapper blocker is the derivative/integration-by-parts/FI boundary in SALD.discreteForwardKlDerivativeObligation / sald.discrete_forward_kl.kl_derivative at appendix.tex:388-413, with the upstream raw KL/weak-FP inputs still at appendix.tex:1358-1387." sourceLabels := [ "thm:forward-KL-discrete", "proof:thm:forward-KL-discrete", "proof:thm:forward-KL-discrete:derivative", "proof:thm:forward-KL-discrete:conditional-fp", "eq:KL-derivative-1-discrete", "eq:KL-derivative-2-discrete", "eq:KL-derivative-3-discrete", "eq:general_KL_derivative_0_discrete", "proof:thm:general-moving-target-SALD-discrete:weak-fp-to-kl-derivative", "sald.discrete_forward_kl.kl_derivative", "SALD.discreteForwardKlDerivativeObligation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] modeDiscipline := [ "faithfulPaper Phase 1 only: keep main_body.tex:299-323 and appendix.tex:260-592 fixed, with sald_version_2.tex excluded.", "Explicit active-backend check: no reviewer blocker moved the shared packet away from sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387; the pressure test only identifies the first downstream blocker for thm:forward-KL-discrete.", "Do not add assumptions to thm:forward-KL-discrete. Any KL differentiability, weak-FP, integration-by-parts, boundary, density, or FI hypotheses remain inside named local backend obligations.", "Use lean-stat-learning-theory only as a local style reference. Do not import it, change Lake dependencies, or claim an SLT theorem is formalized." ] nonGoals := [ "Do not mark thm:forward-KL-discrete, sald.discrete_forward_kl.kl_derivative, sald.discrete_forward_kl.em_interpolation_fp, sald.general_moving_target_discrete.em_interpolation_fp, LSI, DV, or Gronwall formalized.", "Do not add another supplied-hypothesis wrapper around hlog, hkl, hfp, hIBP, or hFI.", "Do not work on display algebra, accumulated-error collection, coefficient-chain re-audit, frozen-delta estimates, or source-index rebaseline unless a reviewer finds a blocking anchor defect.", "Do not replace the paper route through appendix.tex:388-491 with a generalized theorem statement or a weakened derivative inequality." ] lowerPacket := [ "Classification: narrows-source-cited-boundary.", "Middle should synchronize the pressure-test result in the conversion window: cycles 51/56/61/66 already supply the scalar derivative-to-DV-to-Gronwall route under explicit analytic inputs; cycles 85--88 narrow the shared EM/KL inputs; the first remaining non-wrapper blocker is appendix.tex:388-413.", "Lower should target the integration-by-parts/Fisher-identification boundary inside SALD.discreteForwardKlDerivativeObligation / sald.discrete_forward_kl.kl_derivative: from the weak-FP log-ratio action and target transport action, prove or isolate the theorem giving eq:KL-derivative-1-discrete and eq:KL-derivative-2-discrete under explicit density, boundary/no-flux, log-ratio admissibility, target-time integrability, and finite-action hypotheses.", "If too large, lower must record exactly one smaller theorem boundary, such as divergence integration by parts for the log-ratio test, target-transport integration by parts, or Fisher-information identification of the first term. The source line and declaration must be named; wrapper-only restatement is rejected." ] reviewerChecklist := [ "Check that the packet records all three upper decisions: no cycle-88 recovery, Phase 1 stable, and the next lower packet is the discrete derivative/IBP/FI boundary reached by the pressure test.", "Check that the active shared EM backend remains sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387 and no theorem-route work escapes into unrelated APIs.", "Verify that the exact blocker is named as SALD.discreteForwardKlDerivativeObligation / sald.discrete_forward_kl.kl_derivative at appendix.tex:388-413, with upstream raw KL/weak-FP dependencies from eq:general_KL_derivative_0_discrete at appendix.tex:1358-1387.", "Reject any new supplied-hypothesis wrapper unless it removes an older supplied IBP/FI/log-action hypothesis or names a strictly smaller theorem boundary.", "No source constants, signs, theorem statements, statuses, SLT imports, or Lake dependencies may change; source-index and ASTIS check must pass." ] status := ProofStatus.obligation /-- Cycle-89 obligation recording the pressure-test blocker. -/ def cycle89DiscreteForwardKlClosurePressureUpperObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle89_closure_pressure_upper" statement := "Cycle 89 upper pressure-tests thm:forward-KL-discrete through the current compiled EM/KL handoffs and existing LSI, DV, Gronwall, and accumulated-error interfaces. The route reaches SALD.discreteForwardKlDerivativeObligation / sald.discrete_forward_kl.kl_derivative, where the first non-wrapper blocker is appendix.tex lines 388-413: turn the weak-FP log-ratio action and target transport action into eq:KL-derivative-1-discrete and eq:KL-derivative-2-discrete by integration by parts and Fisher-information identification. Upstream raw KL differentiation, mass conservation, log-ratio admissibility, target-time integrability, and weak conditional FP remain source-cited under appendix.tex lines 1358-1387. Classification: narrows-source-cited-boundary." source := saldForwardKlDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle89DiscreteForwardKlClosurePressureUpperPacket", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlDerivativeObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation", "SALD.cycle66DiscreteForwardKlAccumulatedDisplayLowerObligation", "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.kl_derivative", "sald.discrete_forward_kl.dv_velocity_bound", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "probability.lsi_to_kl_fi", "lem:dv_variation", "lem:gronwall", "eq:LSI-KL-FI" ] note := "Upper pressure-test record only. It identifies the next analytic blocker and lower packet; it does not prove KL differentiability, weak FP, integration by parts, FI identification, LSI, DV, Gronwall, or theorem closure." /-- Cycle-89 middle source map for the discrete forward-KL pressure test. -/ def cycle89DiscreteForwardKlClosurePressureMiddleObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle89_closure_pressure_middle" statement := "Cycle 89 middle synchronizes the pressure-test route for thm:forward-KL-discrete. The current EM/KL handoffs from cycles 85--88 and the existing LSI, DV, Gronwall, and accumulated-error interfaces route the theorem skeleton up to SALD.discreteForwardKlDerivativeObligation / sald.discrete_forward_kl.kl_derivative. The next lower packet is the non-wrapper derivative boundary at appendix.tex lines 388-413, with target-transport companion appendix.tex lines 414-436: prove or isolate the integration-by-parts and Fisher-information theorem turning the weak-FP log-ratio action and target transport action into eq:KL-derivative-1-discrete and eq:KL-derivative-2-discrete. Classification: narrows-source-cited-boundary." source := saldForwardKlDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle89DiscreteForwardKlClosurePressureUpperObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlDerivativeObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation", "SALD.cycle66DiscreteForwardKlAccumulatedDisplayLowerObligation", "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.kl_derivative", "sald.discrete_forward_kl.dv_velocity_bound", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "eq:KL-derivative-1-discrete", "eq:KL-derivative-2-discrete", "eq:general_KL_derivative_0_discrete", "eq:LSI-KL-FI", "lem:dv_variation", "lem:gronwall" ] note := "Middle source-map record only. It preserves the theorem statement and constants, keeps the shared EM backend active, and selects the appendix.tex:388-413 IBP/FI boundary as lower work; no new supplied-hypothesis wrapper or formalized analytic closure is introduced." /-- Cycle-89 lower handoff for the discrete derivative IBP/FI split. -/ def cycle89DiscreteForwardKlClosurePressureLowerObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle89_derivative_ibp_lower" statement := "Cycle 89 lower narrows the selected derivative blocker by compiling SALD.discreteForwardKlDerivativeSplitOfRawIbpsScalar and SALD.discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar. The old opaque scalar input dK=-FI+frozenCross+movingCross is replaced by a raw KL derivative split with an explicit mass term, a named first-term IBP/FI identity for eq:KL-derivative-1-discrete, and a named target-transport IBP identity for eq:KL-derivative-2-discrete. The remaining analytic theorem boundary is exactly: prove hfirst from weak-FP log-ratio action plus divergence integration by parts and Fisher identification, prove htarget from target transport integration by parts, and prove hmass from mass conservation under the density/boundary hypotheses. Classification: narrows-source-cited-boundary." source := saldForwardKlDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle89DiscreteForwardKlClosurePressureMiddleObligation", "SALD.discreteForwardKlDerivativeSplitOfRawIbpsScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfKlFiScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundScalar", "SALD.discreteForwardKlDerivativeObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "eq:KL-derivative-0-discrete", "eq:KL-derivative-1-discrete", "eq:KL-derivative-2-discrete", "eq:KL-derivative-5-discrete", "eq:LSI-KL-FI" ] note := "The compiled theorem is local scalar bookkeeping only. It removes the older supplied post-IBP derivative display as a single hypothesis and exposes the smaller missing analytic facts: mass conservation, first-term divergence IBP/FI identification, and target-transport IBP. It does not prove weak FP, log-ratio admissibility, integration by parts, Fisher information, LSI, DV, Gronwall, or theorem closure." /-- Cycle-89 proof-DAG pane for the discrete theorem closure pressure test. -/ def cycle89DiscreteForwardKlClosurePressureDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle89.global_phase_judgment" interface := "Cycle 89 judgment: cycle 88 passed and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single packet that reduces the largest proof risk after the pressure test is the discrete derivative/IBP/FI boundary reached by thm:forward-KL-discrete." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation", "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.kl_derivative" ] reusedBy := ["thm:forward-KL-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle89.active_em_backend_check" interface := "Before assigning lower work, confirm that the active shared backend still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387. The pressure test does not move the backend; it identifies the downstream discrete derivative blocker reached after the current EM/KL handoffs." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] reusedBy := ["thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle89_pressure_route" interface := "Pressure-test route: thm:forward-KL-discrete consumes current EM/KL wrappers, SALD.discreteForwardKlPostLsiDerivativeBoundOfKlFiScalar, discrete DV velocity, cycle-56 Gronwall input, and cycle-61/cycle-66 accumulated-error displays; the first non-wrapper blocker is the analytic derivative action before those scalar handoffs." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle89DiscreteForwardKlClosurePressureUpperObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlDerivativeObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation", "SALD.cycle66DiscreteForwardKlAccumulatedDisplayLowerObligation", "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation", "probability.lsi_to_kl_fi", "sald.discrete_forward_kl.dv_velocity_bound", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.accumulated_error_bridge" ] reusedBy := ["thm:forward-KL-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle89_middle_route_audit" interface := "Middle route audit: synchronize the pressure-test result with the source-to-Lean DAG. Cycles 85--88 supply the current EM/KL handoffs, cycles 51/56/61/66 supply theorem-specific scalar route interfaces, and the first remaining non-wrapper blocker is the derivative/IBP/FI action at appendix.tex:388-413 plus target transport at appendix.tex:414-436." source := saldForwardKlDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle89DiscreteForwardKlClosurePressureMiddleObligation", "SALD.cycle89DiscreteForwardKlClosurePressureUpperObligation", "SALD.discreteForwardKlDerivativeObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "sald.discrete_forward_kl.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "sald.discrete_forward_kl.kl_derivative", "thm:forward-KL-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle89_next_blocker" interface := "Next blocker: appendix.tex:388-413, eq:KL-derivative-1-discrete, plus appendix.tex:414-436 for the target transport action. Exact Lean declaration: SALD.discreteForwardKlDerivativeObligation / sald.discrete_forward_kl.kl_derivative. Lower should prove or isolate the integration-by-parts and Fisher-information theorem boundary, not add a supplied-hypothesis wrapper." source := saldForwardKlDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.discreteForwardKlDerivativeObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.em_conditional_fokker_planck", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "eq:general_KL_derivative_0_discrete", "eq:LSI-KL-FI" ] reusedBy := [ "sald.discrete_forward_kl.kl_derivative", "thm:forward-KL-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle89_lower_derivative_ibp_split" interface := "Lower proof-producing split: replace the opaque post-IBP derivative display by raw KL differentiation plus three named analytic facts: mass conservation, first-term divergence IBP/FI identification for eq:KL-derivative-1-discrete, and target-transport IBP for eq:KL-derivative-2-discrete. The scalar route to eq:KL-derivative-5-discrete now compiles through SALD.discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar." source := saldForwardKlDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle89DiscreteForwardKlClosurePressureLowerObligation", "SALD.discreteForwardKlDerivativeSplitOfRawIbpsScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfKlFiScalar", "SALD.discreteForwardKlDerivativeObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "eq:KL-derivative-0-discrete", "eq:KL-derivative-1-discrete", "eq:KL-derivative-2-discrete", "eq:LSI-KL-FI" ] reusedBy := [ "sald.discrete_forward_kl.kl_derivative", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "thm:forward-KL-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle89.reviewer_pressure_check" interface := "Reviewer check: accept only if the pressure test keeps theorem constants and source labels fixed, names SALD.discreteForwardKlDerivativeObligation / sald.discrete_forward_kl.kl_derivative as the exact next blocker, and prevents wrapper churn or status promotion." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle89DiscreteForwardKlClosurePressureUpperObligation", "SALD.discreteForwardKlStatementContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.kl_derivative" ] reusedBy := ["thm:forward-KL-discrete"] status := ProofStatus.obligation } ] /-- Cycle-90 upper packet for the reviewed discrete KL mass-conservation blocker. The active EM backend remains the shared `appendix.tex:1358-1387` route, but cycle 89's reviewer accepted a theorem-route blocker for `thm:forward-KL-discrete`: the raw discrete KL derivative still has an explicit mass term. This upper packet selects the smallest reviewed sub-boundary, `hmass : massTerm = 0`, before the larger first-term IBP/FI and target-transport IBP identities. -/ def cycle90DiscreteForwardKlMassConservationUpperPacket : DiscreteForwardKlUpperPacket where objective := "Global phase judgment: cycle 89 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the active EM backend still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, but the reviewer identified a theorem-route blocker after the pressure test. The single lower packet that now reduces the largest proof risk is the mass-conservation sub-boundary hmass : massTerm = 0 inside SALD.discreteForwardKlDerivativeObligation / sald.discrete_forward_kl.kl_derivative, sourced at eq:KL-derivative-0-discrete and appendix.tex:338-388, before the remaining hfirst and htarget IBP/FI identities." sourceLabels := [ "thm:forward-KL-discrete", "proof:thm:forward-KL-discrete", "proof:thm:forward-KL-discrete:derivative", "eq:KL-derivative-0-discrete", "eq:KL-derivative-1-discrete", "eq:KL-derivative-2-discrete", "eq:general_KL_derivative_0_discrete", "SALD.discreteForwardKlDerivativeObligation", "SALD.discreteForwardKlDerivativeSplitOfRawIbpsScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar", "sald.discrete_forward_kl.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp" ] modeDiscipline := [ "faithfulPaper Phase 1 only: keep main_body.tex:299-323, appendix.tex:260-592, and sald_version_2.tex exclusion fixed.", "Active-backend check: the shared EM packet remains sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387; cycle 90 leaves it only because the cycle-89 reviewer identified hmass/hfirst/htarget as the next theorem-route blocker.", "Do not add assumptions to thm:forward-KL-discrete or replace the paper derivative route. The selected work is the source statement since integral partial_s hat rho_s dx = 0 in eq:KL-derivative-0-discrete.", "Use Mathlib probability-measure, Measure.map, and derivative-under-integral APIs as local candidates; lean-stat-learning-theory remains a style reference only and is not imported." ] nonGoals := [ "Do not mark thm:forward-KL-discrete, sald.discrete_forward_kl.kl_derivative, the EM interpolation backend, LSI, DV, or Gronwall formalized.", "Do not start the broader hfirst divergence IBP/FI identity or htarget target-transport IBP unless the mass-conservation theorem is discharged or precisely blocked.", "Do not spend this cycle on a broad LSI/DV/Gronwall fallback; no named active-EM Mathlib blocker requires that fallback after the cycle-89 reviewer route.", "Do not add a wrapper that merely renames hmass. Either remove the supplied hmass hypothesis from a local route or name the smaller missing theorem with imports and hypotheses." ] lowerPacket := [ "Classification target: discharges-supplied-hypothesis if lower compiles a local theorem proving the mass term is zero and consumes it in the cycle-89 raw derivative route; otherwise narrows-source-cited-boundary with one exact missing theorem. Wrapper-only output is rejected-wrapper-churn.", "Lower should target exactly hmass : massTerm = 0 in SALD.discreteForwardKlDerivativeSplitOfRawIbpsScalar / SALD.discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar and the source line since int partial_s hat rho_s dx = 0 in eq:KL-derivative-0-discrete.", "The intended theorem boundary is: from hat rho_s = Law(hat X_s), probability mass one for each s, differentiability of the density/law pairing, and admissible constant test 1, prove integral partial_s hat rho_s dx = 0 without changing signs or constants.", "If Mathlib blocks the proof, middle/lower must name the exact missing API, for example derivative of total mass for a differentiable family of probability laws, differentiation under an integral for the constant weak test, or density-to-measure mass preservation." ] reviewerChecklist := [ "Check that the packet records all three upper decisions: no cycle-89 recovery, Phase 1 stable, and hmass mass conservation is the single lower packet after the reviewed pressure test.", "Check the explicit active-backend exception: sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387 remains active, but reviewer identified the theorem-route blocker hmass/hfirst/htarget.", "Reject any LSI, DV, Gronwall, display algebra, accumulated-error, or broad theorem-route work unless the mass-conservation boundary is first discharged or exactly blocked.", "Reject a supplied-hypothesis wrapper unless it removes the older hmass input or exposes a strictly smaller missing theorem with source, imports, and hypotheses.", "No source constants, theorem statements, statuses, SLT imports, Lake dependencies, or source labels may change; source-index and ASTIS check must pass." ] status := ProofStatus.obligation /-- Cycle-90 obligation selecting the mass-conservation lower boundary. -/ def cycle90DiscreteForwardKlMassConservationUpperObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle90_mass_conservation_upper" statement := "Cycle 90 upper records the reviewed theorem-route exception after the cycle-89 pressure test. The shared EM backend remains sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex lines 1358-1387, but the next lower packet is the smallest blocker inside SALD.discreteForwardKlDerivativeObligation / sald.discrete_forward_kl.kl_derivative: prove or sharply isolate hmass : massTerm = 0 from eq:KL-derivative-0-discrete, where the paper drops integral partial_s hat rho_s dx by mass conservation. Lower must either discharge this supplied hypothesis using local Mathlib/probability-law ingredients or name one exact missing theorem; wrappers around hmass are rejected." source := saldForwardKlDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle90DiscreteForwardKlMassConservationUpperPacket", "SALD.cycle89DiscreteForwardKlClosurePressureLowerObligation", "SALD.discreteForwardKlDerivativeSplitOfRawIbpsScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar", "SALD.discreteForwardKlDerivativeObligation", "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "sald.discrete_forward_kl.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "eq:KL-derivative-0-discrete", "eq:general_KL_derivative_0_discrete", "appendix.tex:338-388", "appendix.tex:1358-1387" ] note := "Upper routing record only. It selects the mass-conservation theorem boundary before hfirst and htarget; it does not prove KL differentiability, weak FP, integration by parts, FI identification, LSI, DV, Gronwall, or theorem closure." /-- Cycle-90 middle obligation for the compiled mass-derivative route. -/ def cycle90DiscreteForwardKlMassConservationMiddleObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle90_mass_conservation_middle" statement := "Cycle 90 middle translates eq:KL-derivative-0-discrete into the smaller Lean boundary for hmass. The compiled theorem SALD.discreteForwardKlMassTermZeroOfTotalMassDerivative proves that a mass term is zero once it is the HasDerivAt derivative of a locally constant total-mass function. SALD.discreteForwardKlDerivativeSplitOfMassDerivativeScalar and SALD.discreteForwardKlPostLsiDerivativeBoundOfMassDerivativeScalar then consume that theorem in the cycle-89 raw discrete KL route, replacing the primitive hmass : massTerm = 0 hypothesis by total-mass normalization plus the derivative-under-integral/constant-test witness. Classification: discharges-supplied-hypothesis for the scalar hmass input, while the remaining source-cited boundary is the analytic identification of massTerm with the derivative of the probability-law total mass." source := saldForwardKlDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle90DiscreteForwardKlMassConservationUpperObligation", "SALD.discreteForwardKlMassTermZeroOfTotalMassDerivative", "SALD.discreteForwardKlDerivativeSplitOfMassDerivativeScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfMassDerivativeScalar", "SALD.discreteForwardKlDerivativeSplitOfRawIbpsScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar", "SALD.discreteForwardKlDerivativeObligation", "sald.discrete_forward_kl.kl_derivative", "eq:KL-derivative-0-discrete", "appendix.tex:338-388", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic" ] note := "Compiled local calculus handoff plus remaining obligation. It removes hmass as a primitive scalar hypothesis under explicit total-mass derivative and local probability-mass-one hypotheses; it does not prove density-to-law mass preservation, differentiation under the integral, weak FP, integration by parts, FI, LSI, DV, Gronwall, or theorem closure." /-- Cycle-90 lower obligation for mapped-law constant-test mass conservation. -/ def cycle90DiscreteForwardKlMassConservationLowerObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle90_mass_conservation_lower" statement := "Cycle 90 lower compiles the mapped-law constant weak-test route for eq:KL-derivative-0-discrete. SALD.discreteForwardKlLawConstantTestTotalMassOne proves that the constant test integrates to one against Measure.map (hatX s) P when P is a probability measure. SALD.discreteForwardKlLawConstantTestHasDerivAtZero uses AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample to show the concrete mapped-law constant-test pairing has derivative zero. SALD.discreteForwardKlMassTermZeroOfLawConstantTestDerivative, SALD.discreteForwardKlDerivativeSplitOfLawConstantTestMassScalar, and SALD.discreteForwardKlPostLsiDerivativeBoundOfLawConstantTestMassScalar then feed this into the cycle-89 raw derivative route without a primitive hmass hypothesis or abstract totalMass/local-normalization input. Classification: discharges-supplied-hypothesis for the local scalar hmass/normalization route; the remaining source-cited boundary is identifying the paper massTerm with the derivative of this concrete constant weak-test pairing for hat rho_s = Law(hat X_s)." source := saldForwardKlDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle90DiscreteForwardKlMassConservationMiddleObligation", "SALD.discreteForwardKlLawConstantTestTotalMassOne", "SALD.discreteForwardKlLawConstantTestHasDerivAtZero", "SALD.discreteForwardKlMassTermZeroOfLawConstantTestDerivative", "SALD.discreteForwardKlDerivativeSplitOfLawConstantTestMassScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfLawConstantTestMassScalar", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "AutoSamplingTheory.lawMapIntegral", "SALD.discreteForwardKlDerivativeSplitOfRawIbpsScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar", "SALD.discreteForwardKlDerivativeObligation", "sald.discrete_forward_kl.kl_derivative", "eq:KL-derivative-0-discrete", "appendix.tex:338-388", "Mathlib.MeasureTheory.Measure.Map", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic" ] note := "Compiled local probability-law and derivative-transport handoff plus remaining obligation. It proves the constant-test mapped-law derivative is zero, but does not prove the raw KL differentiability theorem, density derivative representation, weak FP, integration by parts, FI, LSI, DV, Gronwall, theorem closure, or any SLT result." /-- Cycle-90 proof-DAG pane for the discrete KL mass-conservation blocker. -/ def cycle90DiscreteForwardKlMassConservationDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle90.global_phase_judgment" interface := "Cycle 90 judgment: cycle 89 passed and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the largest remaining proof risk after reviewer pressure is the discrete KL derivative mass-conservation hypothesis hmass before hfirst/htarget." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle89DiscreteForwardKlClosurePressureLowerObligation", "SALD.discreteForwardKlDerivativeSplitOfRawIbpsScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar", "sald.discrete_forward_kl.kl_derivative" ] reusedBy := ["thm:forward-KL-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle90.active_em_backend_exception_check" interface := "Confirm that the active shared backend remains sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387. Cycle 90 leaves direct EM work only because the cycle-89 reviewer identified the theorem-route blocker hmass/hfirst/htarget." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation", "SALD.cycle89DiscreteForwardKlClosurePressureLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.kl_derivative" ] reusedBy := ["thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle90_middle_mass_derivative_route" interface := "Middle source map: replace the primitive hmass scalar input by the smaller theorem boundary totalMass'(s0)=massTerm plus totalMass=1 near s0, then use the compiled HasDerivAt local-constant lemma to feed the cycle-89 raw KL route." source := saldForwardKlDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle90DiscreteForwardKlMassConservationMiddleObligation", "SALD.discreteForwardKlMassTermZeroOfTotalMassDerivative", "SALD.discreteForwardKlDerivativeSplitOfMassDerivativeScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfMassDerivativeScalar", "SALD.discreteForwardKlDerivativeSplitOfRawIbpsScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar", "eq:KL-derivative-0-discrete" ] reusedBy := [ "sald.discrete_forward_kl.kl_derivative", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "thm:forward-KL-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle90_lower_law_constant_test_mass" interface := "Lower compiled route: specialize totalMass to the concrete mapped-law constant weak-test pairing for hat rho_s=Law(hat X_s), prove its law integral is one and its derivative is zero using Measure.map probability normalization and AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample, then feed the cycle-89 raw KL scalar route. Remaining boundary: identify the paper massTerm with this concrete derivative." source := saldForwardKlDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle90DiscreteForwardKlMassConservationLowerObligation", "SALD.discreteForwardKlLawConstantTestTotalMassOne", "SALD.discreteForwardKlLawConstantTestHasDerivAtZero", "SALD.discreteForwardKlMassTermZeroOfLawConstantTestDerivative", "SALD.discreteForwardKlDerivativeSplitOfLawConstantTestMassScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfLawConstantTestMassScalar", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.discreteForwardKlDerivativeSplitOfRawIbpsScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar", "eq:KL-derivative-0-discrete" ] reusedBy := [ "sald.discrete_forward_kl.kl_derivative", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "thm:forward-KL-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle90_mass_conservation_lower_packet" interface := "Selected lower packet: prove or isolate massTerm = 0 from the source claim int partial_s hat rho_s dx = 0 in eq:KL-derivative-0-discrete, using the probability-law normalization of hat rho_s and derivative-under-integral/constant-test semantics. Cycle 90 lower has discharged the local constant-test derivative part for a concrete Measure.map law; the remaining boundary is the raw KL massTerm-to-constant-test-derivative identification." source := saldForwardKlDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle90DiscreteForwardKlMassConservationUpperObligation", "SALD.cycle90DiscreteForwardKlMassConservationMiddleObligation", "SALD.cycle90DiscreteForwardKlMassConservationLowerObligation", "SALD.discreteForwardKlMassTermZeroOfTotalMassDerivative", "SALD.discreteForwardKlMassTermZeroOfLawConstantTestDerivative", "SALD.discreteForwardKlPostLsiDerivativeBoundOfMassDerivativeScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfLawConstantTestMassScalar", "SALD.discreteForwardKlDerivativeSplitOfRawIbpsScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar", "SALD.discreteForwardKlDerivativeObligation", "eq:KL-derivative-0-discrete", "eq:general_KL_derivative_0_discrete", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "Mathlib.Analysis.Calculus.ParametricIntegral" ] reusedBy := [ "sald.discrete_forward_kl.kl_derivative", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "thm:forward-KL-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle90.reviewer_mass_conservation_check" interface := "Reviewer check: accept only if cycle 90 either removes the supplied hmass hypothesis or names a strictly smaller mass-conservation theorem boundary; reject LSI/DV/Gronwall drift, wrapper churn, and status promotion." source := saldForwardKlDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle90DiscreteForwardKlMassConservationUpperObligation", "SALD.cycle90DiscreteForwardKlMassConservationMiddleObligation", "SALD.discreteForwardKlDerivativeObligation", "sald.discrete_forward_kl.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := ["thm:forward-KL-discrete"] status := ProofStatus.obligation } ] /-! ### Cycle 91: conditional-kernel component-field backfill -/ /-- Cycle-91 middle obligation for the conditional-kernel theorem boundary. This records the post-cycle-90 return to the active EM backend: `appendix.tex:1368-1377`, where `bar b_{k,s}` is defined by conditioning on `\hat X_s=x`. The cycle keeps the lower packet on the Mathlib `condDistrib`/named-law component-field boundary instead of adding another abstract supplied-hypothesis wrapper. -/ def cycle91GeneralMovingTargetDiscreteConditionalKernelMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle91_conditional_kernel_middle" statement := "Cycle 91 middle records the global phase judgment: cycle 90 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet that reduces the largest remaining proof risk is the conditional-kernel/named-law field boundary for sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex lines 1368-1377. The source map targets hat rho_s = Law(hat X_s), ProbabilityTheory.condDistrib X_k^eta hat X_s P, named component fields condC_{k,s} and condScore_{k,s}, and measurable/integrable bar b_{k,s}." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle90DiscreteForwardKlMassConservationLowerObligation", "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryLowerObligation", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "appendix.tex:1368-1377" ] note := "Middle workflow record only. It keeps theorem statements, constants, source labels, and statuses fixed; the selected proof-producing target is the concrete condDistrib-to-barB regularity theorem below." /-- Cycle-91 lower/backfill obligation for the compiled concrete `condDistrib` drift-regularity theorem. The new theorem removes the older supplied component regularity hypotheses for the canonical conditional-integral route: after the SALD-specific a.e. version equalities for `condC` and `condScore` are supplied, Mathlib regularity of `condDistrib` integrals and the existing component-combination wrapper produce regularity of `barB`. -/ def cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle91_conditional_kernel_lower" statement := "Cycle 91 lower compiles SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions, after SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularity. Under hatRhoS = Measure.map hatXAtS P, finite-measure/standard-Borel hypotheses, AEMeasurable hatXAtS and X_k^eta, AEStronglyMeasurable and Integrable frozen guide and score integrands on the joint law Measure.map (fun omega => (hatXAtS omega, X_k^eta omega)) P, measurable equality sets, and sample-space a.e. equalities between the canonical condDistrib component integrals composed with hatXAtS and condC/condScore composed with hatXAtS, the theorem proves AEStronglyMeasurable and Integrable barB under hatRhoS. Classification: discharges-supplied-hypothesis for the law-space component-version hypotheses and component-field/barB regularity hypotheses behind SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents and SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff. Remaining source-cited boundary: prove the sample-space conditional-expectation/disintegration equalities and equality-set measurability for the SALD fields, plus conditional-kernel compatibility." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelMiddleObligation", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularity", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "Mathlib.Probability.Kernel.CondDistrib", "proof:thm:general-moving-target-SALD-discrete:conditional-drift", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing local theorem plus remaining obligation. It transports sample-space component version equalities through hatRhoS = Law(hatXAtS) using Mathlib ae_map_iff. It does not construct the conditional law, prove the sample-space conditional-expectation equalities, weak Fokker-Planck identity, KL derivative, LSI, DV, Gronwall, theorem closure, or any SLT result." /-- Cycle-91 proof-DAG pane for the conditional-kernel component-field backfill. -/ def cycle91GeneralMovingTargetDiscreteConditionalKernelDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle91.global_phase_judgment" interface := "Cycle 91 judgment: cycle 90 passed and needs no recovery; Phase 1 theorem skeletons are stable enough for cited-theory backfill; the largest remaining proof risk is the conditional-kernel/named-law field theorem at appendix.tex:1368-1377." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelMiddleObligation", "SALD.cycle90DiscreteForwardKlMassConservationLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle91.middle_conditional_kernel_source_map" interface := "Middle source map: keep the TeX window on hat rho_s=Law(hat X_s), condDistrib X_k^eta hat X_s P, component fields condC/condScore, and the bar b_{k,s} combination; reject wrapper churn around abstract KernelIntegralField regularity." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelMiddleObligation", "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryLowerObligation", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle91.lower_condDistrib_named_drift_regular" interface := "Lower proof-producing theorem: SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions derives AEStronglyMeasurable and Integrable barB under hatRhoS from sample-space component version equalities, ae_map_iff transport, canonical condDistrib component integral regularity, and the pointwise barB component formula." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularity", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "Mathlib.Probability.Kernel.CondDistrib" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle91.remaining_conditional_kernel_boundary" interface := "Remaining exact theorem boundary: prove the SALD-specific sample-space conditional-expectation/disintegration equalities for condC/condScore after composing with hatXAtS, the measurable equality-set side conditions needed by ae_map_iff, and the conditional-kernel compatibility theorem; then route endpoint weak-FP readiness through the existing endpoint compatibility wrappers." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularity", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff", "Mathlib.Probability.Kernel.Condexp", "Mathlib.Probability.Kernel.CondDistrib" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-! ### Cycle 92: generator-to-law weak-FP split-generator backfill -/ /-- Cycle-92 middle obligation for the weak generator-to-law boundary. This keeps the conversion window on `appendix.tex:1379-1387`: the paper invokes the Fokker--Planck equation for the frozen EM interpolation, and the current Lean route should remove or narrow one supplied input to the existing sample-generator law-transport handoff. -/ def cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle92_weak_fp_split_generator_middle" statement := "Cycle 92 middle records the active backend judgment: cycle 91 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet remains sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to appendix.tex:1379-1387. The lower target is the generator-to-law weak Fokker-Planck boundary consumed by SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff. To avoid wrapper churn, the selected proof-producing refinement must remove one supplied input; this cycle targets hgeneratorSplit by using a definitionally split generator action driftAction + diffusionAction while keeping sample-path HasDerivAt, law weak derivative, drift source action, and diffusion source action as explicit remaining analytic boundaries." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation", "SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle source-to-Lean synchronization only. The source signs and sigma_eta^2/2 coefficient are fixed; no weak Fokker-Planck theorem, KL derivative, theorem closure, SLT result, or Lake dependency is promoted." /-- Cycle-92 lower obligation for the split-generator law-transport handoff. -/ def cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle92_weak_fp_split_generator_lower" statement := "Cycle 92 lower compiles SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff and SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff for appendix.tex:1379-1387. The first theorem removes the supplied hgeneratorSplit premise from SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff by setting generatorAction phi definitionally to driftAction phi + diffusionAction phi. The second theorem removes the separate hlawDerivative input for the direct weak-test derivative route: it transports the sample-space split-generator HasDerivAt result through AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample and rewrites the derivative value to the source signs. Classification: discharges-supplied-hypothesis for hgeneratorSplit and narrows-source-cited-boundary for hlawDerivative. Remaining exact analytic boundaries are sample-path/parametric-integral differentiation of the split generator sum, drift source-action identification through bar b_{k,s}, diffusion source-action identification with coefficient sigma_eta^2/2, density/time regularity, admissible-test closure, and boundary hypotheses." source := saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitMiddleObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "AutoSamplingTheory.lawMapIntegral", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing local theorems plus remaining obligation. They reduce supplied generator-split/law-derivative inputs but do not prove the EM generator theorem, parametric integral differentiation, source-action identifications, conditional law compatibility, KL derivative, LSI, DV, Gronwall, theorem closure, or any SLT result." /-- Cycle-92 proof-DAG pane for the split-generator weak-FP boundary. -/ def cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle92.middle_generator_to_law_source_map" interface := "Middle source map: appendix.tex:1379-1387 is the Fokker-Planck invocation for the frozen EM interpolation. Reduce one input to the existing sample-generator law-transport handoff rather than adding another source-sign wrapper; choose the generator split because the source generator is drift plus diffusion before the drift/divergence and Laplacian source actions are identified." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitMiddleObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation", "SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle92.lower_sample_split_generator_handoff" interface := "Lower proof-producing handoff: SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff removes hgeneratorSplit by making generatorAction the definitional sum driftAction+diffusionAction, then applies the existing sample-space derivative to mapped-law weak derivative transport and source-sign packaging." source := saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle92.lower_law_derivative_of_sample_split_generator" interface := "Lower proof-producing derivative handoff: SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff transports the sample-space split-generator derivative to the mapped-law weak-test integral and rewrites the derivative value to the paper's drift/divergence and diffusion/Laplacian source signs, so the direct weak derivative route no longer needs a separate hlawDerivative premise." source := saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle92.remaining_generator_to_law_boundary" interface := "Remaining exact boundary after the split-generator and direct law-derivative handoffs: prove the sample-path/Bochner parametric-integral derivative of the split generator sum, connect driftAction to -div(hat rho_s*bar b_{k,s}) through the conditional drift, and connect diffusionAction to +(sigma_eta^2/2) Delta hat rho_s with boundary hypotheses. A named partialS route can still use a supplied law-derivative witness, but the direct weak-test HasDerivAt route no longer needs hlawDerivative." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-! ### Cycle 93: KL/log-ratio mass-derivative backfill -/ /-- Cycle-93 middle obligation for the KL/log-ratio mass derivative boundary. This returns to `appendix.tex:1358-1366` after cycle 92's accepted weak-FP split-generator handoff. The selected middle packet removes one supplied input behind the raw KL-to-weak-FP handoff: the primitive scalar mass-conservation premise is replaced by the mapped-law constant weak-test derivative for `hat rho_s = Law(hat X_s)`. -/ def cycle93GeneralMovingTargetDiscreteKlMassDerivativeMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle93_kl_mass_derivative_middle" statement := "Cycle 93 middle records the active backend judgment: cycle 92 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet remains sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to appendix.tex:1358-1366. The selected boundary is the KL/log-ratio mass derivative inside eq:general_KL_derivative_0_discrete: replace the primitive hmass : massTerm=0 input behind SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction by the concrete mapped-law constant-test derivative for hat rho_s=Law(hat X_s). Classification: discharges-supplied-hypothesis for hmass; raw KL differentiability, target-time integrability, admissible llr closure, and integration-by-parts/FI identification remain source-cited boundaries." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation", "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryLowerObligation", "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlMassTermZeroOfLawConstantTestDerivative", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "eq:general_KL_derivative_0_discrete", "appendix.tex:1358-1366", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Middle source-to-Lean synchronization only. It selects hmass as the supplied hypothesis to discharge by a local mapped-law theorem; it does not prove raw KL differentiability, target-time derivative integrability, log-ratio admissibility closure, weak FP, integration by parts, FI, theorem closure, SLT reuse, or any Lake dependency." /-- Cycle-93 lower obligation for the compiled mapped-law mass handoff. -/ def cycle93GeneralMovingTargetDiscreteKlMassDerivativeLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle93_kl_mass_derivative_lower" statement := "Cycle 93 lower compiles SALD.generalMovingTargetDiscreteKlMassTermZeroOfLawConstantTestDerivative, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfLawConstantTestMassAndSourceSignsWithLogAction, and SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfFiniteKlLlrLawConstantTestMassWithLogAction for appendix.tex:1358-1366. The new handoffs remove the primitive hmass : massTerm=0 premise when massTerm is identified as the HasDerivAt derivative of the constant weak-test pairing integral 1 d Law(hat X_s), and remove the primitive hlog premise for the exact Mathlib llr hatRho tildePi test by using finite KL plus the cycle-88 admissibility closure package. They use AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample, derivative uniqueness for a constant sample-space integral, and SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure. Classification: discharges-supplied-hypothesis for primitive hmass and primitive hlog in the llr-specialized handoff; narrows-source-cited-boundary for the remaining massTerm-to-constant-test-derivative identification and closure package. Remaining exact analytic boundaries are the raw KL differentiability formula hklRaw, target-time derivative identification, the closure package internals, weak-FP source signs, integration by parts, and FI identification." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeMiddleObligation", "SALD.generalMovingTargetDiscreteKlMassTermZeroOfLawConstantTestDerivative", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfLawConstantTestMassAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfFiniteKlLlrLawConstantTestMassWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "eq:general_KL_derivative_0_discrete", "appendix.tex:1358-1366", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Proof-producing local theorem plus remaining obligation. It removes hmass as a primitive scalar input for the general moving-target KL handoff, but does not prove the raw KL derivative display, target-time term, admissible log-ratio closure, weak FP, integration by parts, FI, LSI, DV, Gronwall, theorem closure, or any SLT result." /-- Cycle-93 proof-DAG pane for the KL/log-ratio mass derivative boundary. -/ def cycle93GeneralMovingTargetDiscreteKlMassDerivativeDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle93.middle_kl_mass_derivative_source_map" interface := "Middle source map: appendix.tex:1358-1366 differentiates KL(hat rho_s||tilde pi_s), keeps the target-time term with the paper's minus sign, and drops int partial_s hat rho_s dx=0. Cycle 93 targets only that mass-drop sub-boundary by using hat rho_s=Law(hat X_s) and the mapped-law constant weak-test derivative." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeMiddleObligation", "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation", "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryLowerObligation", "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation", "eq:general_KL_derivative_0_discrete" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle93.lower_law_constant_test_mass" interface := "Lower proof-producing theorem: SALD.generalMovingTargetDiscreteKlMassTermZeroOfLawConstantTestDerivative proves massTerm=0 from the derivative of integral 1 d Measure.map (hatX s) P; SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfLawConstantTestMassAndSourceSignsWithLogAction then feeds that result into the raw KL plus weak-FP source-sign handoff without a primitive hmass premise." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeLowerObligation", "SALD.generalMovingTargetDiscreteKlMassTermZeroOfLawConstantTestDerivative", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfLawConstantTestMassAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample" ] reusedBy := [ "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "sald.general_moving_target_discrete.kl_derivative", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle93.lower_finite_kl_llr_law_mass_handoff" interface := "Lower proof-producing theorem: SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfFiniteKlLlrLawConstantTestMassWithLogAction specializes the law-constant mass handoff to the exact Mathlib llr hatRho tildePi weak test. Finite KL plus SALD.GeneralMovingTargetDiscreteKlLogRatioAdmissibilityClosure supplies admissibility, so this composite no longer needs primitive hlog or hmass inputs; raw KL differentiation, target-time identification, the closure package internals, weak FP, IBP, and FI remain explicit boundaries." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeLowerObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfFiniteKlLlrLawConstantTestMassWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfLawConstantTestMassAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlMassTermZeroOfLawConstantTestDerivative", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.GeneralMovingTargetDiscreteKlLogRatioAdmissibilityClosure", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle93.remaining_raw_kl_target_time_boundary" interface := "Remaining exact boundary after the finite-KL llr and law-mass handoff: prove hklRaw by differentiating KL(hat rho_s||tilde pi_s) under the integral at the Mathlib llr weak test, identify massTerm with the constant-test derivative, and prove the target-time derivative formula int (hat rho_s/tilde pi_s) partial_s tilde pi_s. The cycle-88 closure package internals, downstream IBP, and FI identification remain separate." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeLowerObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfFiniteKlLlrLawConstantTestMassWithLogAction", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.Analysis.Calculus.ParametricIntegral", "appendix.tex:1358-1366" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "sald.discrete_forward_kl.kl_derivative", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] /-! ### Cycle 94: conditional-drift weak action boundary -/ /-- Cycle-94 middle obligation for the conditional-drift source-action boundary. This returns to the weak Fokker--Planck invocation at `appendix.tex:1379-1387` after the cycle-93 KL/log-ratio mass handoff. The selected supplied hypothesis is the primitive `hdriftSource` input used by the cycle-77/82/92 weak-FP source routes. The new lower-ready boundary factors that input through the paper's conditional drift `bar b_{k,s}`: first identify the frozen drift generator action with the weak test-gradient pairing against `barB`, then prove the source-cited no-boundary divergence identity for that pairing. -/ def cycle94GeneralMovingTargetDiscreteWeakFpDriftActionMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle94_weak_fp_drift_action_middle" statement := "Cycle 94 middle records the active backend judgment: cycle 93 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet remains sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to appendix.tex:1379-1387. The selected supplied hypothesis is hdriftSource behind the weak conditional Fokker-Planck source-sign and direct law-derivative handoffs. Replace it by the smaller conditional-drift weak-action boundary: driftAction phi is the weak test-gradient pairing of the named conditional drift barB with phi, and that pairing equals -(driftDiv phi) by the source-cited divergence/no-boundary theorem. Classification: narrows-source-cited-boundary." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeLowerObligation", "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation", "SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle source-to-Lean synchronization only. It does not prove the conditional-expectation generator identity, divergence/integration-by-parts theorem, diffusion source action, weak Fokker-Planck equation, KL derivative, theorem closure, SLT result, or any Lake dependency." /-- Cycle-94 lower obligation for the compiled `barB` drift-action handoff. -/ def cycle94GeneralMovingTargetDiscreteWeakFpDriftActionLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle94_weak_fp_drift_action_lower" statement := "Cycle 94 lower compiles SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction, SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff, and SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff for appendix.tex:1379-1387. The handoffs remove the primitive hdriftSource premise from both the direct sample-split generator law-derivative route and the normalized weak-FP source-sign route. In its place they require two narrower source-cited facts: the frozen EM drift generator action equals the weak pairing of barB with the test gradient, and the barB weak pairing equals -(driftDiv phi) by the divergence/no-boundary theorem. Remaining exact analytic boundaries are the conditional-expectation generator action through condDistrib, the divergence/integration-by-parts theorem for hatRhoS*barB, diffusion source-action identification, sample-path parametric-integral differentiation, density/time regularity, admissible-test closure, and boundary hypotheses." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionMiddleObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.ParametricIntegral", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] note := "Proof-producing local handoff plus remaining obligation. It narrows hdriftSource for the law-derivative and source-sign consumers but does not claim the barB conditional-drift generator theorem or the divergence/no-boundary theorem is formalized." /-- Cycle-94 proof-DAG pane for the conditional-drift weak action boundary. -/ def cycle94GeneralMovingTargetDiscreteWeakFpDriftActionDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle94.middle_weak_fp_drift_action_source_map" interface := "Middle source map: appendix.tex:1379-1387 invokes the Fokker-Planck equation for hat rho_s with drift source -div(hat rho_s*bar b_{k,s}); after cycle 91 has named barB and cycle 92 has split the generator, replace primitive hdriftSource by the barB weak-pairing plus divergence/no-boundary theorem boundary." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionMiddleObligation", "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation", "SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle94.lower_barB_drift_action_handoff" interface := "Lower proof-producing handoff: SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction derives the old hdriftSource shape from two smaller source-cited facts, then SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff and SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff feed that derived drift source into the cycle-92 direct mapped-law derivative and normalized source-sign routes." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff", "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle94.remaining_barB_divergence_boundary" interface := "Remaining exact boundary: prove driftAction phi = integral pairing of barB with grad phi from the condDistrib conditional expectation defining bar b_{k,s}, then prove that this pairing equals -(driftDiv phi) by the divergence/integration-by-parts/no-boundary theorem for hatRhoS*barB." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1368-1377", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-! ### Cycle 95: discrete theorem pressure test after the `barB` handoff -/ /-- Cycle-95 upper packet for the discrete forward-KL closure pressure test. This is an upper-role route record only. It rechecks `thm:forward-KL-discrete` after the cycle-94 `barB` weak-action handoff and selects the next non-wrapper blocker without changing the theorem statement or promoting any analytic backend. -/ def cycle95DiscreteForwardKlClosurePressureUpperPacket : DiscreteForwardKlUpperPacket where objective := "Global phase judgment: cycle 94 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet that now reduces the largest proof risk remains the active EM backend sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to the conditional-expectation generator pairing and barB divergence/no-boundary theorem at appendix.tex:1368-1387. The thm:forward-KL-discrete pressure test routes through the current EM wrappers, KL/log-ratio mass handoffs, LSI, DV, Gronwall, and accumulated-error interfaces; the first non-wrapper blocker is ASTIS.SALD.cycle94.remaining_barB_divergence_boundary, reused by sald.discrete_forward_kl.em_interpolation_fp before theorem closure." sourceLabels := [ "thm:forward-KL-discrete", "proof:thm:forward-KL-discrete", "sald.discrete_forward_kl.em_interpolation_fp", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "ASTIS.SALD.cycle94.remaining_barB_divergence_boundary", "SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionLowerObligation", "eq:general_KL_derivative_0_discrete", "proof:thm:general-moving-target-SALD-discrete:weak-fp-to-kl-derivative", "lem:dv_variation", "lem:gronwall", "eq:LSI-KL-FI" ] modeDiscipline := [ "faithfulPaper Phase 1 only: keep main_body.tex:299-323, appendix.tex:260-592, appendix.tex:1358-1387, and sald_version_2.tex exclusion fixed.", "Active-backend check: reviewer did not move the packet away from sald.general_moving_target_discrete.em_interpolation_fp; cycle 95 pressure-tests thm:forward-KL-discrete only to locate the next actual blocker.", "Do not add assumptions to thm:forward-KL-discrete or replace the paper route through EM weak FP, KL derivative, LSI, DV, Gronwall, and accumulated-error display matching.", "Use Mathlib conditional-kernel, conditional-expectation, Measure.map, Bochner integral, and divergence-theorem APIs as local candidates; lean-stat-learning-theory remains a style reference only and is not imported." ] nonGoals := [ "Do not mark thm:forward-KL-discrete, sald.discrete_forward_kl.em_interpolation_fp, sald.general_moving_target_discrete.em_interpolation_fp, weak FP, KL derivative, LSI, DV, or Gronwall formalized.", "Do not add another wrapper around hdriftSource, hsourceSigns, hklRaw, hlog, hmass, hfirst, htarget, or the accumulated-error display.", "Do not work on display algebra, source-index rebaseline, broad reusable APIs, LSI/DV/Gronwall fallback, or unrelated theorem-route audits while the barB divergence boundary remains open.", "Do not weaken the source signs: keep -div(hat rho_s * bar b_{k,s}) and +(sigma_eta^2/2) Delta hat rho_s exactly as in appendix.tex:1379-1387." ] lowerPacket := [ "Classification: narrows-source-cited-boundary; discharges-supplied-hypothesis only if lower proves one of the two remaining barB facts and removes the corresponding supplied input from the cycle-94 handoff.", "Middle should synchronize the pressure-test result: current compiled wrappers carry thm:forward-KL-discrete up to the shared EM weak-FP source-sign backend, and the next exact blocker is ASTIS.SALD.cycle94.remaining_barB_divergence_boundary.", "Lower should target exactly the conditional-expectation generator pairing and divergence/no-boundary theorem behind SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction: prove driftAction phi equals the weak test-gradient pairing against barB from condDistrib, or prove that pairing equals -(driftDiv phi) for hatRhoS * barB.", "If blocked, lower must name one smaller theorem with source, imports, and hypotheses, such as the condDistrib/condexp generator identity for barB, the weak divergence integration-by-parts theorem, or the boundary/no-flux condition needed by that theorem." ] reviewerChecklist := [ "Check that the upper packet states all three global decisions: no cycle-94 recovery, Phase 1 stable enough for cited-theory backfill, and the barB divergence boundary is the single lower packet.", "Verify that the active lower packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, not display algebra or broad theorem-route churn.", "Confirm the pressure-test blocker is named exactly as ASTIS.SALD.cycle94.remaining_barB_divergence_boundary / SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction with source appendix.tex:1368-1387.", "Reject a new supplied-hypothesis wrapper unless it removes a cycle-94 barB weak-action hypothesis or exposes a strictly smaller source-cited theorem boundary.", "No source constants, signs, theorem statements, statuses, SLT imports, Lake dependencies, or source labels may change; source-index and ASTIS check must pass." ] status := ProofStatus.obligation /-- Cycle-95 obligation recording the discrete theorem pressure-test blocker. -/ def cycle95DiscreteForwardKlClosurePressureUpperObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle95_closure_pressure_upper" statement := "Cycle 95 upper pressure-tests thm:forward-KL-discrete after the cycle-94 barB weak-action handoff. The existing theorem route already has compiled scalar wrappers for the KL/log-ratio mass handoff, LSI, DV, Gronwall, and accumulated-error display under named analytic inputs. The first non-wrapper blocker is not a new display wrapper: it is ASTIS.SALD.cycle94.remaining_barB_divergence_boundary, namely the conditional-expectation generator identity producing the weak pairing against barB and the divergence/no-boundary theorem identifying that pairing with -driftDiv at appendix.tex lines 1368-1387. Classification: narrows-source-cited-boundary." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle95DiscreteForwardKlClosurePressureUpperPacket", "SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfFiniteKlLlrLawConstantTestMassWithLogAction", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.discreteForwardKlPostLsiDerivativeBoundOfLawConstantTestMassScalar", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "SALD.discreteForwardKlMainDisplayBoundScalar", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.kl_derivative", "sald.discrete_forward_kl.dv_velocity_bound", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.accumulated_error_bridge", "ASTIS.SALD.cycle94.remaining_barB_divergence_boundary", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "main_body.tex:299-323" ] note := "Upper routing record only. It selects the next lower packet and does not prove conditional expectation, divergence integration by parts, weak FP, KL differentiability, LSI, DV, Gronwall, accumulated-error closure, or either discrete theorem." /-- Cycle-95 middle source map for the discrete forward-KL pressure test. -/ def cycle95DiscreteForwardKlClosurePressureMiddleObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle95_closure_pressure_middle" statement := "Cycle 95 middle synchronizes the thm:forward-KL-discrete pressure test with the active EM backend. The current compiled route passes through the cycle-94 barB weak-action handoff, the cycle-93 KL mass/log-ratio handoff, the existing LSI/DV/Gronwall interfaces, and the accumulated-error display wrappers, then stops at the same non-wrapper blocker selected by upper: ASTIS.SALD.cycle94.remaining_barB_divergence_boundary. The lower-ready boundary is the source-cited pair of facts at appendix.tex lines 1368-1387: prove the conditional-expectation generator pairing for barB via condDistrib/condexp, or prove the divergence/no-boundary theorem identifying that weak pairing with -driftDiv for hatRhoS * barB. Classification: narrows-source-cited-boundary." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle95DiscreteForwardKlClosurePressureUpperObligation", "SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionLowerObligation", "SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfFiniteKlLlrLawConstantTestMassWithLogAction", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.discreteForwardKlPostLsiDerivativeBoundOfLawConstantTestMassScalar", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "SALD.discreteForwardKlMainDisplayBoundScalar", "ASTIS.SALD.forward_KL_discrete.cycle95_pressure_route", "ASTIS.SALD.forward_KL_discrete.cycle95_next_blocker", "ASTIS.SALD.cycle94.remaining_barB_divergence_boundary", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "main_body.tex:299-323", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.kl_derivative" ] note := "Middle source-map record only. It keeps the discrete theorem statement, constants, signs, source labels, and backend statuses unchanged, and rejects wrapper churn unless a later lower packet removes one of the two remaining barB weak-action inputs." /-- Cycle-95 lower obligation for the component-pairing reduction of the `barB` drift-action blocker. -/ def cycle95DiscreteForwardKlClosurePressureLowerObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle95_closure_pressure_lower" statement := "Cycle 95 lower compiles SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings and SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBComponentPairings. This pressure-test result does not add another theorem-route wrapper: it narrows the first half of ASTIS.SALD.cycle94.remaining_barB_divergence_boundary by replacing the direct supplied fact driftAction phi = weakGradPairing barB phi with component conditional-drift pairings for condC and condScore plus linearity/congruence of the weak gradient pairing and the existing barB component formula. The next non-wrapper analytic blocker is now sharper: prove the component conditional-expectation generator pairings from condDistrib/condexp at appendix.tex lines 1368-1377, and still prove the no-boundary divergence theorem weakGradPairing barB phi = -(driftDiv phi) for hatRhoS * barB at appendix.tex lines 1379-1387. Classification: narrows-source-cited-boundary." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle95DiscreteForwardKlClosurePressureMiddleObligation", "SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] note := "Proof-producing local algebra plus remaining obligation. It removes the direct barB action premise in favor of component conditional-expectation pairings and weak-pairing linearity, but it does not prove those component pairings, the divergence/no-boundary theorem, weak FP, KL differentiability, LSI, DV, Gronwall, accumulated-error closure, or either discrete theorem." /-- Cycle-95 proof-DAG pane for the post-cycle-94 pressure test. -/ def cycle95DiscreteForwardKlClosurePressureDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle95.global_phase_judgment" interface := "Cycle 95 judgment: cycle 94 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet is the barB conditional-expectation generator and divergence/no-boundary theorem behind the active EM weak-FP backend." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionLowerObligation", "SALD.cycle95DiscreteForwardKlClosurePressureUpperPacket", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] reusedBy := ["thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle95_pressure_route" interface := "Pressure-test route: thm:forward-KL-discrete can be routed through the current EM, KL/log-ratio mass, LSI, DV, Gronwall, and accumulated-error wrappers only under the shared weak-FP source-sign input; after cycle 94, that source-sign input is blocked by the barB divergence boundary, not by another scalar theorem wrapper." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle95DiscreteForwardKlClosurePressureUpperObligation", "SALD.cycle89DiscreteForwardKlClosurePressureLowerObligation", "SALD.cycle90DiscreteForwardKlMassConservationLowerObligation", "SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeLowerObligation", "SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionLowerObligation", "SALD.discreteForwardKlPostLsiDerivativeBoundOfLawConstantTestMassScalar", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "SALD.discreteForwardKlMainDisplayBoundScalar", "sald.discrete_forward_kl.dv_velocity_bound", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.accumulated_error_bridge" ] reusedBy := ["thm:forward-KL-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle95_middle_route_audit" interface := "Middle route audit: synchronize the pressure test after cycle 94. Current compiled wrappers carry thm:forward-KL-discrete to the shared weak-FP source-sign input, and the first non-wrapper blocker is the barB conditional-expectation generator pairing plus divergence/no-boundary theorem, not a new KL, LSI, DV, Gronwall, or display wrapper." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle95DiscreteForwardKlClosurePressureMiddleObligation", "SALD.cycle95DiscreteForwardKlClosurePressureUpperObligation", "SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionLowerObligation", "SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff", "SALD.discreteForwardKlPostLsiDerivativeBoundOfLawConstantTestMassScalar", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "SALD.discreteForwardKlMainDisplayBoundScalar", "ASTIS.SALD.cycle94.remaining_barB_divergence_boundary" ] reusedBy := [ "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle95_lower_barB_component_pairing" interface := "Lower proof-producing reduction: SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings proves driftAction phi = weakGradPairing barB phi from the component formula barB = dotTk*condC + sigmaCoeff*condScore, weak-pairing additivity/smul/congruence, and component drift pairings. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBComponentPairings then feeds that into the cycle-94 drift-source handoff. Remaining blocker is the component condDistrib/condexp generator pairing plus the divergence/no-boundary theorem." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle95DiscreteForwardKlClosurePressureLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "ASTIS.SALD.cycle94.remaining_barB_divergence_boundary" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.forward_KL_discrete.cycle95_next_blocker" interface := "Next non-wrapper blocker after the lower component-pairing reduction: prove the component conditional-expectation generator pairings for condC and condScore using the source definition of bar b_{k,s}, and prove weakGradPairing barB phi = -(driftDiv phi) by divergence/integration-by-parts/no-boundary for hatRhoS * barB." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1368-1377", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle95.reviewer_pressure_check" interface := "Reviewer check: accept only a proof-producing discharge of one barB weak-action input or a strictly smaller theorem boundary with source lines and imports. Reject wrapper churn around hdriftSource, source signs, KL mass/log-ratio inputs, LSI/DV/Gronwall, or display algebra." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle95DiscreteForwardKlClosurePressureUpperObligation", "ASTIS.SALD.cycle94.remaining_barB_divergence_boundary", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := ["cycle 95 reviewer", "cycle 96 lower packet"] status := ProofStatus.obligation } ] /-! ### Cycle 96: middle packet for `condDistrib`/`condexp` generator pairings -/ /-- Cycle-96 middle obligation for the active EM conditional-law backend. The upper packet rejected the non-EM LSI/DV/Gronwall fallback because the EM backend still has named work at `appendix.tex:1368-1387`. This middle record therefore keeps lower work on the conditional-law side: turn the Mathlib `condDistrib`/`condexp` representation of the source conditional drift components into the generator weak-action pairings consumed by the cycle-95 `barB` component theorem. -/ def cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle96_condexp_generator_pairing_middle" statement := "Cycle 96 middle rejects the non-EM LSI/DV/Gronwall fallback because cycle 96 upper found active named EM blockers, not an exhausted Mathlib/theory gap. The lower-ready boundary remains sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex lines 1358-1387, narrowed to appendix.tex lines 1368-1377: prove one component conditional-expectation generator pairing for condC or condScore from Mathlib condDistrib/condexp, named hatRhoS = Law(hatX_s), the joint law of (hatX_s, X_k^eta), the existing component conditional-integral regularity helpers, and the weak test-gradient pairing used by SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings. This removes or strictly narrows the component-pairing supplied input before any divergence/no-boundary theorem or non-EM backend is attempted. Classification: narrows-source-cited-boundary." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle95DiscreteForwardKlClosurePressureLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "ProbabilityTheory.condExp_ae_eq_integral_condExpKernel", "lean-stat-learning-theory/SLT/EfronStein.lean", "lean-stat-learning-theory/SLT/GaussianMeasure.lean", "ASTIS.SALD.forward_KL_discrete.cycle95_next_blocker", "ASTIS.SALD.cycle94.remaining_barB_divergence_boundary", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] note := "Middle source-map record only. It adds no supplied-hypothesis wrapper, proves no component pairing, and promotes no theorem/backend status. Lower should either discharge one component generator-pairing input used by the cycle-95 barB component route, or record one smaller missing Mathlib conditional-expectation theorem with exact hypotheses." /-- Cycle-96 lower obligation for the compiled one-component pairing handoff. -/ def cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle96_condexp_generator_pairing_lower" statement := "Cycle 96 lower compiles SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion for appendix.tex lines 1368-1377. This one-component handoff narrows the supplied condC/condScore generator-pairing premise: if the named component field is hatRhoS-a.e. equal to the canonical condDistrib integral and the weak test-gradient pairing is congruent under that hatRhoS-a.e. equality, then it is enough to prove the generator action against the canonical conditional-integral field. Classification: narrows-source-cited-boundary. The remaining smaller theorem is the Mathlib/local-analysis conditional-expectation generator identity for that canonical condDistrib/condexp integral; divergence/no-boundary for hatRhoS * barB remains separate." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingMiddleObligation", "SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] note := "Lower proof-producing boundary narrowing only. It proves the local a.e.-version-to-named-field weak-pairing handoff, but it does not prove the canonical condDistrib/condexp generator theorem, regular conditional law, weak FP equation, KL derivative, divergence/no-boundary theorem, theorem closure, SLT import, or Lake dependency change." /-- Cycle-96 proof-DAG pane for the condexp generator-pairing middle packet. -/ def cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle96.middle_non_em_fallback_rejected" interface := "Cycle 96 middle gate: the non-EM LSI/DV/Gronwall fallback is not allowed because the active EM boundary still has named conditional-law and divergence/no-boundary blockers. Keep the packet on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle95DiscreteForwardKlClosurePressureLowerObligation", "ASTIS.SALD.forward_KL_discrete.cycle95_next_blocker", "ASTIS.SALD.cycle94.remaining_barB_divergence_boundary" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle96.middle_condexp_generator_pairing_boundary" interface := "Lower-ready boundary: prove one component conditional-expectation generator pairing for condC or condScore. The theorem should start from Mathlib condDistrib/condexp for X_k^eta | hat X_s=x, named hatRhoS = Law(hatX_s), and the existing component conditional-integral regularity/versioning helpers, then produce the weak test-gradient pairing input consumed by the cycle-95 barB component route." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingMiddleObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "ProbabilityTheory.condExp_ae_eq_integral_condExpKernel", "appendix.tex:1368-1377" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBComponentPairings", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle96.lower_condDistrib_component_pairing_handoff" interface := "Lower proof-producing handoff: SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion turns a named condC/condScore component pairing into the smaller canonical conditional-integral action theorem, using hatRhoS-a.e. equality with the condDistrib integral and weak-pairing a.e. congruence. Remaining work is the canonical condDistrib/condexp generator identity for one component." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingLowerObligation", "SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "appendix.tex:1368-1377" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBComponentPairings", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle96.lower_packet.condexp_component_generator_pairing" interface := "Accepted lower packet classification: this cycle narrows-source-cited-boundary by compiling the named-field a.e.-version handoff and exposing the smaller canonical condDistrib/condexp generator identity. It is discharges-supplied-hypothesis only after that canonical component action theorem is proved and removes one component pairing premise from the cycle-95 barB component route. Repackaging source signs, KL, LSI, DV, Gronwall, or display assumptions is rejected-wrapper-churn." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingMiddleObligation", "SALD.cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingLowerObligation", "SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion", "ASTIS.SALD.cycle96.middle_condexp_generator_pairing_boundary", "ASTIS.SALD.cycle96.lower_condDistrib_component_pairing_handoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings", "ASTIS.SALD.forward_KL_discrete.cycle95_next_blocker" ] reusedBy := ["cycle 96 lower", "cycle 96 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 97: canonical `condDistrib` disintegration pairing -/ /-- Cycle-97 middle obligation for the canonical conditional-integral pairing. This cycle stops adding supplied-hypothesis wrappers and proves the Mathlib-style map-law disintegration theorem that lower needs for the canonical `condDistrib` component action. -/ def cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle97_canonical_condDistrib_pairing_middle" statement := "Cycle 97 middle keeps the active backend on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex lines 1358-1387, narrowed to appendix.tex lines 1368-1377. It compiles AutoSamplingTheory.condDistribIntegralMapIntegral and AutoSamplingTheory.condDistribIntegralNamedLawIntegral: for X=hatX_s, Y=X_k^eta, finite P, hatRhoS=P.map hatX_s, and an integrable paired component test f on the joint law of (hatX_s,X_k^eta), the law-space integral of the canonical condDistrib conditional integral equals the sample-space paired component integral. This discharges the Mathlib disintegration/pullout sub-hypothesis behind the canonical condDistrib component generator action, but it does not prove weak FP, divergence/no-boundary, log-ratio admissibility, or theorem closure. Classification: discharges-supplied-hypothesis for the map-law disintegration part of the cycle-96 hcanonical boundary." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "AutoSamplingTheory.condDistribIntegralMapIntegral", "AutoSamplingTheory.condDistribIntegralNamedLawIntegral", "SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion", "SALD.cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "ProbabilityTheory.compProd_map_condDistrib", "MeasureTheory.Measure.integral_compProd", "MeasureTheory.integral_map", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] note := "Exact lower theorem boundary after this proof: expand componentAction and weakGradPairing to the paired Bochner integrals used by AutoSamplingTheory.condDistribIntegralNamedLawIntegral, prove the required integrability for that paired test-gradient integrand, and then feed hcanonical in SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion. The divergence/no-boundary theorem for hatRhoS * barB remains separate." /-- Cycle-97 lower-ready obligation after the compiled disintegration theorem. -/ def cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle97_canonical_condDistrib_pairing_lower_ready" statement := "Cycle 97 lower compiles SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfIntegralAction for appendix.tex lines 1368-1377. The theorem uses AutoSamplingTheory.condDistribIntegralNamedLawIntegral with X=hatX_s, Y=X_k^eta, hatRhoS=Law(hatX_s), and f=(test-gradient pairing with one frozen component integrand), then feeds SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion. It removes the raw cycle-96 hcanonical premise under explicit paired-integrand integrability, componentAction definition, weakGradPairing canonical-integral definition, hfieldAe, and hweakAeCongr hypotheses. Classification: narrows-source-cited-boundary; the remaining exact boundary is proving those SALD definition-alignment and paired-integrability facts from the paper weak-test-gradient semantics." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingMiddleObligation", "SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfIntegralAction", "AutoSamplingTheory.condDistribIntegralNamedLawIntegral", "SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBComponentPairings", "ASTIS.SALD.cycle96.lower_condDistrib_component_pairing_handoff", "appendix.tex:1368-1377", "appendix.tex:1379-1387" ] note := "This is not a fresh supplied-hypothesis wrapper. The compiled theorem removes hcanonical as a primitive assumption and leaves smaller obligations: paired-integrand integrability, sample component-action definition, canonical weak-pairing definition, field a.e. version, and a.e.-congruence for the weak pairing." /-- Cycle-97 proof-DAG pane for the canonical `condDistrib` pairing packet. -/ def cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle97.global_phase_judgment" interface := "Cycle 96 passed reviewer/build, so no recovery is needed. Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill. The single lower packet that reduces the largest remaining risk is the canonical condDistrib disintegration pairing behind appendix.tex:1368-1377, feeding the cycle-96 hcanonical component-action boundary before divergence/no-boundary work." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingLowerObligation", "SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion", "ASTIS.SALD.forward_KL_discrete.cycle95_next_blocker" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle97.compiled_condDistrib_disintegration_pairing" interface := "Compiled local Mathlib-style theorem: AutoSamplingTheory.condDistribIntegralMapIntegral, plus the named-law variant AutoSamplingTheory.condDistribIntegralNamedLawIntegral, disintegrates an integrable paired component test through condDistrib Y X mu and rewrites the conditioning marginal as hatRhoS=mu.map X." source := saldGeneralMovingTargetDiscreteCondDistribIntegralMathlibSource targetLean := "AutoSamplingTheory/Probability.lean" dependsOn := [ "AutoSamplingTheory.condDistribIntegralMapIntegral", "AutoSamplingTheory.condDistribIntegralNamedLawIntegral", "ProbabilityTheory.compProd_map_condDistrib", "MeasureTheory.Measure.integral_compProd", "MeasureTheory.integral_map", "Mathlib.Probability.Kernel.CondDistrib" ] reusedBy := [ "SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle97.lower_packet.canonical_condDistrib_component_pairing" interface := "Compiled lower handoff: SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfIntegralAction instantiates AutoSamplingTheory.condDistribIntegralNamedLawIntegral with f equal to the weak test-gradient pairing against condC or condScore, removes hcanonical as a primitive premise, and feeds SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion. Remaining obligations are paired-integrand integrability and the concrete componentAction/weakGradPairing definition equalities." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingLowerObligation", "SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfIntegralAction", "AutoSamplingTheory.condDistribIntegralNamedLawIntegral", "SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings", "appendix.tex:1368-1377" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBComponentPairings", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle97.rejected_wrapper_churn_guard" interface := "Reject fresh wrappers that merely rename hcanonical, hdriftSource, source signs, KL mass/log-ratio, LSI, DV, Gronwall, or theorem displays. Accept only a compiled use of the condDistrib disintegration theorem, a paired-integrability proof, or one exact missing theorem needed to align the weak-pairing definitions." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "AutoSamplingTheory.condDistribIntegralNamedLawIntegral", "ASTIS.SALD.cycle96.lower_packet.condexp_component_generator_pairing", "ASTIS.SALD.cycle94.remaining_barB_divergence_boundary" ] reusedBy := ["cycle 97 lower", "cycle 97 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 98: `barB` divergence/no-boundary integral boundary -/ /-- Cycle-98 middle obligation for the `barB` weak-divergence boundary. The active source span is the Fokker--Planck source-sign line `appendix.tex:1379-1387`. Cycle 98 keeps the packet on the divergence half of `ASTIS.SALD.cycle94.remaining_barB_divergence_boundary`: replace the primitive `hbarBWeakDivergence` input by a law-integral weak-pairing definition and a no-boundary integration-by-parts theorem for `hatRhoS * barB`. -/ def cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle98_barB_divergence_no_boundary_middle" statement := "Cycle 98 middle keeps the active backend on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex lines 1358-1387, narrowed to the generator-to-law weak-FP drift source signs at appendix.tex lines 1379-1387. The selected lower packet is the barB divergence/no-boundary half of ASTIS.SALD.cycle94.remaining_barB_divergence_boundary: replace the primitive hbarBWeakDivergence input by two smaller source-cited facts, namely weakGradPairing barB phi is the hatRhoS-law integral of the test-gradient contraction with barB, and driftDiv phi is the negative of that same integral by the divergence/integration-by-parts/no-boundary theorem for hatRhoS * barB. Classification: narrows-source-cited-boundary." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral", "SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionLowerObligation", "SALD.cycle95DiscreteForwardKlClosurePressureLowerObligation", "SALD.cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingLowerObligation", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] note := "Middle source-map record plus lower-ready target. It adds no theorem-status promotion and does not prove the analytic divergence theorem, boundary decay/no-flux hypotheses, admissible-test closure, density regularity, or weak Fokker-Planck equation." /-- Cycle-98 lower-ready obligation for the compiled integral no-boundary handoff. -/ def cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle98_barB_divergence_no_boundary_lower" statement := "Cycle 98 compiles SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral, SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral, SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound, SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing, and SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral. The first two remove the raw hbarBWeakDivergence premise from the cycle-94 drift-source route under explicit law-integral/no-boundary hypotheses. The lower bounded-pairing theorem discharges the paired-integrability supplied hypothesis from Integrable barB hatRhoS plus an a.e. test-gradient contraction bound. Classification: discharges-supplied-hypothesis for hpairIntegrable, while the analytic divergence/no-boundary identity itself remains source-cited." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryMiddleObligation", "SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "appendix.tex:1358-1387" ] note := "Proof-producing local handoff plus remaining source-cited boundary. The hpairIntegrable input is no longer primitive when barB is integrable and the test-gradient contraction has the recorded norm bound. Lower/reviewer should now target the concrete no-boundary IBP theorem, the weakGradPairing/driftDiv definition equalities, or the concrete test-gradient bound, not a new wrapper around hbarBWeakDivergence." /-- Cycle-98 proof-DAG pane for the `barB` no-boundary integral packet. -/ def cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle98.middle_barB_divergence_no_boundary_source_map" interface := "Middle source map: appendix.tex:1379-1387 uses the source sign -div(hat rho_s*bar b_{k,s}). After cycles 94-97, the remaining divergence half should be proved as an integration-by-parts/no-boundary identity for the law integral of the test-gradient contraction with barB." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryMiddleObligation", "ASTIS.SALD.cycle94.remaining_barB_divergence_boundary", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle98.lower_packet.barB_no_boundary_integral" interface := "Compiled local handoff: SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral proves the old hbarBWeakDivergence shape from the law-integral weak-pairing definition and the no-boundary divergence identity; SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral feeds it into the cycle-94 drift-source route. The bounded-pairing variants additionally remove hpairIntegrable by proving it from Integrable barB hatRhoS and an a.e. test-gradient contraction bound." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryLowerObligation", "SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle98.remaining_exact_no_boundary_theorem" interface := "Remaining exact theorem: for admissible weak tests, prove the concrete a.e. norm bound for the test-gradient contraction from barB regularity/test admissibility, align weakGradPairing with that Bochner integral, and prove driftDiv phi equals its negative by the Mathlib/local divergence theorem under the paper boundary/no-flux hypotheses. The abstract hpairIntegrable premise has been discharged by SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound once the bound is supplied." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-! ### Cycle 99: raw KL derivative finite-KL `llr` boundary -/ /-- Cycle-99 middle obligation for the KL/log-ratio analytic boundary. This returns to `appendix.tex:1358-1366` and narrows the remaining primitive `hklRaw` display to a source-cited raw-KL theorem at the exact Mathlib `llr hatRho tildePi` weak test. It reuses cycle-87 finite-KL log-ratio regularity, cycle-88 admissibility closure, and cycle-93 mapped-law mass handoffs instead of adding another broad wrapper. -/ def cycle99GeneralMovingTargetDiscreteRawKlDerivativeMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle99_raw_kl_derivative_middle" statement := "Cycle 99 middle records the active KL/log-ratio packet over appendix.tex:1358-1366. Classification: narrows-source-cited-boundary. The exact missing theorem boundary is no longer a primitive scalar hklRaw; it is SALD.GeneralMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlr, a finite-KL llr raw-differentiability package exposing density/path regularity, endpoint-safe differentiation under the KL integral, llr weak-action integrability, target-time term integrability and formula, and the mapped-law constant-test mass derivative. The compiled handoff SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlBoundaryAtFiniteKlLlrWithLogAction then feeds that package into the existing finite-KL llr, law-mass, and weak-FP source-sign route. Theorem statements, source signs, coefficients, and statuses remain unchanged." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeLowerObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfFiniteKlLlrLawConstantTestMassWithLogAction", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.GeneralMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlr", "SALD.generalMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlrHklRaw", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlBoundaryAtFiniteKlLlrWithLogAction", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "eq:general_KL_derivative_0_discrete", "appendix.tex:1358-1366", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] note := "Middle source-to-Lean synchronization only. It rejects wrapper churn around hklRaw and records the smaller source-cited KL differentiability theorem with explicit Mathlib/local hypotheses. No KL differentiation theorem, weak FP theorem, IBP/FI identity, theorem closure, SLT result, or Lake dependency is promoted." /-- Cycle-99 lower-ready obligation for the finite-KL `llr` raw-KL package. -/ def cycle99GeneralMovingTargetDiscreteRawKlDerivativeLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle99_raw_kl_derivative_lower" statement := "Cycle 99 lower-ready interface compiles SALD.generalMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlrHklRaw, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlBoundaryAtFiniteKlLlrWithLogAction, and the stricter no-mass route SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfNoMassRawKlBoundaryAtFiniteKlLlrWithLogAction. The source-cited analytic package SALD.GeneralMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlr removes the mapped-law mass-derivative field from the remaining raw-KL boundary by proving the constant mapped-law weak-test derivative locally from AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample. The remaining appendix.tex:1358-1366 theorem boundary is the endpoint-safe no-mass KL differentiation display at the finite-KL llr test with target-time integrability/formula; cycle-88 admissibility closure and downstream weak-FP/IBP/FI remain separate." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle99GeneralMovingTargetDiscreteRawKlDerivativeMiddleObligation", "SALD.GeneralMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlr", "SALD.generalMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlrHklRaw", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlBoundaryAtFiniteKlLlrWithLogAction", "SALD.GeneralMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlr", "SALD.generalMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlrHkl", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfNoMassRawKlBoundaryAtFiniteKlLlrWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfFiniteKlLlrLawConstantTestMassWithLogAction", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1358-1366" ] note := "The compiled no-mass handoff is local composition plus the concrete mapped-law constant-test derivative. It discharges the mass-derivative field for this finite-KL llr route, but the no-mass raw KL differentiability theorem remains source-cited/obligation-level and must be proved before marking hklRaw discharged." /-- Cycle-99 proof-DAG pane for the raw KL finite-KL `llr` boundary. -/ def cycle99GeneralMovingTargetDiscreteRawKlDerivativeDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle99.middle_raw_kl_derivative_source_map" interface := "Middle source map: appendix.tex:1358-1366 differentiates KL(hat rho_s||tilde pi_s) at the log-ratio weak test. Cycle 99 narrows the old hklRaw scalar display to a source-cited raw-KL finite-KL llr package with explicit regularity, target-time, and mass-derivative fields." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle99GeneralMovingTargetDiscreteRawKlDerivativeMiddleObligation", "SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeLowerObligation", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "eq:general_KL_derivative_0_discrete" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle99.raw_kl_derivative_at_finite_kl_llr" interface := "Source-cited analytic theorem boundary: prove SALD.GeneralMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlr from Mathlib KL/llr regularity, ParametricIntegral/Bochner differentiation under the KL integral, target-time derivative integrability and formula, and the mapped-law constant-test mass derivative for hat rho_s=Law(hat X_s)." source := saldGeneralMovingTargetDiscreteKlLogRatioMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.GeneralMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlr", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "appendix.tex:1358-1366" ] reusedBy := [ "SALD.generalMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlrHklRaw", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlBoundaryAtFiniteKlLlrWithLogAction", "sald.general_moving_target_discrete.kl_derivative" ] status := ProofStatus.sourceCited }, { id := "ASTIS.SALD.cycle99.no_mass_raw_kl_derivative_at_finite_kl_llr" interface := "Strictly smaller source-cited analytic theorem boundary: prove SALD.GeneralMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlr, i.e. the no-mass derivative display dK = partialS (llr hatRho tildePi) - targetTimeTerm from finite-KL llr regularity, endpoint-safe KL integral differentiation, and target-time integrability/formula. The mapped-law constant-test derivative is proved locally in the compiled handoff." source := saldGeneralMovingTargetDiscreteKlLogRatioMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.GeneralMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlr", "SALD.generalMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlrHkl", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1358-1366" ] reusedBy := [ "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfNoMassRawKlBoundaryAtFiniteKlLlrWithLogAction", "sald.general_moving_target_discrete.kl_derivative" ] status := ProofStatus.sourceCited }, { id := "ASTIS.SALD.cycle99.lower_packet.raw_kl_llr_handoff" interface := "Compiled local handoff: derive hklRaw from the source-cited finite-KL llr package and compose it with the existing finite-KL admissibility, mapped-law mass, and weak-FP source-sign theorem. The no-mass variant additionally proves the mapped-law constant-test derivative internally and removes the mass-derivative field from the exact llr route. It does not prove the KL differentiation package fields." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle99GeneralMovingTargetDiscreteRawKlDerivativeLowerObligation", "SALD.generalMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlrHklRaw", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlBoundaryAtFiniteKlLlrWithLogAction", "SALD.generalMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlrHkl", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfNoMassRawKlBoundaryAtFiniteKlLlrWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfFiniteKlLlrLawConstantTestMassWithLogAction", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "sald.discrete_forward_kl.kl_derivative", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized } ] /-! ### Cycle 100: `barB` weak-pairing definition alignment -/ /-- Cycle-100 middle obligation for the `barB` weak-pairing definition alignment inside the no-boundary drift source route. -/ def cycle100GeneralMovingTargetDiscreteBarBWeakGradDefMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle100_barB_weakGrad_def_middle" statement := "Cycle 100 middle keeps the active backend on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex lines 1379-1387. Classification: narrows-source-cited-boundary. The selected supplied hypothesis is hweakGradIntegral in the cycle-98 bounded no-boundary route. The lower packet should remove that premise by specializing weakGradPairing to the law-integral definition fun f phi => int x, fieldPairing phi f x d hatRhoS, so weakGradPairing barB phi is definitionally the hatRhoS integral of the barB test-gradient contraction. The concrete contraction bound and the no-boundary driftDiv identity remain the exact source-cited analytic hypotheses." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing", "SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound", "ASTIS.SALD.cycle98.remaining_exact_no_boundary_theorem", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] note := "Middle source synchronization only. This rejects a new wrapper around hbarBWeakDivergence and targets one definition-alignment premise inside the existing no-boundary theorem boundary." /-- Cycle-100 lower-ready obligation for the compiled weak-pairing definition alignment handoff. -/ def cycle100GeneralMovingTargetDiscreteBarBWeakGradDefLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle100_barB_weakGrad_def_lower" statement := "Cycle 100 compiles SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef and SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryWeakGradDef. These local theorems remove hweakGradIntegral from SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral by making the weakGradPairing definition explicit as the hatRhoS-law integral of fieldPairing. Classification: narrows-source-cited-boundary for the generator-to-law weak-FP drift source route; remaining blockers are the concrete a.e. contraction bound hpairNormBound and the no-boundary divergence theorem hdivNoBoundary for hatRhoS * barB." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefMiddleObligation", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryWeakGradDef", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing", "SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387" ] note := "Proof-producing local definition-alignment handoff. It does not prove the contraction bound, the divergence/no-boundary theorem, weak FP, KL derivative, theorem closure, SLT import, or any Lake dependency." /-- Cycle-100 proof-DAG pane for the `barB` weak-pairing definition alignment packet. -/ def cycle100GeneralMovingTargetDiscreteBarBWeakGradDefDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle100.middle_barB_weakGrad_def_source_map" interface := "Middle source map: appendix.tex:1379-1387 uses the drift source sign -div(hat rho_s*bar b_{k,s}). After cycle 98, align weakGradPairing with the hatRhoS law integral by taking weakGradPairing f phi to be int x, fieldPairing phi f x d hatRhoS." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefMiddleObligation", "ASTIS.SALD.cycle98.remaining_exact_no_boundary_theorem", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle100.lower_packet.barB_weakGrad_definition_alignment" interface := "Compiled local handoff: SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef and SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryWeakGradDef remove hweakGradIntegral by definitional specialization of weakGradPairing. hpairNormBound and hdivNoBoundary remain exact analytic boundaries." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefLowerObligation", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryWeakGradDef", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle100.remaining_no_boundary_after_weakGrad_def" interface := "Remaining exact theorem after the weakGrad definition alignment: prove the concrete a.e. norm bound for fieldPairing phi barB and prove driftDiv phi = -int x, fieldPairing phi barB x d hatRhoS by the no-boundary divergence theorem for hatRhoS * barB." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-- Cycle-100 lower-ready obligation for the inner-gradient contraction handoff. -/ def cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle100_barB_inner_gradient_bound_lower" statement := "Cycle 100 lower compiles SALD.generalMovingTargetDiscreteBarBPairNormBoundOfInnerGradientBound, SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryInnerGradientBound, and SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound. These local theorems remove the supplied hpairNormBound premise from the cycle-100 weakGrad-definition route when fieldPairing is the real inner product of the weak-test gradient with barB. Cauchy--Schwarz reduces the contraction boundary to the smaller source-cited weak-test gradient norm estimate hgradNormBound, while hdivNoBoundary remains the no-boundary divergence theorem for hatRhoS * barB." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefLowerObligation", "SALD.generalMovingTargetDiscreteBarBPairNormBoundOfInnerGradientBound", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryInnerGradientBound", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryWeakGradDef", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] note := "Proof-producing contraction handoff only. It does not prove the weak-test gradient bound, no-boundary divergence theorem, weak FP, KL derivative, theorem closure, SLT import, or any Lake dependency." /-- Cycle-100 proof-DAG pane for the inner-gradient contraction packet. -/ def cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle100.lower_packet.barB_inner_gradient_contraction" interface := "Compiled local handoff: Cauchy--Schwarz proves the hpairNormBound contraction for fieldPairing phi barB x = inner (testGrad phi x) (barB x), then routes through the cycle-100 weakGrad law-integral definition alignment. The old hpairNormBound supplied premise is replaced by the smaller hgradNormBound weak-test gradient estimate." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundLowerObligation", "SALD.generalMovingTargetDiscreteBarBPairNormBoundOfInnerGradientBound", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryInnerGradientBound", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "Mathlib.Analysis.InnerProductSpace.Basic" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle100.remaining_no_boundary_after_inner_gradient" interface := "Remaining exact theorem after the inner-gradient contraction: prove the weak-test gradient norm bound ||testGrad phi x|| <= pairBound phi and prove driftDiv phi = -int x, inner (testGrad phi x) (barB x) d hatRhoS by the no-boundary divergence theorem for hatRhoS * barB." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-! ### Cycle 101: post-cycle-100 discrete pressure-test synchronization -/ /-- Cycle-101 middle synchronization for the discrete forward-KL closure pressure test. This is intentionally not another broad theorem-route wrapper. Cycles 89 and 95 already record the route through the discrete theorem wrappers. After the cycle-100 weak-pairing and inner-gradient handoffs, the pressure test stops at the exact no-boundary/weak-test-gradient boundary consumed by the shared EM weak-FP backend. -/ def cycle101DiscreteForwardKlClosurePressureMiddleObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle101_closure_pressure_middle" statement := "Cycle 101 middle pressure-tests thm:forward-KL-discrete through the currently compiled EM wrappers and existing LSI/DV/Gronwall/accumulated-error interfaces. Classification: narrows-source-cited-boundary. No new theorem-route wrapper is needed: cycles 89 and 95 already carry the theorem route, and cycle 100 compiles the weakGrad law-integral definition alignment plus the Cauchy-Schwarz inner-gradient contraction. The exact next non-wrapper blocker is ASTIS.SALD.cycle100.remaining_no_boundary_after_inner_gradient, equivalently the hgradNormBound and hdivNoBoundary premises of SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound. The source anchor is appendix.tex lines 1379-1387, with barB supplied by appendix.tex lines 1368-1377. Lower should prove hdivNoBoundary from a precise no-boundary divergence theorem for hatRhoS * barB, or strictly narrow it to one Mathlib/local theorem with imports and hypotheses, while carrying hgradNormBound explicitly." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle95DiscreteForwardKlClosurePressureMiddleObligation", "SALD.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefLowerObligation", "SALD.cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound", "ASTIS.SALD.cycle100.remaining_no_boundary_after_inner_gradient", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.kl_derivative", "probability.lsi_to_kl_fi", "lem:dv_variation", "lem:gronwall", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "main_body.tex:301-323", "appendix.tex:260-592", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] note := "Middle pressure-test synchronization only. It identifies the next exact non-wrapper blocker after the compiled EM wrappers, LSI/DV/Gronwall interfaces, and cycle-100 handoffs; it does not prove weak FP, KL differentiation, no-boundary IBP, the weak-test gradient estimate, theorem closure, SLT import, or any Lake dependency." /-- Cycle-101 lower product-rule handoff for the no-boundary `barB` drift boundary. -/ def cycle101DiscreteForwardKlNoBoundaryProductRuleLowerObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle101_no_boundary_product_rule_lower" statement := "Cycle 101 lower compiles SALD.generalMovingTargetDiscreteDriftDivNoBoundaryOfProductRule and SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientProductRuleBoundary. Classification: narrows-source-cited-boundary. The old monolithic hdivNoBoundary premise in SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound is replaced by three smaller source-facing inputs: product-rule expansion of div(hatRhoS * barB * phi), a Mathlib divergence-theorem boundary-flux identity, and zero boundary flux for the admissible test. The weak-test gradient estimate hgradNormBound remains explicit." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle101DiscreteForwardKlClosurePressureMiddleObligation", "SALD.generalMovingTargetDiscreteDriftDivNoBoundaryOfProductRule", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientProductRuleBoundary", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable'", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] note := "Proof-producing algebraic handoff only. It does not prove the Euclidean product rule for the weighted field, instantiate Mathlib's box divergence theorem, prove zero boundary flux, prove hgradNormBound, weak FP, KL differentiation, theorem closure, SLT import, or any Lake dependency." /-- Cycle-101 proof-DAG pane for the pressure test after the cycle-100 inner-gradient handoff. -/ def cycle101DiscreteForwardKlClosurePressureDag : List ProofDagBlock := [ { id := "ASTIS.SALD.forward_KL_discrete.cycle101_middle_pressure_sync" interface := "Pressure-test synchronization: thm:forward-KL-discrete routes through the compiled EM/KL scalar handoffs, LSI, DV, Gronwall, accumulated-error interfaces, and the cycle-100 weakGrad/inner-gradient handoffs. The route now stops at the no-boundary divergence theorem for hatRhoS * barB, with hgradNormBound carried as the smaller companion weak-test gradient estimate." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle101DiscreteForwardKlClosurePressureMiddleObligation", "SALD.cycle95DiscreteForwardKlClosurePressureMiddleObligation", "SALD.cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound", "ASTIS.SALD.cycle100.remaining_no_boundary_after_inner_gradient" ] reusedBy := [ "thm:forward-KL-discrete", "sald.discrete_forward_kl.em_interpolation_fp", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle101_lower_product_rule_handoff" interface := "Compiled local handoff: replace the single hdivNoBoundary input by product-rule total divergence, the Mathlib divergence-theorem boundary-flux identity, and zero boundary flux for hatRhoS * barB. The remaining lower theorem is the Euclidean weighted-field product rule plus no-boundary flux hypotheses at appendix.tex:1379-1387; hgradNormBound is still carried separately." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle101DiscreteForwardKlNoBoundaryProductRuleLowerObligation", "SALD.generalMovingTargetDiscreteDriftDivNoBoundaryOfProductRule", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientProductRuleBoundary", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "cycle 101 lower packet", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.forward_KL_discrete.cycle101_next_non_wrapper_blocker" interface := "Next lower packet after the product-rule handoff: prove the Euclidean weighted-field product rule for div(hatRhoS * barB * phi), instantiate the Mathlib divergence theorem to the resulting boundary-flux identity, and discharge zero boundary flux for admissible tests at appendix.tex:1379-1387. The hgradNormBound weak-test gradient estimate remains explicit. A wrapper that merely restates hdivNoBoundary or hgradNormBound is rejected-wrapper-churn." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientProductRuleBoundary", "ASTIS.SALD.cycle100.remaining_no_boundary_after_inner_gradient", "ASTIS.SALD.forward_KL_discrete.cycle101_lower_product_rule_handoff", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "cycle 101 lower packet", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-! ### Cycle 102: zero boundary flux narrowed to trace product -/ /-- Cycle-102 middle/lower-ready handoff for the remaining zero-boundary-flux piece of the `hatRhoS * barB` no-boundary theorem. The cycle stays on the active EM backend and does not open the non-EM LSI/DV/Gronwall fallback. It narrows the cycle-101 raw `hzeroBoundary : boundaryFlux phi = 0` input to a boundary-integral representation plus an a.e. zero trace-product condition. -/ def cycle102DiscreteForwardKlZeroFluxTraceBoundaryMiddleObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle102_zero_flux_trace_boundary_middle" statement := "Cycle 102 compiles SALD.generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero and SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundary. Classification: narrows-source-cited-boundary. The cycle-101 raw hzeroBoundary premise in SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientProductRuleBoundary is replaced by two smaller source-facing inputs for appendix.tex:1379-1387: boundaryFlux phi is the boundary integral of the admissible test trace times the normal trace of hatRhoS * barB, and that trace product is zero a.e. on the boundary by compact support, decay, or zero normal trace. The product-rule identity, Mathlib divergence-theorem identity, and weak-test gradient bound hgradNormBound remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle101DiscreteForwardKlNoBoundaryProductRuleLowerObligation", "SALD.generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundary", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientProductRuleBoundary", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound", "MeasureTheory.integral_congr_ae", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] note := "Proof-producing trace-boundary handoff only. It does not prove the Euclidean product rule, instantiate the divergence theorem, prove the boundary-flux integral representation, prove compact-support/decay/zero-normal-trace hypotheses, prove hgradNormBound, close weak FP/KL/theorem status, import SLT, or change Lake dependencies." /-- Cycle-102 lower handoff reducing the trace-product condition to zero test trace on the boundary. This removes the supplied `htraceProductZero` premise from the trace-boundary route when admissible tests have zero boundary trace a.e. The analytic theorem that supplies that trace fact remains an obligation. -/ def cycle102DiscreteForwardKlTraceZeroLowerObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle102_trace_zero_lower" statement := "Cycle 102 lower compiles SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero and SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero. Classification: narrows-source-cited-boundary. The cycle-102 htraceProductZero premise is discharged from the smaller source-facing condition that admissible tests have zero boundary trace a.e.; hboundaryFluxIntegral, hproductRule, hdivergenceTheorem, and hgradNormBound remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle102DiscreteForwardKlZeroFluxTraceBoundaryMiddleObligation", "SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundary", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] note := "Proof-producing lower handoff only. It does not prove the boundary-flux integral representation, the zero-trace/compact-support theorem for admissible tests, the Euclidean product rule, hgradNormBound, weak FP/KL closure, SLT import, or Lake dependency changes." /-- Cycle-102 proof-DAG pane for the trace-product zero-flux packet. -/ def cycle102DiscreteForwardKlZeroFluxTraceBoundaryDag : List ProofDagBlock := [ { id := "ASTIS.SALD.forward_KL_discrete.cycle102_zero_flux_trace_boundary" interface := "Compiled local handoff: replace hzeroBoundary by a concrete boundary-flux integral boundaryFlux phi = int_y testTrace phi y * normalFluxTrace y d boundaryMeasure plus the source-cited a.e. trace-product zero condition. This targets the no-boundary part of appendix.tex:1379-1387 after the cycle-101 product-rule handoff." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle102DiscreteForwardKlZeroFluxTraceBoundaryMiddleObligation", "SALD.generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundary", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientProductRuleBoundary", "MeasureTheory.integral_congr_ae", "appendix.tex:1379-1387" ] reusedBy := [ "cycle 102 lower packet", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.forward_KL_discrete.cycle102_test_trace_zero_lower" interface := "Compiled local lower handoff: discharge htraceProductZero from the smaller condition testTrace phi = 0 boundaryMeasure-a.e. for admissible tests. This captures the compact-support/zero-boundary-trace route while leaving the actual trace theorem and boundary-flux integral representation explicit." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle102DiscreteForwardKlTraceZeroLowerObligation", "SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundary", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.forward_KL_discrete.cycle102_next_trace_boundary_blocker" interface := "Next exact boundary: prove the boundary-flux integral representation for hatRhoS * barB from the Mathlib divergence theorem/trace setup, and prove zero boundary trace a.e. for admissible tests from compact support or decay. Keep hproductRule, hdivergenceTheorem, and hgradNormBound explicit." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero", "SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundary", "SALD.generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "cycle 102 lower packet", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-! ### Cycle 103: conditional-kernel component-version boundary -/ /-- Cycle-103 middle obligation for the `condC` conditional-kernel component-version theorem. This returns to the conditional drift definition at `appendix.tex:1368-1377` and narrows the cycle-91 remaining boundary to one selected component. The packet deliberately does not add a theorem wrapper around `hguideComp`; it records the exact Mathlib/local theorem still missing behind that sample-space a.e. equality. -/ def cycle103GeneralMovingTargetDiscreteConditionalKernelVersionMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle103_conditional_kernel_component_version_middle" statement := "Cycle 103 middle narrows ASTIS.SALD.cycle91.remaining_conditional_kernel_boundary for appendix.tex lines 1368-1377 to one selected component, condC. Classification: narrows-source-cited-boundary. The lower proof-producing bridge is SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfCondExpKernelMap, backed by AutoSamplingTheory.condDistribIntegralSampleAeEqOfCondExpKernelMap: it derives the old condDistrib sample-space equality from a measure-valued a.e. alignment of condDistrib X_k^eta hatXAtS P (hatXAtS omega) with condExpKernel P ((inferInstance : MeasurableSpace State).comap hatXAtS).map X_k^eta omega, plus the selected condExpKernel-map version of condC and the equality-set measurability consumed by ae_map_iff. The remaining exact theorem boundary is proving those measure-valued kernel alignment/version-selection facts from Mathlib condDistrib/condExpKernel, not restating hguideComp." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "ASTIS.SALD.cycle91.remaining_conditional_kernel_boundary", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfCondExpKernelMap", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularity", "AutoSamplingTheory.condDistribAeEqCondExpKernelMap", "AutoSamplingTheory.condDistribIntegralSampleAeEqOfCondExpKernelMap", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "ProbabilityTheory.condDistrib_apply_ae_eq_condExpKernel_map", "ProbabilityTheory.condExp_ae_eq_integral_condExpKernel", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle source-to-Lean synchronization plus lower bridge registration. It consulted the local SLT EfronStein conditional-expectation/product-measure idiom as reference style but imports no SLT theorem, changes no Lake dependency, and promotes no weak-FP/KL/theorem status. Lower has narrowed the component-version theorem to measure-valued kernel a.e. equality, condExpKernel-map field version selection, and equality-set measurability." /-- Cycle-103 lower obligation after the compiled `condExpKernel.map` bridge. -/ def cycle103GeneralMovingTargetDiscreteConditionalKernelVersionLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle103_conditional_kernel_component_version_lower" statement := "Cycle 103 lower compiles AutoSamplingTheory.condDistribIntegralSampleAeEqOfCondExpKernelMap and SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfCondExpKernelMap for appendix.tex lines 1368-1377. These declarations do not assume the old hguideComp sample-space equality; they derive it from two narrower inputs: (1) a measure-valued P-a.e. equality between condDistrib X_k^eta hatXAtS P (hatXAtS omega) and condExpKernel P ((inferInstance : MeasurableSpace State).comap hatXAtS).map X_k^eta omega, and (2) the selected condExpKernel-map conditional-expectation version of the named condC field, plus the measurable equality set used by ae_map_iff. Classification: narrows-source-cited-boundary. Remaining exact boundary: prove the measure-valued kernel equality and selected condExpKernel-map field version from Mathlib condDistrib/condExpKernel facts under finite-measure, standard-Borel, measurability, and Bochner integrability hypotheses." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "AutoSamplingTheory.condDistribIntegralSampleAeEqOfCondExpKernelMap", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfCondExpKernelMap", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample", "AutoSamplingTheory.condDistribAeEqCondExpKernelMap", "ProbabilityTheory.condDistrib_apply_ae_eq_condExpKernel_map", "ProbabilityTheory.condExp_ae_eq_integral_condExpKernel", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1368-1377" ] note := "Proof-producing lower bridge only. It narrows hguideComp but does not construct the regular conditional kernel, prove measure-valued uniqueness from setwise a.e. equality, choose condC, prove weak FP/KL/no-boundary, import SLT, or promote theorem status." /-- Cycle-103 proof-DAG pane for the one-component conditional-kernel versioning packet. -/ def cycle103GeneralMovingTargetDiscreteConditionalKernelVersionDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle103.middle_conditional_kernel_component_version" interface := "Middle source map: appendix.tex:1368-1377 defines bar b_{k,s} by conditioning the frozen guide and score summands on hat X_s=x. Cycle 103 selects only the condC guide component and lower narrows hguideComp to a condExpKernel.map bridge plus remaining measure-valued kernel/version-selection facts." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle103GeneralMovingTargetDiscreteConditionalKernelVersionMiddleObligation", "SALD.cycle103GeneralMovingTargetDiscreteConditionalKernelVersionLowerObligation", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfCondExpKernelMap", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "AutoSamplingTheory.condDistribAeEqCondExpKernelMap", "AutoSamplingTheory.condDistribIntegralSampleAeEqOfCondExpKernelMap", "appendix.tex:1368-1377" ] reusedBy := [ "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle103.lower_condExpKernel_map_version_bridge" interface := "Compiled lower bridge: a measure-valued a.e. equality condDistrib X_k^eta hatXAtS P (hatXAtS omega) = condExpKernel P (mState.comap hatXAtS).map X_k^eta omega, together with the selected condExpKernel-map version of condC, supplies the condDistrib sample-space equality consumed by cycle 91 and ae_map_iff." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "AutoSamplingTheory.condDistribIntegralSampleAeEqOfCondExpKernelMap", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfCondExpKernelMap", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample", "appendix.tex:1368-1377" ] reusedBy := [ "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "ASTIS.SALD.cycle91.remaining_conditional_kernel_boundary", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle103.condC_condDistrib_condExpKernel_sample_version" interface := "Remaining exact theorem boundary after the compiled lower bridge: prove the measure-valued P-a.e. kernel equality condDistrib X_k^eta hatXAtS P (hatXAtS omega) = condExpKernel P (mState.comap hatXAtS).map X_k^eta omega, prove the selected condExpKernel-map guide-component version for condC, and prove the equality set is measurable for ae_map_iff. Imports: Mathlib.Probability.Kernel.CondDistrib, Mathlib.Probability.Kernel.Condexp, Mathlib.MeasureTheory.Integral.Bochner.Basic." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "ASTIS.SALD.cycle103.lower_condExpKernel_map_version_bridge", "AutoSamplingTheory.condDistribAeEqCondExpKernelMap", "AutoSamplingTheory.condDistribIntegralSampleAeEqOfCondExpKernelMap", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfCondExpKernelMap", "ProbabilityTheory.condDistrib_apply_ae_eq_condExpKernel_map", "ProbabilityTheory.condExp_ae_eq_integral_condExpKernel", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "Mathlib.MeasureTheory.Integral.Bochner.Basic" ] reusedBy := [ "ASTIS.SALD.cycle91.remaining_conditional_kernel_boundary", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle103.rejected_wrapper_churn_guard" interface := "Reject wrappers that take the same hguideComp, hfieldAe, hcanonical, weak-FP, KL, LSI, DV, Gronwall, or no-boundary hypotheses as premises. Accept only a proof of the condC condDistrib/condExpKernel sample-version theorem or a strictly smaller named Mathlib theorem needed by it." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "ASTIS.SALD.cycle103.condC_condDistrib_condExpKernel_sample_version", "ASTIS.SALD.cycle91.remaining_conditional_kernel_boundary" ] reusedBy := ["cycle 103 lower", "cycle 103 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 104: named-law generator-to-law weak-FP transport -/ /-- Cycle-104 upper packet for the named-law generator-to-law weak-FP transport boundary. This cycle stays on the active EM conditional-law/Fokker--Planck backend over `appendix.tex:1358-1387`, narrowed to `appendix.tex:1379-1387`. It does not add a broad wrapper around the source-sign route; it removes the named-law bookkeeping premise between `hatRhoS s = Law(hatX s)` and the cycle-92 `Measure.map` derivative handoff. -/ def cycle104GeneralMovingTargetDiscreteWeakFpNamedLawTransportUpperObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle104_named_law_transport_upper" statement := "Cycle 104 upper records the global phase judgment: cycle 103 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the active lower packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387. Classification: discharges-supplied-hypothesis. The selected lower packet is the named-law generator-to-law weak-FP transport at appendix.tex:1379-1387: remove a separate named-law weak derivative/hlawDerivative premise once hatRhoS s = Measure.map (hatX s) P and the sample-space split-generator derivative are supplied. This uses the existing cycle-79 lawMapIntegral helpers and does not create another source-sign wrapper." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "AutoSamplingTheory.lawMapIntegral", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndSample", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfSampleSplitGeneratorHandoff", "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Upper packet plus compiled local transport theorem. It rejects wrapper churn around hgenerator, hdriftSource, hdivNoBoundary, hgradNormBound, KL, LSI, DV, or Gronwall assumptions; the remaining analytic boundaries are sample-path/Bochner generator differentiation, drift/no-boundary source action, diffusion source action, admissible tests, density/time regularity, and conditional-law compatibility." /-- Cycle-104 lower obligation for the compiled named-law weak derivative transport theorem. -/ def cycle104GeneralMovingTargetDiscreteWeakFpNamedLawTransportLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle104_named_law_transport_lower" statement := "Cycle 104 lower compiles AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndSample and SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfSampleSplitGeneratorHandoff. The generic helper transports a supplied sample-space derivative to a named law path rho s when rho s = Measure.map (X s) P; the SALD theorem specializes it to hatRhoS and the split generator driftAction + diffusionAction, yielding the weak-test derivative with the paper source signs. Classification: discharges-supplied-hypothesis for a primitive named-law hlawDerivative/named-law rewrite premise in the generator-to-law weak-FP boundary. Remaining exact boundaries are the sample-path split-generator derivative, drift source through barB/no-boundary, diffusion source action, weak-test measurability, density/time regularity, and conditional-law compatibility." source := saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource status := ProofStatus.obligation dependsOn := [ "AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndSample", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfSampleSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff", "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing transport handoff only. It does not prove the EM generator theorem, parametric-integral differentiation, drift/no-boundary theorem, diffusion/Laplacian source action, weak FP closure, KL differentiation, theorem closure, SLT import, or any Lake dependency." /-- Cycle-104 proof-DAG pane for the named-law generator-to-law weak-FP transport packet. -/ def cycle104GeneralMovingTargetDiscreteWeakFpNamedLawTransportDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle104.global_phase_judgment" interface := "Cycle 104 judgment: cycle 103 passed and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for single-backend cited-theory backfill; the risk-reducing packet is the named-law generator-to-law weak-FP transport for appendix.tex:1379-1387, still under sald.general_moving_target_discrete.em_interpolation_fp." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle104GeneralMovingTargetDiscreteWeakFpNamedLawTransportUpperObligation", "SALD.cycle103GeneralMovingTargetDiscreteConditionalKernelVersionLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp", "appendix.tex:1379-1387" ] reusedBy := ["cycle 104 lower packet", "cycle 104 reviewer"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle104.lower_packet.named_law_generator_to_law_transport" interface := "Compiled lower handoff: if hatRhoS s = Measure.map (hatX s) P and the sample-space split-generator integral has derivative driftAction phi + diffusionAction phi, then the named-law weak-test integral against hatRhoS has derivative -(driftDiv phi) + (sigma_eta^2/2) smul laplacian phi once the drift and diffusion source-action equalities are supplied. This removes a primitive named-law hlawDerivative premise rather than wrapping it." source := saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle104GeneralMovingTargetDiscreteWeakFpNamedLawTransportLowerObligation", "AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndSample", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfSampleSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle104.remaining_generator_to_law_after_named_transport" interface := "Remaining exact theorem boundary: prove the sample-path/Bochner parametric-integral derivative of the split generator sum for the frozen EM interpolation; prove driftAction through the named conditional drift barB and the no-boundary divergence theorem for hatRhoS * barB; prove the diffusion/Laplacian source action; keep admissible-test, density/time regularity, and conditional-law compatibility explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "ASTIS.SALD.cycle92.remaining_generator_to_law_boundary", "ASTIS.SALD.cycle100.remaining_no_boundary_after_inner_gradient", "ASTIS.SALD.cycle103.condC_condDistrib_condExpKernel_sample_version", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle104.reviewer_named_transport_check" interface := "Reviewer checklist: accept only if the generic named-law transport helper and SALD named-law derivative theorem compile, the packet is classified as discharges-supplied-hypothesis, appendix.tex:1379-1387 remains fixed, and no source-sign/no-boundary/KL/LSI/DV/Gronwall wrapper churn, theorem-status promotion, SLT import, Lake dependency change, fake closure, or sald_version_2 use is introduced." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfSampleSplitGeneratorHandoff", "AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndSample", "appendix.tex:1379-1387" ] reusedBy := ["cycle 104 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 105: pure no-mass KL/log-ratio differentiability boundary -/ /-- Cycle-105 middle obligation for the pure no-mass KL/log-ratio boundary. This cycle returns to `appendix.tex:1358-1366` and narrows the cycle-99 no-mass package. Once the mass term has been removed, the remaining KL differentiability theorem should not carry sample-space law data. -/ def cycle105GeneralMovingTargetDiscretePureRawKlDerivativeMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle105_pure_raw_kl_derivative_middle" statement := "Cycle 105 middle keeps the active packet on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387 and narrows the KL/log-ratio analytic boundary at appendix.tex:1358-1366. Classification: narrows-source-cited-boundary. The exact missing theorem is now the pure no-mass finite-KL llr KL-differentiability statement: finite KL supplies the Mathlib llr representative and its regularity, while endpoint-safe KL differentiation, llr weak-action integrability, target-time integrability, and the target-time derivative formula imply dK = partialS (llr hatRho tildePi) - targetTimeTerm. The boundary no longer carries sample-space P, hatX, s0, hatX a.e. measurability, or a mapped-law mass derivative." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.GeneralMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlr", "SALD.GeneralMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlr", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "eq:general_KL_derivative_0_discrete", "appendix.tex:1358-1366", "sald.general_moving_target_discrete.kl_derivative" ] note := "Middle synchronization only. It rejects wrappers around hklRaw, hlog, hmass, weak-FP source signs, barB no-boundary, LSI, DV, or Gronwall. The theorem statements and backend statuses are unchanged." /-- Cycle-105 lower obligation for the compiled pure no-mass KL handoff. -/ def cycle105GeneralMovingTargetDiscretePureRawKlDerivativeLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle105_pure_raw_kl_derivative_lower" statement := "Cycle 105 lower compiles SALD.generalMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlrHkl and SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfPureNoMassRawKlBoundaryAtFiniteKlLlrWithLogAction. The compiled handoff extracts the no-mass display from the pure finite-KL llr source-cited package, uses the existing finite-KL admissibility closure to select the admissible log-ratio weak test, and feeds the result into the existing source-sign-with-log-action theorem. Classification: narrows-source-cited-boundary because the remaining missing theorem is smaller than cycle 99: it is measure-path KL differentiability only, with no sample-space law or mass-derivative fields." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.GeneralMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlr", "SALD.generalMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlrHkl", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfPureNoMassRawKlBoundaryAtFiniteKlLlrWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.cycle99GeneralMovingTargetDiscreteRawKlDerivativeLowerObligation", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "eq:general_KL_derivative_0_discrete", "appendix.tex:1358-1366", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing equality handoff plus source-cited analytic boundary. It does not prove endpoint-safe KL differentiation, target-time derivative, weak FP, log-ratio closure internals, IBP/FI, theorem closure, SLT import, or any Lake dependency." /-- Cycle-105 proof-DAG pane for the pure no-mass KL/log-ratio boundary. -/ def cycle105GeneralMovingTargetDiscretePureRawKlDerivativeDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle105.middle_pure_raw_kl_derivative_source_map" interface := "Middle source map: after finite KL has selected Mathlib llr and the mass term has been removed, isolate the remaining appendix.tex:1358-1366 theorem as pure no-mass KL differentiability for the measure path, without sample-space law or mapped-law mass data." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle105GeneralMovingTargetDiscretePureRawKlDerivativeMiddleObligation", "SALD.cycle99GeneralMovingTargetDiscreteRawKlDerivativeLowerObligation", "eq:general_KL_derivative_0_discrete" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle105.lower_packet.pure_raw_kl_llr_handoff" interface := "Compiled lower handoff: extract dK = partialS (llr hatRho tildePi) - targetTimeTerm from the pure source-cited package, derive admissibility of that exact llr test through the cycle-88 closure package, and apply the existing weak-FP source-sign-with-log-action theorem." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.GeneralMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlr", "SALD.generalMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlrHkl", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfPureNoMassRawKlBoundaryAtFiniteKlLlrWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle105.remaining_pure_raw_kl_boundary" interface := "Remaining exact theorem: prove the pure no-mass finite-KL llr KL differentiability package from Mathlib finite-KL/log-likelihood-ratio regularity, parametric-integral/Bochner differentiation under the KL integral, target-time derivative integrability and formula, and density/time regularity." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1358-1366" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-! ### Cycle 106: canonical conditional-integral drift regularity -/ /-- Cycle-106 middle obligation for canonical conditional-integral drift regularity. This cycle returns to the conditional-drift line `appendix.tex:1368-1377` and chooses the canonical `condDistrib` representative of `bar b_{k,s}` rather than adding another supplied-hypothesis wrapper. -/ def cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle106_canonical_condDistrib_drift_middle" statement := "Cycle 106 middle keeps the active packet on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387 and targets the conditional integral regularity hypothesis used by cycles 80-84. Classification: discharges-supplied-hypothesis. The exact supplied hypothesis discharged is the primitive component conditional-integral regularity premise for the canonical barB representative: when barB is defined as dotTk times the condDistrib guide integral plus sigmaCoeff times the condDistrib score integral, Mathlib/local conditional-integral lemmas should prove AEStronglyMeasurable and Integrable under hatRhoS = Law(hatXAtS). This does not claim a separately named paper representative has been selected; that a.e. version-selection fact remains a smaller boundary." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularity", "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity", "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "lean-stat-learning-theory/SLT/EfronStein.lean", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle synchronization plus packet guard. Local SLT EfronStein.lean was consulted for product-measure/conditional-integral proof style only; no SLT theorem is imported or marked formalized." /-- Cycle-106 lower obligation for the compiled canonical `condDistrib` drift regularity theorem. -/ def cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle106_canonical_condDistrib_drift_lower" statement := "Cycle 106 lower compiles SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity for appendix.tex lines 1368-1377. The theorem proves AEStronglyMeasurable and Integrable for the canonical conditional drift field x |-> dotTk • ∫ guideIntegrand (x,y) d condDistrib Xk hatXAtS P x + sigmaCoeff • ∫ scoreIntegrand (x,y) d condDistrib Xk hatXAtS P x, using hatRhoS = Measure.map hatXAtS P and the local Mathlib wrappers AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable and AutoSamplingTheory.condDistribIntegralNamedLawIntegrable. The companion SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq proves that any named barB representative a.e. equal to this canonical field inherits the same regularity, so the remaining named-representative boundary is only the source-specific hatRhoS-a.e. equality. Classification: discharges-supplied-hypothesis for the old canonical component conditional-integral regularity premise and narrows-source-cited-boundary for the named barB versioning side condition. Weak FP, no-boundary, diffusion, KL/log-ratio, LSI, DV, and Gronwall remain separate." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing local theorem only. It does not construct the EM process, prove the full regular conditional-law theorem for an arbitrary named representative, prove weak Fokker-Planck source signs, close no-boundary identities, promote theorem status, import SLT, or change Lake dependencies." /-- Cycle-106 proof-DAG pane for canonical conditional-integral drift regularity. -/ def cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle106.middle_canonical_condDistrib_drift_source_map" interface := "Middle source map: appendix.tex:1368-1377 defines bar b_{k,s} by conditioning the frozen guide and score summands on hat X_s=x. Cycle 106 chooses the canonical condDistrib representative and targets only its measurable/integrable conditional-integral regularity." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftMiddleObligation", "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity", "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle106.lower_packet.canonical_condDistrib_drift_regularity" interface := "Compiled lower theorem: the canonical guide and score conditional integrals against condDistrib are each AEStronglyMeasurable and Integrable under the named law hatRhoS, and the dotTk/sigmaCoeff linear combination inherits both properties." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity", "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable" ] reusedBy := [ "ASTIS.SALD.cycle91.remaining_conditional_kernel_boundary", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle106.lower_packet.named_barB_canonical_ae_regular" interface := "Compiled versioning bridge: if a downstream named barB is hatRhoS-a.e. equal to the canonical condDistrib guide-plus-score field, SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq transfers AEStronglyMeasurable and Integrable from the canonical theorem. The equality itself remains the exact source-cited boundary." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity", "appendix.tex:1368-1377" ] reusedBy := [ "ASTIS.SALD.cycle106.remaining_named_barB_version_boundary", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle106.remaining_named_barB_version_boundary" interface := "Remaining exact boundary after the compiled canonical and named-version regularity theorems: if the paper route keeps a separately named barB field, prove it is hatRhoS-a.e. equal to the canonical condDistrib guide-plus-score field. Do not replace this by another broad hcondC/hcondScore or weak-FP wrapper." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample", "ASTIS.SALD.cycle103.condC_condDistrib_condExpKernel_sample_version", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-! ### Cycle 107: boundary-flux integral via Mathlib box divergence theorem -/ /-- Cycle-107 lower obligation for the Mathlib box divergence theorem specialization behind the boundary-flux integral representation. This packet acts on the `hboundaryFluxIntegral` premise consumed by `generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero`. It keeps the product rule, `hdivergenceTheorem`, weak-test gradient bound, and zero boundary trace as separate inputs. -/ def cycle107DiscreteForwardKlBoundaryFluxIntegralLowerObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle107_boundary_flux_integral_lower" statement := "Cycle 107 lower compiles SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox. Classification: narrows-source-cited-boundary. The old hboundaryFluxIntegral premise in SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero is narrowed to a Mathlib box divergence theorem package: continuity of the weighted field on the box, Frechet differentiability off a countable interior set, integrability of the divergence, identification of boundaryFlux with the box divergence integral, and identification of the signed face sum with int_y testTrace phi y * normalFluxTrace y d boundaryMeasure. The theorem invokes MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable directly and leaves hproductRule, hdivergenceTheorem, hgradNormBound, and htestTraceZero explicit." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero", "SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero", "SALD.generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] note := "Proof-producing Mathlib-backed theorem only. It does not prove the Euclidean weighted-field product rule, the paper-specific boundaryFlux/divergence identification, the face-to-trace parametrization, compact-support/zero-trace theorem, hgradNormBound, weak FP/KL closure, theorem-status promotion, SLT import, Lake dependency change, or sald_version_2 use." /-- Cycle-107 proof-DAG pane for the boundary-flux integral packet. -/ def cycle107DiscreteForwardKlBoundaryFluxIntegralDag : List ProofDagBlock := [ { id := "ASTIS.SALD.forward_KL_discrete.cycle107.lower_packet.boundary_flux_integral_box" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox converts a box divergence integral for the weighted field hatRhoS * barB to the signed face sum by MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable, then uses the supplied face-to-trace identification to produce the exact hboundaryFluxIntegral premise." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle107DiscreteForwardKlBoundaryFluxIntegralLowerObligation", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.forward_KL_discrete.cycle107.remaining_box_trace_instantiation_boundary" interface := "Remaining exact boundary after the compiled box theorem: instantiate weightedField with the concrete hatRhoS * barB flux, prove continuity on the source box, off-countable Frechet differentiability, divergence integrability, boundaryFlux equals the interior divergence integral, and the signed Mathlib face sum equals the existing testTrace/normalFluxTrace boundary integral. Keep hproductRule, hdivergenceTheorem, hgradNormBound, and htestTraceZero separate." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "ASTIS.SALD.cycle106.remaining_named_barB_version_boundary", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-! ### Cycle 108: concrete `hatRhoS * barB` product-flux continuity -/ /-- Cycle-108 lower obligation for the concrete product-flux instantiation of the cycle-107 Mathlib box theorem. This packet stays on the active EM backend. It acts only on the continuity piece of `ASTIS.SALD.forward_KL_discrete.cycle107.remaining_box_trace_instantiation_boundary` for the weighted field `hatRhoS * barB`; the derivative, divergence integrability, boundaryFlux/interior-divergence, and signed-face/trace identifications remain explicit. -/ def cycle108DiscreteForwardKlHatRhoBarBFluxContinuityLowerObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle108_hatRho_barB_flux_continuity_lower" statement := "Cycle 108 lower compiles SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldContinuousOnBox and SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox. Classification: narrows-source-cited-boundary. The remaining cycle-107 box-trace instantiation for weightedField=hatRhoS*barB is narrowed by discharging the generic hcontinuous input from separate continuity of the density representative hatRhoDensity and the conditional drift barB on the source box. The Mathlib box divergence theorem is still consumed through SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox, while off-countable Frechet differentiability, divergence integrability, boundaryFlux equals the interior divergence integral, signed-face-to-testTrace identification, hproductRule, hdivergenceTheorem, hgradNormBound, and htestTraceZero remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldContinuousOnBox", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] note := "Proof-producing continuity/product-flux handoff only. It does not prove the Frechet derivative of hatRhoS*barB, divergence integrability, boundaryFlux/divergence identification, signed-face trace parametrization, product rule, hgradNormBound, zero trace, weak FP/KL closure, theorem-status promotion, SLT import, Lake dependency change, or sald_version_2 use." /-- Cycle-108 proof-DAG pane for the concrete `hatRhoS * barB` continuity piece of the box-trace boundary. -/ def cycle108DiscreteForwardKlHatRhoBarBFluxContinuityDag : List ProofDagBlock := [ { id := "ASTIS.SALD.forward_KL_discrete.cycle108.lower_packet.hatRho_barB_flux_continuity" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldContinuousOnBox proves continuity of x |-> hatRhoDensity x • barB x on the Mathlib source box from separate density and barB continuity, and SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox feeds that concrete product flux into the cycle-107 box theorem." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle108DiscreteForwardKlHatRhoBarBFluxContinuityLowerObligation", "SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldContinuousOnBox", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "Mathlib.Topology.Algebra.MulAction", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.forward_KL_discrete.cycle108.remaining_hatRho_barB_box_trace_boundary" interface := "Remaining exact boundary after the compiled product-flux continuity handoff: prove the Frechet derivative of x |-> hatRhoDensity x • barB x off a countable interior set, divergence integrability, boundaryFlux equals the interior divergence integral, and the signed Mathlib face sum equals the existing testTrace/normalFluxTrace boundary integral. Keep hproductRule, hdivergenceTheorem, hgradNormBound, and htestTraceZero separate." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "ASTIS.SALD.forward_KL_discrete.cycle107.remaining_box_trace_instantiation_boundary", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-! ### Cycle 108: concrete `hatRhoS * barB` product-flux derivative -/ /-- Cycle-108 lower obligation for the concrete product-flux Frechet derivative sub-boundary. This packet stays below the active EM no-boundary backend. It acts only on the Frechet differentiability piece of `ASTIS.SALD.forward_KL_discrete.cycle108.remaining_hatRho_barB_box_trace_boundary`: the derivative of `hatRhoDensity x • barB x` is obtained from separate density and `barB` derivatives, off the union of their countable exception sets. -/ def cycle108DiscreteForwardKlHatRhoBarBFluxDerivativeLowerObligation : ProofObligation where id := "sald.discrete_forward_kl.cycle108_hatRho_barB_flux_derivative_lower" statement := "Cycle 108 lower compiles SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAt, SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAtOffUnion, and SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBoxProductDeriv. Classification: narrows-source-cited-boundary. The remaining cycle-108 box-trace instantiation for weightedField=hatRhoS*barB is narrowed by discharging the generic off-countable Frechet differentiability input from separate density and barB derivatives off their own countable exception sets. Divergence integrability, boundaryFlux equals the interior divergence integral, signed-face-to-testTrace identification, hproductRule, hdivergenceTheorem, hgradNormBound, and htestTraceZero remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAt", "SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAtOffUnion", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBoxProductDeriv", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero", "Mathlib.Analysis.Calculus.FDeriv.Mul", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] note := "Proof-producing product-derivative handoff only. It does not prove divergence integrability, boundaryFlux/divergence identification, signed-face trace parametrization, product rule, hgradNormBound, zero trace, weak FP/KL closure, theorem-status promotion, SLT import, Lake dependency change, or sald_version_2 use." /-- Cycle-108 proof-DAG pane for the concrete `hatRhoS * barB` derivative piece of the box-trace boundary. -/ def cycle108DiscreteForwardKlHatRhoBarBFluxDerivativeDag : List ProofDagBlock := [ { id := "ASTIS.SALD.forward_KL_discrete.cycle108.lower_packet.hatRho_barB_flux_product_derivative" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAt proves the pointwise product derivative for x |-> hatRhoDensity x • barB x; SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAtOffUnion lifts it off the union of separate density and drift exception sets; SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBoxProductDeriv feeds that derivative into the concrete box theorem." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle108DiscreteForwardKlHatRhoBarBFluxDerivativeLowerObligation", "SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAt", "SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAtOffUnion", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBoxProductDeriv", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox", "Mathlib.Analysis.Calculus.FDeriv.Mul", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.forward_KL_discrete.cycle108.remaining_hatRho_barB_box_trace_after_product_derivative" interface := "Remaining exact boundary after the compiled product-flux derivative handoff: prove divergence integrability for the product derivative, boundaryFlux equals the interior divergence integral, and signed Mathlib faces equal the existing testTrace/normalFluxTrace boundary integral. Keep hproductRule, hdivergenceTheorem, hgradNormBound, and htestTraceZero separate." source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBoxProductDeriv", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "ASTIS.SALD.forward_KL_discrete.cycle108.remaining_hatRho_barB_box_trace_boundary", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-! ### Cycle 109: named `barB` source-definition boundary -/ /-- Cycle-109 exact missing theorem for the named `barB` source definition. This is the lower-ready boundary behind `appendix.tex:1368-1377` after the canonical `condDistrib` regularity theorem from cycle 106. The target is not another direct a.e. equality hypothesis for `barB`; it is the source theorem that selects the paper's conditional-expectation representative and aligns it with Mathlib's conditional-distribution kernel. -/ def cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefBoundary : ProofObligation where id := "sald.general_moving_target_discrete.cycle109_named_barB_source_def_boundary" statement := "Cycle 109 records the exact missing theorem behind appendix.tex:1368-1377. Classification: narrows-source-cited-boundary. Proposed theorem boundary: prove the source-definition version of SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpKernelSourceDef from Mathlib conditional-kernel ingredients, with imports Mathlib.Probability.Kernel.CondDistrib, Mathlib.Probability.Kernel.Condexp, Mathlib.MeasureTheory.Integral.Bochner.Basic, AutoSamplingTheory.Probability, and AutoSamplingTheory.SALD. Hypotheses: a finite/probability common law P on a standard-Borel sample space, standard-Borel state space, measurable or a.e.-measurable hatXAtS and XkEta, hhatRhoS : hatRhoS = Measure.map hatXAtS P, the guide and score integrands for dot t_k c_{t_k} and (sigma_eta^2/2) nabla log pi_{t_k}, equality-set measurability for the canonical guide-plus-score field, the measure-valued a.e. kernel alignment condDistrib XkEta hatXAtS P (hatXAtS omega) = condExpKernel P (mState.comap hatXAtS).map XkEta omega, and the paper source definition of barB as the corresponding condExpKernel.map conditional expectation. Conclusion: barB is hatRhoS-a.e. equal to the canonical condDistrib guide-plus-score field consumed by SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource status := ProofStatus.sourceCited dependsOn := [ "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpKernelSourceDef", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity", "AutoSamplingTheory.condDistribAeEqCondExpKernelMap", "AutoSamplingTheory.condDistribIntegralSampleAeEqOfCondExpKernelMap", "ProbabilityTheory.condDistrib_apply_ae_eq_condExpKernel_map", "ProbabilityTheory.condExp_ae_eq_integral_condExpKernel", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "This is a source-cited theorem boundary, not a broad wrapper. It does not promote the conditional law, weak FP, KL/log-ratio, no-boundary, LSI, DV, Gronwall, theorem status, SLT reuse status, or Lake dependencies." /-- Cycle-109 middle packet for the named `barB` source-definition bridge. -/ def cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle109_named_barB_source_def_middle" statement := "Cycle 109 middle keeps the active packet on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to the named barB source-definition boundary at appendix.tex:1368-1377. Classification: narrows-source-cited-boundary. The compiled local theorem SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpKernelSourceDef replaces a direct hbarBAe equality for barB with two smaller source-facing inputs: the measure-valued sample-space alignment between condDistrib XkEta hatXAtS P and condExpKernel P (mState.comap hatXAtS).map XkEta, and the source conditional-expectation definition of barB using that condExpKernel.map representative. The remaining exact theorem is SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefBoundary, with imports and hypotheses listed explicitly." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefBoundary", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpKernelSourceDef", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfCondExpKernelMap", "SALD.cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftLowerObligation", "SALD.cycle103GeneralMovingTargetDiscreteConditionalKernelVersionLowerObligation", "AutoSamplingTheory.condDistribAeEqCondExpKernelMap", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle synchronization plus one compiled source-definition bridge. Local SLT/EfronStein.lean was consulted for conditional-expectation and product-measure proof style only; no SLT theorem is imported or marked formalized." /-- Cycle-109 lower packet using Mathlib's product conditional-expectation identity for the named `barB` source definition. -/ def cycle109GeneralMovingTargetDiscreteNamedBarBCondExpSourceLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle109_named_barB_condExp_source_lower" statement := "Cycle 109 lower compiles SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef for appendix.tex:1368-1377. Classification: narrows-source-cited-boundary. The theorem uses Mathlib ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib to turn the paper's sample-space conditional-expectation definition of the frozen guide-plus-score drift into the downstream hatRhoS-a.e. canonical condDistrib equality consumed by SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq. This bypasses the middle condExpKernel.map route's measure-valued kernel-alignment hypothesis for this source-definition path; the remaining exact theorem is to identify the selected paper representative hbarBCondExp : P[dotTk • guideIntegrand(hatXAtS omega, XkEta omega) + sigmaCoeff • scoreIntegrand(hatXAtS omega, XkEta omega) | mState.comap hatXAtS] = barB (hatXAtS omega) a.e. and prove equality-set measurability for ae_map_iff under explicit finite/probability P, measurable hatXAtS, a.e.-measurable XkEta, standard-Borel state, complete vector space, and guide/score Bochner integrability hypotheses." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity", "SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefBoundary", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing local theorem from Mathlib conditional expectation and Bochner integral linearity. It does not prove weak FP, box trace, KL/log-ratio, no-boundary, theorem status, SLT import, Lake dependency change, or sald_version_2 use." /-- Cycle-109 proof-DAG pane for the named `barB` source-definition boundary. -/ def cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle109.middle_named_barB_source_def_boundary" interface := "Middle source map: appendix.tex:1368-1377 defines barB by conditioning the guide-plus-score frozen drift on hat X_s=x. The current theorem boundary is the named representative equality between that source conditional expectation and the canonical condDistrib guide-plus-score field under hatRhoS = Law(hat X_s)." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefMiddleObligation", "SALD.cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftLowerObligation", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle109.compiled_named_barB_source_def_bridge" interface := "Compiled local bridge: SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpKernelSourceDef transports the source condExpKernel.map definition of barB through the sample-space kernel alignment and hatRhoS = Measure.map hatXAtS P to produce the downstream hatRhoS-a.e. canonical condDistrib equality." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpKernelSourceDef", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp" ] reusedBy := [ "ASTIS.SALD.cycle106.remaining_named_barB_version_boundary", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle109.lower_named_barB_condExp_source_bridge" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef applies Mathlib condExp_prod_ae_eq_integral_condDistrib to the frozen guide-plus-score summand, uses Bochner integral add/smul linearity along condDistrib fibers, and transports the resulting sample-space equality through hatRhoS = Measure.map hatXAtS." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle109GeneralMovingTargetDiscreteNamedBarBCondExpSourceLowerObligation", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1368-1377" ] reusedBy := [ "ASTIS.SALD.cycle106.remaining_named_barB_version_boundary", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle109.remaining_condExp_source_representative_for_named_barB" interface := "Remaining exact theorem after the lower condExp source bridge: prove the paper's selected representative hbarBCondExp, namely the sample-space a.e. equality between the Mathlib conditional expectation of dotTk • guideIntegrand(hatXAtS omega, XkEta omega) + sigmaCoeff • scoreIntegrand(hatXAtS omega, XkEta omega) given mState.comap hatXAtS and barB (hatXAtS omega), plus equality-set measurability for ae_map_iff. Required hypotheses are finite/probability P, measurable hatXAtS, a.e.-measurable XkEta, standard-Borel state, CompleteSpace Vec, and guide/score Bochner integrability." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "SALD.cycle109GeneralMovingTargetDiscreteNamedBarBCondExpSourceLowerObligation", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle109.remaining_condExpKernel_source_def_and_kernel_alignment" interface := "Remaining exact theorem for the older condExpKernel.map route: prove the measure-valued a.e. alignment condDistrib XkEta hatXAtS P (hatXAtS omega) = condExpKernel P (mState.comap hatXAtS).map XkEta omega and prove the source condExpKernel.map conditional-expectation representative for barB. The lower condExp source bridge gives a smaller alternative route through ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib, so this node is no longer the only source-definition path." source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefBoundary", "AutoSamplingTheory.condDistribAeEqCondExpKernelMap", "AutoSamplingTheory.condDistribIntegralSampleAeEqOfCondExpKernelMap", "ProbabilityTheory.condDistrib_apply_ae_eq_condExpKernel_map", "ProbabilityTheory.condExp_ae_eq_integral_condExpKernel", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle109.reviewer_named_barB_source_def_check" interface := "Reviewer check: accept only as narrows-source-cited-boundary if the packet names either the remaining condExp source-representative theorem or the older condExpKernel source-definition theorem with imports and hypotheses, avoids a direct hbarBAe wrapper, preserves appendix.tex:1368-1377 and the EM backend, and leaves weak FP, box-trace, KL, LSI, DV, Gronwall, theorem status, SLT import, and Lake dependencies unchanged." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefMiddleObligation", "SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefBoundary", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpKernelSourceDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef", "ASTIS.SALD.cycle109.remaining_condExp_source_representative_for_named_barB", "appendix.tex:1368-1377" ] reusedBy := ["cycle 109 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 110: named barB equality-set measurability -/ /-- Cycle-110 middle packet for the selected named `barB` representative. This keeps the post-cycle-109 lower packet on the source conditional-drift definition. The only supplied side condition discharged here is the equality-set measurability required by `ae_map_iff` in the cycle-109 product-conditional-expectation bridge; the source representative equality itself remains the exact lower theorem. -/ def cycle110GeneralMovingTargetDiscreteNamedBarBEqMeasMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle110_named_barB_eq_meas_middle" statement := "Cycle 110 middle keeps the active EM conditional-law/Fokker-Planck backend over appendix.tex:1358-1387 narrowed to the selected named barB source representative at appendix.tex:1368-1377, because appendix.tex:1379-1387 weak-FP source signs consume that named drift. Classification: discharges-supplied-hypothesis. The compiled lower target removes the supplied hbarBEqMeas premise from SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef under strongly measurable canonical and named barB representatives; the remaining exact source theorem is hbarBCondExp, the sample-space conditional-expectation representative equality." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef", "SALD.cycle109GeneralMovingTargetDiscreteNamedBarBCondExpSourceLowerObligation", "ASTIS.SALD.cycle109.remaining_condExp_source_representative_for_named_barB", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle synchronization only plus a local measurable-set theorem. It rejects another named-law generator wrapper because cycle 104 already discharged that premise with lawMapIntegral helpers. It does not prove the conditional expectation representative, weak FP, box trace, KL/log-ratio, LSI, DV, Gronwall, theorem closure, SLT import, Lake dependency change, or sald_version_2 use." /-- Cycle-110 lower obligation for equality-set measurability of named `barB`. -/ def cycle110GeneralMovingTargetDiscreteNamedBarBEqMeasLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle110_named_barB_eq_meas_lower" statement := "Cycle 110 lower compiles SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable. Classification: discharges-supplied-hypothesis for hbarBEqMeas in SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef. The theorem applies Mathlib's StronglyMeasurable.measurableSet_eq_fun to the canonical condDistrib guide-plus-score field and the selected named barB representative. Remaining exact theorem: prove hbarBCondExp, namely P[dotTk • guideIntegrand(hatXAtS omega, XkEta omega) + sigmaCoeff • scoreIntegrand(hatXAtS omega, XkEta omega) | mState.comap hatXAtS] = barB (hatXAtS omega) a.e., plus the strong measurability of the chosen representatives if not already supplied by the source regularity backend." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef", "MeasureTheory.StronglyMeasurable.measurableSet_eq_fun", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing measurable-set helper only. It does not replace the source conditional-expectation representative with a direct hbarBAe wrapper, and it does not promote weak FP, no-boundary, KL/log-ratio, theorem status, SLT reuse status, or dependencies." /-- Cycle-110 proof-DAG pane for the named `barB` equality-set packet. -/ def cycle110GeneralMovingTargetDiscreteNamedBarBEqMeasDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle110.middle_named_barB_eq_meas_boundary" interface := "Middle source map: after cycle 109, the exact removable supplied side condition is hbarBEqMeas for the canonical guide-plus-score field and named barB. This supports the source representative route needed before appendix.tex:1379-1387 weak-FP source signs use barB." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle110GeneralMovingTargetDiscreteNamedBarBEqMeasMiddleObligation", "SALD.cycle109GeneralMovingTargetDiscreteNamedBarBCondExpSourceLowerObligation", "appendix.tex:1368-1377", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle110.lower_named_barB_eq_meas" interface := "Compiled lower theorem: strongly measurable canonical condDistrib guide-plus-score field and strongly measurable named barB imply the equality set required by ae_map_iff is measurable." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle110GeneralMovingTargetDiscreteNamedBarBEqMeasLowerObligation", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable", "MeasureTheory.StronglyMeasurable.measurableSet_eq_fun", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef" ] reusedBy := [ "ASTIS.SALD.cycle109.remaining_condExp_source_representative_for_named_barB", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle110.remaining_condExp_source_representative_after_eq_meas" interface := "Remaining exact theorem after equality-set measurability: prove the selected sample-space conditional-expectation representative hbarBCondExp for the guide-plus-score frozen drift, and supply strong measurability of the canonical and named representatives if the lower route uses the cycle-110 measurable-set helper." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "ASTIS.SALD.cycle109.remaining_condExp_source_representative_for_named_barB", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle110.reviewer_named_barB_eq_meas_check" interface := "Reviewer check: accept only if the compiled theorem removes hbarBEqMeas under measurable representatives, the packet is classified as discharges-supplied-hypothesis, and no direct hbarBAe wrapper, generator-to-law wrapper churn, weak-FP/no-boundary/KL/LSI/DV/Gronwall promotion, theorem-status change, SLT import, Lake dependency change, fake closure, or sald_version_2 use appears." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle110GeneralMovingTargetDiscreteNamedBarBEqMeasLowerObligation", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable", "appendix.tex:1368-1377" ] reusedBy := ["cycle 110 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 110: dominated generator-to-law weak-FP transport -/ /-- Cycle-110 lower obligation for the dominated parametric-integral generator-to-law step. This is the assigned appendix.tex:1379-1387 packet: it does not add another source-sign wrapper. Instead it derives the sample-space weak-test derivative from Mathlib's dominated parametric-integral theorem, then reuses the cycle-79 law-map transport and cycle-104 named-law route to remove the supplied integral-level `hsampleGenerator` premise. -/ def cycle110GeneralMovingTargetDiscreteWeakFpDominatedGeneratorLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle110_weak_fp_dominated_generator_lower" statement := "Cycle 110 lower compiles AutoSamplingTheory.lawMapIntegralHasDerivAtOfDominated, AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndDominated, and SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff for appendix.tex:1379-1387. Classification: discharges-supplied-hypothesis for the integral-level hsampleGenerator premise in SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfSampleSplitGeneratorHandoff. The new theorem proves that premise from pointwise sample-path HasDerivAt, local neighborhood, a.e. measurability, Bochner integrability, and an integrable derivative bound, then transports the result through Measure.map/hatRhoS = Law(hatX_s) and rewrites the drift/diffusion source signs." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "AutoSamplingTheory.lawMapIntegralHasDerivAtOfDominated", "AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndDominated", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfSampleSplitGeneratorHandoff", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Remaining exact analytic boundary: prove the EM interpolation pointwise derivative and domination package for each admissible weak test, identify the derivative integral with driftAction+diffusionAction, then use the existing barB/no-boundary and diffusion source-action backends. No theorem status, KL/log-ratio, LSI, DV, Gronwall, SLT import, Lake dependency, source-index rebaseline, or sald_version_2 use is promoted." /-- Cycle-110 proof-DAG pane for the dominated generator-to-law weak-FP transport packet. -/ def cycle110GeneralMovingTargetDiscreteWeakFpDominatedGeneratorDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle110.lower_packet.dominated_generator_to_law_transport" interface := "Compiled lower theorem: pointwise sample-path derivatives plus a local dominated derivative-under-integral package imply the named-law weak-test derivative for hatRhoS=Law(hatX_s), with source signs after the existing drift/diffusion action rewrites. This discharges the older integral-level hsampleGenerator premise instead of adding another weak-FP wrapper." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle110GeneralMovingTargetDiscreteWeakFpDominatedGeneratorLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff", "AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndDominated", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfDominated", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle110.remaining_parametric_generator_boundary_after_dominated_transport" interface := "Remaining exact theorem after the dominated transport: for every admissible weak test, prove the EM interpolation pointwise HasDerivAt and integrable bound/dominated convergence package, identify the resulting derivative integral with driftAction+diffusionAction, and keep the separate barB/no-boundary and diffusion source-action identifications." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff", "ASTIS.SALD.cycle104.remaining_generator_to_law_after_named_transport", "ASTIS.SALD.cycle100.remaining_no_boundary_after_inner_gradient", "ASTIS.SALD.cycle110.remaining_condExp_source_representative_after_eq_meas", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle110.reviewer_dominated_generator_to_law_check" interface := "Reviewer check: accept only if the compiled theorem removes the integral-level hsampleGenerator premise through Mathlib parametric-integral plus existing lawMapIntegral helpers, and no broad wrapper churn, theorem-status promotion, non-EM fallback, SLT import, Lake dependency change, fake closure, or sald_version_2 use appears." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle110GeneralMovingTargetDiscreteWeakFpDominatedGeneratorLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff", "AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndDominated", "appendix.tex:1379-1387" ] reusedBy := ["cycle 110 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 111: KL target-time derivative at finite-KL `llr` -/ /-- Cycle-111 middle obligation for the KL target-time subboundary. The active source slice is still `appendix.tex:1358-1366`. This packet does not restate the whole pure raw-KL package; it isolates the target-density time derivative term inside `eq:general_KL_derivative_0_discrete`. -/ def cycle111GeneralMovingTargetDiscreteTargetTimeDerivativeMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle111_target_time_derivative_middle" statement := "Cycle 111 keeps the active KL/log-ratio packet on appendix.tex:1358-1366 and narrows ASTIS.SALD.cycle105.remaining_pure_raw_kl_boundary. Classification: narrows-source-cited-boundary. The exact missing theorem is now the target-time subterm inside SALD.GeneralMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlr: prove targetTimeTermIntegrable and targetTimeDerivativeFormula for - int (hat rho_s / tilde pi_s) * partial_s tilde pi_s dx from finite-KL llr regularity, a concrete density-ratio representative, pointwise HasDerivAt for the target density, and a local dominated parametric-integral hypothesis. The packet adds no sample-space P, hatX, massTerm, mapped-law mass derivative, hbarB, weak-FP source-sign, LSI, DV, or Gronwall field." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.GeneralMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlr", "SALD.generalMovingTargetDiscretePureRawKlTargetTimeFieldsOfDominated", "SALD.generalMovingTargetDiscreteTargetTimeDerivativeOfDominated", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "eq:general_KL_derivative_0_discrete", "appendix.tex:1358-1366" ] note := "Middle synchronization plus a lower-ready target-time theorem. It rejects broad hkl/hlog/hmass wrappers, route audits, source-index rebaselines, theorem-status promotion, SLT import, Lake dependency changes, and sald_version_2 use." /-- Cycle-111 lower obligation for the dominated target-time theorem. -/ def cycle111GeneralMovingTargetDiscreteTargetTimeDerivativeLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle111_target_time_derivative_lower" statement := "Cycle 111 lower compiles SALD.generalMovingTargetDiscreteTargetTimeDerivativeOfDominated, SALD.generalMovingTargetDiscreteTargetTimeDerivativeSourceRatioCongr, and SALD.generalMovingTargetDiscretePureRawKlTargetTimeFieldsOfDominated for appendix.tex:1358-1366. Classification: narrows-source-cited-boundary. The first theorem proves the concrete dominated target-time derivative and weighted-derivative integrability by Mathlib parametric integral. The source-ratio congruence theorem transfers the target-time integral, integrability, and HasDerivAt formula from the chosen fixed weight to the paper's density-ratio representative under a single a.e. equality. The field handoff consumes finite-KL llr regularity and feeds those concrete results into targetTimeTermIntegrable and targetTimeDerivativeFormula. Remaining exact boundary: prove that a.e. source density-ratio equality, verify target-density pointwise derivative and domination, bridge the concrete Mathlib HasDerivAt/integrability statements to the package fields, and still prove endpoint-safe first-term KL differentiation." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteTargetTimeDerivativeOfDominated", "SALD.generalMovingTargetDiscreteTargetTimeDerivativeSourceRatioCongr", "SALD.generalMovingTargetDiscretePureRawKlTargetTimeFieldsOfDominated", "SALD.GeneralMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlr", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "MeasureTheory.integral_congr_ae", "MeasureTheory.Integrable.congr", "appendix.tex:1358-1366", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing target-time handoff only. It does not prove the source density-ratio a.e. equality, endpoint-safe KL differentiation, weak FP, hbarBCondExp, no-boundary, diffusion source-action, IBP/FI, theorem closure, SLT reuse, or any Lake dependency." /-- Cycle-111 proof-DAG pane for the target-time KL derivative packet. -/ def cycle111GeneralMovingTargetDiscreteTargetTimeDerivativeDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle111.middle_target_time_derivative_boundary" interface := "Middle source map: inside appendix.tex:1358-1366, isolate the target-time term -int (hat rho_s / tilde pi_s) * partial_s tilde pi_s dx from the remaining pure no-mass finite-KL llr KL-differentiability package." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle111GeneralMovingTargetDiscreteTargetTimeDerivativeMiddleObligation", "ASTIS.SALD.cycle105.remaining_pure_raw_kl_boundary", "eq:general_KL_derivative_0_discrete" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle111.lower_packet.target_time_dominated_derivative" interface := "Compiled lower theorem: a fixed density-ratio weight, pointwise target-density derivative, local derivative domination, and target-time term identification imply weighted derivative integrability and the HasDerivAt formula for the weighted target integral." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle111GeneralMovingTargetDiscreteTargetTimeDerivativeLowerObligation", "SALD.generalMovingTargetDiscreteTargetTimeDerivativeOfDominated", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "appendix.tex:1358-1366" ] reusedBy := [ "ASTIS.SALD.cycle105.remaining_pure_raw_kl_boundary", "sald.general_moving_target_discrete.kl_derivative" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle111.lower_packet.target_time_source_ratio_congr" interface := "Compiled lower bridge: if the chosen target-time weight agrees a.e. with the paper's source density-ratio representative, then the target-time integral, weighted-derivative integrability, and HasDerivAt formula transfer to that source ratio by a.e. integral congruence." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteTargetTimeDerivativeSourceRatioCongr", "MeasureTheory.integral_congr_ae", "MeasureTheory.Integrable.congr", "appendix.tex:1358-1366" ] reusedBy := [ "ASTIS.SALD.cycle111.remaining_pure_raw_kl_after_target_time", "SALD.generalMovingTargetDiscretePureRawKlTargetTimeFieldsOfDominated", "sald.general_moving_target_discrete.kl_derivative" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle111.lower_packet.target_time_fields_for_pure_raw_kl" interface := "Compiled field handoff: finite KL supplies Mathlib llr regularity, and the dominated target-time theorem supplies the two target-time fields consumed by SALD.GeneralMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlr through explicit source-specific bridges." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscretePureRawKlTargetTimeFieldsOfDominated", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.generalMovingTargetDiscreteTargetTimeDerivativeOfDominated", "SALD.GeneralMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlr" ] reusedBy := [ "SALD.generalMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlrHkl", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfPureNoMassRawKlBoundaryAtFiniteKlLlrWithLogAction", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle111.remaining_pure_raw_kl_after_target_time" interface := "Remaining exact theorem after the target-time narrowing: prove the a.e. equality identifying the fixed weight with the source density-ratio representative hat rho_s / tilde pi_s, prove the target-density pointwise derivative/domination package and source bridges, and separately prove endpoint-safe first-term KL differentiation at the finite-KL llr weak test." source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "ASTIS.SALD.cycle105.remaining_pure_raw_kl_boundary", "SALD.generalMovingTargetDiscretePureRawKlTargetTimeFieldsOfDominated", "SALD.generalMovingTargetDiscreteTargetTimeDerivativeSourceRatioCongr", "SALD.GeneralMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlr", "appendix.tex:1358-1366" ] reusedBy := [ "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-! ### Cycle 112: named `barB` conditional-expectation representative -/ /-- Cycle-112 middle obligation for the selected named `barB` representative. The active source slice is `appendix.tex:1368-1377`. This packet narrows the remaining `hbarBCondExp` premise after cycle 110 by replacing it with the standard uniqueness characterization of conditional expectation. -/ def cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle112_named_barB_condExp_representative_middle" statement := "Cycle 112 keeps the active EM conditional-law/Fokker-Planck backend over appendix.tex:1358-1387 narrowed to the selected named barB source representative at appendix.tex:1368-1377. Classification: narrows-source-cited-boundary. The exact supplied hypothesis narrowed is hbarBCondExp in SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef. The lower-ready theorem replaces that direct conditional-expectation a.e. equality by Mathlib's conditional-expectation uniqueness boundary: barB(hatXAtS omega) is mState.comap hatXAtS-a.e. strongly measurable and integrable, and its Bochner set integrals over every hatXAtS-measurable finite-measure set equal the set integrals of the frozen guide-plus-score drift." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable", "MeasureTheory.ae_eq_condExp_of_forall_setIntegral_eq", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle synchronization plus a compiled conditional-expectation uniqueness handoff. It rejects hbarBCondExp wrapper churn, weak-FP/no-boundary/KL/LSI/DV/Gronwall promotion, broad route audits, theorem-status changes, SLT imports, Lake dependency changes, source-index rebaselines, and sald_version_2 use." /-- Cycle-112 lower obligation for the compiled conditional-expectation representative handoff. -/ def cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle112_named_barB_condExp_representative_lower" statement := "Cycle 112 lower compiles SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq, SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef, SALD.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents, and SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef for appendix.tex:1368-1377. Classification: narrows-source-cited-boundary. The first theorem derives the old hbarBCondExp premise from the conditional-expectation uniqueness criterion: guide/score integrability gives integrability of the frozen drift, while the selected barB representative supplies a.e. measurability, integrability, and matching set integrals on all hatXAtS-measurable finite-measure sets. The state-event bridge narrows that set-integral condition to measurable state events {omega | hatXAtS omega in t}, matching the source conditioning on hat X_s=x. The downstream theorems plug the derived hbarBCondExp into the existing cycle-109 product-condExp source bridge, so the canonical condDistrib equality no longer needs hbarBCondExp as a primitive premise. Remaining exact boundary: prove candidate regularity plus the source state-event set-integral characterization for the selected paper representative barB(hatXAtS omega)." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef", "SALD.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable", "MeasureTheory.ae_eq_condExp_of_forall_setIntegral_eq", "MeasurableSpace.measurableSet_comap", "MeasureTheory.Integrable.comp_aemeasurable", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] note := "Proof-producing boundary narrowing only. It does not prove the state-event set-integral characterization, conditional law construction, weak Fokker-Planck equation, no-boundary theorem, diffusion source action, KL/log-ratio derivative, theorem closure, SLT reuse, or any Lake dependency." /-- Cycle-112 proof-DAG pane for the named `barB` conditional-expectation representative boundary. -/ def cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle112.middle_named_barB_condExp_representative" interface := "Middle source map: appendix.tex:1368-1377 names barB as the conditional expectation of the frozen guide-plus-score drift given hat X_s=x. After cycle 110 removed equality-set measurability, the remaining removable primitive is hbarBCondExp." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeMiddleObligation", "ASTIS.SALD.cycle110.remaining_condExp_source_representative_after_eq_meas", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle112.lower_packet.named_barB_condExp_uniqueness" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq derives hbarBCondExp from Mathlib conditional-expectation uniqueness under candidate measurability, candidate integrability, guide/score integrability, and equality of Bochner set integrals on all hatXAtS-measurable finite-measure sets." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq", "MeasureTheory.ae_eq_condExp_of_forall_setIntegral_eq", "MeasureTheory.Integrable.comp_aemeasurable", "appendix.tex:1368-1377" ] reusedBy := [ "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef", "ASTIS.SALD.cycle112.remaining_named_barB_set_integral_characterization" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle112.lower_packet.named_barB_source_bridge_without_hbarBCondExp" interface := "Compiled downstream handoff: SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef feeds the derived hbarBCondExp into the existing product-condExp source bridge, replacing the primitive hbarBCondExp input with the set-integral characterization boundary." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef", "SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib" ] reusedBy := [ "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle112.lower_packet.named_barB_state_event_set_integral" interface := "Compiled lower bridge: equality of Bochner set integrals on source-facing state events {omega | hatXAtS omega in t} implies the all-comap set-integral hypothesis used by conditional-expectation uniqueness, via MeasurableSpace.measurableSet_comap." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef", "MeasurableSpace.measurableSet_comap", "appendix.tex:1368-1377" ] reusedBy := [ "ASTIS.SALD.cycle112.remaining_named_barB_set_integral_characterization", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle112.remaining_named_barB_set_integral_characterization" interface := "Remaining exact theorem after the cycle-112 lower narrowing: prove the paper-selected representative barB(hatXAtS omega) is a.e. measurable for mState.comap hatXAtS, integrable under P, and has the same Bochner set integrals as the frozen guide-plus-score drift over source-facing events {omega | hatXAtS omega in t} for measurable state sets t." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "ASTIS.SALD.cycle110.remaining_condExp_source_representative_after_eq_meas", "SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq", "SALD.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef", "MeasureTheory.ae_eq_condExp_of_forall_setIntegral_eq", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-! ### Cycle 113: named `barB` state-field regularity pullback -/ /-- Cycle-113 lower obligation for pulling the selected named `barB` regularity back from the state marginal. The discrete theorem pressure test reaches the cycle-112 named `barB` representative boundary. This lower packet discharges the sample-space candidate-regularity inputs of that boundary from source-facing state-field regularity, leaving the state-event set-integral characterization as the next non-wrapper blocker. -/ def cycle113GeneralMovingTargetDiscreteNamedBarBStateFieldRegularityLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle113_named_barB_state_field_regularity_lower" statement := "Cycle 113 lower compiles SALD.generalMovingTargetDiscreteNamedBarBComapRegularityOfStateField and SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateFieldSetIntegralDef for appendix.tex:1368-1377. Classification: discharges-supplied-hypothesis. The exact supplied hypotheses discharged from SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef are hbarBMeas and hbarBInt: if the selected named state-field barB is StronglyMeasurable on State and Integrable under hatRhoS, with hatRhoS = Measure.map hatXAtS P, then barB(hatXAtS omega) is AEStronglyMeasurable for mState.comap hatXAtS and Integrable under P. The downstream theorem feeds those derived regularity facts into the existing cycle-112 state-event set-integral bridge. Remaining exact non-wrapper blocker after the forward-KL-discrete pressure test: prove the source Bochner set-integral characterization over events {omega | hatXAtS omega in t} for measurable state sets t, plus any source proof of state-field Integrable barB hatRhoS not already supplied by the selected representative." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteNamedBarBComapRegularityOfStateField", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateFieldSetIntegralDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef", "SALD.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents", "MeasureTheory.Integrable.comp_measurable", "Measurable.of_comap_le", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete" ] note := "Proof-producing lower theorem only. It does not prove the state-event set-integral characterization, conditional law construction, weak Fokker-Planck equation, KL derivative, LSI, DV, Gronwall, theorem closure, SLT reuse, Lake changes, source-index rebaseline, or sald_version_2 content." /-- Cycle-113 proof-DAG pane for the named `barB` regularity pullback. -/ def cycle113GeneralMovingTargetDiscreteNamedBarBStateFieldRegularityDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle113.lower_packet.named_barB_state_field_regularity" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteNamedBarBComapRegularityOfStateField derives the cycle-112 hbarBMeas and hbarBInt candidate-regularity inputs from StronglyMeasurable barB, Integrable barB hatRhoS, hatRhoS=Law(hatXAtS), and Measurable hatXAtS." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteNamedBarBComapRegularityOfStateField", "MeasureTheory.Integrable.comp_measurable", "Measurable.of_comap_le", "appendix.tex:1368-1377" ] reusedBy := [ "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateFieldSetIntegralDef", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle113.lower_packet.named_barB_state_field_set_integral_bridge" interface := "Compiled downstream handoff: SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateFieldSetIntegralDef feeds the derived regularity facts into the cycle-112 state-event set-integral bridge, replacing hbarBMeas and hbarBInt with source-facing state-field hypotheses." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateFieldSetIntegralDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef", "SALD.generalMovingTargetDiscreteNamedBarBComapRegularityOfStateField", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle113.remaining_named_barB_state_event_set_integral" interface := "Remaining exact non-wrapper blocker after cycle 113: prove the source Bochner set-integral characterization for the selected representative over events {omega | hatXAtS omega in t}; if the selected state representative is not already known integrable under hatRhoS, prove that state-field integrability from the same source construction." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "ASTIS.SALD.cycle112.remaining_named_barB_set_integral_characterization", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateFieldSetIntegralDef", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.obligation } ] /-! ### Cycle 114: canonical `barB` state-event set integral -/ /-- Cycle-114 lower obligation for the canonical state-event set-integral part of the remaining named `barB` boundary. This packet stays on the dynamic leaf `ASTIS.SALD.cycle113.remaining_named_barB_state_event_set_integral` for `appendix.tex:1368-1377`. It does not assume the paper's separately named representative is already the canonical Mathlib `condDistrib` field; instead it compiles the canonical set-integral theorem and leaves only the smaller source version-selection/integrability boundary for a non-canonical selected representative. -/ def cycle114GeneralMovingTargetDiscreteCanonicalBarBStateEventSetIntegralLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle114_canonical_barB_state_event_set_integral_lower" statement := "Cycle 114 lower compiles SALD.generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib for appendix.tex:1368-1377. Classification: narrows-source-cited-boundary. The theorem proves the source state-event Bochner set-integral characterization for the canonical Mathlib condDistrib representative of barB: over every measurable state set t, the set integral of dotTk times the conditional guide integral plus sigmaCoeff times the conditional score integral, pulled back by hatXAtS, equals the set integral of the frozen guide-plus-score sample drift on {omega | hatXAtS omega in t}. This strictly narrows hbarBStateSetIntegral / ASTIS.SALD.cycle113.remaining_named_barB_state_event_set_integral to the remaining source version-selection theorem that the paper's selected named barB is the canonical condDistrib representative under hatRhoS, plus Integrable barB hatRhoS if a separate version is used." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib'", "MeasureTheory.setIntegral_condExp", "MeasureTheory.setIntegral_congr_ae", "MeasureTheory.Integrable.condDistrib_ae", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "ASTIS.SALD.cycle113.remaining_named_barB_state_event_set_integral" ] note := "Dynamic-leaf worker packet only. It rejects wrapper churn around hbarBStateSetIntegral: the compiled theorem discharges the canonical condDistrib state-event integral part and leaves the smaller paper-version boundary. It does not prove weak FP, KL derivative, LSI, DV, Gronwall, theorem closure, SLT reuse, Lake changes, source-index rebaseline, or sald_version_2 content." /-- Cycle-114 proof-DAG pane for the canonical state-event set-integral narrowing. -/ def cycle114GeneralMovingTargetDiscreteCanonicalBarBStateEventSetIntegralDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle114.lower_packet.canonical_barB_state_event_set_integral" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib proves hbarBStateSetIntegral for the canonical condDistrib representative on all source-facing state events {omega | hatXAtS omega in t}, using Mathlib's product conditional-expectation theorem and setIntegral_condExp." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib'", "MeasureTheory.setIntegral_condExp", "MeasureTheory.setIntegral_congr_ae", "appendix.tex:1368-1377" ] reusedBy := [ "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateFieldSetIntegralDef", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle114.remaining_named_barB_version_after_canonical_state_event_integral" interface := "Remaining exact theorem after the canonical state-event narrowing: prove the paper-selected named barB is the canonical condDistrib representative hatRhoS-a.e., or instantiate the downstream EM route with the canonical representative; if a separate version is kept, also prove Integrable barB hatRhoS from the source construction." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "ASTIS.SALD.cycle113.remaining_named_barB_state_event_set_integral", "SALD.generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateFieldSetIntegralDef", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.obligation } ] /-! ### Cycle 115: selected named `barB` version after canonical state events -/ /-- Cycle-115 middle obligation for the selected-version boundary left after the canonical state-event set-integral theorem. This packet stays on `ASTIS.SALD.cycle114.remaining_named_barB_version_after_canonical_state_event_integral` for `appendix.tex:1368-1377`. It compiles the local theorem that turns the smaller `hatRhoS`-a.e. selected-version equality into the old `hbarBStateSetIntegral` input and into `Integrable barB hatRhoS`; it does not prove the source selected-version equality itself. -/ def cycle115GeneralMovingTargetDiscreteNamedBarBSelectedVersionMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle115_named_barB_selected_version_middle" statement := "Cycle 115 middle compiles SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq for appendix.tex:1368-1377. Classification: narrows-source-cited-boundary. The theorem shows that once the paper-selected named barB is hatRhoS-a.e. equal to the canonical Mathlib condDistrib guide-plus-score representative, the remaining hbarBStateSetIntegral premise follows on all state events by pulling the a.e. equality back through hatXAtS and applying MeasureTheory.setIntegral_congr_ae to the cycle-114 canonical state-event theorem. The same selected-to-canonical a.e. equality also yields Integrable barB hatRhoS from SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq. Remaining exact boundary: prove the source selected-version equality between the paper conditional-expectation representative and the canonical condDistrib field." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "MeasureTheory.ae_eq_comp", "MeasureTheory.setIntegral_congr_ae", "ASTIS.SALD.cycle114.remaining_named_barB_version_after_canonical_state_event_integral", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area refiner packet. It rejects wrapper churn: the compiled theorem derives the old state-event set-integral and integrability inputs from the smaller selected-to-canonical a.e. equality, leaving that equality as the only selected-version theorem boundary. No weak FP, KL derivative, LSI, DV, Gronwall, theorem-status promotion, SLT import, Lake change, source-index rebaseline, or sald_version_2 content." /-- Cycle-115 lower obligation for the source conditional-expectation version of the selected `barB` bridge. -/ def cycle115GeneralMovingTargetDiscreteNamedBarBCondExpSourceLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle115_named_barB_condExp_source_lower" statement := "Cycle 115 lower compiles SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCondExpSourceDef for appendix.tex:1368-1377. Classification: discharges-supplied-hypothesis. The exact supplied hypothesis discharged is hbarBAe in SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq: the theorem derives the selected-to-canonical hatRhoS-a.e. equality from SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef and the paper's sample-space conditional-expectation representative hbarBCondExp, then reuses the cycle-115 middle theorem to obtain Integrable barB hatRhoS and hbarBStateSetIntegral. Remaining exact boundary: prove hbarBCondExp for the paper-selected conditional-expectation representative, with hbarBEqMeas supplied by the existing cycle-110 measurable-representative route if not source-supplied." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCondExpSourceDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable", "ASTIS.SALD.cycle115.remaining_named_barB_selected_canonical_ae_eq", "ASTIS.SALD.cycle110.remaining_condExp_source_representative_after_eq_meas", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower theorem only. It removes the supplied hbarBAe equality from the selected-version bridge using the already source-cited conditional-expectation definition route. It does not prove weak FP, KL derivative, LSI, DV, Gronwall, theorem-status promotion, SLT import, Lake change, source-index rebaseline, or sald_version_2 content." /-- Cycle-115 proof-DAG pane for the selected named `barB` version boundary. -/ def cycle115GeneralMovingTargetDiscreteNamedBarBSelectedVersionDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle115.middle_packet.named_barB_selected_version_to_state_events" interface := "Compiled middle theorem: SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq derives Integrable barB hatRhoS and hbarBStateSetIntegral from the selected paper barB being hatRhoS-a.e. equal to the canonical condDistrib guide-plus-score field." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "MeasureTheory.ae_eq_comp", "MeasureTheory.setIntegral_congr_ae", "appendix.tex:1368-1377" ] reusedBy := [ "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateFieldSetIntegralDef", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle115.lower_packet.named_barB_condExp_source_to_state_events" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCondExpSourceDef discharges the supplied hbarBAe equality in the selected-version bridge by deriving it from the paper sample-space conditional-expectation representative hbarBCondExp, then returns Integrable barB hatRhoS and hbarBStateSetIntegral." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCondExpSourceDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib'", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle115.remaining_named_barB_condExp_source_representative" interface := "Remaining exact theorem after the cycle-115 lower packet: prove hbarBCondExp, the paper-selected sample-space conditional-expectation representative equality for barB(hatXAtS omega); if not supplied directly, also discharge hbarBEqMeas through the cycle-110 measurable-representative theorem." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "ASTIS.SALD.cycle115.remaining_named_barB_selected_canonical_ae_eq", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCondExpSourceDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable", "ASTIS.SALD.cycle114.remaining_named_barB_version_after_canonical_state_event_integral", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.obligation } ] /-! ### Cycle 116: canonical `barB` conditional-expectation representative -/ /-- Cycle-116 lower obligation for the canonical conditional-expectation representative of the named `barB` drift. This packet stays on `ASTIS.SALD.cycle115.remaining_named_barB_condExp_source_representative` for `appendix.tex:1368-1377`. It compiles the standalone canonical theorem that proves the old `hbarBCondExp` conclusion when `barB` is chosen as the Mathlib `condDistrib` guide-plus-score field. The remaining source boundary is only the paper-selected named-version equality to that canonical representative. -/ def cycle116GeneralMovingTargetDiscreteCanonicalBarBCondExpLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle116_canonical_barB_condExp_lower" statement := "Cycle 116 lower compiles SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib and SALD.generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib for appendix.tex:1368-1377. Classification: narrows-source-cited-boundary. The first theorem proves hbarBCondExp for the canonical Mathlib condDistrib guide-plus-score representative: the conditional expectation of the frozen sample drift given mState.comap hatXAtS equals the canonical state field composed with hatXAtS, using ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib' and Bochner integral add/smul linearity. The second theorem specializes the cycle-115 selected barB bridge to that canonical representative, discharging both hbarBCondExp and the equality-set measurability premise in the canonical case and returning Integrable canonical barB hatRhoS plus hbarBStateSetIntegral. Remaining exact boundary: identify the paper-selected named barB with the canonical condDistrib field, or instantiate the downstream EM route with the canonical representative." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib", "SALD.generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib", "SALD.generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCondExpSourceDef", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib'", "MeasureTheory.integral_add", "MeasureTheory.integral_smul", "ASTIS.SALD.cycle115.remaining_named_barB_condExp_source_representative", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet. It narrows the hbarBCondExp boundary to the paper-selected version step by proving the canonical representative case directly from Mathlib conditional distributions. It does not prove weak FP, KL derivative, LSI, DV, Gronwall, theorem-status promotion, SLT import, Lake change, source-index rebaseline, wrapper churn, or sald_version_2 content." /-- Cycle-116 proof-DAG pane for the canonical `barB` conditional-expectation representative. -/ def cycle116GeneralMovingTargetDiscreteCanonicalBarBCondExpDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle116.lower_packet.canonical_barB_condExp" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib proves hbarBCondExp for the canonical condDistrib guide-plus-score representative composed with hatXAtS." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib'", "MeasureTheory.integral_add", "MeasureTheory.integral_smul", "appendix.tex:1368-1377" ] reusedBy := [ "SALD.generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCondExpSourceDef", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle116.lower_packet.canonical_barB_condExp_to_state_events" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib specializes the cycle-115 selected bridge to the canonical condDistrib field, discharging hbarBCondExp and hbarBEqMeas for that representative and returning Integrable canonical barB hatRhoS plus hbarBStateSetIntegral." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib", "SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCondExpSourceDef", "SALD.generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle116.remaining_named_barB_selected_version_after_canonical_condExp" interface := "Remaining exact theorem after the canonical hbarBCondExp narrowing: identify the paper-selected named barB with the canonical condDistrib guide-plus-score field, or instantiate downstream EM obligations with the canonical representative." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "ASTIS.SALD.cycle115.remaining_named_barB_condExp_source_representative", "SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.obligation } ] /-! ### Cycle 117: selected named `barB` version-selection boundary -/ /-- Cycle-117 middle obligation for the selected paper `barB` version step. The refreshed blueprint illness area is not another conditional-expectation wrapper. Cycle 116 already proves the old `hbarBCondExp` input for the canonical Mathlib `condDistrib` representative. The remaining lower-ready boundary is the faithful source version-selection step: either identify the paper-selected named `barB` with that canonical field, or use the canonical representative downstream if the paper's definition supports that choice. -/ def cycle117GeneralMovingTargetDiscreteNamedBarBVersionSelectionMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle117_named_barB_version_selection_middle" statement := "Cycle 117 middle follows the refreshed blueprint illness area for appendix.tex:1368-1377. Classification: narrows-source-cited-boundary. It rejects another hbarBCondExp wrapper: cycle 116 already proves hbarBCondExp and the state-event interface for the canonical condDistrib guide-plus-score field. The exact lower-ready theorem is the paper-selected named barB version-selection boundary, hbarBAe: canonicalCondDistribBarB =ae[hatRhoS] barB, or a source-supported downstream instantiation of sald.general_moving_target_discrete.em_interpolation_fp with the canonical representative. This narrows ASTIS.SALD.cycle116.remaining_named_barB_selected_version_after_canonical_condExp / ASTIS.SALD.cycle115.remaining_named_barB_condExp_source_representative to selected representative choice only; hbarBEqMeas remains available via cycle110 if a separate selected barB is retained." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "ASTIS.SALD.cycle116.remaining_named_barB_selected_version_after_canonical_condExp", "ASTIS.SALD.cycle115.remaining_named_barB_condExp_source_representative", "SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib", "SALD.generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalPointwiseEq", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCondExpSourceDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area refiner packet. It narrows the remaining named barB boundary to the representative-selection theorem after the canonical hbarBCondExp proof; it does not add a broad wrapper, route audit, SLT import, Lake change, theorem-status promotion, weak-FP/KL proof, source-index rebaseline, or sald_version_2 content." /-- Cycle-117 lower obligation for the source-supported pointwise canonical representative choice. The lower packet compiles the bridge from the source's pointwise definition of `\bar b_{k,s}` as the canonical conditional-distribution guide-plus-score field to the downstream state-event package. This is a canonical-representative instantiation of the selected-version boundary, not another `hbarBCondExp` wrapper. -/ def cycle117GeneralMovingTargetDiscreteNamedBarBVersionSelectionLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle117_named_barB_version_selection_lower" statement := "Cycle 117 lower compiles SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalPointwiseEq for appendix.tex:1368-1377. Classification: narrows-source-cited-boundary. The exact boundary narrowed is ASTIS.SALD.cycle117.lower_ready.selected_named_barB_canonical_ae_eq / ASTIS.SALD.cycle116.remaining_named_barB_selected_version_after_canonical_condExp: if the paper-selected barB is chosen pointwise as the canonical condDistrib guide-plus-score field, then hbarBAe is discharged by a.e. reflexivity and the existing cycle-115 selected-version bridge returns Integrable barB hatRhoS plus the source state-event set-integral identity. Remaining exact source choice: either use this canonical representative downstream in the EM weak-FP route, or, if a separate named version is retained, prove the pointwise/canonical equality for that version." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalPointwiseEq", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib", "ASTIS.SALD.cycle117.lower_ready.selected_named_barB_canonical_ae_eq", "ASTIS.SALD.cycle116.remaining_named_barB_selected_version_after_canonical_condExp", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower packet inside the refreshed illness area. It proves only the pointwise canonical-representative handoff for barB version selection; it does not prove weak FP, KL derivative, LSI, DV, Gronwall, theorem closure, SLT reuse, Lake changes, source-index rebaseline, broad route audit, wrapper churn, or sald_version_2 content." /-- Cycle-117 proof-DAG pane for the selected paper `barB` version boundary. -/ def cycle117GeneralMovingTargetDiscreteNamedBarBVersionSelectionDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle117.middle_packet.named_barB_version_selection" interface := "Middle packet: the current lower-ready boundary is the source version-selection theorem identifying the paper-selected named barB with the canonical condDistrib guide-plus-score representative, not another hbarBCondExp wrapper." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "ASTIS.SALD.cycle116.remaining_named_barB_selected_version_after_canonical_condExp", "SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib", "SALD.generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle117.lower_packet.selected_barB_pointwise_canonical" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalPointwiseEq discharges the selected-to-canonical hbarBAe premise when the paper-selected barB is pointwise the canonical condDistrib guide-plus-score field, then reuses the cycle-115 selected-version bridge to return Integrable barB hatRhoS and the state-event set-integral identity." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalPointwiseEq", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle117.lower_ready.selected_named_barB_canonical_ae_eq" interface := "Remaining source choice after the lower packet: either use the canonical condDistrib guide-plus-score representative directly downstream, as supported by the pointwise definition at appendix.tex:1368-1377, or prove the pointwise/canonical equality for a separately retained named barB version." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalPointwiseEq", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib'", "Mathlib.Probability.Kernel.CondDistrib", "appendix.tex:1368-1377" ] reusedBy := [ "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCondExpSourceDef", "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.obligation } ] /-! ### Cycle 118: direct canonical `barB` downstream instantiation -/ /-- Cycle-118 middle obligation for direct canonical downstream use. The refreshed illness area is no longer a request for another `hbarBCondExp` bridge. The source definition at `appendix.tex:1368-1377` allows the lower route to choose the canonical Mathlib `condDistrib` guide-plus-score field as the `barB` representative consumed by the EM state-event interface at `appendix.tex:1379-1387`. If a separate named representative is retained, the only remaining boundary is its pointwise equality to that canonical field. -/ def cycle118GeneralMovingTargetDiscreteCanonicalBarBDownstreamMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle118_canonical_barB_downstream_middle" statement := "Cycle 118 middle follows the refreshed blueprint illness area for appendix.tex:1368-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: ASTIS.SALD.cycle117.lower_ready.selected_named_barB_canonical_ae_eq / ASTIS.SALD.cycle116.remaining_named_barB_selected_version_after_canonical_condExp is now the direct downstream canonical representative instantiation: choose barB to be the canonical condDistrib guide-plus-score field in sald.general_moving_target_discrete.em_interpolation_fp and consume SALD.generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib. A separately retained named barB still has only the pointwise/canonical equality fallback handled by SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalPointwiseEq. Another hbarBCondExp wrapper is rejected because SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib already proves the canonical representative case." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "ASTIS.SALD.cycle117.lower_ready.selected_named_barB_canonical_ae_eq", "ASTIS.SALD.cycle116.remaining_named_barB_selected_version_after_canonical_condExp", "SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib", "SALD.generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalPointwiseEq", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib'", "MeasureTheory.setIntegral_condExp", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area refiner packet. It narrows the active barB representative boundary to direct canonical downstream instantiation of the EM state-event interface; it does not add a wrapper around hbarBCondExp, a route audit, a non-EM fallback, a theorem-status promotion, a Lake change, an SLT import, a source-index rebaseline, or sald_version_2 content." /-- Cycle-118 lower obligation for the direct canonical EM state-event interface. This lower packet compiles the concrete representative handoff requested by the refreshed illness area: the downstream interface may take `barB` to be the canonical conditional-distribution guide-plus-score field, with integrability and all source-facing state-event set integrals supplied by the canonical package. -/ def cycle118GeneralMovingTargetDiscreteCanonicalBarBDownstreamLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle118_canonical_barB_downstream_lower" statement := "Cycle 118 lower compiles SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface for appendix.tex:1368-1387. Classification: narrows-source-cited-boundary. The exact boundary narrowed is ASTIS.SALD.cycle118.lower_ready.direct_canonical_barB_em_state_event_interface / ASTIS.SALD.cycle117.lower_ready.selected_named_barB_canonical_ae_eq: the theorem existentially instantiates the EM state-event interface with barB equal to the canonical condDistrib guide-plus-score field and consumes SALD.generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib to return Integrable barB hatRhoS plus the state-event set-integral identity. A separate named barB remains only as the pointwise/canonical equality fallback." source := saldGeneralMovingTargetDiscreteConditionalDriftSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface", "SALD.generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib", "SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib", "ASTIS.SALD.cycle118.lower_ready.direct_canonical_barB_em_state_event_interface", "ASTIS.SALD.cycle117.lower_ready.selected_named_barB_canonical_ae_eq", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area refiner lower packet. It uses the source-supported canonical representative directly and rejects another hbarBCondExp wrapper, broad route audit, non-EM fallback, SLT import, Lake change, theorem-status promotion, source-index rebaseline, or sald_version_2 content." /-- Cycle-118 proof-DAG pane for the direct canonical `barB` downstream route. -/ def cycle118GeneralMovingTargetDiscreteCanonicalBarBDownstreamDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle118.middle_packet.direct_canonical_barB_downstream" interface := "Middle packet: use the source definition of bar b_{k,s} at appendix.tex:1368-1377 to instantiate the downstream EM state-event interface with the canonical condDistrib guide-plus-score representative, rather than adding another hbarBCondExp wrapper." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "ASTIS.SALD.cycle117.lower_ready.selected_named_barB_canonical_ae_eq", "SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib", "SALD.generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib", "appendix.tex:1368-1377", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle118.lower_ready.direct_canonical_barB_em_state_event_interface" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface instantiates the EM conditional-drift/state-event consumer with barB := canonicalCondDistribBarB and consumes SALD.generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib; only if a separate named barB is retained should lower prove pointwise equality to the canonical field via SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalPointwiseEq." source := saldGeneralMovingTargetDiscreteConditionalDriftSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface", "SALD.generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib", "SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalPointwiseEq", "ASTIS.SALD.cycle117.lower_ready.selected_named_barB_canonical_ae_eq", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete" ] status := ProofStatus.formalized } ] /-! ### Cycle 119: canonical `barB` weak-FP consumer boundary -/ /-- Cycle-119 middle obligation for consuming the canonical `barB` witness in the weak-FP generator/source-sign path. The refreshed illness area after cycle 118 is no longer the representative choice for `barB`. This packet fixes `barB` to the canonical `condDistrib` guide-plus-score field supplied by `generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface` and narrows the remaining weak-FP boundary to the EM path derivative, domination, and source-action identities needed by the dominated generator handoff. -/ def cycle119GeneralMovingTargetDiscreteCanonicalBarBWeakFpConsumerMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle119_canonical_barB_weak_fp_consumer_middle" statement := "Cycle 119 middle follows the refreshed dynamic leaf for appendix.tex:1368-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: ASTIS.SALD.cycle110.remaining_parametric_generator_boundary_after_dominated_transport is now the canonical-barB weak-FP consumer boundary. The lower route should instantiate barB by SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface, keep the canonical condDistrib guide-plus-score field from appendix.tex:1368-1377, and consume SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff plus the cycle-94 barB source-sign handoffs. The remaining named theorem is not hbarBCondExp: it is the EM pointwise path-derivative/local-domination package and the canonical source-action identities driftAction = weakGradPairing canonicalBarB, weakGradPairing canonicalBarB = -driftDiv by no-boundary divergence, diffusionAction = sigmaCoeff • laplacian, and, for the normalized source-sign consumer, the corresponding law-derivative/partialS identification." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "ASTIS.SALD.cycle118.lower_ready.direct_canonical_barB_em_state_event_interface", "ASTIS.SALD.cycle110.remaining_parametric_generator_boundary_after_dominated_transport", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "Mathlib.Probability.Kernel.CondDistrib", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area refiner middle packet. It rejects another hbarBCondExp or named-representative wrapper, broad source-index or theorem-route audit, non-EM fallback, SLT import, Lake dependency change, theorem-status promotion, fake closure, and sald_version_2 content." /-- Cycle-119 lower obligation for the canonical `barB` weak-FP consumer. This is a proof-producing lower packet, not a theorem-status promotion. It compiles the consumer that takes the canonical conditional-drift/state-event package accepted in cycle 118 and uses it inside the dominated named-law weak derivative route. The remaining source-cited theorem boundary is still the EM path derivative/domination and source-action package. -/ def cycle119GeneralMovingTargetDiscreteCanonicalBarBWeakFpConsumerLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle119_canonical_barB_weak_fp_consumer_lower_ready" statement := "Cycle 119 lower compiles SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated for appendix.tex:1368-1387. Classification: narrows-source-cited-boundary. The theorem consumes SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface, fixes barB to the canonical condDistrib guide-plus-score field, derives Integrable canonicalBarB (hatRhoS s0) locally, feeds SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound, and then applies SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff. The supplied canonical-barB integrability input is no longer part of the weak-FP consumer. Remaining exact boundary: ASTIS.SALD.cycle119.remaining_em_path_derivative_domination_and_source_actions, namely the EM sample-path HasDerivAt/local-domination package, derivative-value split, canonical drift weak-action identity, no-boundary divergence identity for hatRhoS * canonicalBarB, diffusion source-action identity, and law-derivative/partialS uniqueness if the normalized source-sign consumer is used." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle119GeneralMovingTargetDiscreteCanonicalBarBWeakFpConsumerMiddleObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff", "SALD.cycle110GeneralMovingTargetDiscreteWeakFpDominatedGeneratorLowerObligation", "SALD.cycle118GeneralMovingTargetDiscreteCanonicalBarBDownstreamLowerObligation", "ASTIS.SALD.cycle119.remaining_em_path_derivative_domination_and_source_actions", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] note := "Proof-producing lower packet. It consumes the cycle-118 canonical barB witness and removes a supplied integrability input from the weak-FP consumer path. It does not prove the EM path derivative, no-boundary theorem, diffusion source action, KL derivative, LSI, DV, Gronwall, source-index rebaseline, SLT reuse, Lake dependency, theorem status, or sald_version_2 content." /-- Cycle-119 proof-DAG pane for the canonical `barB` weak-FP consumer. -/ def cycle119GeneralMovingTargetDiscreteCanonicalBarBWeakFpConsumerDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle119.middle_packet.canonical_barB_weak_fp_consumer" interface := "Middle packet: keep barB fixed to the canonical condDistrib guide-plus-score representative from appendix.tex:1368-1377 and route it into the weak-FP generator/source-sign consumer at appendix.tex:1379-1387. This rejects reopening hbarBCondExp or a selected-representative equality unless a separate named barB is deliberately retained." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle119GeneralMovingTargetDiscreteCanonicalBarBWeakFpConsumerMiddleObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff", "appendix.tex:1368-1377", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle119.lower_ready.canonical_barB_weak_fp_consumer_boundary" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated consumes the cycle-118 canonical barB witness in the dominated named-law weak-FP route, derives Integrable canonicalBarB (hatRhoS s0), feeds the inner-gradient no-boundary drift-source handoff, and leaves only the EM path-derivative/domination/source-action theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle119GeneralMovingTargetDiscreteCanonicalBarBWeakFpConsumerLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff", "ASTIS.SALD.cycle110.remaining_parametric_generator_boundary_after_dominated_transport" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle119.remaining_em_path_derivative_domination_and_source_actions" interface := "Remaining exact theorem: for each admissible weak test, prove the EM interpolation pointwise HasDerivAt and local dominated derivative-under-integral package, identify the derivative integral with driftAction + diffusionAction for the canonical condDistrib barB, prove driftAction = weakGradPairing canonicalBarB, prove weakGradPairing canonicalBarB = -driftDiv through the no-boundary divergence theorem for hatRhoS * canonicalBarB, prove diffusionAction = sigmaCoeff • laplacian, and supply the law-derivative/partialS uniqueness step if the normalized source-sign consumer is used." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndDominated", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface", "ASTIS.SALD.cycle100.remaining_no_boundary_after_inner_gradient", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle119.reviewer_canonical_barB_weak_fp_consumer_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, the target remains the EM conditional-drift/weak-FP backend at appendix.tex:1368-1387, the canonical barB witness is consumed rather than wrapped, and python3 tools/astis.py check passes with no fake closures, non-EM fallback, SLT/Lake change, source-index rebaseline, theorem-status promotion, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle119GeneralMovingTargetDiscreteCanonicalBarBWeakFpConsumerLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff", "appendix.tex:1368-1387" ] reusedBy := ["cycle 119 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 120: EM sample-path derivative and domination boundary -/ /-- Cycle-120 middle obligation for the EM sample-path derivative and domination inputs. After the cycle-119 canonical `barB` weak-FP consumer compiled, the remaining source-cited theorem was still a bundle. This middle packet narrows that bundle to the seven parametric-integral hypotheses required by `generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated`. The derivative-value split and source-action identities remain separate boundary facts. -/ def cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle120_em_path_derivative_domination_middle" statement := "Cycle 120 middle follows the blueprint illness area over appendix.tex:1379-1387 after the cycle-119 canonical barB weak-FP consumer. Classification: narrows-source-cited-boundary. Exact boundary narrowed: ASTIS.SALD.cycle119.remaining_em_path_derivative_domination_and_source_actions is split so lower first targets the seven EM sample-path derivative/domination inputs consumed by SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated: hsampleNeighborhood, hsampleMeas, hsampleInt, hsampleDerivMeas, hsampleDerivBound, hboundInt, and hpathDeriv. Keep hderivValue, canonical drift weak-action, no-boundary divergence, diffusion source action, and optional law-derivative/partialS uniqueness explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff", "AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndDominated", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area refiner middle packet. It is not an hbarBCondExp wrapper, source-index rebaseline, theorem-route audit, non-EM fallback, SLT import, Lake change, theorem-status promotion, fake closure, or sald_version_2 use." /-- Cycle-120 lower-ready obligation for the concrete EM path derivative/domination package. The lower theorem should be proved from the frozen interpolation formula and admissible-test regularity. It should not absorb the source-action identities: those are downstream facts tied to the Fokker--Planck drift and diffusion terms. -/ def cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle120_em_path_derivative_domination_lower_ready" statement := "Cycle 120 lower-ready packet narrows ASTIS.SALD.cycle119.remaining_em_path_derivative_domination_and_source_actions to ASTIS.SALD.cycle120.lower_ready.em_sample_path_derivative_domination. SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalDominated now discharges the local-neighborhood input hsampleNeighborhood by specializing to the source EM interval. The remaining lower-ready facts are, for each admissible weak test, sample measurability, sample integrability, derivative measurability, local derivative bound, bound integrability, and pointwise HasDerivAt inputs for the frozen EM interpolation. The expected source route is the path formula eq:general_moving_target_SALD_frozen_interp and appendix.tex:1379-1387, with test regularity and common-space assumptions exposed. Remaining exact boundary after this packet: hderivValue identifying the sample-derivative integral with driftAction + diffusionAction, canonical barB drift weak-action/pairing measurability and gradient bound, no-boundary divergence, diffusion source action, and law-derivative/partialS uniqueness if the normalized consumer is used." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationMiddleObligation", "ASTIS.SALD.cycle119.remaining_em_path_derivative_domination_and_source_actions", "ASTIS.SALD.cycle120.lower_ready.em_sample_path_derivative_domination", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated", "AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndDominated", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] note := "Lower-ready source-cited theorem boundary only. It rejects wrapper churn around the already consumed canonical barB witness and leaves source-action equalities outside this packet." /-- Cycle-120 proof-DAG pane for the EM sample-path derivative/domination subboundary. -/ def cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle120.middle_em_path_derivative_domination" interface := "Middle packet: split the cycle-119 remaining theorem so lower targets only the EM path derivative and local domination hypotheses behind Mathlib's dominated derivative-under-integral theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle119GeneralMovingTargetDiscreteCanonicalBarBWeakFpConsumerLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated", "AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndDominated", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle120.lower_ready.em_sample_path_derivative_domination" interface := "Lower-ready exact theorem after the interval-neighborhood lower packet: from the frozen EM interpolation and admissible-test regularity, prove hsampleMeas, hsampleInt, hsampleDerivMeas, hsampleDerivBound, hboundInt, and hpathDeriv for each admissible weak test. Do not include hderivValue or the drift/diffusion source-action equalities in this theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationLowerObligation", "eq:general_moving_target_SALD_frozen_interp", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle120.lower_packet.em_interval_neighborhood" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalDominated specializes the canonical-barB dominated weak-FP route to the source EM interval Set.Ioo sLeft sRight and discharges hsampleNeighborhood from hs0Interval; hsampleMeas, hsampleInt, hsampleDerivMeas, hsampleDerivBound, hboundInt, hpathDeriv, hderivValue, and source-action identities remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated", "Mathlib.Topology.Instances.Real", "appendix.tex:1379-1387", "eq:general_moving_target_SALD_frozen_interp" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle120.remaining_source_actions_after_path_derivative" interface := "Remaining exact theorem after the path-derivative/domination packet: identify the sample-derivative integral as driftAction + diffusionAction, prove the canonical barB weak-action and pairing regularity inputs, prove the no-boundary divergence identity, prove the sigmaCoeff diffusion source action, and supply law-derivative/partialS uniqueness only if the normalized source-sign consumer is used." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "ASTIS.SALD.cycle120.lower_ready.em_sample_path_derivative_domination", "SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound", "ASTIS.SALD.cycle100.remaining_no_boundary_after_inner_gradient", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle120.reviewer_em_path_derivative_domination_check" interface := "Reviewer check: accept only if the packet classification remains narrows-source-cited-boundary, the target is the EM conditional-drift/weak-FP backend at appendix.tex:1368-1387, no hbarBCondExp wrapper or non-EM fallback is introduced, and python3 tools/astis.py check passes." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationMiddleObligation", "SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationLowerObligation", "SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationDag", "appendix.tex:1368-1387" ] reusedBy := ["cycle 120 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 121: EM sample measurability from named-law test measurability -/ /-- Cycle-121 lower obligation for discharging the sample measurability input from the EM interval dominated packet. This packet stays inside the cycle-120 EM path-derivative/domination boundary. It removes only the supplied `hsampleMeas` hypothesis by using the named-law identity `hatRhoS s = Measure.map (hatX s) P`, law-space test measurability, and sample-path a.e. measurability. It leaves the source-specific EM integrability, derivative, domination, derivative-value, and source-action facts explicit. -/ def cycle121GeneralMovingTargetDiscreteEmSampleMeasLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle121_em_sample_meas_lower" statement := "Cycle 121 lower compiles SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasDominated for appendix.tex:1379-1387. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: hsampleMeas from SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalDominated. The theorem derives sample-space AEStronglyMeasurable (fun omega => testEval phi (hatX s omega)) from htestMeas on the named law hatRhoS s, hhatRhoS s = Measure.map (hatX s) P, and hhatX s, then reuses the cycle-120 source EM interval route. Remaining exact boundary: hsampleInt, hsampleDerivMeas, hsampleDerivBound, hboundInt, hpathDeriv, hderivValue, canonical drift weak-action/pairing regularity, no-boundary divergence, diffusion source action, and optional law-derivative/partialS uniqueness." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalDominated", "MeasureTheory.AEStronglyMeasurable.comp_aemeasurable", "MeasureTheory.Measure.map", "ASTIS.SALD.cycle120.lower_ready.em_sample_path_derivative_domination", "ASTIS.SALD.cycle120.remaining_source_actions_after_path_derivative", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area refiner lower packet. It rejects hbarBCondExp wrapper churn, selected-representative replay, broad source-index or theorem-route audits, non-EM fallback, SLT import, Lake changes, theorem-status promotion, fake closure, and sald_version_2 content." /-- Cycle-121 proof-DAG pane for sample measurability discharge inside the EM path-derivative/domination boundary. -/ def cycle121GeneralMovingTargetDiscreteEmSampleMeasDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle121.lower_packet.em_sample_meas_from_law_test_meas" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasDominated derives hsampleMeas from law-space htestMeas plus hatRhoS s = Measure.map (hatX s) P and hhatX s, then reuses the cycle-120 EM interval theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle121GeneralMovingTargetDiscreteEmSampleMeasLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalDominated", "MeasureTheory.AEStronglyMeasurable.comp_aemeasurable", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle121.remaining_em_path_derivative_domination_after_sample_meas" interface := "Remaining exact theorem after hsampleMeas is discharged: prove hsampleInt, hsampleDerivMeas, hsampleDerivBound, hboundInt, and hpathDeriv for each admissible weak test on the source EM interval, then separately prove hderivValue and the canonical barB source-action identities." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasDominated", "ASTIS.SALD.cycle120.remaining_source_actions_after_path_derivative", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle121.reviewer_em_sample_meas_check" interface := "Reviewer check: accept only if the packet classification is discharges-supplied-hypothesis for hsampleMeas, the proof uses named-law Measure.map measurability rather than a new EM wrapper, and python3 tools/astis.py check passes with no hbarBCondExp wrapper churn, non-EM fallback, SLT/Lake change, source-index rebaseline, theorem-status promotion, fake closure, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle121GeneralMovingTargetDiscreteEmSampleMeasLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasDominated", "appendix.tex:1368-1387" ] reusedBy := ["cycle 121 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 122: EM sample integrability from named-law test integrability -/ /-- Cycle-122 lower obligation for discharging the sample integrability input from the EM interval measurable dominated packet. This packet stays inside the cycle-121 EM path-derivative/domination boundary. It removes only the supplied `hsampleInt` hypothesis by using the named-law identity `hatRhoS s0 = Measure.map (hatX s0) P`, law-space test integrability, and sample-path a.e. measurability. It leaves the source-specific derivative, domination, derivative-value, and source-action facts explicit. -/ def cycle122GeneralMovingTargetDiscreteEmSampleIntLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle122_em_sample_int_lower" statement := "Cycle 122 lower compiles SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDominated for appendix.tex:1379-1387. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: hsampleInt from SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasDominated. The theorem derives sample-space Integrable (fun omega => testEval phi (hatX s0 omega)) P from law-space htestInt on the named law hatRhoS s0, hhatRhoS s0 = Measure.map (hatX s0) P, and hhatX s0, then reuses the cycle-121 source EM interval route. Remaining exact boundary: hsampleDerivMeas, hsampleDerivBound, hboundInt, hpathDeriv, hderivValue, canonical drift weak-action/pairing regularity, no-boundary divergence, diffusion source action, and optional law-derivative/partialS uniqueness." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasDominated", "MeasureTheory.Integrable.comp_aemeasurable", "MeasureTheory.integrable_map_measure", "MeasureTheory.Measure.map", "ASTIS.SALD.cycle121.remaining_em_path_derivative_domination_after_sample_meas", "ASTIS.SALD.cycle120.remaining_source_actions_after_path_derivative", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker lower packet. It rejects hbarBCondExp wrapper churn, selected-representative replay, broad source-index or theorem-route audits, non-EM fallback, SLT import, Lake changes, theorem-status promotion, fake closure, and sald_version_2 content." /-- Cycle-122 proof-DAG pane for sample integrability discharge inside the EM path-derivative/domination boundary. -/ def cycle122GeneralMovingTargetDiscreteEmSampleIntDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle122.lower_packet.em_sample_int_from_law_test_int" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDominated derives hsampleInt from law-space htestInt plus hatRhoS s0 = Measure.map (hatX s0) P and hhatX s0, then reuses the cycle-121 EM interval theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle122GeneralMovingTargetDiscreteEmSampleIntLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasDominated", "MeasureTheory.Integrable.comp_aemeasurable", "MeasureTheory.integrable_map_measure", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle122.remaining_em_path_derivative_domination_after_sample_int" interface := "Remaining exact theorem after hsampleMeas and hsampleInt are discharged: prove hsampleDerivMeas, hsampleDerivBound, hboundInt, and hpathDeriv for each admissible weak test on the source EM interval, then separately prove hderivValue and the canonical barB source-action identities." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDominated", "ASTIS.SALD.cycle120.remaining_source_actions_after_path_derivative", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle122.reviewer_em_sample_int_check" interface := "Reviewer check: accept only if the packet classification is discharges-supplied-hypothesis for hsampleInt, the proof uses named-law Measure.map integrability transport rather than a new EM wrapper, and python3 tools/astis.py check passes with no hbarBCondExp wrapper churn, non-EM fallback, SLT/Lake change, source-index rebaseline, theorem-status promotion, fake closure, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle122GeneralMovingTargetDiscreteEmSampleIntLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDominated", "appendix.tex:1368-1387" ] reusedBy := ["cycle 122 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 123: EM sample derivative measurability from concrete derivative representative -/ /-- Cycle-123 lower obligation for discharging the sample derivative measurability input from the EM interval measurable/integrable dominated packet. This packet stays inside the cycle-122 EM path-derivative/domination boundary. It removes only the supplied `hsampleDerivMeas` hypothesis by requiring a source-facing concrete derivative representative, `fun omega => deriv (fun t => testEval phi (hatX t omega)) s0`, its a.e. strong measurability, and its a.e. equality to the chosen `sampleDeriv phi s0`. It leaves local domination, bound integrability, pointwise path derivatives, derivative-value splitting, and source-action facts explicit. -/ def cycle123GeneralMovingTargetDiscreteEmSampleDerivMeasLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle123_em_sample_deriv_meas_lower" statement := "Cycle 123 lower compiles SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasDominated for appendix.tex:1379-1387. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: hsampleDerivMeas from SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDominated. The theorem derives sample-space AEStronglyMeasurable (sampleDeriv phi s0) P from a concrete EM derivative representative fun omega => deriv (fun t => testEval phi (hatX t omega)) s0, its AEStronglyMeasurable proof, and an ae-equality to sampleDeriv phi s0 via AEStronglyMeasurable.congr, then reuses the cycle-122 source EM interval route. Remaining exact boundary: hsampleDerivBound, hboundInt, hpathDeriv, hderivValue, canonical drift weak-action/pairing regularity, no-boundary divergence, diffusion source action, and optional law-derivative/partialS uniqueness." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDominated", "MeasureTheory.AEStronglyMeasurable.congr", "Mathlib.Analysis.Calculus.FDeriv.Measurable.aestronglyMeasurable_deriv_with_param", "ASTIS.SALD.cycle122.remaining_em_path_derivative_domination_after_sample_int", "ASTIS.SALD.cycle120.remaining_source_actions_after_path_derivative", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker lower packet. It rejects hbarBCondExp wrapper churn, broad source-index or theorem-route audits, non-EM fallback, SLT import, Lake changes, theorem-status promotion, fake closure, and sald_version_2 content. The derivative-with-parameter Mathlib theorem was consulted as the eventual route for the concrete representative, but no extra topological assumptions are added here." /-- Cycle-123 proof-DAG pane for derivative measurability discharge inside the EM path-derivative/domination boundary. -/ def cycle123GeneralMovingTargetDiscreteEmSampleDerivMeasDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle123.lower_packet.em_sample_deriv_meas_from_concrete_deriv" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasDominated derives hsampleDerivMeas from AEStronglyMeasurable of the concrete derivative fun omega => deriv (fun t => testEval phi (hatX t omega)) s0 plus ae equality to sampleDeriv phi s0, then reuses the cycle-122 EM interval theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle123GeneralMovingTargetDiscreteEmSampleDerivMeasLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDominated", "MeasureTheory.AEStronglyMeasurable.congr", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle123.remaining_em_path_derivative_domination_after_deriv_meas" interface := "Remaining exact theorem after hsampleMeas, hsampleInt, and hsampleDerivMeas are discharged: prove hsampleDerivBound, hboundInt, and hpathDeriv for each admissible weak test on the source EM interval, then separately prove hderivValue and the canonical barB source-action identities." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasDominated", "ASTIS.SALD.cycle120.remaining_source_actions_after_path_derivative", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle123.reviewer_em_sample_deriv_meas_check" interface := "Reviewer check: accept only if the packet classification is discharges-supplied-hypothesis for hsampleDerivMeas, the proof uses a concrete EM derivative representative and ae-congruence rather than a wrapper around hsampleDerivMeas, and python3 tools/astis.py check passes with no hbarBCondExp wrapper churn, non-EM fallback, SLT/Lake change, source-index rebaseline, theorem-status promotion, fake closure, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle123GeneralMovingTargetDiscreteEmSampleDerivMeasLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasDominated", "appendix.tex:1368-1387" ] reusedBy := ["cycle 123 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 124: EM sample derivative bound from concrete derivative representative -/ /-- Cycle-124 lower obligation for discharging the sample derivative bound input from the EM interval measurable/integrable/derivative-measurable dominated packet. This packet stays inside the cycle-123 EM path-derivative/domination dynamic leaf. It removes only the supplied `hsampleDerivBound` hypothesis by requiring a source-facing concrete derivative bound on the EM interval and an a.e. interval equality between the concrete derivative representative and the chosen `sampleDeriv` representative. It leaves bound integrability, pointwise path derivatives, derivative-value splitting, and source-action facts explicit. -/ def cycle124GeneralMovingTargetDiscreteEmSampleDerivBoundLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle124_em_sample_deriv_bound_lower" statement := "Cycle 124 lower compiles SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundDominated for appendix.tex:1379-1387. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: hsampleDerivBound from SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasDominated. The theorem derives the selected sample-derivative norm bound from a concrete EM derivative representative bound and an interval a.e. equality to sampleDeriv, while also reusing that interval equality at s0 for the cycle-123 derivative-measurability bridge. Remaining exact boundary: hboundInt, hpathDeriv, hderivValue, canonical drift weak-action/pairing regularity, no-boundary divergence, diffusion source action, and optional law-derivative/partialS uniqueness." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasDominated", "ASTIS.SALD.cycle123.remaining_em_path_derivative_domination_after_deriv_meas", "ASTIS.SALD.cycle120.remaining_source_actions_after_path_derivative", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker lower packet. It rejects hbarBCondExp wrapper churn, broad source-index or theorem-route audits, non-EM fallback, SLT import, Lake changes, theorem-status promotion, fake closure, and sald_version_2 content. No new Mathlib theorem is promoted; this is an a.e.-congruence/norm-bound transport around the concrete EM derivative representative." /-- Cycle-124 proof-DAG pane for derivative-bound discharge inside the EM path-derivative/domination boundary. -/ def cycle124GeneralMovingTargetDiscreteEmSampleDerivBoundDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle124.lower_packet.em_sample_deriv_bound_from_concrete_deriv" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundDominated derives hsampleDerivBound from a concrete EM derivative bound and interval a.e. equality to sampleDeriv, then reuses the cycle-123 EM interval theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle124GeneralMovingTargetDiscreteEmSampleDerivBoundLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasDominated", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle124.remaining_em_path_derivative_domination_after_deriv_bound" interface := "Remaining exact theorem after hsampleMeas, hsampleInt, hsampleDerivMeas, and hsampleDerivBound are discharged: prove hboundInt and hpathDeriv for each admissible weak test on the source EM interval, then separately prove hderivValue and the canonical barB source-action identities." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundDominated", "ASTIS.SALD.cycle120.remaining_source_actions_after_path_derivative", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle124.reviewer_em_sample_deriv_bound_check" interface := "Reviewer check: accept only if the packet classification is discharges-supplied-hypothesis for hsampleDerivBound, the proof uses a concrete EM derivative bound plus interval a.e. equality rather than a wrapper around hsampleDerivBound, and python3 tools/astis.py check passes with no hbarBCondExp wrapper churn, non-EM fallback, SLT/Lake change, source-index rebaseline, theorem-status promotion, fake closure, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle124GeneralMovingTargetDiscreteEmSampleDerivBoundLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundDominated", "appendix.tex:1368-1387" ] reusedBy := ["cycle 124 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 124: EM joint-law bound integrability transport -/ /-- Cycle-124 lower obligation for discharging the bound-integrability input from the EM interval derivative-bound dominated packet. This packet stays inside the EM path-derivative/domination dynamic leaf. It removes only the supplied `hboundInt` hypothesis by transporting integrability from a source-facing joint-law representative of the dominating bound on `(hatX s0, Xk)` and an a.e. equality to the selected sample-space bound. -/ def cycle124GeneralMovingTargetDiscreteEmBoundIntLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle124_em_bound_int_lower" statement := "Cycle 124 lower compiles SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntDominated for appendix.tex:1379-1387. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: hboundInt from SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundDominated. The theorem derives sample-space Integrable (bound phi) P from an integrable joint-law bound on Measure.map (fun omega => (hatX s0 omega, Xk omega)) P plus an a.e. equality to the selected sample-space bound, using AEMeasurable.prodMk, Integrable.comp_aemeasurable, and Integrable.congr. Remaining exact boundary: hpathDeriv, hderivValue, canonical drift weak-action/pairing regularity, no-boundary divergence, diffusion source action, and optional law-derivative/partialS uniqueness." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundDominated", "ASTIS.SALD.cycle124.remaining_em_path_derivative_domination_after_deriv_bound", "ASTIS.SALD.cycle120.remaining_source_actions_after_path_derivative", "MeasureTheory.AEMeasurable.prodMk", "MeasureTheory.Integrable.comp_aemeasurable", "MeasureTheory.Integrable.congr", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker lower packet. It rejects wrapper churn around hboundInt, broad source-index or theorem-route audits, non-EM fallback, SLT import, Lake changes, theorem-status promotion, fake closure, and sald_version_2 content. The concrete joint-law moment estimate remains source-facing; this compiled theorem only transports that integrability to the selected sample-space bound." /-- Cycle-124 proof-DAG pane for bound-integrability transport inside the EM path-derivative/domination boundary. -/ def cycle124GeneralMovingTargetDiscreteEmBoundIntDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle124.lower_packet.em_bound_int_from_joint_law" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntDominated derives hboundInt from a joint-law integrable bound and a.e. equality to the selected sample-space bound, then reuses the cycle-124 derivative-bound theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle124GeneralMovingTargetDiscreteEmBoundIntLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundDominated", "MeasureTheory.Integrable.comp_aemeasurable", "MeasureTheory.Integrable.congr", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle124.remaining_em_path_derivative_domination_after_bound_int" interface := "Remaining exact theorem after hsampleMeas, hsampleInt, hsampleDerivMeas, hsampleDerivBound, and hboundInt are discharged: prove hpathDeriv for each admissible weak test on the source EM interval, then separately prove hderivValue and the canonical barB source-action identities." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntDominated", "ASTIS.SALD.cycle120.remaining_source_actions_after_path_derivative", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle124.reviewer_em_bound_int_check" interface := "Reviewer check: accept only if the packet classification is discharges-supplied-hypothesis for hboundInt, the proof transports a joint-law integrable bound rather than restating hboundInt, and python3 tools/astis.py check passes with no wrapper churn, non-EM fallback, SLT/Lake change, source-index rebaseline, theorem-status promotion, fake closure, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle124GeneralMovingTargetDiscreteEmBoundIntLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntDominated", "appendix.tex:1368-1387" ] reusedBy := ["cycle 124 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 125: EM path derivative from concrete differentiability -/ /-- Cycle-125 lower obligation for discharging the pointwise path-derivative input from the EM interval bound-integrability dominated packet. This packet stays inside the EM path-derivative/domination dynamic leaf. It removes only the supplied `hpathDeriv` hypothesis by deriving the selected `HasDerivAt` input from a.e. differentiability of the concrete weak-test EM path and the already exposed interval a.e. equality between `sampleDeriv` and `deriv (fun t => testEval phi (hatX t omega))`. -/ def cycle125GeneralMovingTargetDiscreteEmPathDerivLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle125_em_path_deriv_lower" statement := "Cycle 125 lower compiles SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated for appendix.tex:1379-1387. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: hpathDeriv from SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntDominated. The theorem derives the selected sample-path HasDerivAt input from a.e. DifferentiableAt Real of the concrete weak-test EM path fun t => testEval phi (hatX t omega), Mathlib DifferentiableAt.hasDerivAt, and the interval a.e. equality sampleDeriv phi s omega = deriv (fun t => testEval phi (hatX t omega)) s. Remaining exact boundary: hderivValue, canonical drift weak-action/pairing regularity, no-boundary divergence, diffusion source action, and optional law-derivative/partialS uniqueness." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntDominated", "ASTIS.SALD.cycle124.remaining_em_path_derivative_domination_after_bound_int", "ASTIS.SALD.cycle120.remaining_source_actions_after_path_derivative", "Mathlib.Analysis.Calculus.Deriv.Basic.DifferentiableAt.hasDerivAt", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker lower packet. It rejects wrapper churn around hpathDeriv, broad source-index or theorem-route audits, non-EM fallback, SLT import, Lake changes, theorem-status promotion, fake closure, and sald_version_2 content. The concrete differentiability proof for the frozen EM weak-test path remains source-facing; this compiled theorem converts it to the dominated parametric-integral input." /-- Cycle-125 proof-DAG pane for path-derivative discharge inside the EM path-derivative/domination boundary. -/ def cycle125GeneralMovingTargetDiscreteEmPathDerivDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle125.lower_packet.em_path_deriv_from_concrete_differentiability" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated derives hpathDeriv from concrete weak-test path differentiability plus the existing interval a.e. equality to sampleDeriv, then reuses the cycle-124 bound-integrability theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle125GeneralMovingTargetDiscreteEmPathDerivLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntDominated", "Mathlib.Analysis.Calculus.Deriv.Basic.DifferentiableAt.hasDerivAt", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle125.remaining_source_actions_after_path_deriv" interface := "Remaining exact theorem after hsampleMeas, hsampleInt, hsampleDerivMeas, hsampleDerivBound, hboundInt, and hpathDeriv are discharged: prove hderivValue and the canonical barB drift weak-action/pairing regularity, no-boundary divergence, diffusion source action, and optional law-derivative/partialS uniqueness." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated", "ASTIS.SALD.cycle120.remaining_source_actions_after_path_derivative", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle125.reviewer_em_path_deriv_check" interface := "Reviewer check: accept only if the packet classification is discharges-supplied-hypothesis for hpathDeriv, the proof uses DifferentiableAt.hasDerivAt and the concrete derivative representative rather than restating hpathDeriv, and python3 tools/astis.py check passes with no wrapper churn, non-EM fallback, SLT/Lake change, source-index rebaseline, theorem-status promotion, fake closure, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle125GeneralMovingTargetDiscreteEmPathDerivLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated", "appendix.tex:1368-1387" ] reusedBy := ["cycle 125 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 126: EM derivative-value split from concrete derivative integral -/ /-- Cycle-126 lower obligation for discharging the derivative-value input from the EM interval path-derivative dominated packet. This packet stays inside the EM conditional-law/Fokker--Planck backend. It removes only the supplied `hderivValue` hypothesis by requiring the smaller source-facing concrete derivative integral split for `deriv (fun t => testEval phi (hatX t omega)) s0`; the existing interval a.e. equality transports that split to the selected `sampleDeriv` representative. -/ def cycle126GeneralMovingTargetDiscreteEmDerivValueLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle126_em_deriv_value_lower" statement := "Cycle 126 lower compiles SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDominated for appendix.tex:1379-1387. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: hderivValue from SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated. The theorem derives the selected sample-derivative integral value from a concrete EM derivative integral split for deriv (fun t : Real => testEval phi (hatX t omega)) s0 plus the interval a.e. equality to sampleDeriv specialized at s0 and MeasureTheory.integral_congr_ae. Remaining exact boundary: canonical drift weak-action/pairing regularity, no-boundary divergence, diffusion source action, and optional law-derivative/partialS uniqueness." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated", "ASTIS.SALD.cycle125.remaining_source_actions_after_path_deriv", "MeasureTheory.integral_congr_ae", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker lower packet. It rejects wrapper churn around hderivValue, broad source-index or theorem-route audits, non-EM fallback, SLT import, Lake changes, theorem-status promotion, fake closure, and sald_version_2 content. The concrete EM generator/Ito integral split remains source-cited and is strictly smaller than the old sampleDeriv-level hderivValue input." /-- Cycle-126 proof-DAG pane for derivative-value discharge inside the EM conditional-drift/weak-Fokker--Planck backend. -/ def cycle126GeneralMovingTargetDiscreteEmDerivValueDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle126.lower_packet.em_deriv_value_from_concrete_integral_split" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDominated derives hderivValue from a concrete EM derivative integral split plus interval a.e. equality to sampleDeriv, then reuses the cycle-125 path theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle126GeneralMovingTargetDiscreteEmDerivValueLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated", "MeasureTheory.integral_congr_ae", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle126.remaining_canonical_source_actions_after_deriv_value" interface := "Remaining exact theorem after hsampleMeas, hsampleInt, hsampleDerivMeas, hsampleDerivBound, hboundInt, hpathDeriv, and hderivValue are discharged: prove the canonical barB drift weak-action/pairing regularity, no-boundary divergence, diffusion source action, and optional law-derivative/partialS uniqueness." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDominated", "ASTIS.SALD.cycle120.remaining_source_actions_after_path_derivative", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle126.reviewer_em_deriv_value_check" interface := "Reviewer check: accept only if the packet classification is discharges-supplied-hypothesis for hderivValue, the proof uses a concrete derivative integral split and MeasureTheory.integral_congr_ae rather than restating hderivValue, and python3 tools/astis.py check passes with no wrapper churn, non-EM fallback, SLT/Lake change, source-index rebaseline, theorem-status promotion, fake closure, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle126GeneralMovingTargetDiscreteEmDerivValueLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDominated", "appendix.tex:1368-1387" ] reusedBy := ["cycle 126 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 127: Canonical barB drift action from component pairings -/ /-- Cycle-127 lower obligation for discharging the canonical `barB` drift weak-action input from the EM interval path-value dominated packet. This packet stays inside the EM conditional-law/Fokker--Planck backend. It removes only the supplied `hdriftBarBAction` continuation by requiring the smaller source-facing guide and score component action pairings for the canonical `condDistrib` fields, plus weak-pairing congruence, additivity, and scalar linearity. Pairing measurability, the gradient bound, no-boundary divergence, diffusion source action, and optional law-derivative/`partialS` uniqueness remain explicit. -/ def cycle127GeneralMovingTargetDiscreteEmDriftActionLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle127_em_drift_action_lower" statement := "Cycle 127 lower compiles SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionDominated for appendix.tex:1368-1387. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: the canonical hdriftBarBAction continuation from SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDominated. The theorem derives driftAction phi equals the weak test-gradient pairing against canonicalBarB from the guideAction/scoreAction component split, the two canonical condDistrib component pairings, and explicit weak-pairing congruence/additivity/scalar-linearity. Remaining exact boundary: canonical pairing measurability, gradient bound, no-boundary divergence, diffusion source action, and optional law-derivative/partialS uniqueness." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDominated", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfIntegralAction", "ASTIS.SALD.cycle126.remaining_canonical_source_actions_after_deriv_value", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet. It rejects wrapper churn around hdriftBarBAction: the old direct canonical barB action continuation is absent from the theorem statement and is reconstructed from guide/score component pairings plus weak-pairing algebra. No broad source-index rebaseline, theorem-route audit, non-EM fallback, SLT import, Lake/toolchain change, theorem-status promotion, fake closure, or sald_version_2 content is introduced." /-- Cycle-127 lower obligation for discharging the raw canonical `barB` pairing-measurability input from the drift-action packet. -/ def cycle127GeneralMovingTargetDiscreteEmPairMeasLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle127_em_pair_meas_lower" statement := "Cycle 127 lower compiles SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasDominated for appendix.tex:1368-1387. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: the raw hpairMeas continuation from SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionDominated. The theorem derives AEStronglyMeasurable (fun x => inner Real (testGrad phi x) (canonicalBarB x)) from separate AEStronglyMeasurable testGrad phi and canonicalBarB hypotheses using Mathlib AEStronglyMeasurable.inner. Remaining exact boundary: separate test-gradient and canonical-field measurability, gradient bound, no-boundary divergence, diffusion source action, and optional law-derivative/partialS uniqueness." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionDominated", "MeasureTheory.AEStronglyMeasurable.inner", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf local refinement after the drift-action packet. It rejects wrapper churn around hpairMeas: the old paired inner-product measurability continuation is absent from the theorem statement and is reconstructed from separate source-facing field measurability plus Mathlib AEStronglyMeasurable.inner. The theorem does not prove the gradient bound, no-boundary theorem, diffusion source action, law-derivative/partialS uniqueness, SLT import, Lake/toolchain change, theorem-status promotion, fake closure, or sald_version_2 content." /-- Cycle-127 proof-DAG pane for the canonical `barB` drift-action discharge inside the EM conditional-drift/weak-Fokker--Planck backend. -/ def cycle127GeneralMovingTargetDiscreteEmDriftActionDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle127.lower_packet.canonical_barB_drift_action_from_components" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionDominated derives the canonical hdriftBarBAction continuation from guideAction/scoreAction component pairings for the canonical condDistrib guide and score fields plus weak-pairing congruence/additivity/scalar-linearity, then reuses the cycle-126 path-value theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle127GeneralMovingTargetDiscreteEmDriftActionLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDominated", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings", "appendix.tex:1368-1377", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle127.lower_packet.canonical_barB_pair_meas_from_fields" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasDominated derives the old hpairMeas continuation from separate AEStronglyMeasurable testGrad phi and canonicalBarB hypotheses using MeasureTheory.AEStronglyMeasurable.inner, then reuses the cycle-127 drift-action theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle127GeneralMovingTargetDiscreteEmPairMeasLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionDominated", "MeasureTheory.AEStronglyMeasurable.inner", "appendix.tex:1368-1377", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle127.remaining_canonical_source_actions_after_drift_action" interface := "Remaining exact theorem after the canonical drift weak-action identity and raw hpairMeas continuation are discharged: prove the separate weak-test-gradient and canonicalBarB measurability inputs, gradient-bound regularity, the no-boundary divergence identity for hatRhoS * canonicalBarB, the diffusion source action, and optional law-derivative/partialS uniqueness." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionDominated", "ASTIS.SALD.cycle126.remaining_canonical_source_actions_after_deriv_value", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle127.reviewer_em_drift_action_check" interface := "Reviewer check: accept only if the packet classification is discharges-supplied-hypothesis for hdriftBarBAction, the old direct driftAction=weakGradPairing canonicalBarB continuation is absent from the theorem statement, the guide/score component pairings and weak-pairing algebra preserve dotTk and sigmaCoeff signs, and python3 tools/astis.py check passes with no wrapper churn, non-EM fallback, SLT/Lake change, source-index rebaseline, theorem-status promotion, fake closure, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle127GeneralMovingTargetDiscreteEmDriftActionLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionDominated", "appendix.tex:1368-1387" ] reusedBy := ["cycle 127 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 128: Canonical barB no-boundary trace refiner -/ /-- Cycle-128 lower obligation for narrowing the direct canonical `barB` no-boundary input in the EM weak-FP consumer. -/ def cycle128GeneralMovingTargetDiscreteEmNoBoundaryTraceLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle128_em_no_boundary_trace_lower" statement := "Cycle 128 lower compiles SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceDominated for appendix.tex:1368-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the direct hdivNoBoundary continuation from SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasDominated. The theorem reconstructs driftDiv phi = - integral inner (testGrad phi) canonicalBarB by product-rule total divergence, divergence-theorem boundary flux, boundary trace integral, zero admissible-test trace, MeasureTheory.integral_congr_ae, and SALD.generalMovingTargetDiscreteDriftDivNoBoundaryOfProductRule. Remaining exact boundary: separate test-gradient and canonical-field measurability, gradient-bound regularity, diffusion source action, and optional law-derivative/partialS uniqueness." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasDominated", "SALD.generalMovingTargetDiscreteDriftDivNoBoundaryOfProductRule", "SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero", "SALD.generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero", "MeasureTheory.integral_congr_ae", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area refiner packet. It rejects wrapper churn around hdivNoBoundary: the old direct no-boundary continuation is absent from the theorem statement and is reconstructed from product-rule/divergence/boundary-trace facts. The theorem keeps htestGradMeas, hcanonicalBarBMeas, hgradNormBound, and hdiffusionSource explicit, and it does not introduce non-EM fallback, SLT import, Lake/toolchain change, theorem-status promotion, fake closure, broad audit, or sald_version_2 content." /-- Cycle-128 lower obligation for discharging the canonical `barB` measurability input after the no-boundary trace refiner. -/ def cycle128GeneralMovingTargetDiscreteEmCanonicalBarBMeasLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle128_em_canonical_barB_meas_lower" statement := "Cycle 128 lower compiles SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDominated for appendix.tex:1368-1387. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: hcanonicalBarBMeas from SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceDominated. The theorem derives AEStronglyMeasurable canonicalBarB (hatRhoS s0) from hhatRhoS s0, hXk, the guide/score measurability and integrability inputs, and SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity. Remaining exact boundary: htestGradMeas, hgradNormBound, hdiffusionSource, and optional law-derivative/partialS uniqueness." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceDominated", "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity", "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower packet after the no-boundary refiner. It rejects wrapper churn around hcanonicalBarBMeas: the old direct canonical-field measurability continuation is absent from the theorem statement and is reconstructed from the existing condDistrib regularity theorem. The theorem keeps htestGradMeas, hgradNormBound, and hdiffusionSource explicit, and it does not introduce non-EM fallback, SLT import, Lake/toolchain change, theorem-status promotion, fake closure, broad audit, or sald_version_2 content." /-- Cycle-128 proof-DAG pane for the canonical `barB` no-boundary trace refinement inside the EM conditional-drift/weak-Fokker--Planck backend. -/ def cycle128GeneralMovingTargetDiscreteEmNoBoundaryTraceDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle128.lower_packet.canonical_barB_no_boundary_trace" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceDominated removes the direct hdivNoBoundary continuation from the post-cycle-127 pair-meas theorem and derives it from product-rule total divergence, divergence-theorem boundary flux, boundary trace integral, zero admissible-test trace, MeasureTheory.integral_congr_ae, and SALD.generalMovingTargetDiscreteDriftDivNoBoundaryOfProductRule." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle128GeneralMovingTargetDiscreteEmNoBoundaryTraceLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasDominated", "SALD.generalMovingTargetDiscreteDriftDivNoBoundaryOfProductRule", "MeasureTheory.integral_congr_ae", "appendix.tex:1368-1377", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle128.lower_packet.canonical_barB_meas_from_condDistrib_regular" interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDominated removes the direct hcanonicalBarBMeas premise from the no-boundary trace theorem by deriving AEStronglyMeasurable canonicalBarB (hatRhoS s0) from SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity and the existing guide/score condDistrib measurability and integrability inputs." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle128GeneralMovingTargetDiscreteEmCanonicalBarBMeasLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceDominated", "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity", "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable", "appendix.tex:1368-1377", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle128.remaining_canonical_source_actions_after_no_boundary_trace" interface := "Remaining exact theorem after the no-boundary trace/product-rule refiner and canonicalBarB measurability discharge: prove the separate weak-test-gradient measurability input if the final consumer still requires it, the gradient-bound regularity, the diffusion source action, and optional law-derivative/partialS uniqueness." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasDominated", "appendix.tex:1379-1387", "appendix.tex:1368-1377" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle128.reviewer_em_no_boundary_trace_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, the new theorem statement lacks the direct hdivNoBoundary premise and old raw hpairMeas premise, no-boundary is reconstructed from product-rule/divergence/boundary-trace facts, hgradNormBound and hdiffusionSource remain explicit, and python3 tools/astis.py check passes with forbidden hits zero." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle128GeneralMovingTargetDiscreteEmNoBoundaryTraceLowerObligation", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceDominated", "appendix.tex:1368-1387" ] reusedBy := ["cycle 128 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 129: EM diffusion source-action boundary -/ /-- Cycle-129 lower-ready obligation for narrowing the remaining diffusion source-action input in the canonical EM weak-FP consumer. -/ def cycle129GeneralMovingTargetDiscreteEmDiffusionSourceLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle129_em_diffusion_source_lower" statement := "Cycle 129 lower compiles SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDiffusionSourceDominated for appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Stale hpathDeriv work is rejected as wrapper churn because cycle 125 compiled SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated, and cycle 126 compiled the derivative-value continuation. The exact boundary narrowed here is the supplied hdiffusionSource input in SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDominated: the compiled continuation removes that direct premise and replaces it with the two source-facing facts consumed by SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianAction, namely the EM/Brownian diffusion generator contribution equals the named weak diffusion action, and the weak Laplacian integration-by-parts theorem identifies that action with sigmaCoeff • laplacian phi." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianAction", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDiffusionSourceDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDominated", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area refiner packet. It does not reassign lower work to hpathDeriv, hderivValue, hdriftBarBAction, hpairMeas, hdivNoBoundary, or hcanonicalBarBMeas, all of which already have compiled follow-on theorems. It proves only the local composition into the canonical consumer. It does not prove the Brownian generator theorem, weak Laplacian integration by parts, gradient-bound regularity, weak-test-gradient measurability, law-derivative/partialS uniqueness, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, or sald_version_2 content." /-- Cycle-129 proof-DAG pane for the remaining diffusion source-action boundary inside the canonical EM weak-FP backend. -/ def cycle129GeneralMovingTargetDiscreteEmDiffusionSourceDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle129.lower_packet.diffusion_source_boundary" interface := "Compiled lower continuation: SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDiffusionSourceDominated removes the direct hdiffusionSource premise from the canonical EM consumer by using SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianAction to compose the EM/Brownian diffusion weak action and weak Laplacian integration-by-parts identity." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle129GeneralMovingTargetDiscreteEmDiffusionSourceLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianAction", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDiffusionSourceDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDominated", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle129.retired_stale_hpathDeriv_leaf" interface := "Stale-leaf retirement: do not assign lower work to hpathDeriv from SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntDominated; cycle 125 already discharged it via the concrete EM differentiability representative and Mathlib DifferentiableAt.hasDerivAt." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated", "SALD.cycle125GeneralMovingTargetDiscreteEmPathDerivDag", "appendix.tex:1379-1387" ] reusedBy := ["cycle 129 lower selection"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle129.reviewer_diffusion_source_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, the new theorem statement lacks the direct hdiffusionSource premise from the canonical consumer, the exact remaining theorem boundary is Brownian diffusion generator action plus weak Laplacian integration by parts, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle129GeneralMovingTargetDiscreteEmDiffusionSourceLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianAction", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDiffusionSourceDominated", "appendix.tex:1379-1387" ] reusedBy := ["cycle 129 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 130: EM weak Laplacian integration-by-parts boundary -/ /-- Cycle-130 lower-ready obligation for narrowing the remaining weak Laplacian action input in the EM diffusion-source helper. -/ def cycle130GeneralMovingTargetDiscreteEmLaplacianIbPLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle130_em_laplacian_ibp_lower" statement := "Cycle 130 middle compiles SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianActionOfIntegrationByParts and SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianIntegrationByParts for appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the direct hlaplacianAction premise consumed by SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianAction is replaced by the source-facing hdiffusionLaplacianTerm and hweakLaplacianIbP facts. The EM/Brownian hdiffusionAction premise remains separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianActionOfIntegrationByParts", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianIntegrationByParts", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianAction", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDiffusionSourceDominated", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area refiner packet. It rejects wrapper churn around the already narrowed hdiffusionSource consumer and targets only the hlaplacianAction sub-boundary. It does not reassign lower work to hsampleInt, hsampleDerivMeas, hsampleDerivBound, hboundInt, hpathDeriv, hderivValue, hdriftBarBAction, raw hpairMeas, hcanonicalBarBMeas, or direct hdiffusionSource. It does not prove Brownian construction, density regularity, full weak FP, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, or sald_version_2 content." /-- Cycle-130 lower_1 scout obligation for narrowing the weak Laplacian IBP identity to the two Green-identity steps visible in the source proof. -/ def cycle130GeneralMovingTargetDiscreteEmGreenLaplacianIbPScoutObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle130_em_green_laplacian_ibp_scout" statement := "Cycle 130 lower_1 compiles SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity and SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfGreenLaplacianIbP for appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the direct hweakLaplacianIbP premise from SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianIntegrationByParts is replaced by first-Green density-Laplacian-to-negative-gradient-pairing, second-Green negative-gradient-pairing-to-test-Laplacian, and test-Laplacian normalization hypotheses. The EM/Brownian hdiffusionAction fact and hdiffusionLaplacianTerm remain separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfGreenLaplacianIbP", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianIntegrationByParts", "SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianActionOfIntegrationByParts", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Natural-language proof-scout packet plus compiled abstract composition. It does not prove the analytic Green identities, but records the exact lower_2 target: instantiate the first and second no-boundary integration-by-parts steps for the source density and admissible weak tests, likely through the existing SALD box-divergence/no-boundary route and Mathlib MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable. No SLT import, non-EM fallback, Lake/toolchain change, theorem-status promotion, fake closure, wrapper churn, or sald_version_2 content." /-- Cycle-130 lower_2 obligation narrowing the first Green identity to no-boundary flux algebra. -/ def cycle130GeneralMovingTargetDiscreteEmFirstGreenNoBoundaryFluxLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle130_em_first_green_no_boundary_flux_lower" statement := "Cycle 130 lower_2 compiles SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfFirstGreenNoBoundaryFlux for appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the direct hfirstGreen premise from SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfGreenLaplacianIbP is replaced by hfirstGreenResidual, hfirstGreenDivergence, and hfirstGreenZeroBoundary, exposing the first Green residual, divergence-theorem boundary-flux identity, and zero boundary-flux condition. The EM/Brownian hdiffusionAction fact, hdiffusionLaplacianTerm, hsecondGreen, and htestLaplacian remain separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfFirstGreenNoBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfGreenLaplacianIbP", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area lower_2 packet. It rejects wrapper churn around hweakLaplacianIbP and targets only the first-Green leaf selected by lower_1. The theorem does not prove second Green, test-Laplacian normalization, Brownian diffusion construction, full weak FP, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, or sald_version_2 content." /-! ### Cycle 131: EM second Green no-boundary flux boundary -/ /-- Cycle-131 middle/lower-ready obligation narrowing the second Green identity to no-boundary flux algebra. -/ def cycle131GeneralMovingTargetDiscreteEmSecondGreenNoBoundaryFluxLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle131_em_second_green_no_boundary_flux_lower" statement := "Cycle 131 middle compiles SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenNoBoundaryFlux for appendix.tex:1379-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the direct hsecondGreen premise left in SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfFirstGreenNoBoundaryFlux is replaced by hsecondGreenResidual, hsecondGreenDivergence, and hsecondGreenZeroBoundary, exposing the second Green residual, divergence-theorem boundary-flux identity, and zero boundary-flux condition. The EM/Brownian hdiffusionAction fact, hdiffusionLaplacianTerm, first-Green residual/divergence/zero-boundary facts, and htestLaplacian remain separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenNoBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfFirstGreenNoBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfGreenLaplacianIbP", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area refiner packet. It rejects stale dynamic leaves and wrapper churn, and targets only the second-Green leaf selected by the cycle-131 upper handoff. The theorem does not prove the analytic second-Green residual/divergence/zero-flux facts, test-Laplacian normalization, Brownian diffusion construction, full weak FP, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, or sald_version_2 content." /-- Cycle-131 lower_1 scout/lower handoff narrowing the second-Green zero boundary flux input to a trace-product no-boundary condition. -/ def cycle131GeneralMovingTargetDiscreteEmSecondGreenTraceBoundaryLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle131_em_second_green_trace_boundary_lower" statement := "Cycle 131 lower_1 compiles SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenTraceBoundary for appendix.tex:1392-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the direct hsecondGreenZeroBoundary premise from SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenNoBoundaryFlux is replaced by hsecondGreenBoundaryFluxIntegral and hsecondGreenTraceProductZero. The second-Green residual/divergence facts, hdiffusionAction, hdiffusionLaplacianTerm, first-Green subfacts, and htestLaplacian remain separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenTraceBoundary", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenNoBoundaryFlux", "MeasureTheory.integral_congr_ae", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing lower_1 handoff plus scout route. It does not prove the analytic boundary-flux integral representation, trace/compact-support theorem, second-Green residual, second-Green divergence, test-Laplacian normalization, full weak FP, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, or sald_version_2 content." /-- Cycle-131 lower_2 obligation narrowing the second-Green boundary-flux integral input to the existing Mathlib box-divergence interface. -/ def cycle131GeneralMovingTargetDiscreteEmSecondGreenBoxBoundaryFluxLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle131_em_second_green_box_boundary_flux_lower" statement := "Cycle 131 lower_2 compiles SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFlux for appendix.tex:1392-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the direct hsecondGreenBoundaryFluxIntegral premise from SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenTraceBoundary is replaced by the source-facing Mathlib box-divergence package: weighted-field continuity on the source box, countable derivative exception set, off-exception Frechet derivative, divergence integrability, secondGreenBoundaryFlux equals the interior divergence integral, and signed-face trace identification. hsecondGreenTraceProductZero, second-Green residual/divergence facts, first-Green subfacts, hdiffusionAction, hdiffusionLaplacianTerm, and htestLaplacian remain separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenTraceBoundary", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing lower_2 handoff. It does not prove the analytic weighted-field regularity, divergence integrability, face-to-trace identification, trace-product-zero theorem, second-Green residual/divergence facts, test-Laplacian normalization, full weak FP, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, or sald_version_2 content." /-- Cycle-131 proof-DAG pane for removing the direct second Green premise from the EM diffusion-source action path. -/ def cycle131GeneralMovingTargetDiscreteEmSecondGreenNoBoundaryFluxDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle131.middle_packet.second_green_no_boundary_flux" interface := "Compiled illness-area refiner: SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenNoBoundaryFlux removes the direct hsecondGreen premise left by SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfFirstGreenNoBoundaryFlux. It derives the second Green identity from hsecondGreenResidual, hsecondGreenDivergence, and hsecondGreenZeroBoundary, while keeping hdiffusionAction, hdiffusionLaplacianTerm, the first-Green subfacts, and htestLaplacian explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle131GeneralMovingTargetDiscreteEmSecondGreenNoBoundaryFluxLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenNoBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfFirstGreenNoBoundaryFlux", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "appendix.tex:1379-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle131.lower_1_packet.second_green_trace_boundary" interface := "Compiled lower_1 handoff: SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenTraceBoundary removes the direct hsecondGreenZeroBoundary premise by deriving it from a boundary-flux integral representation and a.e. zero second-Green trace product. It keeps hsecondGreenResidual, hsecondGreenDivergence, first-Green subfacts, hdiffusionAction, hdiffusionLaplacianTerm, and htestLaplacian explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle131GeneralMovingTargetDiscreteEmSecondGreenTraceBoundaryLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenTraceBoundary", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenNoBoundaryFlux", "MeasureTheory.integral_congr_ae", "appendix.tex:1392-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle131.lower_2_packet.second_green_box_boundary_flux" interface := "Compiled lower_2 continuation: SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFlux removes the direct hsecondGreenBoundaryFluxIntegral premise from the lower_1 trace-boundary theorem by deriving it from SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox. It keeps hsecondGreenTraceProductZero, second-Green residual/divergence facts, first-Green subfacts, hdiffusionAction, hdiffusionLaplacianTerm, and htestLaplacian explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle131GeneralMovingTargetDiscreteEmSecondGreenBoxBoundaryFluxLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenTraceBoundary", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "appendix.tex:1392-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle131.remaining_laplacian_ibp_analysis" interface := "Remaining exact theorem boundary after cycle 131: prove the density-Laplacian weak term for the EM/Brownian diffusion contribution, instantiate the first-Green residual/divergence/zero-flux facts, instantiate the second-Green residual/divergence facts, prove the box-divergence regularity/face-trace facts behind the second-Green boundary flux, prove trace-product-zero for source admissible tests, and prove the test-Laplacian normalization." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "appendix.tex:1379-1387", "appendix.tex:1392-1427", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenTraceBoundary", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenNoBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfFirstGreenNoBoundaryFlux", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle131.reviewer_second_green_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, the new theorem statement lacks a direct hsecondGreen premise, hdiffusionAction, hdiffusionLaplacianTerm, first-Green subfacts, and htestLaplacian remain separate, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle131GeneralMovingTargetDiscreteEmSecondGreenNoBoundaryFluxLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenNoBoundaryFlux", "appendix.tex:1379-1427" ] reusedBy := ["cycle 131 reviewer"] status := ProofStatus.obligation } ] /-- Cycle-132 lower packet narrowing the remaining second-Green trace-product zero input to the source-facing zero-test-trace condition for admissible weak tests. -/ def cycle132GeneralMovingTargetDiscreteEmSecondGreenTestTraceZeroLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle132_em_second_green_test_trace_zero_lower" statement := "Cycle 132 middle compiles SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTestTraceZero for appendix.tex:1392-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hsecondGreenTraceProductZero premise from SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFlux is replaced by hsecondGreenTestTraceZero, using SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero. The box-divergence regularity/face-trace facts, second-Green residual/divergence facts, first-Green subfacts, hdiffusionAction, hdiffusionLaplacianTerm, and htestLaplacian remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFlux", "SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area refiner packet. It removes only the supplied second-Green trace-product-zero input by deriving it from the admissible-test zero trace already used by the source no-boundary weak-test setup. It does not prove the analytic trace theorem, weighted-field regularity, divergence integrability, face-to-trace identification, second-Green residual/divergence facts, test-Laplacian normalization, full weak FP, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, or sald_version_2 content." /-- Cycle-132 lower_1 scout continuation narrowing the second-Green zero-test trace input to trace identification plus the existing admissible-test zero trace boundary. -/ def cycle132GeneralMovingTargetDiscreteEmSecondGreenTraceEqTestTraceZeroScoutObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle132_em_second_green_trace_eq_test_trace_zero_lower_1" statement := "Cycle 132 lower_1 compiles SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTraceEqTestTraceZero for appendix.tex:1392-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the direct hsecondGreenTestTraceZero premise exposed by SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTestTraceZero is replaced by hsecondGreenTestTraceEq, identifying the second-Green test trace with the standard admissible-test boundary trace a.e., and htestTraceZero, the existing zero boundary trace condition for admissible weak tests. The theorem keeps box-divergence regularity/face-trace facts, second-Green residual/divergence facts, first-Green subfacts, hdiffusionAction, hdiffusionLaplacianTerm, and htestLaplacian explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTraceEqTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTestTraceZero", "SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero", "ASTIS.SALD.forward_KL_discrete.cycle102_test_trace_zero_lower", "appendix.tex:1392-1427", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing lower_1 scout packet. It does not prove the analytic compact-support/zero-trace theorem, the second-Green trace-identification theorem, weighted-field regularity, divergence integrability, face-to-trace identification, second-Green residual/divergence facts, test-Laplacian normalization, full weak FP, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, or sald_version_2 content." /-- Cycle-132 lower_2 continuation narrowing second-Green trace identification from an a.e. statement to pointwise selected-trace equality. -/ def cycle132GeneralMovingTargetDiscreteEmSecondGreenPointwiseTraceEqLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle132_em_second_green_pointwise_trace_eq_lower_2" statement := "Cycle 132 lower_2 compiles SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqTestTraceZero for appendix.tex:1392-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hsecondGreenTestTraceEq premise from SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTraceEqTestTraceZero is replaced by hsecondGreenTestTracePointwiseEq, a pointwise source trace-definition equality between secondGreenTestTrace and the standard admissible-test boundary trace. The theorem derives the a.e. equality via Filter.Eventually.of_forall and keeps htestTraceZero, box-divergence regularity/face-trace facts, second-Green residual/divergence facts, first-Green subfacts, hdiffusionAction, hdiffusionLaplacianTerm, and htestLaplacian explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTraceEqTestTraceZero", "Filter.Eventually.of_forall", "appendix.tex:1392-1427", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing lower_2 packet. It does not prove the analytic compact-support/zero-trace theorem, the pointwise source trace-definition equality, weighted-field regularity, divergence integrability, face-to-trace identification, second-Green residual/divergence facts, test-Laplacian normalization, full weak FP, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, or sald_version_2 content." /-- Cycle-132 proof-DAG pane for the second-Green zero-test-trace narrowing. -/ def cycle132GeneralMovingTargetDiscreteEmSecondGreenTestTraceZeroDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle132.lower_packet.second_green_test_trace_zero" interface := "Compiled illness-area refiner: SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTestTraceZero removes the direct hsecondGreenTraceProductZero premise by deriving it from hsecondGreenTestTraceZero via SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero. It keeps the cycle-131 box-divergence package, second-Green residual/divergence facts, first-Green subfacts, hdiffusionAction, hdiffusionLaplacianTerm, and htestLaplacian explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle132GeneralMovingTargetDiscreteEmSecondGreenTestTraceZeroLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFlux", "SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero", "appendix.tex:1392-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle132.lower_1_packet.second_green_trace_eq_test_trace_zero" interface := "Compiled lower_1 scout continuation: SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTraceEqTestTraceZero removes the direct hsecondGreenTestTraceZero premise by deriving it from hsecondGreenTestTraceEq plus the existing admissible-test zero boundary trace condition htestTraceZero. It keeps the cycle-131 box-divergence package, second-Green residual/divergence facts, first-Green subfacts, hdiffusionAction, hdiffusionLaplacianTerm, and htestLaplacian explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle132GeneralMovingTargetDiscreteEmSecondGreenTraceEqTestTraceZeroScoutObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTraceEqTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTestTraceZero", "SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero", "appendix.tex:1392-1427", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle132.lower_2_packet.second_green_pointwise_trace_eq" interface := "Compiled lower_2 continuation: SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqTestTraceZero removes the a.e. hsecondGreenTestTraceEq premise by deriving it from hsecondGreenTestTracePointwiseEq via Filter.Eventually.of_forall, while keeping htestTraceZero, the cycle-131 box-divergence package, second-Green residual/divergence facts, first-Green subfacts, hdiffusionAction, hdiffusionLaplacianTerm, and htestLaplacian explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle132GeneralMovingTargetDiscreteEmSecondGreenPointwiseTraceEqLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTraceEqTestTraceZero", "Filter.Eventually.of_forall", "appendix.tex:1392-1427", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle132.remaining_laplacian_ibp_analysis" interface := "Remaining exact theorem boundary after cycle 132: prove the density-Laplacian weak term for the EM/Brownian diffusion contribution, instantiate the first-Green residual/divergence/zero-flux facts, instantiate the second-Green residual/divergence facts, prove the box-divergence regularity/face-trace facts behind the second-Green boundary flux, prove the pointwise source trace-definition equality identifying the second-Green test trace with the admissible-test boundary trace, prove the analytic zero-test-trace theorem for source admissible tests, and prove htestLaplacian." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "appendix.tex:1379-1387", "appendix.tex:1392-1427", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTraceEqTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTestTraceZero", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle132.reviewer_second_green_trace_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, the new theorem statement lacks a direct hsecondGreenTraceProductZero premise, hdiffusionAction, hdiffusionLaplacianTerm, first-Green subfacts, second-Green residual/divergence facts, box-divergence inputs, and htestLaplacian remain separate, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle132GeneralMovingTargetDiscreteEmSecondGreenTestTraceZeroLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTestTraceZero", "appendix.tex:1392-1427" ] reusedBy := ["cycle 132 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 133: EM second-Green pointwise test-trace zero boundary -/ /-- Cycle-133 middle packet narrowing the admissible-test zero boundary trace input from an a.e. statement to a pointwise source theorem. -/ def cycle133GeneralMovingTargetDiscreteEmSecondGreenPointwiseTestTraceZeroLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle133_em_second_green_pointwise_test_trace_zero_lower" statement := "Cycle 133 middle compiles SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero for appendix.tex:1392-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the htestTraceZero premise from SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqTestTraceZero is replaced by htestTracePointwiseZero, a pointwise boundary-trace vanishing theorem for source admissible tests. The theorem derives the old a.e. boundary trace fact by Filter.Eventually.of_forall and keeps hsecondGreenTestTracePointwiseEq, the box-divergence regularity/face-trace facts, second-Green residual/divergence facts, first-Green subfacts, hdiffusionAction, hdiffusionLaplacianTerm, and htestLaplacian explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqTestTraceZero", "Filter.Eventually.of_forall", "appendix.tex:1392-1427", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area refiner packet. It rejects stale dynamic leaves and wrapper churn by removing the direct a.e. htestTraceZero premise and exposing the smaller pointwise zero-trace source theorem. It does not prove compact support/decay, pointwise second-Green trace identification, weighted-field regularity, divergence integrability, face-to-trace identification, second-Green residual/divergence facts, test-Laplacian normalization, full weak FP, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, or sald_version_2 content." /-- Cycle-133 lower_1 scout packet narrowing the test-Laplacian normalization input from the broad weak-FP context to a test-local theorem. -/ def cycle133GeneralMovingTargetDiscreteEmTestLaplacianNormalizationScoutObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle133_em_test_laplacian_normalization_lower_1" statement := "Cycle 133 lower_1 compiles SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfTestLaplacianNormalization for appendix.tex:1379-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the broad htestLaplacian premise from SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero is replaced by the test-local premise htestLaplacianLocal : ∀ φ, Admissible φ → testRegular → testLaplacianAction φ = laplacian φ. The theorem ignores commonSpace, conditionalKernel, driftRegular, densityRegular, and boundaryBehavior when reconstructing the old htestLaplacian shape, and keeps hsecondGreenTestTracePointwiseEq, htestTracePointwiseZero, box-divergence regularity/face-trace facts, second-Green residual/divergence facts, first-Green subfacts, hdiffusionAction, and hdiffusionLaplacianTerm explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfTestLaplacianNormalization", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Natural-language proof-scout packet plus compiled local composition. It does not prove the analytic source theorem that the chosen test-Laplacian action equals the abstract laplacian for admissible regular tests; that is the remaining test-local boundary. It does not prove compact support/decay, pointwise second-Green trace identification, weighted-field regularity, divergence integrability, face-to-trace identification, second-Green residual/divergence facts, Brownian diffusion construction, full weak FP, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, or sald_version_2 content." /-- Cycle-133 lower_2 packet narrowing the test-local Laplacian normalization to an operator-level source identity for the regular test calculus. -/ def cycle133GeneralMovingTargetDiscreteEmTestLaplacianOperatorNormalizationLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle133_em_test_laplacian_operator_normalization_lower_2" statement := "Cycle 133 lower_2 compiles SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization for appendix.tex:1379-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the test-local htestLaplacianLocal premise left by lower_1 is derived from the operator-level source identity htestLaplacianOperator : testRegular → testLaplacianAction = laplacian. The admissibility argument is no longer part of the analytic normalization leaf; it is inert once the selected test calculus is regular." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfTestLaplacianNormalization", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Proof-producing lower_2 packet. It does not prove the operator-level source identity for the selected regular test calculus; that is the remaining analytic boundary. It does not prove compact support/decay, pointwise second-Green trace identification, weighted-field regularity, divergence integrability, face-to-trace identification, second-Green residual/divergence facts, Brownian diffusion construction, full weak FP, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, or sald_version_2 content." /-- Cycle-133 proof-DAG pane for the pointwise admissible-test trace boundary. -/ def cycle133GeneralMovingTargetDiscreteEmSecondGreenPointwiseTestTraceZeroDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle133.middle_packet.second_green_pointwise_test_trace_zero" interface := "Compiled illness-area refiner: SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero removes the direct a.e. htestTraceZero premise by deriving it from htestTracePointwiseZero via Filter.Eventually.of_forall. It keeps hsecondGreenTestTracePointwiseEq, the cycle-131 box-divergence package, second-Green residual/divergence facts, first-Green subfacts, hdiffusionAction, hdiffusionLaplacianTerm, and htestLaplacian explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle133GeneralMovingTargetDiscreteEmSecondGreenPointwiseTestTraceZeroLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqTestTraceZero", "Filter.Eventually.of_forall", "appendix.tex:1392-1427", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle133.lower_1_packet.test_laplacian_normalization" interface := "Compiled lower_1 scout continuation: SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfTestLaplacianNormalization removes the broad htestLaplacian premise by deriving it from the test-local htestLaplacianLocal premise, depending only on Admissible phi and testRegular. It keeps hsecondGreenTestTracePointwiseEq, htestTracePointwiseZero, the cycle-131 box-divergence package, second-Green residual/divergence facts, first-Green subfacts, hdiffusionAction, and hdiffusionLaplacianTerm explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle133GeneralMovingTargetDiscreteEmTestLaplacianNormalizationScoutObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfTestLaplacianNormalization", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero", "appendix.tex:1379-1387", "appendix.tex:1392-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle133.lower_2_packet.test_laplacian_operator_normalization" interface := "Compiled lower_2 continuation: SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization narrows htestLaplacianLocal to the operator-level source identity htestLaplacianOperator : testRegular → testLaplacianAction = laplacian. It removes admissibility from the analytic normalization boundary while leaving the selected regular test calculus identity explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle133GeneralMovingTargetDiscreteEmTestLaplacianOperatorNormalizationLowerObligation", "SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization", "SALD.cycle133GeneralMovingTargetDiscreteEmTestLaplacianNormalizationScoutObligation", "appendix.tex:1379-1387", "appendix.tex:1392-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle133.remaining_laplacian_ibp_analysis" interface := "Remaining exact theorem boundary after cycle 133: prove the density-Laplacian weak term for the EM/Brownian diffusion contribution, instantiate the first-Green residual/divergence/zero-flux facts, instantiate the second-Green residual/divergence facts, prove the box-divergence regularity/face-trace facts behind the second-Green boundary flux, prove the pointwise source trace-definition equality identifying the second-Green test trace with the admissible-test boundary trace, prove pointwise zero boundary trace for source admissible tests, and prove the operator-level test-Laplacian normalization htestLaplacianOperator." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "appendix.tex:1379-1387", "appendix.tex:1392-1427", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfTestLaplacianNormalization", "SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqTestTraceZero", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle133.reviewer_pointwise_test_trace_zero_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, the new theorem statement lacks direct a.e. htestTraceZero, the lower_1 theorem statement lacks broad htestLaplacian and replaces it only with test-local htestLaplacianLocal, the lower_2 theorem narrows htestLaplacianLocal to htestLaplacianOperator, hsecondGreenTestTracePointwiseEq remains explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle133GeneralMovingTargetDiscreteEmSecondGreenPointwiseTestTraceZeroLowerObligation", "SALD.cycle133GeneralMovingTargetDiscreteEmTestLaplacianNormalizationScoutObligation", "SALD.cycle133GeneralMovingTargetDiscreteEmTestLaplacianOperatorNormalizationLowerObligation", "SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfTestLaplacianNormalization", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero", "appendix.tex:1392-1427" ] reusedBy := ["cycle 133 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 134: EM test-Laplacian source pullback boundary -/ /-- Cycle-134 dynamic-leaf packet narrowing the operator-level test-Laplacian normalization to shared source-pullback definitions. -/ def cycle134GeneralMovingTargetDiscreteEmTestLaplacianSourcePullbackLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle134_em_test_laplacian_source_pullback_lower" statement := "Cycle 134 middle compiles SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback and SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfSourceLaplacianPullback for appendix.tex:1379-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the direct htestLaplacianOperator premise left by SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization is replaced by two source-definition facts against one shared selected weak-test Laplacian action: htestLaplacianActionDef and hweakFpLaplacianDef. The theorem keeps hsecondGreenTestTracePointwiseEq, htestTracePointwiseZero, box-divergence regularity/face-trace facts, second-Green residual/divergence facts, first-Green subfacts, hdiffusionAction, and hdiffusionLaplacianTerm explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfSourceLaplacianPullback", "SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfTestLaplacianNormalization", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet. It does not introduce a theorem equivalent to htestLaplacianOperator under a pointwise or funext name; instead it records the smaller source-calculus definition boundary that the selected test-Laplacian action and weak-FP Laplacian action are pullbacks of the same source Laplacian. It does not prove those source definitions, compact support/decay, pointwise second-Green trace identification, weighted-field regularity, divergence integrability, face-to-trace identification, second-Green residual/divergence facts, Brownian diffusion construction, full weak FP, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, or sald_version_2 content." /-- Cycle-134 lower_1 proof-scout obligation for the Mathlib source formula below the shared source-pullback Laplacian boundary. -/ def cycle134GeneralMovingTargetDiscreteEmSourceLaplacianStdBasisScoutObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle134_em_source_laplacian_std_basis_scout" statement := "Cycle 134 lower_1 compiles SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv for appendix.tex:1379-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining source-definition facts htestLaplacianActionDef and hweakFpLaplacianDef should no longer target an unspecified sourceLaplacianAction; the lower_2-ready route is to instantiate the selected source test as a smooth real-valued function on a finite-dimensional real inner-product space and identify both abstract actions with Mathlib's Laplacian, whose standard-basis iterated-derivative formula is now a compiled local theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback", "Mathlib.Analysis.InnerProductSpace.Laplacian", "InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis", "appendix.tex:1379-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf proof-scout packet. It produces a compiled Mathlib-backed local theorem for the selected weak-test Laplacian formula, but does not claim the remaining source definitions are proved. The next implementation step is one of htestLaplacianActionDef or hweakFpLaplacianDef by showing the corresponding abstract action is the Mathlib Laplacian/standard-basis second-derivative action on the selected source test. No SLT import, non-EM fallback, source-index rebaseline, theorem-status promotion, fake closure, broad audit, wrapper churn, or sald_version_2 content is introduced." /-- Cycle-134 lower_2 obligation narrowing the weak-FP Laplacian definition leaf to the standard-basis source formula. -/ def cycle134GeneralMovingTargetDiscreteEmWeakFpLaplacianStdBasisLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle134_em_weak_fp_laplacian_std_basis_lower" statement := "Cycle 134 lower_2 compiles SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula for appendix.tex:1379-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hweakFpLaplacianDef no longer targets an unspecified sourceLaplacianAction; when the selected weak test is represented in a finite-dimensional real inner-product source space, it follows from hweakFpStdBasisDef, the statement that the weak-FP abstract laplacian action is the standard-basis second-derivative source formula. The sibling htestLaplacianActionDef and all second-Green, box-divergence, trace, diffusion, and first-Green leaves remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis", "appendix.tex:1379-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet. It narrows only hweakFpLaplacianDef to the concrete hweakFpStdBasisDef source formula after instantiating the source Laplacian action by Mathlib's Laplacian passed through the source action functional. It does not claim htestLaplacianActionDef, second-Green, box-divergence, trace, diffusion, first-Green, KL differentiation, LSI, DV, Gronwall, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, or sald_version_2 content." /-- Cycle-134 proof-DAG pane for the source-pullback test-Laplacian boundary. -/ def cycle134GeneralMovingTargetDiscreteEmTestLaplacianSourcePullbackDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle134.middle_packet.test_laplacian_source_pullback" interface := "Compiled dynamic-leaf worker packet: SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfSourceLaplacianPullback removes the direct htestLaplacianOperator premise by deriving it from SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback. The new source-facing leaves are htestLaplacianActionDef and hweakFpLaplacianDef, both identifying their abstract actions with the same selected weak-test source Laplacian pullback over appendix.tex:1379-1427." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle134GeneralMovingTargetDiscreteEmTestLaplacianSourcePullbackLowerObligation", "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfSourceLaplacianPullback", "SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization", "appendix.tex:1379-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle134.lower_1_packet.source_laplacian_std_basis_formula" interface := "Compiled lower_1 proof-scout theorem: SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv identifies Mathlib's source Laplacian for a selected real-valued finite-dimensional test function with the standard-basis sum of second iterated Frechet derivatives. This narrows htestLaplacianActionDef and hweakFpLaplacianDef to action-definition equalities against Mathlib's Laplacian, instead of an unspecified sourceLaplacianAction." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle134GeneralMovingTargetDiscreteEmSourceLaplacianStdBasisScoutObligation", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1427" ] reusedBy := [ "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle134.lower_2_packet.weak_fp_laplacian_std_basis_definition" interface := "Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula narrows hweakFpLaplacianDef to hweakFpStdBasisDef, a source formula saying the weak-FP abstract laplacian action is the standard-basis second-derivative action on the selected weak test. The theorem uses the lower_1 Mathlib Laplacian formula and leaves htestLaplacianActionDef plus all second-Green, box-divergence, trace, diffusion, and first-Green leaves separate." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle134GeneralMovingTargetDiscreteEmWeakFpLaplacianStdBasisLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1427" ] reusedBy := [ "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle134.remaining_source_laplacian_definitions" interface := "Remaining exact theorem boundary after cycle 134 lower_2: instantiate the selected weak-test representation and prove htestLaplacianActionDef for the test-calculus action, and prove hweakFpStdBasisDef for the weak-FP action's standard-basis second-derivative source formula; all second-Green, box-divergence, trace, diffusion, and first-Green leaves remain separate." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "appendix.tex:1379-1387", "appendix.tex:1392-1427", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfSourceLaplacianPullback", "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle134.reviewer_test_laplacian_source_pullback_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, the weak-FP source-definition leaf is narrowed from hweakFpLaplacianDef to hweakFpStdBasisDef through SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula, htestLaplacianActionDef remains explicit, hsecondGreenTestTracePointwiseEq and htestTracePointwiseZero remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle134GeneralMovingTargetDiscreteEmWeakFpLaplacianStdBasisLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula", "SALD.cycle134GeneralMovingTargetDiscreteEmTestLaplacianSourcePullbackLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfSourceLaplacianPullback", "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback", "appendix.tex:1379-1427" ] reusedBy := ["cycle 134 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 135: EM test-Laplacian action standard-basis definition -/ /-- Cycle-135 lower obligation narrowing the test-calculus Laplacian action definition leaf to the standard-basis source formula. -/ def cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle135_em_test_laplacian_std_basis_lower" statement := "Cycle 135 compiles SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula and SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula for appendix.tex:1379-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: htestLaplacianActionDef no longer targets an unspecified sourceLaplacianAction; when the selected weak test is represented in a finite-dimensional real inner-product source space, it follows from htestLaplacianStdBasisDef, the statement that the test-calculus abstract laplacian action is the standard-basis second-derivative source formula. The operator leaf can now be derived directly from htestLaplacianStdBasisDef plus hweakFpStdBasisDef. All second-Green, box-divergence, trace, diffusion, and first-Green leaves remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis", "appendix.tex:1379-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet. It narrows htestLaplacianActionDef to the concrete htestLaplacianStdBasisDef source formula after instantiating the source Laplacian action by Mathlib's Laplacian passed through the source action functional. The lower_1 operator scout theorem also composes htestLaplacianStdBasisDef with hweakFpStdBasisDef to derive the old htestLaplacianOperator boundary. It does not claim the source-facing htestLaplacianStdBasisDef or hweakFpStdBasisDef, second-Green, box-divergence, trace, diffusion, first-Green, KL differentiation, LSI, DV, Gronwall, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, or sald_version_2 content." /-- Cycle-135 lower_2 downstream consumer for the standard-basis test-Laplacian source formulas. -/ def cycle135GeneralMovingTargetDiscreteEmSecondGreenStdBasisConsumerLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle135_em_second_green_std_basis_consumer_lower" statement := "Cycle 135 lower_2 compiles SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula for appendix.tex:1379-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the downstream second-Green diffusion-source consumer no longer takes htestLaplacianActionDef and hweakFpLaplacianDef; it derives the old test-Laplacian normalization from htestLaplacianStdBasisDef plus hweakFpStdBasisDef through the cycle-135 standard-basis operator bridge. All second-Green, box-divergence, trace, diffusion, and first-Green leaves remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfTestLaplacianNormalization", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet. It does not prove htestLaplacianStdBasisDef or hweakFpStdBasisDef; it removes the older htestLaplacianActionDef/hweakFpLaplacianDef supplied hypotheses from the downstream second-Green consumer by deriving the required local test-Laplacian normalization from the two smaller standard-basis source formulas. No SLT import, non-EM fallback, broad audit, theorem-status promotion, fake closure, wrapper churn, Lake/toolchain change, or sald_version_2 content is introduced." /-- Cycle-135 proof-DAG pane for the test-calculus standard-basis boundary. -/ def cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle135.lower_packet.test_laplacian_std_basis_definition" interface := "Compiled dynamic-leaf worker theorem: SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula narrows htestLaplacianActionDef to htestLaplacianStdBasisDef, a source formula saying the test-calculus abstract laplacian action is the standard-basis second-derivative action on the selected weak test. The theorem uses the cycle-134 Mathlib Laplacian formula and leaves hweakFpStdBasisDef plus all second-Green, box-divergence, trace, diffusion, and first-Green leaves separate." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisLowerObligation", "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1427" ] reusedBy := [ "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle135.lower_1_packet.operator_std_basis_composition" interface := "Compiled lower_1 scout theorem: SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula derives the old htestLaplacianOperator boundary directly from the two smaller source-facing leaves htestLaplacianStdBasisDef and hweakFpStdBasisDef. It composes the cycle-134 weak-FP standard-basis bridge, the cycle-135 test-action standard-basis bridge, and the cycle-134 source-pullback operator theorem; it does not discharge either source-facing standard-basis leaf." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisLowerObligation", "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback", "InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1392-1427" ] reusedBy := [ "SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfSourceLaplacianPullback" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle135.lower_2_packet.second_green_std_basis_consumer" interface := "Compiled lower_2 continuation: SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula derives the downstream second-Green diffusion-source conclusion from htestLaplacianStdBasisDef and hweakFpStdBasisDef, so the consumer no longer takes htestLaplacianActionDef or hweakFpLaplacianDef. The theorem keeps the two source-facing standard-basis formulas and all second-Green, box-divergence, trace, diffusion, and first-Green leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle135GeneralMovingTargetDiscreteEmSecondGreenStdBasisConsumerLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization", "appendix.tex:1379-1387", "appendix.tex:1392-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle135.remaining_source_laplacian_definitions" interface := "Remaining exact theorem boundary after cycle 135 lower_2: prove the source-facing htestLaplacianStdBasisDef for the local test-calculus action and hweakFpStdBasisDef for the weak-FP action's standard-basis source formula. Once both are supplied, SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula feeds the downstream second-Green consumer; all second-Green, box-divergence, trace, diffusion, and first-Green leaves remain separate." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "appendix.tex:1379-1387", "appendix.tex:1392-1427", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle135.reviewer_test_laplacian_std_basis_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, the downstream consumer removes htestLaplacianActionDef and hweakFpLaplacianDef in favor of htestLaplacianStdBasisDef plus hweakFpStdBasisDef through SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula, hsecondGreenTestTracePointwiseEq and htestTracePointwiseZero remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisLowerObligation", "SALD.cycle135GeneralMovingTargetDiscreteEmSecondGreenStdBasisConsumerLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula", "appendix.tex:1379-1427" ] reusedBy := ["cycle 135 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 136: EM weak-FP standard-basis source-density formula -/ /-- Cycle-136 lower obligation narrowing the weak-FP standard-basis source formula to a density-Laplacian action formula. -/ def cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle136_em_weak_fp_std_basis_source_density_lower" statement := "Cycle 136 compiles SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula, SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula, and SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField for appendix.tex:1379-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining hweakFpStdBasisDef source formula is no longer the primitive weak-FP input; it follows from a smaller weak-FP-to-density-Laplacian action equality plus a source density-Laplacian standard-basis formula for the selected weak test. The density-Laplacian standard-basis formula is further narrowed to a source action definition plus a pointwise standard-basis source field equality, and lower_2 narrows that pointwise standard-basis field equality to hsourceDensityLaplacianEqLaplacian, the source-field identification with Mathlib's Laplacian of the selected weak test. The theorem does not reintroduce htestLaplacianOperator, htestLaplacianActionDef, or hweakFpLaplacianDef, and all second-Green, box-divergence, trace, diffusion, and first-Green leaves remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula", "SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula", "SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet. It narrows only hweakFpStdBasisDef to the source density-Laplacian standard-basis formula and the equality selecting that density-Laplacian action as the weak-FP abstract Laplacian; lower_1 further narrows hdensityLaplacianStdBasisDef to a source action definition plus a pointwise standard-basis source formula; lower_2 narrows that pointwise formula to hsourceDensityLaplacianEqLaplacian, the source-field equality to Mathlib's Laplacian of the selected weak test. It does not claim hweakFpDensityLaplacianAction, hdensityLaplacianActionDef, htestLaplacianStdBasisDef, second-Green, box-divergence, trace, diffusion, first-Green, KL differentiation, LSI, DV, Gronwall, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-136 proof-DAG pane for the weak-FP standard-basis source-density boundary. -/ def cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle136.middle_packet.weak_fp_std_basis_source_density" interface := "Compiled dynamic-leaf worker theorem: SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula derives hweakFpStdBasisDef from hweakFpDensityLaplacianAction plus hdensityLaplacianStdBasisDef. It keeps htestLaplacianStdBasisDef and the second-Green, box-divergence, trace, diffusion, and first-Green leaves explicit, while avoiding the older htestLaplacianOperator, htestLaplacianActionDef, and hweakFpLaplacianDef boundaries." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula", "appendix.tex:1379-1387", "appendix.tex:1392-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle136.lower_1_packet.density_laplacian_pointwise_source_formula" interface := "Compiled lower_1 proof-scout theorem: SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula derives hdensityLaplacianStdBasisDef from hdensityLaplacianActionDef, the source functional representation of the density-Laplacian weak action, plus hsourceDensityLaplacianStdBasis, the pointwise standard-basis second-derivative source formula for the selected weak test." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation", "SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula", "appendix.tex:1379-1387", "appendix.tex:1392-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle136.lower_2_packet.source_density_laplacian_laplacian_source_field" interface := "Compiled lower_2 Lean theorem: SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField derives hsourceDensityLaplacianStdBasis from hsourceDensityLaplacianEqLaplacian, the source-field equality identifying the named density-Laplacian source field with Mathlib's Laplacian of the selected weak test. The standard-basis second-derivative formula is supplied by SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation", "SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField", "SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1387", "appendix.tex:1392-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle136.remaining_weak_fp_source_density_formula" interface := "Remaining exact theorem boundary after cycle 136 lower_2: prove hweakFpDensityLaplacianAction, selecting the source density-Laplacian weak action as the weak-FP abstract Laplacian; prove hdensityLaplacianActionDef, representing the density-Laplacian action as the source functional applied to a named pointwise source field; and prove hsourceDensityLaplacianEqLaplacian, identifying that pointwise source field with Mathlib's Laplacian of the selected weak test. The sibling htestLaplacianStdBasisDef and all second-Green, box-divergence, trace, diffusion, and first-Green leaves remain separate." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "appendix.tex:1379-1387", "appendix.tex:1392-1427", "SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField", "SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle136.reviewer_weak_fp_std_basis_source_density_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hweakFpStdBasisDef is narrowed to the density-Laplacian standard-basis source formula through SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula, the old htestLaplacianOperator/htestLaplacianActionDef/hweakFpLaplacianDef leaves are not reintroduced, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula", "appendix.tex:1379-1427" ] reusedBy := ["cycle 136 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 137: EM weak-FP density-Laplacian pointwise IBP -/ /-- Cycle-137 lower obligation narrowing the weak-FP density-Laplacian action boundary to pointwise weak Laplacian integration by parts. -/ def cycle137GeneralMovingTargetDiscreteEmWeakFpDensityLaplacianActionLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle137_em_weak_fp_density_laplacian_action_lower" statement := "Cycle 137 compiles SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP and SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP for appendix.tex:1379-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining hweakFpDensityLaplacianAction function-equality input from cycle 136 is no longer primitive; it follows from the pointwise weak Laplacian integration-by-parts identity densityLaplacianAction phi = laplacian phi for each selected weak test. The downstream standard-basis consumer can now use that pointwise IBP equality plus hdensityLaplacianStdBasisDef, while hdensityLaplacianActionDef, hsourceDensityLaplacianEqLaplacian, htestLaplacianStdBasisDef, and all second-Green, box-divergence, trace, diffusion, and first-Green leaves remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula", "SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet. It narrows only hweakFpDensityLaplacianAction to a pointwise weak Laplacian IBP theorem boundary and feeds that smaller boundary into the cycle-136 weak-FP standard-basis source formula. It does not prove the underlying Green identity, density-Laplacian source action definition, source-field Laplacian equality, htestLaplacianStdBasisDef, second-Green, box-divergence, trace, diffusion, first-Green, KL differentiation, LSI, DV, Gronwall, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-137 lower_1 scout obligation splitting pointwise weak Laplacian IBP into the three Green/test-calculus identities from the source proof. -/ def cycle137GeneralMovingTargetDiscreteEmPointwiseGreenIbPScoutObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle137_em_pointwise_green_ibp_scout" statement := "Cycle 137 lower_1 compiles SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity and SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity for appendix.tex:1379-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the pointwise weak Laplacian IBP theorem densityLaplacianAction phi = laplacian phi is no longer a single supplied theorem boundary; it follows from three pointwise source-facing leaves: first-Green densityLaplacianAction phi = negativeGradientPairAction phi, second-Green negativeGradientPairAction phi = testLaplacianAction phi, and test-Laplacian normalization testLaplacianAction phi = laplacian phi. The density-Laplacian source-action definition, source-field Laplacian equality, htestLaplacianStdBasisDef, box-divergence, trace, diffusion, and first/second Green analytic instantiations remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP", "SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 proof-scout packet. It gives lower_2 a one-theorem implementation target: prove the three pointwise Green/test-calculus leaves for the selected weak tests, then feed them through SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity. It does not prove the analytic Green identities, source density-Laplacian definitions, htestLaplacianStdBasisDef, box-divergence, trace, diffusion, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-137 lower_2 narrowing of the second-Green pointwise leaf to the box-divergence and pointwise-trace boundary. -/ def cycle137GeneralMovingTargetDiscreteEmSecondGreenPointwiseBoxBoundaryFluxLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle137_em_second_green_pointwise_box_boundary_flux_lower" statement := "Cycle 137 lower_2 compiles SALD.generalMovingTargetDiscreteSecondGreenPointwiseOfBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero and SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux for appendix.tex:1392-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the direct hsecondGreenPointwise leaf from SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity is no longer primitive; it follows from the second-Green residual and divergence identities, the Mathlib box-divergence regularity/derivative/integrability package, signed-face trace identification, pointwise equality with the admissible test trace, and pointwise zero test trace. First-Green pointwise, test-Laplacian pointwise normalization, hdensityLaplacianActionDef, hsourceDensityLaplacianEqLaplacian, and htestLaplacianStdBasisDef remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteSecondGreenPointwiseOfBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "Filter.Eventually.of_forall", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 worker packet. It narrows only the pointwise second-Green leaf under the cycle-137 pointwise Green split, replacing the direct hsecondGreenPointwise premise by the already source-facing box-divergence and pointwise trace-zero boundary. It does not prove first-Green pointwise, test-Laplacian pointwise normalization, density-Laplacian source definitions, diffusion-source closure, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-137 proof-DAG pane for the weak-FP density-Laplacian pointwise integration-by-parts boundary. -/ def cycle137GeneralMovingTargetDiscreteEmWeakFpDensityLaplacianActionDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle137.middle_packet.weak_fp_density_laplacian_pointwise_ibp" interface := "Compiled dynamic-leaf worker theorem: SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP derives hweakFpDensityLaplacianAction from the pointwise weak Laplacian IBP equality densityLaplacianAction phi = laplacian phi. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP then feeds that smaller pointwise boundary plus hdensityLaplacianStdBasisDef into the existing cycle-136 standard-basis source formula." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle137GeneralMovingTargetDiscreteEmWeakFpDensityLaplacianActionLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula", "appendix.tex:1379-1387", "appendix.tex:1392-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle137.lower_1_packet.pointwise_green_ibp_scout" interface := "Compiled lower_1 proof-scout theorem: SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity derives the pointwise weak Laplacian IBP equality from first-Green, second-Green, and test-Laplacian pointwise identities. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity feeds those three leaves plus hdensityLaplacianStdBasisDef into the cycle-137 standard-basis consumer." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle137GeneralMovingTargetDiscreteEmPointwiseGreenIbPScoutObligation", "SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP", "SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity", "appendix.tex:1379-1387", "appendix.tex:1392-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle137.lower_2_packet.second_green_pointwise_box_boundary_flux" interface := "Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteSecondGreenPointwiseOfBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero derives the pointwise second-Green identity negativeGradientPairAction phi = testLaplacianAction phi from residual/divergence facts, the Mathlib box-divergence package, signed-face trace identification, pointwise second-Green trace equality, and pointwise zero test trace. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux feeds that derived second-Green leaf into the lower_1 pointwise Green standard-basis consumer while leaving first-Green and test-Laplacian pointwise leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle137GeneralMovingTargetDiscreteEmSecondGreenPointwiseBoxBoundaryFluxLowerObligation", "SALD.generalMovingTargetDiscreteSecondGreenPointwiseOfBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "Filter.Eventually.of_forall", "appendix.tex:1392-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle137.remaining_density_laplacian_source_action" interface := "Remaining exact theorem boundary after cycle 137 lower_2: prove the first-Green pointwise identity densityLaplacianAction phi = negativeGradientPairAction phi and the test-Laplacian pointwise normalization testLaplacianAction phi = laplacian phi. The direct second-Green pointwise leaf is narrowed to residual/divergence, box-divergence regularity/face-trace, pointwise second-Green trace equality, and pointwise zero test trace. The already separate hdensityLaplacianActionDef and hsourceDensityLaplacianEqLaplacian source-field facts from cycle 136 also remain, and htestLaplacianStdBasisDef plus diffusion leaves stay explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "appendix.tex:1379-1387", "appendix.tex:1392-1427", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux", "SALD.generalMovingTargetDiscreteSecondGreenPointwiseOfBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity", "SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP", "SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP", "SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity", "SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula", "SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle137.reviewer_weak_fp_density_laplacian_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hweakFpDensityLaplacianAction is narrowed to the pointwise weak Laplacian IBP equality through SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP, the standard-basis consumer uses SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle137GeneralMovingTargetDiscreteEmWeakFpDensityLaplacianActionLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP", "appendix.tex:1379-1427" ] reusedBy := ["cycle 137 reviewer"] status := ProofStatus.obligation } ] /-- Cycle-138 narrowing of the first-Green pointwise leaf to boundary-flux cancellation facts. -/ def cycle138GeneralMovingTargetDiscreteEmFirstGreenPointwiseBoundaryFluxLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle138_em_first_green_pointwise_boundary_flux_lower" statement := "Cycle 138 middle compiles SALD.generalMovingTargetDiscreteFirstGreenPointwiseOfBoundaryFluxZero and SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfFirstGreenBoundaryFluxAndSecondGreenBoxBoundaryFlux for appendix.tex:1392-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the direct hfirstGreenPointwise leaf from SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux is no longer primitive; it follows from hfirstGreenResidual, hfirstGreenDivergence, and hfirstGreenZeroBoundary, exposing the first-Green residual, divergence/boundary-flux identity, and zero-boundary fact. The theorem keeps htestLaplacianPointwise, hdensityLaplacianActionDef, hsourceDensityLaplacianEqLaplacian, htestLaplacianStdBasisDef, diffusion leaves, and the second-Green box-divergence/trace leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteFirstGreenPointwiseOfBoundaryFluxZero", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfFirstGreenBoundaryFluxAndSecondGreenBoxBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity", "SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet. It narrows only the first-Green pointwise leaf under the cycle-137 pointwise Green split, replacing direct hfirstGreenPointwise by residual/divergence/zero-boundary facts. It does not prove test-Laplacian pointwise normalization, density-Laplacian source definitions, second-Green residual/divergence, box-divergence, trace, diffusion-source closure, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-138 lower_1 scout bridge narrowing pointwise test-Laplacian normalization to shared source-pullback definitions. -/ def cycle138GeneralMovingTargetDiscreteEmTestLaplacianPointwiseSourcePullbackScoutObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle138_em_test_laplacian_pointwise_source_pullback_scout" statement := "Cycle 138 lower_1 compiles SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfSourcePullback for appendix.tex:1379-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the direct htestLaplacianPointwise leaf from SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity is derived from htestLaplacianActionDef and hweakFpLaplacianDef, the source-facing facts that the test-calculus action and weak-FP abstract Laplacian are both pullbacks of the same selected source-Laplacian action." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfSourcePullback", "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback", "SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 proof-scout packet. It does not prove htestLaplacianActionDef or hweakFpLaplacianDef, nor their standard-basis refinements; it only removes the direct pointwise normalization as a primitive source theorem. It does not close the density-Laplacian source action, second-Green residual/divergence, box-divergence, trace, diffusion, KL differentiation, theorem closure, SLT import, Lake/toolchain change, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-138 lower_2 bridge from the test-calculus standard-basis source formula to the pointwise test-Laplacian normalization leaf. -/ def cycle138GeneralMovingTargetDiscreteEmTestLaplacianPointwiseStdBasisLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle138_em_test_laplacian_pointwise_std_basis_lower" statement := "Cycle 138 lower_2 compiles SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula for appendix.tex:1379-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the htestLaplacianActionDef side of the pointwise test-Laplacian normalization is no longer primitive; it follows from htestLaplacianStdBasisDef via the Mathlib Laplacian standard-basis bridge. The weak-FP side remains the explicit hweakFpLaplacianDef source boundary, because deriving it from hweakFpStdBasisDef would reuse the standard-basis conclusion being proved by the Green route." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfSourcePullback", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet. It implements the valid non-circular part of the lower_1 handoff: htestLaplacianActionDef is instantiated from htestLaplacianStdBasisDef for the pointwise Green route. It deliberately does not instantiate hweakFpLaplacianDef from hweakFpStdBasisDef or hweakFpDensityLaplacianAction, since either would feed a target or ancestor target back into the pointwise proof. It does not close hweakFpLaplacianDef, density-Laplacian source definitions, second-Green residual/divergence, box-divergence, trace, diffusion, KL differentiation, theorem closure, SLT import, Lake/toolchain change, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-138 proof-DAG pane for the first-Green pointwise boundary-flux sub-boundary. -/ def cycle138GeneralMovingTargetDiscreteEmFirstGreenPointwiseBoundaryFluxDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle138.middle_packet.first_green_pointwise_boundary_flux" interface := "Compiled dynamic-leaf theorem: SALD.generalMovingTargetDiscreteFirstGreenPointwiseOfBoundaryFluxZero derives hfirstGreenPointwise from hfirstGreenResidual, hfirstGreenDivergence, and hfirstGreenZeroBoundary. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfFirstGreenBoundaryFluxAndSecondGreenBoxBoundaryFlux feeds this derived first-Green leaf into the existing cycle-137 second-Green box-boundary standard-basis consumer." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle138GeneralMovingTargetDiscreteEmFirstGreenPointwiseBoundaryFluxLowerObligation", "SALD.generalMovingTargetDiscreteFirstGreenPointwiseOfBoundaryFluxZero", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfFirstGreenBoundaryFluxAndSecondGreenBoxBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux", "appendix.tex:1392-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle138.lower_1_packet.test_laplacian_pointwise_source_pullback" interface := "Compiled lower_1 scout theorem: SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfSourcePullback derives the pointwise test-Laplacian normalization testLaplacianAction phi = laplacian phi from htestLaplacianActionDef and hweakFpLaplacianDef, both identifying the two abstract actions with the same selected source-Laplacian pullback. The source-definition leaves themselves remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle138GeneralMovingTargetDiscreteEmTestLaplacianPointwiseSourcePullbackScoutObligation", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfSourcePullback", "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback", "SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity", "appendix.tex:1379-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle138.lower_2_packet.test_laplacian_pointwise_std_basis" interface := "Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula derives the pointwise test-Laplacian normalization testLaplacianAction phi = laplacian phi from htestLaplacianStdBasisDef plus the still-explicit hweakFpLaplacianDef. This removes htestLaplacianActionDef from the pointwise Green route while avoiding the circular use of hweakFpStdBasisDef." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle138GeneralMovingTargetDiscreteEmTestLaplacianPointwiseStdBasisLowerObligation", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfSourcePullback", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle138.remaining_pointwise_green_after_first_green" interface := "Remaining exact theorem boundary after cycle 138 lower_2: prove the source-facing htestLaplacianStdBasisDef for the local test-calculus action and prove hweakFpLaplacianDef from a non-circular weak-FP source-Laplacian definition, then use SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula to obtain the pointwise test-Laplacian normalization. The density-Laplacian source-action/source-field facts and the explicit second-Green residual/divergence, box-divergence, signed-face trace, pointwise trace equality, and pointwise zero-trace leaves remain separate. The first-Green pointwise leaf is narrowed to residual/divergence/zero-boundary facts, and htestLaplacianActionDef is no longer primitive for the pointwise route." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfSourcePullback", "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfFirstGreenBoundaryFluxAndSecondGreenBoxBoundaryFlux", "SALD.generalMovingTargetDiscreteFirstGreenPointwiseOfBoundaryFluxZero", "SALD.generalMovingTargetDiscreteSecondGreenPointwiseOfBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero", "SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula", "SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField", "appendix.tex:1379-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle138.reviewer_first_green_pointwise_boundary_flux_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hfirstGreenPointwise is narrowed to first-Green residual/divergence/zero-boundary facts through SALD.generalMovingTargetDiscreteFirstGreenPointwiseOfBoundaryFluxZero, htestLaplacianPointwise is narrowed first to source-pullback definitions through SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfSourcePullback and then on the test side to htestLaplacianStdBasisDef through SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula, hweakFpLaplacianDef remains explicit rather than being derived from hweakFpStdBasisDef, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle138GeneralMovingTargetDiscreteEmFirstGreenPointwiseBoundaryFluxLowerObligation", "SALD.cycle138GeneralMovingTargetDiscreteEmTestLaplacianPointwiseSourcePullbackScoutObligation", "SALD.cycle138GeneralMovingTargetDiscreteEmTestLaplacianPointwiseStdBasisLowerObligation", "SALD.generalMovingTargetDiscreteFirstGreenPointwiseOfBoundaryFluxZero", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfSourcePullback", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfFirstGreenBoundaryFluxAndSecondGreenBoxBoundaryFlux", "appendix.tex:1392-1427" ] reusedBy := ["cycle 138 reviewer"] status := ProofStatus.obligation } ] /-- Cycle-139 weak-FP source-Laplacian field split for the pointwise test-Laplacian route. -/ def cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle139_em_weak_fp_source_laplacian_lower" statement := "Cycle 139 middle compiles SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField and SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField for appendix.tex:1379-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hweakFpLaplacianDef premise left by SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula is no longer primitive; it is split into hweakFpSourceActionDef, identifying the weak-FP abstract Laplacian action with the source functional applied to a named source field, and hweakFpSourceFieldEqLaplacian, identifying that source field with Mathlib's Laplacian of the selected weak-test representative. The route keeps htestLaplacianStdBasisDef, density-Laplacian source-action/source-field facts, second-Green residual/divergence, box-divergence, signed-face trace, pointwise trace equality, pointwise zero trace, and diffusion leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet. It narrows only the non-circular weak-FP source-Laplacian definition needed by the pointwise test-Laplacian route. It does not derive hweakFpLaplacianDef from hweakFpStdBasisDef or hweakFpDensityLaplacianAction, and it does not reintroduce htestLaplacianActionDef or htestLaplacianOperator. It does not close htestLaplacianStdBasisDef, density-Laplacian source facts, second-Green residual/divergence, box-divergence, trace, diffusion, KL differentiation, theorem closure, SLT import, Lake/toolchain change, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-139 lower_1 state-integral scout for `hweakFpSourceActionDef`. -/ def cycle139GeneralMovingTargetDiscreteEmWeakFpSourceActionIntegralScoutObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle139_em_weak_fp_source_action_integral_scout" statement := "Cycle 139 lower_1 compiles SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral for appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hweakFpSourceActionDef is no longer an opaque function equality; it follows from the EM state-law map identity hatRhoS = Measure.map hatXAtS P, the sourceLaplacianFunctional definition as integration against hatRhoS, measurability of weakFpLaplacianSourceField under hatRhoS, and the source state/sample integral formula for laplacian phi. The theorem exposes the lower_2-ready sample pullback integral boundary without using hweakFpStdBasisDef, hweakFpDensityLaplacianAction, htestLaplacianOperator, or a non-EM fallback." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral", "MeasureTheory.integral_map", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 proof-scout packet. It produces a compiled local theorem for the state-law integral transport below hweakFpSourceActionDef, but does not prove the remaining source analytic facts: the sample-space Laplacian action formula, source functional definition, source-field measurability, hweakFpSourceFieldEqLaplacian, htestLaplacianStdBasisDef, density-Laplacian source facts, Green/trace/box-divergence leaves, or diffusion construction. No SLT import, Lake/toolchain change, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content is introduced." /-- Cycle-139 lower_2 source-Laplacian state-integral narrowing. -/ def cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianStateIntegralLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle139_em_weak_fp_source_laplacian_state_integral_lower" statement := "Cycle 139 lower_2 compiles SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral for appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the lower_1 hweakFpFieldMeas and hlaplacianStateIntegral inputs are no longer stated for an opaque weakFpLaplacianSourceField; they are reduced to sourceLaplacianFieldMeas and hlaplacianSourceStateIntegral for Mathlib's Laplacian of the selected weak-test representative, together with hweakFpSourceFieldEqLaplacian. The theorem then reuses the compiled EM state-law integral_map bridge." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "MeasureTheory.integral_map", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 packet. It proves a compiled local theorem below hweakFpSourceActionDef and narrows the source-action state-integral inputs to the concrete source Laplacian field. It does not prove hweakFpSourceFieldEqLaplacian, sourceLaplacianFieldMeas, the source sample-space Laplacian integral formula, htestLaplacianStdBasisDef, density-Laplacian source facts, Green/trace/box-divergence leaves, diffusion construction, KL differentiation, theorem closure, SLT import, Lake/toolchain change, theorem-status promotion, fake closure, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-139 proof-DAG pane for the weak-FP source-Laplacian field split. -/ def cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle139.middle_packet.weak_fp_source_laplacian_field" interface := "Compiled dynamic-leaf theorem: SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField derives the weak-FP source-Laplacian pullback hweakFpLaplacianDef from hweakFpSourceActionDef and hweakFpSourceFieldEqLaplacian. This exposes the named source field behind the weak-FP abstract Laplacian action over appendix.tex:1379-1427 without using the circular hweakFpStdBasisDef route." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle139.middle_packet.test_laplacian_pointwise_weak_fp_source_field" interface := "Compiled downstream theorem: SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField feeds the weak-FP source-field split into SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula. The test side still uses htestLaplacianStdBasisDef, while the weak-FP side uses hweakFpSourceActionDef and hweakFpSourceFieldEqLaplacian." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianLowerObligation", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula", "appendix.tex:1379-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle139.lower_1_packet.weak_fp_source_action_state_integral" interface := "Compiled lower_1 theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral derives hweakFpSourceActionDef from the EM state-law integral route. Inputs are hatRhoS = Measure.map hatXAtS P, sourceLaplacianFunctional as the hatRhoS integral, AEStronglyMeasurable weakFpLaplacianSourceField under hatRhoS, and the sample-space pullback integral formula for laplacian phi; Mathlib integral_map transports the sample integral to the named law." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceActionIntegralScoutObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral", "MeasureTheory.integral_map", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle139.lower_2_packet.source_laplacian_state_integral" interface := "Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral feeds the lower_1 state-law integral bridge after replacing the opaque weakFpLaplacianSourceField measurability and state-integral inputs by the concrete Mathlib source Laplacian field of selectedTest, using hweakFpSourceFieldEqLaplacian for the transport." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianStateIntegralLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "MeasureTheory.integral_map", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle139.remaining_weak_fp_source_laplacian_field" interface := "Remaining exact theorem boundary after cycle 139 lower_2: prove hweakFpSourceFieldEqLaplacian, sourceLaplacianFieldMeas for Laplacian.laplacian (selectedTest phi) under hatRhoS, and the source sample-space pullback integral formula hlaplacianSourceStateIntegral for that Laplacian field; keep the source functional definition, EM map-law measurability, htestLaplacianStdBasisDef, density-Laplacian source-action/source-field facts, second-Green residual/divergence, box-divergence, signed-face trace, pointwise trace equality, pointwise zero trace, and diffusion leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula", "appendix.tex:1379-1427" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle139.reviewer_weak_fp_source_laplacian_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hweakFpLaplacianDef is narrowed to hweakFpSourceActionDef plus hweakFpSourceFieldEqLaplacian through SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField, hweakFpSourceActionDef is further narrowed by SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral to sourceLaplacianFieldMeas and hlaplacianSourceStateIntegral for the selected source Laplacian field, the downstream pointwise route uses SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField, htestLaplacianOperator is not reintroduced, hweakFpStdBasisDef and hweakFpDensityLaplacianAction are not used to close this weak-FP source definition, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianLowerObligation", "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceActionIntegralScoutObligation", "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianStateIntegralLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "appendix.tex:1379-1427" ] reusedBy := ["cycle 139 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 140: EM source-Laplacian state integral via frozen generator -/ /-- Cycle-140 narrowing of the source-Laplacian state-integral leaf to the frozen EM generator component. -/ def cycle140GeneralMovingTargetDiscreteEmLaplacianSourceStateIntegralLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle140_em_laplacian_source_state_integral_lower" statement := "Cycle 140 middle compiles SALD.generalMovingTargetDiscreteLaplacianSourceStateIntegralOfEmGeneratorStateIntegral and SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStateIntegral for appendix.tex:984-995 and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hlaplacianSourceStateIntegral premise under SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral is no longer primitive; it follows from a source-cited equality identifying the weak-FP Laplacian action with the frozen EM generator's Laplacian component and a source integral formula for that generator component against the sample path hatXAtS. The theorem keeps hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, htestLaplacianStdBasisDef, density-Laplacian source facts, Green/trace/box-divergence, and diffusion leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteLaplacianSourceStateIntegralOfEmGeneratorStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area refiner packet. It narrows only hlaplacianSourceStateIntegral to the smaller EM frozen-generator action/state-integral interface. It does not prove hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, the source functional definition, htestLaplacianStdBasisDef, density-Laplacian source facts, second-Green residual/divergence, box-divergence, trace, diffusion, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-140 lower_1 law-integral scout for the frozen EM generator Laplacian component. -/ def cycle140GeneralMovingTargetDiscreteEmGeneratorLawIntegralScoutObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle140_em_generator_law_integral_scout" statement := "Cycle 140 lower_1 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral and SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorLawIntegral for appendix.tex:984-995 and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorStateIntegral premise is no longer primitive; it follows from the law-space source fact hemGeneratorLawIntegral against hatRhoS, the map-law identity hatRhoS = Measure.map hatXAtS P, AEMeasurable hatXAtS, and source-Laplacian field measurability, using Mathlib MeasureTheory.integral_map. The remaining implementer theorem is the law-space frozen-generator Laplacian identity, while hlaplacianEqEmGenerator, hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, htestLaplacianStdBasisDef, density-Laplacian source facts, Green/trace/box-divergence, and diffusion leaves stay explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorLawIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStateIntegral", "MeasureTheory.integral_map", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 proof-scout packet inside the cycle-140 illness area. It proves only law-to-sample integral transport for the frozen generator Laplacian component and the downstream consumer theorem. It does not prove the law-space generator identity, the generator/action equality, source-field equality, source-field measurability, source-functional definition, test standard-basis formula, density-Laplacian source facts, Green/trace/box-divergence leaves, diffusion construction, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-140 lower_2 source-functional narrowing for the frozen EM generator Laplacian law integral. -/ def cycle140GeneralMovingTargetDiscreteEmGeneratorSourceFunctionalLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle140_em_generator_source_functional_lower" statement := "Cycle 140 lower_2 compiles SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional for appendix.tex:984-995 and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the law-space hemGeneratorLawIntegral premise under SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorLawIntegral is no longer primitive; it follows from a source-cited definition of the frozen EM generator Laplacian action as sourceLaplacianFunctional applied to Mathlib's selected-test Laplacian, together with the existing source-functional law integral definition. The theorem keeps hlaplacianEqEmGenerator, hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, htestLaplacianStdBasisDef, density-Laplacian source facts, Green/trace/box-divergence, and diffusion leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorLawIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Lower_2 proof-producing packet inside the cycle-140 illness area. It narrows only the law-space frozen EM generator Laplacian integral to the smaller source-functional action definition plus the existing source functional as integration against hatRhoS. It does not prove hlaplacianEqEmGenerator, source-field equality, source-field measurability, source-functional construction, test standard-basis formula, density-Laplacian source facts, Green/trace/box-divergence leaves, diffusion construction, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-140 proof-DAG pane for the source-Laplacian state-integral boundary. -/ def cycle140GeneralMovingTargetDiscreteEmLaplacianSourceStateIntegralDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle140.middle_packet.laplacian_source_state_integral_em_generator" interface := "Compiled illness-area theorem: SALD.generalMovingTargetDiscreteLaplacianSourceStateIntegralOfEmGeneratorStateIntegral narrows hlaplacianSourceStateIntegral to two source-cited EM facts: laplacian phi equals the frozen generator Laplacian component, and that component is the sample-space integral of Mathlib's Laplacian of selectedTest phi along hatXAtS." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle140GeneralMovingTargetDiscreteEmLaplacianSourceStateIntegralLowerObligation", "SALD.generalMovingTargetDiscreteLaplacianSourceStateIntegralOfEmGeneratorStateIntegral", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle140.middle_packet.weak_fp_source_action_em_generator_state_integral" interface := "Compiled consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStateIntegral feeds the cycle-140 source-integral bridge into the cycle-139 weak-FP source-action theorem, replacing only hlaplacianSourceStateIntegral while keeping hweakFpSourceFieldEqLaplacian and hsourceLaplacianFieldMeas explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle140GeneralMovingTargetDiscreteEmLaplacianSourceStateIntegralLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStateIntegral", "SALD.generalMovingTargetDiscreteLaplacianSourceStateIntegralOfEmGeneratorStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle140.lower_1_packet.em_generator_laplacian_law_integral" interface := "Compiled lower_1 theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral derives hemGeneratorStateIntegral from the law-space source fact hemGeneratorLawIntegral against hatRhoS by hatRhoS = Measure.map hatXAtS P and Mathlib MeasureTheory.integral_map. SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorLawIntegral feeds this into the existing cycle-140 weak-FP source-action consumer." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle140GeneralMovingTargetDiscreteEmGeneratorLawIntegralScoutObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorLawIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStateIntegral", "MeasureTheory.integral_map", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle140.lower_2_packet.em_generator_source_functional" interface := "Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional derives lower_1's law-space hemGeneratorLawIntegral from the smaller source-functional action definition for the frozen EM generator Laplacian component and the existing sourceLaplacianFunctional law-integral definition. The law-to-sample transport and weak-FP source-action consumer remain the compiled lower_1 route." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle140GeneralMovingTargetDiscreteEmGeneratorSourceFunctionalLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorLawIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle140.remaining_em_generator_laplacian_state_integral" interface := "Remaining exact theorem boundary after cycle 140 lower_2: prove hlaplacianEqEmGenerator for the selected weak-FP Laplacian action and prove the source-functional definition of the frozen EM generator Laplacian component. The law-space hemGeneratorLawIntegral now follows by SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional, and the sample-space hemGeneratorStateIntegral follows by SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral. The sibling hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, htestLaplacianStdBasisDef, density-Laplacian source facts, Green/trace/box-divergence, and diffusion leaves remain separate." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorLawIntegral", "SALD.generalMovingTargetDiscreteLaplacianSourceStateIntegralOfEmGeneratorStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle140.reviewer_em_generator_state_integral_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hlaplacianSourceStateIntegral is narrowed to hlaplacianEqEmGenerator plus the source-functional frozen EM generator Laplacian action definition through SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional, hweakFpSourceFieldEqLaplacian and hsourceLaplacianFieldMeas remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle140GeneralMovingTargetDiscreteEmGeneratorSourceFunctionalLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional", "SALD.cycle140GeneralMovingTargetDiscreteEmGeneratorLawIntegralScoutObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorLawIntegral", "SALD.cycle140GeneralMovingTargetDiscreteEmLaplacianSourceStateIntegralLowerObligation", "SALD.generalMovingTargetDiscreteLaplacianSourceStateIntegralOfEmGeneratorStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStateIntegral", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := ["cycle 140 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 141: EM generator source action via standard-basis formula -/ /-- Cycle-141 narrowing of the frozen EM generator source-action leaf. -/ def cycle141GeneralMovingTargetDiscreteEmGeneratorStdBasisSourceLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle141_em_generator_std_basis_source_lower" statement := "Cycle 141 middle compiles SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula and SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStdBasisSourceFormula for appendix.tex:984-995 and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining hemGeneratorSourceActionDef premise under SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional is no longer primitive; it follows from hemGeneratorStdBasisDef, the source-cited frozen EM generator standard-basis second-derivative action formula, plus the existing Mathlib selected-test Laplacian bridge. The theorem keeps hlaplacianEqEmGenerator, hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, htestLaplacianStdBasisDef, density-Laplacian source facts, Green/trace/box-divergence leaves, and diffusion leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area refiner packet. It narrows only hemGeneratorSourceActionDef to the smaller standard-basis source formula for the frozen EM generator Laplacian component. It does not prove hemGeneratorStdBasisDef, hlaplacianEqEmGenerator, source-field equality, source-field measurability, source-functional construction, test standard-basis formula, density-Laplacian source facts, Green/trace/box-divergence leaves, diffusion construction, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-141 lower_1 scout narrowing below the EM generator standard-basis leaf. -/ def cycle141GeneralMovingTargetDiscreteEmGeneratorTraceFieldSourceScoutObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle141_em_generator_trace_field_source_scout" statement := "Cycle 141 lower_1 compiles SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField and SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceFieldSourceFormula for appendix.tex:984-995 and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining hemGeneratorStdBasisDef premise under SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStdBasisSourceFormula is no longer primitive; it follows from hemGeneratorTraceActionDef, the source-cited frozen EM generator trace-action definition, plus htraceFieldStdBasis, the pointwise standard-basis trace formula for the named EM generator trace field. The theorem keeps hlaplacianEqEmGenerator, hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, htestLaplacianStdBasisDef, density-Laplacian source facts, Green/trace/box-divergence leaves, and diffusion leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceFieldSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStdBasisSourceFormula", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 proof-scout packet inside the cycle-141 illness area. It narrows only hemGeneratorStdBasisDef to the smaller named trace-field action definition and pointwise standard-basis trace-field formula. It does not prove either analytic trace-field fact, hlaplacianEqEmGenerator, source-field equality, source-field measurability, source-functional construction, test standard-basis formula, density-Laplacian source facts, Green/trace/box-divergence leaves, diffusion construction, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-141 lower_2 law-integral narrowing below the trace-action leaf. -/ def cycle141GeneralMovingTargetDiscreteEmGeneratorTraceLawIntegralLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle141_em_generator_trace_law_integral_lower" statement := "Cycle 141 lower_2 compiles SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula for appendix.tex:984-995 and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining hemGeneratorTraceActionDef premise under SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceFieldSourceFormula is no longer primitive; it follows from the law-space source fact hemGeneratorTraceLawIntegral against hatRhoS plus the already explicit hsourceLaplacianFunctional definition. The theorem keeps hlaplacianEqEmGenerator, htraceFieldStdBasis, hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, htestLaplacianStdBasisDef, density-Laplacian source facts, Green/trace/box-divergence leaves, and diffusion leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceFieldSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 packet inside the cycle-141 illness area. It removes only the direct hemGeneratorTraceActionDef hypothesis from the downstream trace-field consumer by exposing the smaller law-space trace integral boundary hemGeneratorTraceLawIntegral. It does not prove that law-integral theorem, htraceFieldStdBasis, hlaplacianEqEmGenerator, source-field equality, source-field measurability, source-functional construction, test standard-basis formula, density-Laplacian source facts, Green/trace/box-divergence leaves, diffusion construction, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-142 state-integral narrowing below the trace-law leaf. -/ def cycle142GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle142_em_generator_trace_state_integral_middle" statement := "Cycle 142 middle compiles SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral and SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceStateIntegralSourceFormula for appendix.tex:984-995 and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining hemGeneratorTraceLawIntegral premise is no longer primitive; it follows from the sample-space trace integral hemGeneratorTraceStateIntegral, the source-law identity hatRhoS = Measure.map hatXAtS P, AEMeasurable hatXAtS, and trace-field measurability under hatRhoS, using Mathlib MeasureTheory.integral_map. The theorem keeps hlaplacianEqEmGenerator, htraceFieldStdBasis, hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, htestLaplacianStdBasisDef, density-Laplacian source facts, Green/trace/box-divergence leaves, and diffusion leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceStateIntegralSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula", "MeasureTheory.integral_map", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf middle packet inside the cycle-141 EM generator trace-field illness area. It narrows only hemGeneratorTraceLawIntegral to hemGeneratorTraceStateIntegral plus the map-law/measurability transport facts for hatRhoS. It does not prove hemGeneratorTraceStateIntegral, htraceFieldMeas, htraceFieldStdBasis, hlaplacianEqEmGenerator, source-field equality, source-field measurability, source-functional construction, test standard-basis formula, density-Laplacian source facts, Green/trace/box-divergence leaves, diffusion construction, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-142 lower_1 scout narrowing below the trace-state integral leaf. -/ def cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianStateIntegralScoutObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle142_em_generator_trace_laplacian_state_integral_scout" statement := "Cycle 142 lower_1 compiles SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldMeasOfSourceLaplacianFieldMeas, SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegral, and SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula for appendix.tex:984-995 and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining trace-specific htraceFieldMeas and hemGeneratorTraceStateIntegral premises are no longer primitive; they follow from hsourceLaplacianFieldMeas, hemGeneratorLaplacianStateIntegral, and htraceFieldStdBasis via SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv. The theorem keeps hlaplacianEqEmGenerator, htraceFieldStdBasis, hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, htestLaplacianStdBasisDef, density-Laplacian source facts, Green/trace/box-divergence leaves, and diffusion leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldMeasOfSourceLaplacianFieldMeas", "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceStateIntegralSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 proof-scout packet inside the cycle-142 EM generator trace-state illness area. It narrows only the trace-specific state integral and measurability side condition to the already named selected-test Laplacian state integral plus source-Laplacian measurability and htraceFieldStdBasis. It does not prove hemGeneratorLaplacianStateIntegral, htraceFieldStdBasis, hlaplacianEqEmGenerator, source-field equality, source-functional construction, density-Laplacian source facts, Green/trace/box-divergence leaves, diffusion construction, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-142 lower_2 law-integral narrowing below the EM Laplacian state-integral leaf. -/ def cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianLawIntegralLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle142_em_generator_trace_laplacian_law_integral_lower" statement := "Cycle 142 lower_2 compiles SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula for appendix.tex:984-995 and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining hemGeneratorLaplacianStateIntegral premise under SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula is no longer primitive; it follows from the law-space selected-test Laplacian generator fact hemGeneratorLaplacianLawIntegral, the source-law identity hatRhoS = Measure.map hatXAtS P, AEMeasurable hatXAtS, and source-Laplacian field measurability by SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral. The theorem keeps hlaplacianEqEmGenerator, htraceFieldStdBasis, hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, htestLaplacianStdBasisDef, density-Laplacian source facts, Green/trace/box-divergence leaves, and diffusion leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "MeasureTheory.integral_map", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 packet inside the cycle-142 EM generator trace-state illness area. It narrows only hemGeneratorLaplacianStateIntegral to the smaller law-space selected-test Laplacian generator integral hemGeneratorLaplacianLawIntegral plus already explicit map-law and measurability transport facts. It does not prove that law integral, htraceFieldStdBasis, hlaplacianEqEmGenerator, source-field equality, source-field measurability, source-functional construction, density-Laplacian source facts, Green/trace/box-divergence leaves, diffusion construction, KL differentiation, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content." /-- Cycle-141 proof-DAG pane for the EM generator source-action split. -/ def cycle141GeneralMovingTargetDiscreteEmGeneratorStdBasisSourceDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle141.middle_packet.em_generator_source_action_std_basis" interface := "Compiled illness-area theorem: SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula derives hemGeneratorSourceActionDef from hemGeneratorStdBasisDef, the source-cited standard-basis second-derivative formula for the frozen EM generator Laplacian component, using SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorStdBasisSourceLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStdBasisSourceFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle141.middle_packet.weak_fp_source_action_em_generator_std_basis" interface := "Compiled consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStdBasisSourceFormula feeds the standard-basis EM generator source-action split into the cycle-140 source-functional route, replacing the direct hemGeneratorSourceActionDef premise while keeping hlaplacianEqEmGenerator, hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, and hsourceLaplacianFunctional explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorStdBasisSourceLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle141.lower_1_packet.em_generator_trace_field_source" interface := "Compiled lower_1 proof-scout theorem: SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField derives hemGeneratorStdBasisDef from two smaller source-cited facts: hemGeneratorTraceActionDef, identifying the frozen EM generator Laplacian component with the source functional applied to a named trace field, and htraceFieldStdBasis, identifying that trace field pointwise with the standard-basis second derivatives of selectedTest." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorTraceFieldSourceScoutObligation", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceFieldSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStdBasisSourceFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle141.lower_1_packet.weak_fp_source_action_em_generator_trace_field" interface := "Compiled lower_1 consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceFieldSourceFormula feeds the trace-field split through the cycle-141 standard-basis EM generator source-action route, replacing the direct hemGeneratorStdBasisDef premise while keeping hlaplacianEqEmGenerator, hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, and hsourceLaplacianFunctional explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorTraceFieldSourceScoutObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceFieldSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStdBasisSourceFormula", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle141.lower_2_packet.em_generator_trace_law_integral" interface := "Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula removes hemGeneratorTraceActionDef from the downstream trace-field consumer by deriving it from hemGeneratorTraceLawIntegral, the law-space trace integral against hatRhoS, and the existing hsourceLaplacianFunctional definition. The remaining analytic EM trace facts are hemGeneratorTraceLawIntegral and htraceFieldStdBasis." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorTraceLawIntegralLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceFieldSourceFormula", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle141.remaining_em_generator_trace_field_source" interface := "Remaining exact theorem boundary after cycle 141 lower_2: prove hlaplacianEqEmGenerator for the selected weak-FP Laplacian action, hemGeneratorTraceLawIntegral for the frozen EM generator Laplacian trace action as a law-space integral against hatRhoS, and htraceFieldStdBasis for the named trace field's pointwise standard-basis second-derivative formula. The direct hemGeneratorTraceActionDef leaf now follows from SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula, hemGeneratorStdBasisDef follows from SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField, and hemGeneratorSourceActionDef follows from SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula. The sibling hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, htestLaplacianStdBasisDef, density-Laplacian source facts, Green/trace/box-divergence, and diffusion leaves remain separate." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceFieldSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStdBasisSourceFormula", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle141.reviewer_em_generator_std_basis_source_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorSourceActionDef is narrowed to hemGeneratorStdBasisDef through SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula, hemGeneratorStdBasisDef is further narrowed to hemGeneratorTraceActionDef plus htraceFieldStdBasis through SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField, and hemGeneratorTraceActionDef is narrowed to hemGeneratorTraceLawIntegral plus hsourceLaplacianFunctional through SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula. Sibling EM/weak-FP source leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorStdBasisSourceLowerObligation", "SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorTraceFieldSourceScoutObligation", "SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorTraceLawIntegralLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceFieldSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := ["cycle 141 reviewer"] status := ProofStatus.obligation } ] /-- Cycle-142 proof-DAG pane for the EM generator trace state-integral split. -/ def cycle142GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle142.middle_packet.em_generator_trace_law_integral_state_transport" interface := "Compiled middle theorem: SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral derives hemGeneratorTraceLawIntegral from hemGeneratorTraceStateIntegral, hatRhoS = Measure.map hatXAtS P, AEMeasurable hatXAtS, and trace-field measurability under hatRhoS, using MeasureTheory.integral_map." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralMiddleObligation", "SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral", "MeasureTheory.integral_map", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceStateIntegralSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle142.middle_packet.weak_fp_source_action_em_generator_trace_state_integral" interface := "Compiled consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceStateIntegralSourceFormula feeds the state-integral trace split through the cycle-141 trace-law route, replacing the direct hemGeneratorTraceLawIntegral premise while keeping hlaplacianEqEmGenerator, htraceFieldStdBasis, hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, and hsourceLaplacianFunctional explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralMiddleObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceStateIntegralSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle142.lower_1_packet.trace_field_meas_from_source_laplacian" interface := "Compiled lower_1 theorem: SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldMeasOfSourceLaplacianFieldMeas derives htraceFieldMeas from hsourceLaplacianFieldMeas and htraceFieldStdBasis by identifying the named trace field with Mathlib's Laplacian through SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianStateIntegralScoutObligation", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldMeasOfSourceLaplacianFieldMeas", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle142.lower_1_packet.em_generator_trace_state_from_laplacian_state" interface := "Compiled lower_1 theorem: SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegral derives hemGeneratorTraceStateIntegral from hemGeneratorLaplacianStateIntegral and htraceFieldStdBasis by replacing the trace-field integrand with Mathlib's selected-test Laplacian." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianStateIntegralScoutObligation", "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegral", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle142.lower_1_packet.weak_fp_source_action_em_generator_trace_laplacian_state" interface := "Compiled lower_1 consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula feeds the Laplacian state-integral split through the cycle-142 trace-state route, replacing direct htraceFieldMeas and hemGeneratorTraceStateIntegral while keeping hlaplacianEqEmGenerator, htraceFieldStdBasis, hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, and hsourceLaplacianFunctional explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianStateIntegralScoutObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceStateIntegralSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldMeasOfSourceLaplacianFieldMeas", "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegral", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle142.lower_2_packet.em_generator_laplacian_law_integral" interface := "Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula derives the trace-Laplacian state-integral consumer from the law-space selected-test Laplacian generator identity hemGeneratorLaplacianLawIntegral, using SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral and MeasureTheory.integral_map to recover hemGeneratorLaplacianStateIntegral." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianLawIntegralLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula", "MeasureTheory.integral_map", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle142.remaining_em_generator_trace_state_integral" interface := "Remaining exact theorem boundary after cycle 142 lower_2: prove hlaplacianEqEmGenerator for the selected weak-FP Laplacian action, hemGeneratorLaplacianLawIntegral for the frozen EM generator selected-test Laplacian action as a law-space integral against hatRhoS, and htraceFieldStdBasis for the named trace field's pointwise standard-basis second-derivative formula. The sample-space hemGeneratorLaplacianStateIntegral now follows from the law integral by SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral; htraceFieldMeas follows from hsourceLaplacianFieldMeas; and hemGeneratorTraceStateIntegral follows from the selected-test Laplacian state integral via SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula", "SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianLawIntegralLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldMeasOfSourceLaplacianFieldMeas", "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceStateIntegralSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle142.reviewer_em_generator_trace_state_integral_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorLaplacianStateIntegral is narrowed to hemGeneratorLaplacianLawIntegral through SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula, hemGeneratorTraceStateIntegral and htraceFieldMeas remain narrowed through the lower_1 trace-Laplacian route, sibling EM/weak-FP source leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianLawIntegralLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianStateIntegralScoutObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldMeasOfSourceLaplacianFieldMeas", "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegral", "SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralMiddleObligation", "SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceStateIntegralSourceFormula", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := ["cycle 142 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 143: EM generator Laplacian law integral via state events -/ /-- Cycle-143 state-event narrowing for the frozen EM generator Laplacian law integral. -/ def cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianStateEventLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle143_em_generator_laplacian_state_event_lower" statement := "Cycle 143 middle compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula and SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateEventFormula for appendix.tex:984-995 and appendix.tex:1368-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorLaplacianLawIntegral premise under SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula is no longer primitive; it follows from a source-cited EM conditional-law/state-event formula: a named frozen-generator Laplacian event field has total state-event action equal to emGeneratorLaplacianAction, and its integrals over every measurable state event agree with the selected-test Laplacian integrals against hatRhoS. The theorem keeps hlaplacianEqEmGenerator, htraceFieldStdBasis, hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, density-Laplacian, Green/trace/box-divergence, and diffusion leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateEventFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet inside the cycle-143 EM generator trace-state illness area. It narrows only hemGeneratorLaplacianLawIntegral to a state-event law-integral boundary tied to the conditional-law source step. It does not reintroduce the cycle-140 source-functional wrapper, prove hlaplacianEqEmGenerator, prove htraceFieldStdBasis, close source-field equality or measurability, promote density-Laplacian/Green/trace/box-divergence/diffusion leaves, import SLT, change Lake/toolchain files, use sald_version_2, close a theorem by fake proof, or take a non-EM fallback." /-- Cycle-143 proof-DAG pane for the state-event law-integral split. -/ def cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianStateEventDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle143.middle_packet.em_generator_laplacian_law_integral_state_event" interface := "Compiled theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula derives hemGeneratorLaplacianLawIntegral by applying the source-cited EM state-event equality to Set.univ. The remaining analytic boundary is the named event field total-action formula plus equality of state-event integrals for every measurable state set." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianStateEventLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateEventFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle143.middle_packet.weak_fp_source_action_em_generator_trace_laplacian_state_event" interface := "Compiled consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateEventFormula feeds the state-event law-integral split through the cycle-142 trace-Laplacian law-integral route, replacing only hemGeneratorLaplacianLawIntegral while keeping sibling source leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianStateEventLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle143.remaining_em_generator_laplacian_state_event_formula" interface := "Remaining exact theorem boundary after cycle 143 middle: prove the source-cited EM conditional-law/state-event formula for the frozen-generator Laplacian event field: total state-event action equals emGeneratorLaplacianAction, and every measurable state-event integral of that event field equals the selected-test Laplacian integral against hatRhoS. Still separate: hlaplacianEqEmGenerator, htraceFieldStdBasis, hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, htestLaplacianStdBasisDef, density-Laplacian, Green/trace/box-divergence, and diffusion leaves." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle143.reviewer_em_generator_laplacian_state_event_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorLaplacianLawIntegral is narrowed to a state-event EM conditional-law formula through SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateEventFormula, cycle-140/141 source-functional wrapper churn is not reintroduced, sibling EM/weak-FP source leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianStateEventLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula", "SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianLawIntegralLowerObligation", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 143 reviewer"] status := ProofStatus.obligation } ] /-- Cycle-143 lower-1 pointwise narrowing for the frozen EM generator Laplacian state-event formula. -/ def cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianPointwiseEventLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle143_em_generator_laplacian_pointwise_event_lower_1" statement := "Cycle 143 lower_1 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise and SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventFormula for appendix.tex:984-995 and appendix.tex:1368-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorLaplacianStateEventEqLaplacian premise from the cycle-143 state-event split is no longer primitive; it follows from the pointwise source definition hemGeneratorLaplacianEventFieldEqLaplacian identifying the named frozen-generator event field with Laplacian.laplacian (selectedTest phi). The total-event action hemGeneratorLaplacianTotalEventIntegral remains explicit, as do hlaplacianEqEmGenerator, htraceFieldStdBasis, source-field/source-functional, density-Laplacian, Green/trace/box-divergence, and diffusion leaves." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 proof-scout packet inside the cycle-143 EM generator trace-state illness area. It narrows only the all-state-events equality to a pointwise event-field definition plus the existing total-event action. It does not prove the total-event action, prove hlaplacianEqEmGenerator, prove htraceFieldStdBasis, close source-field/source-functional leaves, promote density-Laplacian/Green/trace/box-divergence/diffusion leaves, import SLT, change Lake/toolchain files, use sald_version_2, close a theorem by fake proof, or take a non-EM fallback." /-- Cycle-143 lower-1 proof-DAG pane for pointwise event-field narrowing. -/ def cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianPointwiseEventDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle143.lower_1_packet.em_generator_laplacian_state_event_pointwise" interface := "Compiled theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise derives the measurable-state-event equality from the pointwise identification emGeneratorLaplacianEventField phi = fun x => Laplacian.laplacian (selectedTest phi) x." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianPointwiseEventLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateEventFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle143.lower_1_packet.weak_fp_source_action_em_generator_trace_laplacian_pointwise_event" interface := "Compiled consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventFormula feeds the pointwise event-field split through the cycle-143 state-event route, replacing only hemGeneratorLaplacianStateEventEqLaplacian while keeping sibling source leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianPointwiseEventLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle143.lower_1_remaining_em_generator_laplacian_pointwise_event_formula" interface := "Remaining exact theorem boundary after cycle 143 lower_1: prove the source-cited EM total-event action for the named frozen-generator Laplacian event field and prove the pointwise event-field identity with Laplacian.laplacian (selectedTest phi). The all-measurable-state-event equality now follows locally." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle143.lower_1_reviewer_em_generator_laplacian_pointwise_event_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorLaplacianStateEventEqLaplacian is narrowed to hemGeneratorLaplacianEventFieldEqLaplacian through SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventFormula, sibling EM/weak-FP source leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianPointwiseEventLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 143 reviewer"] status := ProofStatus.obligation } ] /-- Cycle-143 lower-2 action-definition narrowing for the frozen EM generator Laplacian total-event formula. -/ def cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianActionDefLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle143_em_generator_laplacian_action_def_lower_2" statement := "Cycle 143 lower_2 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef and SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula for appendix.tex:984-995 and appendix.tex:1368-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorLaplacianTotalEventIntegral premise from the lower_1 pointwise event split is no longer primitive; it follows from the source-facing action definition hemGeneratorLaplacianActionDef, namely emGeneratorLaplacianAction = fun phi => integral x in univ, emGeneratorLaplacianEventField phi x d hatRhoS. The pointwise event-field identity hemGeneratorLaplacianEventFieldEqLaplacian remains explicit, as do hlaplacianEqEmGenerator, htraceFieldStdBasis, source-field/source-functional, density-Laplacian, Green/trace/box-divergence, and diffusion leaves." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 worker packet inside the cycle-143 EM generator trace-state illness area. It narrows only the total-event action premise to the function-level source action definition for the named frozen EM generator Laplacian event field. It does not prove that source action definition, prove the pointwise event-field identity, prove hlaplacianEqEmGenerator, prove htraceFieldStdBasis, close source-field/source-functional leaves, promote density-Laplacian/Green/trace/box-divergence/diffusion leaves, import SLT, change Lake/toolchain files, use sald_version_2, close a theorem by fake proof, or take a non-EM fallback." /-- Cycle-143 lower-2 proof-DAG pane for the action-definition narrowing. -/ def cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianActionDefDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle143.lower_2_packet.em_generator_laplacian_total_event_action_def" interface := "Compiled theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef derives the per-test total-event formula from the function-level source action definition hemGeneratorLaplacianActionDef by applying congrFun." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianActionDefLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef", "appendix.tex:984-995", "appendix.tex:1368-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle143.lower_2_packet.weak_fp_source_action_em_generator_trace_laplacian_action_def" interface := "Compiled consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula feeds the action-definition split through the lower_1 pointwise event route, replacing only hemGeneratorLaplacianTotalEventIntegral while keeping sibling source leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianActionDefLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle143.lower_2_remaining_em_generator_laplacian_action_def_pointwise_event_formula" interface := "Remaining exact theorem boundary after cycle 143 lower_2: prove the source action definition hemGeneratorLaplacianActionDef for the named frozen-generator Laplacian event field and prove the pointwise event-field identity with Laplacian.laplacian (selectedTest phi). The older per-test total-event formula now follows locally." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle143.lower_2_reviewer_em_generator_laplacian_action_def_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorLaplacianTotalEventIntegral is narrowed to hemGeneratorLaplacianActionDef through SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula, sibling EM/weak-FP source leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianActionDefLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 143 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 144: EM generator Laplacian event-field standard-basis boundary -/ /-- Cycle-144 standard-basis event-field narrowing for the frozen EM generator Laplacian pointwise event formula. -/ def cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisEventLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle144_em_generator_laplacian_std_basis_event_lower" statement := "Cycle 144 middle compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula and SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula for appendix.tex:984-995 and appendix.tex:1368-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorLaplacianEventFieldEqLaplacian premise left by SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula is no longer primitive; it follows from the source-facing standard-basis event-field definition hemGeneratorLaplacianEventFieldStdBasisDef through SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula and SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv. hemGeneratorLaplacianActionDef, hlaplacianEqEmGenerator, htraceFieldStdBasis, source-field/source-functional leaves, density-Laplacian, Green/trace/box-divergence, and diffusion leaves remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet inside the cycle-142/143 EM generator trace-state illness area. It rejects wrapper churn around hemGeneratorLaplacianEventFieldEqLaplacian by replacing the direct Mathlib-Laplacian field equality with the source standard-basis event-field definition used by the frozen EM generator. It does not prove hemGeneratorLaplacianActionDef, hlaplacianEqEmGenerator, htraceFieldStdBasis, source-field/source-functional leaves, density-Laplacian, Green/trace/box-divergence, diffusion leaves, SLT imports, Lake/toolchain changes, theorem-status promotion, fake closure, non-EM fallback, broad audits, or sald_version_2 content." /-- Cycle-144 proof-DAG pane for the standard-basis frozen EM event-field narrowing. -/ def cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisEventDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle144.middle_packet.em_generator_laplacian_event_field_std_basis" interface := "Compiled theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula derives hemGeneratorLaplacianEventFieldEqLaplacian from the source-facing standard-basis definition hemGeneratorLaplacianEventFieldStdBasisDef by reusing SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisEventLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle144.middle_packet.weak_fp_source_action_em_generator_trace_laplacian_std_basis_event" interface := "Compiled consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula feeds the standard-basis event-field split through the cycle-143 action-definition route, replacing only hemGeneratorLaplacianEventFieldEqLaplacian while keeping hemGeneratorLaplacianActionDef and sibling source leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisEventLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle144.remaining_em_generator_laplacian_std_basis_event_and_action_def" interface := "Remaining exact theorem boundary after cycle 144: prove the source action definition hemGeneratorLaplacianActionDef and the source standard-basis event-field definition hemGeneratorLaplacianEventFieldStdBasisDef for the named frozen EM Laplacian event field. The older pointwise equality with Mathlib Laplacian now follows locally." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle144.reviewer_em_generator_laplacian_std_basis_event_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorLaplacianEventFieldEqLaplacian is narrowed to hemGeneratorLaplacianEventFieldStdBasisDef through SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula, hemGeneratorLaplacianActionDef remains explicit, sibling EM/weak-FP source leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisEventLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula", "SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianActionDefLowerObligation", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 144 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 144 lower_2: EM generator standard-basis action boundary -/ /-- Cycle-144 lower-2 standard-basis action narrowing for the frozen EM generator Laplacian action definition. -/ def cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle144_em_generator_laplacian_std_basis_action_lower_2" statement := "Cycle 144 lower_2 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionFormula and SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorLaplacianActionDef premise left by SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula is no longer primitive; it follows from the source-facing standard-basis action integral hemGeneratorLaplacianStdBasisActionDef together with the already exposed standard-basis event-field definition hemGeneratorLaplacianEventFieldStdBasisDef. The bridge rewrites only the event-field integrand inside the Set.univ integral. hlaplacianEqEmGenerator, htraceFieldStdBasis, source-field/source-functional leaves, density-Laplacian, Green/trace/box-divergence, diffusion leaves, and the two standard-basis EM source definitions remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 worker packet inside the cycle-142/143/144 EM generator trace-state illness area. It rejects wrapper churn by replacing the broad action definition with the smaller source standard-basis action integral plus the named event-field standard-basis definition. It does not prove either standard-basis source definition, prove hlaplacianEqEmGenerator, prove htraceFieldStdBasis, close source-field/source-functional leaves, promote density-Laplacian/Green/trace/box-divergence/diffusion leaves, import SLT, change Lake/toolchain files, use sald_version_2, close a theorem by fake proof, or take a non-EM fallback." /-- Cycle-144 lower-2 proof-DAG pane for the standard-basis action narrowing. -/ def cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle144.lower_2_packet.em_generator_laplacian_action_std_basis_action" interface := "Compiled theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionFormula derives hemGeneratorLaplacianActionDef from the source-facing standard-basis action integral hemGeneratorLaplacianStdBasisActionDef and the event-field standard-basis definition hemGeneratorLaplacianEventFieldStdBasisDef by rewriting the integrand under the Set.univ state integral." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle144.lower_2_packet.weak_fp_source_action_em_generator_trace_laplacian_std_basis_action" interface := "Compiled consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula feeds the standard-basis action split through the cycle-144 standard-basis event consumer, replacing only hemGeneratorLaplacianActionDef while keeping hemGeneratorLaplacianEventFieldStdBasisDef and sibling source leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle144.lower_2_remaining_em_generator_laplacian_std_basis_action_and_event_defs" interface := "Remaining exact theorem boundary after cycle 144 lower_2: prove the source standard-basis action integral hemGeneratorLaplacianStdBasisActionDef and the source standard-basis event-field definition hemGeneratorLaplacianEventFieldStdBasisDef for the named frozen EM Laplacian event field. The older action definition and pointwise Mathlib-Laplacian equality now follow locally." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle144.lower_2_reviewer_em_generator_laplacian_std_basis_action_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorLaplacianActionDef is narrowed to hemGeneratorLaplacianStdBasisActionDef plus hemGeneratorLaplacianEventFieldStdBasisDef through SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula, sibling EM/weak-FP source leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionFormula", "SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisEventLowerObligation", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 144 lower_2 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 145: EM generator law-integral to standard-basis action boundary -/ /-- Cycle-145 law-integral narrowing for the frozen EM generator standard-basis Laplacian action definition. -/ def cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisLawLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle145_em_generator_laplacian_std_basis_law_lower" statement := "Cycle 145 middle compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula and SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventLawIntegralFormula for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorLaplacianStdBasisActionDef premise left by SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula is no longer primitive; it follows from hemGeneratorLaplacianLawIntegral by rewriting Mathlib's Laplacian into the source standard-basis second-derivative formula through SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv. hemGeneratorLaplacianEventFieldStdBasisDef, hlaplacianEqEmGenerator, htraceFieldStdBasis, source-field/source-functional leaves, density-Laplacian, Green/trace/box-divergence, and diffusion leaves remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet inside the cycle-142/143/144 EM generator trace-state illness area. It rejects wrapper churn by replacing the standard-basis action integral with the smaller law-space EM Laplacian action integral plus the local Mathlib standard-basis rewrite. It does not prove hemGeneratorLaplacianLawIntegral, hemGeneratorLaplacianEventFieldStdBasisDef, hlaplacianEqEmGenerator, htraceFieldStdBasis, source-field/source-functional leaves, density-Laplacian, Green/trace/box-divergence, diffusion leaves, SLT imports, Lake/toolchain changes, theorem-status promotion, fake closure, non-EM fallback, broad audits, or sald_version_2 content." /-- Cycle-145 proof-DAG pane for the law-integral to standard-basis action narrowing. -/ def cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisLawDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle145.middle_packet.em_generator_laplacian_std_basis_action_from_law_integral" interface := "Compiled theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula derives hemGeneratorLaplacianStdBasisActionDef from hemGeneratorLaplacianLawIntegral by rewriting the law-space Mathlib Laplacian integral to the Set.univ standard-basis source action." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisLawLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle145.middle_packet.weak_fp_source_action_em_generator_trace_laplacian_law_integral" interface := "Compiled consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventLawIntegralFormula feeds the law-integral split through the cycle-144 standard-basis action consumer, replacing only hemGeneratorLaplacianStdBasisActionDef while keeping hemGeneratorLaplacianEventFieldStdBasisDef and sibling source leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisLawLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle145.remaining_em_generator_law_integral_and_std_basis_event_defs" interface := "Remaining exact theorem boundary after cycle 145: prove the source law-space EM generator Laplacian action integral hemGeneratorLaplacianLawIntegral and the source standard-basis event-field definition hemGeneratorLaplacianEventFieldStdBasisDef for the named frozen EM Laplacian event field. The old standard-basis action integral now follows locally." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle145.reviewer_em_generator_laplacian_law_integral_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorLaplacianStdBasisActionDef is narrowed to hemGeneratorLaplacianLawIntegral plus the local Mathlib standard-basis Laplacian rewrite, hemGeneratorLaplacianEventFieldStdBasisDef remains explicit, sibling EM/weak-FP source leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisLawLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula", "SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionLowerObligation", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 145 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 145 lower_2: EM generator trace-field event boundary -/ /-- Cycle-145 lower_2 narrowing for the frozen EM generator Laplacian event-field standard-basis definition. -/ def cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianTraceEventLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle145_em_generator_laplacian_trace_event_lower" statement := "Cycle 145 lower_2 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField and SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventLawIntegralFormula for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorLaplacianEventFieldStdBasisDef premise left by SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventLawIntegralFormula is no longer primitive; it follows from the source-facing equality hemGeneratorLaplacianEventFieldEqTraceField identifying the named frozen-generator Laplacian event field with the already tracked trace field, together with htraceFieldStdBasis. hemGeneratorLaplacianLawIntegral remains explicit, as do hlaplacianEqEmGenerator, source-field/source-functional leaves, density-Laplacian, Green/trace/box-divergence, and diffusion leaves." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 worker packet inside the cycle-142/143/144/145 EM generator trace-state illness area. It rejects wrapper churn by replacing the event-field standard-basis definition with the smaller source equality between the named Laplacian event field and the trace field plus the already explicit htraceFieldStdBasis. It does not prove hemGeneratorLaplacianLawIntegral, prove htraceFieldStdBasis, prove hemGeneratorLaplacianEventFieldEqTraceField, close source-field/source-functional leaves, promote density-Laplacian/Green/trace/box-divergence/diffusion leaves, import SLT, change Lake/toolchain files, use sald_version_2, close a theorem by fake proof, or take a non-EM fallback." /-- Cycle-145 lower_2 proof-DAG pane for the trace-field event narrowing. -/ def cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianTraceEventDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle145.lower_2_packet.em_generator_laplacian_event_field_std_basis_from_trace_field" interface := "Compiled theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField derives hemGeneratorLaplacianEventFieldStdBasisDef from hemGeneratorLaplacianEventFieldEqTraceField and htraceFieldStdBasis by transitivity of function equality." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianTraceEventLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventLawIntegralFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle145.lower_2_packet.weak_fp_source_action_em_generator_trace_event_law_integral" interface := "Compiled consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventLawIntegralFormula feeds the trace-field event split through the cycle-145 law-integral route, replacing only hemGeneratorLaplacianEventFieldStdBasisDef while keeping hemGeneratorLaplacianLawIntegral and sibling source leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianTraceEventLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventLawIntegralFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle145.lower_2_remaining_em_generator_law_integral_and_trace_event_equality" interface := "Remaining exact theorem boundary after cycle 145 lower_2: prove hemGeneratorLaplacianLawIntegral and the source equality hemGeneratorLaplacianEventFieldEqTraceField identifying the named frozen EM Laplacian event field with the already tracked trace field. The old event-field standard-basis definition now follows locally from that equality and htraceFieldStdBasis." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle145.lower_2_reviewer_trace_event_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorLaplacianEventFieldStdBasisDef is narrowed to hemGeneratorLaplacianEventFieldEqTraceField plus htraceFieldStdBasis through SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventLawIntegralFormula, hemGeneratorLaplacianLawIntegral remains explicit, sibling EM/weak-FP source leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianTraceEventLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField", "SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisLawLowerObligation", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 145 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 146: EM generator trace-event total-event boundary -/ /-- Cycle-146 narrowing for the law-space generator Laplacian integral under the latest trace-event route. -/ def cycle146GeneralMovingTargetDiscreteEmGeneratorTraceEventTotalEventLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle146_em_generator_trace_event_total_event_lower" statement := "Cycle 146 middle compiles SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventFormula for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorLaplacianLawIntegral premise left by SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventLawIntegralFormula is no longer primitive; it follows from hemGeneratorLaplacianTotalEventIntegral, the total state-event formula for the named frozen-generator Laplacian event field, plus hemGeneratorLaplacianEventFieldEqTraceField and htraceFieldStdBasis. The compiled route derives the measurable-state-event Laplacian equality through SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField, SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula, and SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise, then applies SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula. hlaplacianEqEmGenerator, hemGeneratorLaplacianTotalEventIntegral, hemGeneratorLaplacianEventFieldEqTraceField, htraceFieldStdBasis, source-field/source-functional leaves, density-Laplacian, Green/trace/box-divergence, and diffusion leaves remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet inside the cycle-142/143/144/145 EM generator trace-state illness area. It rejects wrapper churn by removing only the direct law-space EM generator Laplacian integral from the current trace-event consumer and replacing it with the smaller total-state-event formula plus the already exposed trace-event equality/trace standard-basis route. It does not prove hemGeneratorLaplacianTotalEventIntegral, hemGeneratorLaplacianEventFieldEqTraceField, htraceFieldStdBasis, hlaplacianEqEmGenerator, source-field/source-functional leaves, density-Laplacian, Green/trace/box-divergence, diffusion leaves, SLT imports, Lake/toolchain changes, theorem-status promotion, fake closure, non-EM fallback, broad audits, or sald_version_2 content." /-- Cycle-146 proof-DAG pane for the trace-event total-event narrowing. -/ def cycle146GeneralMovingTargetDiscreteEmGeneratorTraceEventTotalEventDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle146.middle_packet.em_generator_trace_event_total_event" interface := "Compiled theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventFormula feeds the total-state-event formula through the current trace-event law-integral route. It derives hemGeneratorLaplacianLawIntegral from hemGeneratorLaplacianTotalEventIntegral, hemGeneratorLaplacianEventFieldEqTraceField, htraceFieldStdBasis, and the local Mathlib standard-basis Laplacian bridge." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorTraceEventTotalEventLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle146.remaining_em_generator_total_event_and_trace_event" interface := "Remaining exact theorem boundary after cycle 146: prove hemGeneratorLaplacianTotalEventIntegral for the named frozen EM Laplacian event field, prove hemGeneratorLaplacianEventFieldEqTraceField identifying that event field with the trace field, and prove htraceFieldStdBasis. The older hemGeneratorLaplacianLawIntegral now follows locally from the total-state-event formula plus the trace-event standard-basis route." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle146.reviewer_trace_event_total_event_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorLaplacianLawIntegral is narrowed to hemGeneratorLaplacianTotalEventIntegral plus hemGeneratorLaplacianEventFieldEqTraceField and htraceFieldStdBasis through SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventFormula, sibling EM/weak-FP source leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorTraceEventTotalEventLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventFormula", "SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianTraceEventLowerObligation", "SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianStateEventLowerObligation", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 146 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 146 lower_1: EM trace-field Laplacian boundary -/ /-- Cycle-146 lower_1 narrowing for the trace-field standard-basis formula inside the latest trace-event total-event route. -/ def cycle146GeneralMovingTargetDiscreteEmGeneratorTraceFieldLaplacianLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle146_em_generator_trace_field_laplacian_lower_1" statement := "Cycle 146 lower_1 compiles SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField and SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventTraceLaplacianFormula for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the htraceFieldStdBasis premise left by SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventFormula is no longer primitive; it follows from the smaller source-facing htraceFieldEqLaplacian identification of the named trace field with Mathlib's selected-test Laplacian plus SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv. hemGeneratorLaplacianTotalEventIntegral, hemGeneratorLaplacianEventFieldEqTraceField, hlaplacianEqEmGenerator, source-field/source-functional leaves, density-Laplacian, Green/trace/box-divergence, and diffusion leaves remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 proof-scout packet inside the cycle-142/143/144/145/146 EM generator trace-state illness area. It rejects wrapper churn by narrowing only the trace-field standard-basis source formula to a smaller field-equals-Laplacian source boundary and the existing local Mathlib standard-basis bridge. It does not prove hemGeneratorLaplacianTotalEventIntegral, prove hemGeneratorLaplacianEventFieldEqTraceField, prove htraceFieldEqLaplacian, prove hlaplacianEqEmGenerator, close source-field/source-functional leaves, promote density-Laplacian/Green/trace/box-divergence/diffusion leaves, import SLT, change Lake/toolchain files, use sald_version_2, close a theorem by fake proof, or take a non-EM fallback." /-- Cycle-146 lower_1 proof-DAG pane for the trace-field Laplacian narrowing. -/ def cycle146GeneralMovingTargetDiscreteEmGeneratorTraceFieldLaplacianDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle146.lower_1_packet.trace_field_std_basis_from_laplacian" interface := "Compiled theorem: SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField derives htraceFieldStdBasis from htraceFieldEqLaplacian by applying the local Mathlib standard-basis Laplacian bridge to each selected test." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorTraceFieldLaplacianLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventTraceLaplacianFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle146.lower_1_packet.weak_fp_source_action_em_generator_trace_laplacian_total_event_trace_laplacian" interface := "Compiled consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventTraceLaplacianFormula feeds the trace-field Laplacian split through the current trace-event total-event route, replacing only htraceFieldStdBasis while keeping hemGeneratorLaplacianTotalEventIntegral and hemGeneratorLaplacianEventFieldEqTraceField explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorTraceFieldLaplacianLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle146.lower_1_remaining_total_event_trace_event_and_trace_laplacian" interface := "Remaining exact theorem boundary after cycle 146 lower_1: prove hemGeneratorLaplacianTotalEventIntegral, prove hemGeneratorLaplacianEventFieldEqTraceField, and prove htraceFieldEqLaplacian. The older htraceFieldStdBasis now follows locally from htraceFieldEqLaplacian plus the Mathlib standard-basis Laplacian theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle146.lower_1_reviewer_trace_field_laplacian_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, htraceFieldStdBasis is narrowed to htraceFieldEqLaplacian through SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventTraceLaplacianFormula, sibling EM/weak-FP source leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorTraceFieldLaplacianLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorTraceEventTotalEventLowerObligation", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 146 lower_1 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 146 lower_2: EM pointwise trace-event boundary -/ /-- Cycle-146 lower_2 narrowing for the event-field/trace-field equality inside the latest trace-event total-event route. -/ def cycle146GeneralMovingTargetDiscreteEmGeneratorPointwiseTraceEventLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle146_em_generator_pointwise_trace_event_lower_2" statement := "Cycle 146 lower_2 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields and SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorLaplacianEventFieldEqTraceField premise left by SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventTraceLaplacianFormula is no longer primitive; it follows from the smaller pointwise event-field Laplacian identity hemGeneratorLaplacianEventFieldEqLaplacian plus htraceFieldEqLaplacian. hemGeneratorLaplacianTotalEventIntegral, hemGeneratorLaplacianEventFieldEqLaplacian, htraceFieldEqLaplacian, hlaplacianEqEmGenerator, source-field/source-functional leaves, density-Laplacian, Green/trace/box-divergence, and diffusion leaves remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 worker packet inside the cycle-142/143/144/145/146 EM generator trace-state illness area. It rejects wrapper churn by replacing only the trace-event equality with two source-cited Laplacian identities for the same named event and trace fields. It does not prove hemGeneratorLaplacianTotalEventIntegral, prove hemGeneratorLaplacianEventFieldEqLaplacian, prove htraceFieldEqLaplacian, prove hlaplacianEqEmGenerator, close source-field/source-functional leaves, promote density-Laplacian/Green/trace/box-divergence/diffusion leaves, import SLT, change Lake/toolchain files, use sald_version_2, close a theorem by fake proof, or take a non-EM fallback." /-- Cycle-146 lower_2 proof-DAG pane for the pointwise trace-event narrowing. -/ def cycle146GeneralMovingTargetDiscreteEmGeneratorPointwiseTraceEventDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle146.lower_2_packet.em_generator_laplacian_event_field_trace_from_laplacian_fields" interface := "Compiled theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields derives hemGeneratorLaplacianEventFieldEqTraceField from hemGeneratorLaplacianEventFieldEqLaplacian and htraceFieldEqLaplacian by transitivity through Mathlib's selected-test Laplacian." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorPointwiseTraceEventLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle146.lower_2_packet.weak_fp_source_action_em_generator_pointwise_event_total_event_trace_laplacian" interface := "Compiled consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula feeds the pointwise event-field Laplacian split through the lower_1 trace-field Laplacian route, replacing only hemGeneratorLaplacianEventFieldEqTraceField while keeping hemGeneratorLaplacianTotalEventIntegral and htraceFieldEqLaplacian explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorPointwiseTraceEventLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventTraceLaplacianFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle146.lower_2_remaining_total_event_pointwise_event_and_trace_laplacian" interface := "Remaining exact theorem boundary after cycle 146 lower_2: prove hemGeneratorLaplacianTotalEventIntegral, prove hemGeneratorLaplacianEventFieldEqLaplacian, and prove htraceFieldEqLaplacian. The older hemGeneratorLaplacianEventFieldEqTraceField now follows locally from the two Laplacian identities." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle146.lower_2_reviewer_pointwise_trace_event_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorLaplacianEventFieldEqTraceField is narrowed to hemGeneratorLaplacianEventFieldEqLaplacian plus htraceFieldEqLaplacian through SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula, sibling EM/weak-FP source leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorPointwiseTraceEventLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields", "SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorTraceFieldLaplacianLowerObligation", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 146 lower_2 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 147: EM pointwise action-definition trace-Laplacian boundary -/ /-- Cycle-147 narrowing for the total-event formula inside the current pointwise trace-event route. -/ def cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseActionDefTraceLaplacianLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle147_em_generator_pointwise_action_def_trace_laplacian" statement := "Cycle 147 middle compiles SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefTraceLaplacianFormula for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorLaplacianTotalEventIntegral premise left by SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula is no longer primitive; it follows from the smaller source-facing action definition hemGeneratorLaplacianActionDef through SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef. hemGeneratorLaplacianActionDef, hemGeneratorLaplacianEventFieldEqLaplacian, htraceFieldEqLaplacian, hlaplacianEqEmGenerator, source-field/source-functional leaves, density-Laplacian, Green/trace/box-divergence, and diffusion leaves remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet inside the cycle-142/143/144/145/146/147 EM generator trace-state illness area. It rejects wrapper churn by replacing only the total-event formula with the source-facing action definition while keeping the pointwise event-field/Laplacian and trace-field/Laplacian identities explicit. It does not prove hemGeneratorLaplacianActionDef, prove hemGeneratorLaplacianEventFieldEqLaplacian, prove htraceFieldEqLaplacian, prove hlaplacianEqEmGenerator, close source-field/source-functional leaves, promote density-Laplacian/Green/trace/box-divergence/diffusion leaves, import SLT, change Lake/toolchain files, use sald_version_2, close a theorem by fake proof, or take a non-EM fallback." /-- Cycle-147 proof-DAG pane for the pointwise action-definition trace-Laplacian narrowing. -/ def cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseActionDefTraceLaplacianDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle147.middle_packet.em_generator_laplacian_total_event_from_action_def_trace_laplacian" interface := "Compiled helper reuse: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef derives hemGeneratorLaplacianTotalEventIntegral from hemGeneratorLaplacianActionDef by applying the function-level action definition to each selected test." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseActionDefTraceLaplacianLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef", "appendix.tex:984-995", "appendix.tex:1368-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefTraceLaplacianFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle147.middle_packet.weak_fp_source_action_em_generator_pointwise_event_action_def_trace_laplacian" interface := "Compiled consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefTraceLaplacianFormula feeds the action-definition split through the current pointwise trace-event total-event route, replacing only hemGeneratorLaplacianTotalEventIntegral while keeping hemGeneratorLaplacianEventFieldEqLaplacian and htraceFieldEqLaplacian explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseActionDefTraceLaplacianLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle147.remaining_action_def_pointwise_event_and_trace_laplacian" interface := "Remaining exact theorem boundary after cycle 147: prove hemGeneratorLaplacianActionDef, prove hemGeneratorLaplacianEventFieldEqLaplacian, and prove htraceFieldEqLaplacian for the named frozen EM Laplacian contribution. The older hemGeneratorLaplacianTotalEventIntegral now follows locally from hemGeneratorLaplacianActionDef." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle147.reviewer_pointwise_action_def_trace_laplacian_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorLaplacianTotalEventIntegral is narrowed to hemGeneratorLaplacianActionDef through SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefTraceLaplacianFormula, sibling EM/weak-FP source leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseActionDefTraceLaplacianLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 147 reviewer"] status := ProofStatus.obligation } ] /-- Cycle-147 lower-1 proof-scout narrowing for the action-definition premise inside the current pointwise event/trace-Laplacian route. -/ def cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseStdBasisActionTraceLaplacianLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle147_em_generator_pointwise_stdbasis_action_trace_laplacian" statement := "Cycle 147 lower_1 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionPointwiseEventFormula and SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStdBasisActionTraceLaplacianFormula for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorLaplacianActionDef premise left by SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefTraceLaplacianFormula is no longer primitive; it follows from the source standard-basis action integral hemGeneratorLaplacianStdBasisActionDef together with the already explicit pointwise event-field Laplacian identity hemGeneratorLaplacianEventFieldEqLaplacian and SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv. hemGeneratorLaplacianStdBasisActionDef, hemGeneratorLaplacianEventFieldEqLaplacian, htraceFieldEqLaplacian, hlaplacianEqEmGenerator, source-field/source-functional leaves, density-Laplacian, Green/trace/box-divergence, and diffusion leaves remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionPointwiseEventFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStdBasisActionTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 proof-scout packet inside the cycle-142/143/144/145/146/147 EM generator trace-state illness area. It rejects wrapper churn by replacing only hemGeneratorLaplacianActionDef with the source standard-basis action formula, using the existing event-field/Laplacian premise and local Mathlib Laplacian standard-basis bridge. It does not prove hemGeneratorLaplacianStdBasisActionDef, prove hemGeneratorLaplacianEventFieldEqLaplacian, prove htraceFieldEqLaplacian, prove hlaplacianEqEmGenerator, close source-field/source-functional leaves, promote density-Laplacian/Green/trace/box-divergence/diffusion leaves, import SLT, change Lake/toolchain files, use sald_version_2, close a theorem by fake proof, or take a non-EM fallback." /-- Cycle-147 lower-1 proof-DAG pane for the standard-basis action narrowing inside the pointwise event/trace-Laplacian route. -/ def cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseStdBasisActionTraceLaplacianDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle147.lower_1_packet.em_generator_laplacian_action_def_from_stdbasis_action_pointwise_event" interface := "Compiled helper: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionPointwiseEventFormula derives hemGeneratorLaplacianActionDef from hemGeneratorLaplacianStdBasisActionDef plus the already explicit hemGeneratorLaplacianEventFieldEqLaplacian, rewriting the standard-basis integrand through SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv and then through the event-field identity." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseStdBasisActionTraceLaplacianLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionPointwiseEventFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStdBasisActionTraceLaplacianFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle147.lower_1_packet.weak_fp_source_action_em_generator_pointwise_event_stdbasis_action_trace_laplacian" interface := "Compiled consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStdBasisActionTraceLaplacianFormula feeds the standard-basis action split through the current pointwise event/trace Laplacian route, replacing only hemGeneratorLaplacianActionDef while keeping hemGeneratorLaplacianEventFieldEqLaplacian and htraceFieldEqLaplacian explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseStdBasisActionTraceLaplacianLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStdBasisActionTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionPointwiseEventFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefTraceLaplacianFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle147.lower_1_remaining_stdbasis_action_pointwise_event_and_trace_laplacian" interface := "Remaining exact theorem boundary after cycle 147 lower_1: prove hemGeneratorLaplacianStdBasisActionDef, prove hemGeneratorLaplacianEventFieldEqLaplacian, and prove htraceFieldEqLaplacian for the named frozen EM Laplacian contribution. The older hemGeneratorLaplacianActionDef now follows locally from the standard-basis action formula plus the event-field/Laplacian identity." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStdBasisActionTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionPointwiseEventFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle147.lower_1_reviewer_stdbasis_action_trace_laplacian_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorLaplacianActionDef is narrowed to hemGeneratorLaplacianStdBasisActionDef plus the already explicit hemGeneratorLaplacianEventFieldEqLaplacian through SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStdBasisActionTraceLaplacianFormula, sibling EM/weak-FP source leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseStdBasisActionTraceLaplacianLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStdBasisActionTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefTraceLaplacianFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 147 lower_1 reviewer"] status := ProofStatus.obligation } ] /-- Cycle-147 lower-2 narrowing for the standard-basis action premise inside the current pointwise event/trace-Laplacian route. -/ def cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseLawIntegralTraceLaplacianLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle147_em_generator_pointwise_law_integral_trace_laplacian" statement := "Cycle 147 lower_2 compiles SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventLawIntegralTraceLaplacianFormula for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorLaplacianStdBasisActionDef premise left by SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStdBasisActionTraceLaplacianFormula is no longer primitive; it follows from the smaller law-space selected-test Laplacian integral hemGeneratorLaplacianLawIntegral through SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula and SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv. hemGeneratorLaplacianLawIntegral, hemGeneratorLaplacianEventFieldEqLaplacian, htraceFieldEqLaplacian, hlaplacianEqEmGenerator, source-field/source-functional leaves, density-Laplacian, Green/trace/box-divergence, and diffusion leaves remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventLawIntegralTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStdBasisActionTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 worker packet inside the cycle-142/143/144/145/146/147 EM generator trace-state illness area. It rejects wrapper churn by replacing only hemGeneratorLaplacianStdBasisActionDef with the law-space selected-test Laplacian integral already tracked by the EM generator route, while keeping the pointwise event-field/Laplacian and trace-field/Laplacian identities explicit. It does not prove hemGeneratorLaplacianLawIntegral, prove hemGeneratorLaplacianEventFieldEqLaplacian, prove htraceFieldEqLaplacian, prove hlaplacianEqEmGenerator, close source-field/source-functional leaves, promote density-Laplacian/Green/trace/box-divergence/diffusion leaves, import SLT, change Lake/toolchain files, use sald_version_2, close a theorem by fake proof, or take a non-EM fallback." /-- Cycle-147 lower-2 proof-DAG pane for the law-integral narrowing inside the pointwise event/trace-Laplacian route. -/ def cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseLawIntegralTraceLaplacianDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle147.lower_2_packet.em_generator_laplacian_stdbasis_action_from_law_integral_pointwise_event" interface := "Compiled helper reuse: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula derives hemGeneratorLaplacianStdBasisActionDef from hemGeneratorLaplacianLawIntegral by rewriting Mathlib's selected-test Laplacian into the source standard-basis second-derivative display." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseLawIntegralTraceLaplacianLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventLawIntegralTraceLaplacianFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle147.lower_2_packet.weak_fp_source_action_em_generator_pointwise_event_law_integral_trace_laplacian" interface := "Compiled consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventLawIntegralTraceLaplacianFormula feeds the law-integral split through the current pointwise event/trace Laplacian route, replacing only hemGeneratorLaplacianStdBasisActionDef while keeping hemGeneratorLaplacianEventFieldEqLaplacian and htraceFieldEqLaplacian explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseLawIntegralTraceLaplacianLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventLawIntegralTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStdBasisActionTraceLaplacianFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle147.lower_2_remaining_law_integral_pointwise_event_and_trace_laplacian" interface := "Remaining exact theorem boundary after cycle 147 lower_2: prove hemGeneratorLaplacianLawIntegral, prove hemGeneratorLaplacianEventFieldEqLaplacian, and prove htraceFieldEqLaplacian for the named frozen EM Laplacian contribution. The older hemGeneratorLaplacianStdBasisActionDef now follows locally from the law integral and the standard-basis Laplacian bridge." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventLawIntegralTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle147.lower_2_reviewer_law_integral_trace_laplacian_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorLaplacianStdBasisActionDef is narrowed to hemGeneratorLaplacianLawIntegral through SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventLawIntegralTraceLaplacianFormula, sibling EM/weak-FP source leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseLawIntegralTraceLaplacianLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventLawIntegralTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStdBasisActionTraceLaplacianFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 147 lower_2 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 148: EM pointwise state-event trace-Laplacian boundary -/ /-- Cycle-148 middle narrowing for the law-integral premise inside the current pointwise event/trace-Laplacian route. -/ def cycle148GeneralMovingTargetDiscreteEmGeneratorPointwiseStateEventTraceLaplacianMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle148_em_generator_pointwise_state_event_trace_laplacian" statement := "Cycle 148 middle compiles SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorLaplacianLawIntegral premise left by SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventLawIntegralTraceLaplacianFormula is no longer primitive; it follows from the source-cited EM conditional-law/state-event formula through SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula. The pointwise event-field Laplacian identity hemGeneratorLaplacianEventFieldEqLaplacian and trace-field Laplacian identity htraceFieldEqLaplacian remain explicit, as do hlaplacianEqEmGenerator, source-field/source-functional leaves, density-Laplacian, Green/trace/box-divergence, and diffusion leaves." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventLawIntegralTraceLaplacianFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf middle packet inside the cycle-142/143/144/145/146/147/148 EM generator trace-state illness area. It rejects wrapper churn by replacing only hemGeneratorLaplacianLawIntegral in the current pointwise event/trace route with the source-cited EM conditional-law/state-event formula. It does not prove hemGeneratorLaplacianTotalEventIntegral, prove hemGeneratorLaplacianStateEventEqLaplacian, prove hemGeneratorLaplacianEventFieldEqLaplacian, prove htraceFieldEqLaplacian, prove hlaplacianEqEmGenerator, close source-field/source-functional leaves, promote density-Laplacian/Green/trace/box-divergence/diffusion leaves, import SLT, change Lake/toolchain files, use sald_version_2, close a theorem by fake proof, or take a non-EM fallback." /-- Cycle-148 proof-DAG pane for the state-event narrowing inside the pointwise event/trace-Laplacian route. -/ def cycle148GeneralMovingTargetDiscreteEmGeneratorPointwiseStateEventTraceLaplacianDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle148.middle_packet.em_generator_laplacian_law_integral_from_state_event_current_pointwise" interface := "Compiled helper reuse: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula derives hemGeneratorLaplacianLawIntegral by applying the source-cited EM state-event equality to Set.univ." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle148GeneralMovingTargetDiscreteEmGeneratorPointwiseStateEventTraceLaplacianMiddleObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle148.middle_packet.weak_fp_source_action_em_generator_pointwise_event_state_event_trace_laplacian" interface := "Compiled consumer theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula feeds the state-event split through the current pointwise event/trace Laplacian route, replacing only hemGeneratorLaplacianLawIntegral while keeping hemGeneratorLaplacianEventFieldEqLaplacian and htraceFieldEqLaplacian explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle148GeneralMovingTargetDiscreteEmGeneratorPointwiseStateEventTraceLaplacianMiddleObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventLawIntegralTraceLaplacianFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle148.remaining_state_event_pointwise_event_and_trace_laplacian" interface := "Remaining exact theorem boundary after cycle 148 middle: prove the source-cited EM conditional-law/state-event formula for the named frozen EM Laplacian contribution, namely hemGeneratorLaplacianTotalEventIntegral and hemGeneratorLaplacianStateEventEqLaplacian, and still prove hemGeneratorLaplacianEventFieldEqLaplacian and htraceFieldEqLaplacian. The older hemGeneratorLaplacianLawIntegral now follows locally from the state-event formula." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle148.reviewer_pointwise_state_event_trace_laplacian_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorLaplacianLawIntegral is narrowed to the source-cited EM conditional-law/state-event formula through SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula, sibling EM/weak-FP source leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle148GeneralMovingTargetDiscreteEmGeneratorPointwiseStateEventTraceLaplacianMiddleObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventLawIntegralTraceLaplacianFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 148 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 148 lower_1: state-event pointwise scout -/ /-- Cycle-148 lower_1 scout for the state-event equality left by the current state-event trace-Laplacian route. -/ def cycle148GeneralMovingTargetDiscreteEmGeneratorStateEventPointwiseScoutObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle148_lower_1_em_generator_state_event_pointwise_scout" statement := "Cycle 148 lower_1 narrows the hemGeneratorLaplacianStateEventEqLaplacian boundary left by SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula. The state-event set-integral equality is not an independent lower_2 analytic target in the current branch: it follows from the already explicit source-cited pointwise identity hemGeneratorLaplacianEventFieldEqLaplacian via compiled SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise, by rewriting the integrand under every measurable state-event integral. The packet therefore rejects a new wrapper that merely restates SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula; lower_2 should spend implementation effort on the remaining genuine EM source leaves hemGeneratorLaplacianTotalEventIntegral, hemGeneratorLaplacianEventFieldEqLaplacian, and htraceFieldEqLaplacian." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 proof-scout packet inside the cycle-142/143/144/145/146/147/148 EM generator trace-state illness area. It uses only local SALD declarations already compiled from the Mathlib set-integral rewrite shape; no SLT import or port claim, sald_version_2 use, non-EM fallback, broad route audit, theorem-status/Lake/toolchain change, fake closure, or new wrapper theorem is introduced." /-- Cycle-148 lower_1 proof-DAG pane for reducing the state-event equality to the pointwise event-field identity. -/ def cycle148GeneralMovingTargetDiscreteEmGeneratorStateEventPointwiseScoutDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle148.lower_1_packet.state_event_eq_from_pointwise" interface := "Proof route: derive hemGeneratorLaplacianStateEventEqLaplacian from hemGeneratorLaplacianEventFieldEqLaplacian by applying SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise. For htests, phi, and measurable state event t, rewrite emGeneratorLaplacianEventField phi to fun x => Laplacian.laplacian (selectedTest phi) x inside the set integral." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle148GeneralMovingTargetDiscreteEmGeneratorStateEventPointwiseScoutObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise", "hemGeneratorLaplacianEventFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle148.lower_1_reject_duplicate_state_event_wrapper" interface := "Do not add a new theorem whose signature is just the cycle-148 state-event consumer with hemGeneratorLaplacianStateEventEqLaplacian replaced by hemGeneratorLaplacianEventFieldEqLaplacian; that duplicates the already compiled pointwise total-event trace-Laplacian consumer. The useful next implementation block is one genuine source leaf, not another consumer wrapper." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 148 lower_2 handoff"] status := ProofStatus.obligation } ] /-! ### Cycle 148 lower_2: total-event source-functional leaf -/ /-- Cycle-148 lower_2 narrowing for the total-event action formula left by the state-event trace-Laplacian route. -/ def cycle148GeneralMovingTargetDiscreteEmGeneratorTotalEventSourceFunctionalLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle148_lower_2_em_generator_total_event_source_functional" statement := "Cycle 148 lower_2 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfSourceFunctional for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorLaplacianTotalEventIntegral premise left by the cycle-148 state-event route is no longer primitive; it follows from the source-functional definition of the frozen EM generator Laplacian action, the source-functional law-integral definition hsourceLaplacianFunctional, and the already explicit pointwise event-field/Laplacian identity hemGeneratorLaplacianEventFieldEqLaplacian. The state-event equality, htraceFieldEqLaplacian, hlaplacianEqEmGenerator, source-field/source-functional leaves, density-Laplacian, Green/trace/box-divergence, and diffusion leaves remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfSourceFunctional", "hemGeneratorSourceActionDef", "hsourceLaplacianFunctional", "hemGeneratorLaplacianEventFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 worker packet inside the cycle-142/143/144/145/146/147/148 EM generator trace-state illness area. It implements one genuine source leaf rather than adding a downstream wrapper: the theorem proves exactly hemGeneratorLaplacianTotalEventIntegral from the source-functional action definition and pointwise event-field identity. It does not prove hemGeneratorLaplacianStateEventEqLaplacian, htraceFieldEqLaplacian, hlaplacianEqEmGenerator, close density-Laplacian/Green/trace/box-divergence/diffusion leaves, import SLT, change Lake/toolchain files, use sald_version_2, close a theorem by fake proof, or take a non-EM fallback." /-- Cycle-148 lower_2 proof-DAG pane for reducing the total-event action formula to the source-functional action definition. -/ def cycle148GeneralMovingTargetDiscreteEmGeneratorTotalEventSourceFunctionalDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle148.lower_2_packet.total_event_from_source_functional" interface := "Compiled source leaf: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfSourceFunctional derives hemGeneratorLaplacianTotalEventIntegral from hemGeneratorSourceActionDef, hsourceLaplacianFunctional, and hemGeneratorLaplacianEventFieldEqLaplacian by rewriting the law integral over Set.univ." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle148GeneralMovingTargetDiscreteEmGeneratorTotalEventSourceFunctionalLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfSourceFunctional", "hemGeneratorSourceActionDef", "hsourceLaplacianFunctional", "hemGeneratorLaplacianEventFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle148.lower_2_remaining_source_functional_state_event_trace" interface := "Remaining exact theorem boundary after cycle 148 lower_2: prove the source-functional action definition for the named frozen EM Laplacian action, prove hemGeneratorLaplacianStateEventEqLaplacian when it is not supplied through the pointwise route, and still prove hemGeneratorLaplacianEventFieldEqLaplacian and htraceFieldEqLaplacian. The total-event formula now follows locally from the source-functional action definition plus the event-field identity." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfSourceFunctional", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-! ### Cycle 149: total-event standard-basis source-functional leaf -/ /-- Cycle-149 narrowing for the source-functional action definition left by the cycle-148 total-event route. -/ def cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceFunctionalLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle149_em_generator_total_event_stdbasis_source_functional" statement := "Cycle 149 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceFunctional for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the direct hemGeneratorSourceActionDef premise in the cycle-148 total-event source-functional route is no longer primitive; it follows from the smaller source-cited standard-basis source formula hemGeneratorStdBasisDef via SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula. The packet keeps hsourceLaplacianFunctional and hemGeneratorLaplacianEventFieldEqLaplacian explicit, and leaves htraceFieldEqLaplacian, state-event equality when not supplied by pointwise rewriting, hlaplacianEqEmGenerator, source-field leaves, density-Laplacian, Green/trace/box-divergence, and diffusion leaves untouched." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "hemGeneratorStdBasisDef", "hsourceLaplacianFunctional", "hemGeneratorLaplacianEventFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet in the EM conditional-law/state-event set-integral illness area. It composes existing local SALD declarations and the already-local Mathlib Laplacian standard-basis bridge; no SLT file was consulted, no SLT import or port claim is made, no sald_version_2 source is used, and no broad route audit, theorem-status/Lake/toolchain change, fake closure, non-EM fallback, or wrapper churn is introduced." /-- Cycle-149 proof-DAG pane for replacing the source-functional action definition by the standard-basis source formula in the total-event route. -/ def cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceFunctionalDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle149.packet.total_event_from_stdbasis_source_functional" interface := "Compiled source-boundary bridge: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceFunctional derives hemGeneratorLaplacianTotalEventIntegral from hemGeneratorStdBasisDef, hsourceLaplacianFunctional, and hemGeneratorLaplacianEventFieldEqLaplacian by first deriving hemGeneratorSourceActionDef through the cycle-141 standard-basis source route, then applying the cycle-148 total-event source-functional theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceFunctionalLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfSourceFunctional", "hemGeneratorStdBasisDef", "hsourceLaplacianFunctional", "hemGeneratorLaplacianEventFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle149.remaining_stdbasis_source_event_trace" interface := "Remaining exact theorem boundary after cycle 149: prove the paper's standard-basis source formula hemGeneratorStdBasisDef for the named frozen EM Laplacian action, plus hemGeneratorLaplacianEventFieldEqLaplacian and htraceFieldEqLaplacian; keep the state-event equality explicit only when it is not supplied by the pointwise rewrite route." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceFunctional", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-- Cycle-149 lower_1 narrowing for the event-field Laplacian identity left by the standard-basis source-functional total-event route. -/ def cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceEventLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle149_em_generator_total_event_stdbasis_source_event" statement := "Cycle 149 lower_1 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceAndEventFormula for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the direct hemGeneratorLaplacianEventFieldEqLaplacian premise in the cycle-149 total-event standard-basis source-functional route is no longer primitive; it follows from the smaller source-cited standard-basis event-field definition hemGeneratorLaplacianEventFieldStdBasisDef via SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula. The packet keeps hemGeneratorStdBasisDef, hsourceLaplacianFunctional, htraceFieldEqLaplacian, and state-event equality when not supplied by pointwise rewriting explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "hemGeneratorStdBasisDef", "hemGeneratorLaplacianEventFieldStdBasisDef", "hsourceLaplacianFunctional", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 worker/scout packet in the EM conditional-law/state-event set-integral illness area. It reuses only local SALD declarations and the already-local Mathlib Laplacian standard-basis bridge; no SLT file was consulted, no SLT import or port claim is made, no sald_version_2 source is used, and no broad route audit, theorem-status/Lake/toolchain change, fake closure, non-EM fallback, or duplicate wrapper churn is introduced." /-- Cycle-149 lower_1 proof-DAG pane for replacing the pointwise event-field Laplacian identity by the standard-basis event-field source formula. -/ def cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceEventDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle149.lower_1_packet.total_event_from_stdbasis_source_and_event" interface := "Compiled bridge: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceAndEventFormula derives hemGeneratorLaplacianTotalEventIntegral from hemGeneratorStdBasisDef, hemGeneratorLaplacianEventFieldStdBasisDef, and hsourceLaplacianFunctional by first deriving hemGeneratorLaplacianEventFieldEqLaplacian through the cycle-144 standard-basis event-field route, then applying the cycle-149 standard-basis source-functional theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceEventLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula", "hemGeneratorStdBasisDef", "hemGeneratorLaplacianEventFieldStdBasisDef", "hsourceLaplacianFunctional", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle149.lower_1_remaining_stdbasis_source_event_trace" interface := "Remaining exact theorem boundary after cycle 149 lower_1: prove the paper's standard-basis source formula hemGeneratorStdBasisDef and standard-basis event-field definition hemGeneratorLaplacianEventFieldStdBasisDef for the named frozen EM Laplacian contribution, plus htraceFieldEqLaplacian; keep state-event equality explicit only when it is not supplied by the pointwise rewrite route." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceAndEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-- Cycle-149 lower_2 narrowing for the standard-basis source and event-field premises left by the lower_1 total-event route. -/ def cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceFieldSourceEventLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle149_em_generator_total_event_trace_field_source_event" statement := "Cycle 149 lower_2 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceFieldSourceAndEventFormula for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundaries narrowed: hemGeneratorStdBasisDef and hemGeneratorLaplacianEventFieldStdBasisDef in the cycle-149 lower_1 total-event route are no longer primitive; they follow from hemGeneratorTraceActionDef, hemGeneratorLaplacianEventFieldEqTraceField, and htraceFieldEqLaplacian through the existing trace-field standard-basis bridges. The packet keeps hsourceLaplacianFunctional explicit and leaves the trace-action, trace-event equality, trace-field Laplacian, state-event equality when not supplied by pointwise rewriting, and sibling EM/weak-FP leaves separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceFieldSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "hemGeneratorTraceActionDef", "hemGeneratorLaplacianEventFieldEqTraceField", "htraceFieldEqLaplacian", "hsourceLaplacianFunctional", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 worker packet in the EM conditional-law/state-event set-integral illness area. It reuses only local SALD trace-field bridges and the already-local Mathlib Laplacian standard-basis bridge; no SLT file was consulted, no SLT import or port claim is made, no sald_version_2 source is used, and no broad route audit, theorem-status/Lake/toolchain change, fake closure, non-EM fallback, or duplicate wrapper churn is introduced." /-- Cycle-149 lower_2 proof-DAG pane for replacing the standard-basis source/event premises by trace-field source identities. -/ def cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceFieldSourceEventDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle149.lower_2_packet.total_event_from_trace_field_source_and_event" interface := "Compiled bridge: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceFieldSourceAndEventFormula derives hemGeneratorLaplacianTotalEventIntegral from hemGeneratorTraceActionDef, hemGeneratorLaplacianEventFieldEqTraceField, htraceFieldEqLaplacian, and hsourceLaplacianFunctional by reconstructing hemGeneratorStdBasisDef and hemGeneratorLaplacianEventFieldStdBasisDef, then applying the cycle-149 lower_1 total-event theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceFieldSourceEventLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceFieldSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "hemGeneratorTraceActionDef", "hemGeneratorLaplacianEventFieldEqTraceField", "htraceFieldEqLaplacian", "hsourceLaplacianFunctional", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle149.lower_2_remaining_trace_action_event_trace_laplacian" interface := "Remaining exact theorem boundary after cycle 149 lower_2: prove hemGeneratorTraceActionDef for the named frozen EM Laplacian action, hemGeneratorLaplacianEventFieldEqTraceField identifying the named event field with the trace field, and htraceFieldEqLaplacian. Keep hsourceLaplacianFunctional and state-event equality explicit when they are not supplied by pointwise rewriting." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceFieldSourceAndEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-! ### Cycle 150: total-event trace law-integral source leaf -/ /-- Cycle-150 middle narrowing for the trace-action definition left by the current total-event trace-field route. -/ def cycle150GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLawIntegralSourceEventMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle150_em_generator_total_event_trace_law_integral_source_event" statement := "Cycle 150 middle compiles SALD.generalMovingTargetDiscreteEmGeneratorTraceActionDefOfTraceLawIntegral and SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLawIntegralSourceAndEventFormula for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorTraceActionDef premise left by SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceFieldSourceAndEventFormula is no longer primitive; it follows from the smaller law-space trace integral hemGeneratorTraceLawIntegral plus the already explicit source-functional definition hsourceLaplacianFunctional. The packet keeps hemGeneratorLaplacianEventFieldEqTraceField and htraceFieldEqLaplacian explicit, and leaves state-event equality when not supplied by pointwise rewriting plus sibling EM/weak-FP leaves separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorTraceActionDefOfTraceLawIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLawIntegralSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceFieldSourceAndEventFormula", "hemGeneratorTraceLawIntegral", "hsourceLaplacianFunctional", "hemGeneratorLaplacianEventFieldEqTraceField", "htraceFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf middle packet in the EM conditional-law/state-event set-integral illness area. It reuses the cycle-141 trace-law/source-functional split in the current total-event trace-field route instead of adding a broad wrapper. It does not prove hemGeneratorTraceLawIntegral, hemGeneratorLaplacianEventFieldEqTraceField, htraceFieldEqLaplacian, hemGeneratorLaplacianStateEventEqLaplacian, hlaplacianEqEmGenerator, source-field leaves, density-Laplacian, Green/trace/box-divergence, diffusion leaves, import SLT, change Lake/toolchain files, use sald_version_2, close a theorem by fake proof, take a non-EM fallback, or promote theorem status." /-- Cycle-150 lower_1 narrowing for the trace law-integral premise left by the current total-event trace-law route. -/ def cycle150GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceStateIntegralSourceEventLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle150_em_generator_total_event_trace_state_integral_source_event" statement := "Cycle 150 lower_1 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceStateIntegralSourceAndEventFormula for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorTraceLawIntegral premise left by SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLawIntegralSourceAndEventFormula is no longer primitive; it follows from the smaller sample-space trace integral hemGeneratorTraceStateIntegral, the law identity hhatRhoS, the measurability hhatX, and htraceFieldMeas via the existing map-integral helper SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral. The packet keeps hsourceLaplacianFunctional, hemGeneratorLaplacianEventFieldEqTraceField, and htraceFieldEqLaplacian explicit, and leaves state-event equality when not supplied by pointwise rewriting plus sibling EM/weak-FP leaves separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceStateIntegralSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLawIntegralSourceAndEventFormula", "hemGeneratorTraceStateIntegral", "hhatRhoS", "hhatX", "htraceFieldMeas", "hsourceLaplacianFunctional", "hemGeneratorLaplacianEventFieldEqTraceField", "htraceFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 proof-scout packet in the EM conditional-law/state-event set-integral illness area. It reuses the existing cycle-142 map-integral transport helper inside the current total-event trace-law route. It does not prove hemGeneratorTraceStateIntegral, htraceFieldMeas, hemGeneratorLaplacianEventFieldEqTraceField, htraceFieldEqLaplacian, hemGeneratorLaplacianStateEventEqLaplacian, hlaplacianEqEmGenerator, source-field leaves, density-Laplacian, Green/trace/box-divergence, diffusion leaves, import SLT, change Lake/toolchain files, use sald_version_2, close a theorem by fake proof, take a non-EM fallback, or promote theorem status." /-- Cycle-150 lower_2 narrowing for the trace-state premise left by the current total-event trace-state route. -/ def cycle150GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLaplacianStateIntegralSourceEventLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle150_em_generator_total_event_trace_laplacian_state_integral_source_event" statement := "Cycle 150 lower_2 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndEventFormula for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorTraceStateIntegral and htraceFieldMeas premises left by SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceStateIntegralSourceAndEventFormula are no longer primitive; they follow from the smaller sample-space selected-test Laplacian integral hemGeneratorLaplacianStateIntegral, source-Laplacian measurability hsourceLaplacianFieldMeas, and the already explicit htraceFieldEqLaplacian through the local trace-field standard-basis and trace-state helper declarations. The packet keeps hsourceLaplacianFunctional, hemGeneratorLaplacianEventFieldEqTraceField, and htraceFieldEqLaplacian explicit, and leaves state-event equality when not supplied by pointwise rewriting plus sibling EM/weak-FP leaves separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldMeasOfSourceLaplacianFieldMeas", "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceStateIntegralSourceAndEventFormula", "hemGeneratorLaplacianStateIntegral", "hsourceLaplacianFieldMeas", "hhatRhoS", "hhatX", "hsourceLaplacianFunctional", "hemGeneratorLaplacianEventFieldEqTraceField", "htraceFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 worker packet in the EM conditional-law/state-event set-integral illness area. It reuses only existing local SALD trace-field/Laplacian bridges and the already-local Mathlib standard-basis Laplacian bridge. It does not prove hemGeneratorLaplacianStateIntegral, hsourceLaplacianFieldMeas, hemGeneratorLaplacianEventFieldEqTraceField, htraceFieldEqLaplacian, hemGeneratorLaplacianStateEventEqLaplacian, hlaplacianEqEmGenerator, source-field leaves, density-Laplacian, Green/trace/box-divergence, diffusion leaves, import SLT, change Lake/toolchain files, use sald_version_2, close a theorem by fake proof, take a non-EM fallback, or promote theorem status." /-- Cycle-150 proof-DAG pane for reducing the total-event trace-action leaf to the law-space trace integral. -/ def cycle150GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLawIntegralSourceEventDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle150.middle_packet.trace_action_from_trace_law_integral" interface := "Compiled helper: SALD.generalMovingTargetDiscreteEmGeneratorTraceActionDefOfTraceLawIntegral derives hemGeneratorTraceActionDef from the law-space source fact hemGeneratorTraceLawIntegral and the already explicit source-functional definition hsourceLaplacianFunctional." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle150GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLawIntegralSourceEventMiddleObligation", "SALD.generalMovingTargetDiscreteEmGeneratorTraceActionDefOfTraceLawIntegral", "hemGeneratorTraceLawIntegral", "hsourceLaplacianFunctional", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLawIntegralSourceAndEventFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle150.middle_packet.total_event_from_trace_law_integral_source_and_event" interface := "Compiled bridge: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLawIntegralSourceAndEventFormula derives hemGeneratorLaplacianTotalEventIntegral from hemGeneratorTraceLawIntegral, hsourceLaplacianFunctional, hemGeneratorLaplacianEventFieldEqTraceField, and htraceFieldEqLaplacian by first recovering hemGeneratorTraceActionDef, then applying the cycle-149 trace-field source/event total-event theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle150GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLawIntegralSourceEventMiddleObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLawIntegralSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorTraceActionDefOfTraceLawIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceFieldSourceAndEventFormula", "hemGeneratorTraceLawIntegral", "hsourceLaplacianFunctional", "hemGeneratorLaplacianEventFieldEqTraceField", "htraceFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle150.lower_1_packet.total_event_from_trace_state_integral_source_and_event" interface := "Compiled bridge: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceStateIntegralSourceAndEventFormula derives hemGeneratorLaplacianTotalEventIntegral from hemGeneratorTraceStateIntegral, hhatRhoS, hhatX, htraceFieldMeas, hsourceLaplacianFunctional, hemGeneratorLaplacianEventFieldEqTraceField, and htraceFieldEqLaplacian by first recovering hemGeneratorTraceLawIntegral through SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral, then applying the cycle-150 trace-law total-event theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle150GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceStateIntegralSourceEventLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceStateIntegralSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLawIntegralSourceAndEventFormula", "hemGeneratorTraceStateIntegral", "hhatRhoS", "hhatX", "htraceFieldMeas", "hsourceLaplacianFunctional", "hemGeneratorLaplacianEventFieldEqTraceField", "htraceFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle150.lower_2_packet.total_event_from_trace_laplacian_state_integral_source_and_event" interface := "Compiled bridge: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndEventFormula derives hemGeneratorLaplacianTotalEventIntegral from hemGeneratorLaplacianStateIntegral, hsourceLaplacianFieldMeas, hhatRhoS, hhatX, hsourceLaplacianFunctional, hemGeneratorLaplacianEventFieldEqTraceField, and htraceFieldEqLaplacian by first recovering htraceFieldMeas and hemGeneratorTraceStateIntegral through the local trace-field/Laplacian helpers, then applying the cycle-150 trace-state total-event theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle150GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLaplacianStateIntegralSourceEventLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldMeasOfSourceLaplacianFieldMeas", "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceStateIntegralSourceAndEventFormula", "hemGeneratorLaplacianStateIntegral", "hsourceLaplacianFieldMeas", "hhatRhoS", "hhatX", "hsourceLaplacianFunctional", "hemGeneratorLaplacianEventFieldEqTraceField", "htraceFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle150.lower_2_remaining_trace_laplacian_state_event_trace_laplacian" interface := "Remaining exact theorem boundary after cycle 150 lower_2: prove hemGeneratorLaplacianStateIntegral for the named frozen EM Laplacian selected-test action as a sample-space integral along hatXAtS, prove hsourceLaplacianFieldMeas when not supplied by the source-Laplacian route, prove hemGeneratorLaplacianEventFieldEqTraceField, and prove htraceFieldEqLaplacian. Keep hsourceLaplacianFunctional and state-event equality explicit when they are not supplied by pointwise rewriting." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle150.lower_1_remaining_trace_state_event_trace_laplacian" interface := "Remaining exact theorem boundary after cycle 150 lower_1: prove hemGeneratorTraceStateIntegral for the named frozen EM Laplacian trace action as a sample-space integral along hatXAtS, prove htraceFieldMeas when not supplied by the trace-field/source-Laplacian route, prove hemGeneratorLaplacianEventFieldEqTraceField, and prove htraceFieldEqLaplacian. Keep hsourceLaplacianFunctional and state-event equality explicit when they are not supplied by pointwise rewriting." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceStateIntegralSourceAndEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle150.remaining_trace_law_event_trace_laplacian" interface := "Remaining exact theorem boundary after cycle 150 middle: prove hemGeneratorTraceLawIntegral for the named frozen EM Laplacian trace action as a law-space integral against hatRhoS, prove hemGeneratorLaplacianEventFieldEqTraceField, and prove htraceFieldEqLaplacian. Keep hsourceLaplacianFunctional and state-event equality explicit when they are not supplied by pointwise rewriting." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLawIntegralSourceAndEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle150.reviewer_trace_law_total_event_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorTraceActionDef is narrowed to hemGeneratorTraceLawIntegral plus hsourceLaplacianFunctional through SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLawIntegralSourceAndEventFormula, sibling EM/weak-FP source leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle150GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLawIntegralSourceEventMiddleObligation", "SALD.generalMovingTargetDiscreteEmGeneratorTraceActionDefOfTraceLawIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLawIntegralSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceFieldSourceAndEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 150 reviewer"] status := ProofStatus.obligation } ] /-- Cycle-151 direct-leaf narrowing for the event-field/trace-field equality left by the current trace-Laplacian state total-event route. -/ def cycle151GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLaplacianStateIntegralPointwiseEventObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle151_em_generator_total_event_trace_laplacian_state_integral_pointwise_event" statement := "Cycle 151 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndPointwiseEventFormula for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hemGeneratorLaplacianEventFieldEqTraceField premise left by SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndEventFormula is no longer primitive; it follows from the smaller pointwise EM event-field identity hemGeneratorLaplacianEventFieldEqLaplacian and the already explicit htraceFieldEqLaplacian through SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields. The packet keeps hemGeneratorLaplacianStateIntegral, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, and htraceFieldEqLaplacian explicit, with state-event and sibling EM leaves separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndPointwiseEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndEventFormula", "hemGeneratorLaplacianEventFieldEqLaplacian", "htraceFieldEqLaplacian", "hemGeneratorLaplacianStateIntegral", "hsourceLaplacianFieldMeas", "hsourceLaplacianFunctional", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet in the EM conditional-law/state-event set-integral illness area. It targets only the direct hemGeneratorLaplacianEventFieldEqTraceField leaf selected by the refreshed blueprint and derives it from the pointwise event-field Laplacian identity plus htraceFieldEqLaplacian. No SLT file was consulted or imported; no Lake/toolchain change, theorem-status promotion, non-EM fallback, broad audit, wrapper churn outside this direct leaf, fake closure, or sald_version_2.tex use." /-- Cycle-151 proof-DAG pane for the direct event-field/trace-field equality leaf in the trace-Laplacian state total-event route. -/ def cycle151GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLaplacianStateIntegralPointwiseEventDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle151.packet.total_event_from_pointwise_event_trace_laplacian" interface := "Compiled bridge: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndPointwiseEventFormula derives the total-event formula from hemGeneratorLaplacianStateIntegral, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, hemGeneratorLaplacianEventFieldEqLaplacian, and htraceFieldEqLaplacian. It first recovers hemGeneratorLaplacianEventFieldEqTraceField through SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields, then applies the cycle-150 trace-Laplacian state total-event theorem." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle151GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLaplacianStateIntegralPointwiseEventObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndPointwiseEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndEventFormula", "hemGeneratorLaplacianEventFieldEqLaplacian", "htraceFieldEqLaplacian", "hemGeneratorLaplacianStateIntegral", "hsourceLaplacianFieldMeas", "hsourceLaplacianFunctional", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle151.remaining_trace_laplacian_state_pointwise_event" interface := "Remaining exact theorem boundary after cycle 151: prove hemGeneratorLaplacianStateIntegral for the named frozen EM Laplacian selected-test action as a sample-space integral along hatXAtS, prove hsourceLaplacianFieldMeas when not supplied by the source-Laplacian route, prove hemGeneratorLaplacianEventFieldEqLaplacian, and prove htraceFieldEqLaplacian. Keep hsourceLaplacianFunctional and state-event equality explicit when they are not supplied by pointwise rewriting." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndPointwiseEventFormula", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle151.reviewer_pointwise_event_trace_laplacian_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorLaplacianEventFieldEqTraceField is narrowed to hemGeneratorLaplacianEventFieldEqLaplacian plus htraceFieldEqLaplacian through SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndPointwiseEventFormula, remaining direct leaves stay explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle151GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLaplacianStateIntegralPointwiseEventObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndPointwiseEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 151 reviewer"] status := ProofStatus.obligation } ] /-- Cycle-151 lower_2 direct-leaf narrowing for the pointwise EM event-field Laplacian identity. -/ def cycle151GeneralMovingTargetDiscreteEmGeneratorPointwiseEventSourceFieldLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle151_em_generator_pointwise_event_source_field_lower_2" statement := "Cycle 151 lower_2 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfWeakFpSourceField for appendix.tex:984-995 and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the pointwise EM event-field identity hemGeneratorLaplacianEventFieldEqLaplacian is no longer primitive; it follows from the smaller source-cited equality hemGeneratorLaplacianEventFieldEqSourceField identifying the named frozen EM Laplacian event field with the weak-FP source Laplacian field, plus the already tracked hweakFpSourceFieldEqLaplacian. The packet is a direct leaf theorem and does not add a total-event/source-functional consumer wrapper." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfWeakFpSourceField", "hemGeneratorLaplacianEventFieldEqSourceField", "hweakFpSourceFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 worker packet in the EM conditional-law/state-event set-integral illness area. It targets only the direct hemGeneratorLaplacianEventFieldEqLaplacian leaf requested after the lower_1 handoff. No SLT file was consulted or imported; no Lake/toolchain change, theorem-status promotion, non-EM fallback, broad audit, downstream wrapper churn, fake closure, or sald_version_2.tex use." /-- Cycle-151 lower_2 proof-DAG pane for the pointwise event-field source-field leaf. -/ def cycle151GeneralMovingTargetDiscreteEmGeneratorPointwiseEventSourceFieldDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle151.lower_2_packet.pointwise_event_from_source_field" interface := "Compiled direct-leaf theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfWeakFpSourceField derives hemGeneratorLaplacianEventFieldEqLaplacian from hemGeneratorLaplacianEventFieldEqSourceField plus hweakFpSourceFieldEqLaplacian." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle151GeneralMovingTargetDiscreteEmGeneratorPointwiseEventSourceFieldLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfWeakFpSourceField", "hemGeneratorLaplacianEventFieldEqSourceField", "hweakFpSourceFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndPointwiseEventFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle151.lower_2_remaining_pointwise_event_source_field" interface := "Remaining exact theorem boundary after cycle 151 lower_2: prove hemGeneratorLaplacianEventFieldEqSourceField for the named frozen EM Laplacian event field and keep hweakFpSourceFieldEqLaplacian, htraceFieldEqLaplacian, hemGeneratorLaplacianStateIntegral, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, state-event, and sibling EM leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfWeakFpSourceField", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-- Cycle-152 direct-leaf narrowing for the EM event-field/source-field equality. -/ def cycle152GeneralMovingTargetDiscreteEmGeneratorEventSourceFieldStdBasisLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle152_em_generator_event_source_field_stdbasis_lower" statement := "Cycle 152 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqSourceFieldOfStdBasisFields for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hemGeneratorLaplacianEventFieldEqSourceField is no longer primitive; it follows from two smaller source-facing standard-basis field definitions, hemGeneratorLaplacianEventFieldStdBasisDef for the named frozen EM Laplacian event field and hweakFpSourceFieldStdBasisDef for weakFpLaplacianSourceField. The packet is a direct equality leaf and does not add a total-event/source-functional consumer wrapper." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqSourceFieldOfStdBasisFields", "hemGeneratorLaplacianEventFieldStdBasisDef", "hweakFpSourceFieldStdBasisDef", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet in the EM conditional-law/state-event set-integral illness area. It targets only hemGeneratorLaplacianEventFieldEqSourceField after the cycle-151 lower_2 source-field split. No SLT file was consulted or imported; no Lake/toolchain change, theorem-status promotion, non-EM fallback, broad audit, downstream wrapper churn, fake closure, or sald_version_2.tex use." /-- Cycle-152 lower_1 scout narrowing for the weak-FP source-field standard-basis leaf exposed by the direct event/source equality split. -/ def cycle152GeneralMovingTargetDiscreteWeakFpSourceFieldStdBasisScoutObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle152_weak_fp_source_field_stdbasis_scout" statement := "Cycle 152 lower_1 compiles SALD.generalMovingTargetDiscreteWeakFpSourceFieldStdBasisDefOfLaplacianField for appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hweakFpSourceFieldStdBasisDef is no longer primitive; it follows from hweakFpSourceFieldEqLaplacian plus the local Mathlib standard-basis Laplacian theorem SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv. The packet keeps hemGeneratorLaplacianEventFieldStdBasisDef, htraceFieldEqLaplacian, hemGeneratorLaplacianStateIntegral, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, state-event, and sibling EM leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceFieldStdBasisDefOfLaplacianField", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "hweakFpSourceFieldEqLaplacian", "hweakFpSourceFieldStdBasisDef", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 proof-scout packet in the same EM source-field illness area. It uses only the existing local Mathlib Laplacian standard-basis bridge; it does not prove the EM event-field standard-basis definition, total-event formulas, source-functional facts, SLT ports, or any non-EM fallback." /-- Cycle-152 lower_2 direct narrowing for the weak-FP source-field/Laplacian equality. -/ def cycle152GeneralMovingTargetDiscreteWeakFpSourceFieldPointwiseLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle152_weak_fp_source_field_pointwise_lower_2" statement := "Cycle 152 lower_2 compiles SALD.generalMovingTargetDiscreteWeakFpSourceFieldEqLaplacianOfPointwise for appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hweakFpSourceFieldEqLaplacian is no longer primitive; it follows from hweakFpSourceFieldPointwiseEqLaplacian, the pointwise source-field identity identifying weakFpLaplacianSourceField phi with Mathlib's Laplacian of the selected weak test. The packet keeps hemGeneratorLaplacianEventFieldStdBasisDef, htraceFieldEqLaplacian, hemGeneratorLaplacianStateIntegral, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, state-event, and sibling EM leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteWeakFpSourceFieldEqLaplacianOfPointwise", "hweakFpSourceFieldEqLaplacian", "hweakFpSourceFieldPointwiseEqLaplacian", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 proof-producing packet in the weak-FP source-field illness area. It only converts the pointwise source field identity into the function-level equality used by the cycle-152 standard-basis scout; it does not prove the pointwise analytic identity, EM event-field standard-basis definition, source-functional facts, SLT ports, theorem closure, or any non-EM fallback." /-- Cycle-152 proof-DAG pane for the source-field equality narrowing. -/ def cycle152GeneralMovingTargetDiscreteEmGeneratorEventSourceFieldStdBasisDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle152.packet.event_source_field_from_stdbasis_fields" interface := "Compiled direct-leaf theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqSourceFieldOfStdBasisFields derives hemGeneratorLaplacianEventFieldEqSourceField from hemGeneratorLaplacianEventFieldStdBasisDef plus hweakFpSourceFieldStdBasisDef." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle152GeneralMovingTargetDiscreteEmGeneratorEventSourceFieldStdBasisLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqSourceFieldOfStdBasisFields", "hemGeneratorLaplacianEventFieldStdBasisDef", "hweakFpSourceFieldStdBasisDef", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfWeakFpSourceField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle152.lower_1_packet.weak_fp_source_field_stdbasis_from_laplacian" interface := "Compiled lower_1 scout theorem: SALD.generalMovingTargetDiscreteWeakFpSourceFieldStdBasisDefOfLaplacianField derives hweakFpSourceFieldStdBasisDef from hweakFpSourceFieldEqLaplacian plus SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle152GeneralMovingTargetDiscreteWeakFpSourceFieldStdBasisScoutObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceFieldStdBasisDefOfLaplacianField", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "hweakFpSourceFieldEqLaplacian", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqSourceFieldOfStdBasisFields", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfWeakFpSourceField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle152.lower_2_packet.weak_fp_source_field_eq_laplacian_pointwise" interface := "Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteWeakFpSourceFieldEqLaplacianOfPointwise derives hweakFpSourceFieldEqLaplacian from hweakFpSourceFieldPointwiseEqLaplacian, the pointwise source-field identity for each selected weak test." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle152GeneralMovingTargetDiscreteWeakFpSourceFieldPointwiseLowerObligation", "SALD.generalMovingTargetDiscreteWeakFpSourceFieldEqLaplacianOfPointwise", "hweakFpSourceFieldPointwiseEqLaplacian", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteWeakFpSourceFieldStdBasisDefOfLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqSourceFieldOfStdBasisFields", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfWeakFpSourceField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle152.remaining_event_source_field_stdbasis_defs" interface := "Remaining exact theorem boundary after cycle 152 lower_2: prove hemGeneratorLaplacianEventFieldStdBasisDef for the named frozen EM Laplacian event field and hweakFpSourceFieldPointwiseEqLaplacian for weakFpLaplacianSourceField; hweakFpSourceFieldEqLaplacian and hweakFpSourceFieldStdBasisDef now follow locally from the latter. Keep htraceFieldEqLaplacian, hemGeneratorLaplacianStateIntegral, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, state-event, and sibling EM leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqSourceFieldOfStdBasisFields", "SALD.generalMovingTargetDiscreteWeakFpSourceFieldStdBasisDefOfLaplacianField", "SALD.generalMovingTargetDiscreteWeakFpSourceFieldEqLaplacianOfPointwise", "hweakFpSourceFieldPointwiseEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation } ] /-! ### Cycle 153: EM generator Laplacian state integral from source functional -/ /-- Cycle-153 direct-leaf narrowing for the selected-test Laplacian state integral. -/ def cycle153GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralSourceFunctionalLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle153_em_generator_laplacian_state_integral_source_functional" statement := "Cycle 153 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfSourceFunctional for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the direct hemGeneratorLaplacianStateIntegral leaf is no longer primitive for the active EM conditional-law/state-event set-integral backend; it follows from the smaller source-functional action definition hemGeneratorSourceActionDef, the explicit hsourceLaplacianFunctional law-integral definition, source-Laplacian measurability, hhatRhoS = Measure.map hatXAtS P, and AEMeasurable hatXAtS, reusing SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral. The packet keeps hsourceLaplacianFunctional and hsourceLaplacianFieldMeas explicit, and leaves hemGeneratorLaplacianEventFieldStdBasisDef, hweakFpSourceFieldPointwiseEqLaplacian, htraceFieldEqLaplacian, state-event, and sibling EM leaves separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "hemGeneratorSourceActionDef", "hsourceLaplacianFunctional", "hsourceLaplacianFieldMeas", "hhatRhoS", "hhatX", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet inside the EM conditional-law/state-event set-integral illness area. It targets only hemGeneratorLaplacianStateIntegral and reuses the already compiled law-to-state integral transport. It does not prove the source-functional action definition, source-Laplacian measurability, source-functional construction, EM event-field standard-basis definition, weak-FP pointwise source field identity, trace-field Laplacian identity, state-event formula, theorem closure, SLT import, Lake/toolchain change, theorem-status promotion, fake closure, broad audit, downstream wrapper churn, non-EM fallback, or sald_version_2.tex content." /-- Cycle-153 lower_2 narrowing from the standard-basis source formula. -/ def cycle153GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralStdBasisSourceFunctionalLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle153_lower2_em_generator_laplacian_state_integral_stdbasis_source_functional" statement := "Cycle 153 lower_2 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the direct hemGeneratorSourceActionDef input under SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfSourceFunctional is no longer primitive; it follows from the smaller source-cited standard-basis source formula hemGeneratorStdBasisDef through SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula. The packet keeps hsourceLaplacianFunctional, hsourceLaplacianFieldMeas, hhatRhoS, and hhatX explicit, and leaves hemGeneratorLaplacianEventFieldStdBasisDef, hweakFpSourceFieldPointwiseEqLaplacian, htraceFieldEqLaplacian, state-event, and sibling EM leaves separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "hemGeneratorStdBasisDef", "hsourceLaplacianFunctional", "hsourceLaplacianFieldMeas", "hhatRhoS", "hhatX", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 worker packet inside the EM conditional-law/state-event set-integral illness area. It targets only the source-action premise needed to recover hemGeneratorLaplacianStateIntegral and reuses the compiled standard-basis-to-Laplacian bridge. It does not prove hemGeneratorStdBasisDef, source-functional construction, source-Laplacian measurability, EM event-field standard-basis definition, weak-FP pointwise source field identity, trace-field Laplacian identity, state-event formula, theorem closure, SLT import, Lake/toolchain change, theorem-status promotion, fake closure, broad audit, downstream wrapper churn, non-EM fallback, or sald_version_2.tex content." /-- Cycle-153 proof-DAG pane for the state-integral/source-functional leaf. -/ def cycle153GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralSourceFunctionalDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle153.packet.laplacian_state_integral_from_source_functional" interface := "Compiled direct-leaf theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfSourceFunctional derives hemGeneratorLaplacianStateIntegral from hemGeneratorSourceActionDef plus hsourceLaplacianFunctional, then reuses SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral to transport the law-space Laplacian integral along hatXAtS." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle153GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralSourceFunctionalLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "hemGeneratorSourceActionDef", "hsourceLaplacianFunctional", "hsourceLaplacianFieldMeas", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndPointwiseEventFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle153.lower_2_packet.laplacian_state_integral_from_stdbasis_source_functional" interface := "Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional derives hemGeneratorLaplacianStateIntegral from hemGeneratorStdBasisDef by first recovering hemGeneratorSourceActionDef through SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula, then reusing SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfSourceFunctional." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle153GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralStdBasisSourceFunctionalLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "hemGeneratorStdBasisDef", "hsourceLaplacianFunctional", "hsourceLaplacianFieldMeas", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndPointwiseEventFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle153.lower_2_remaining_laplacian_state_integral_stdbasis_source_functional" interface := "Remaining exact theorem boundary after cycle 153 lower_2: prove hemGeneratorStdBasisDef for the named frozen EM Laplacian action as the sourceLaplacianFunctional applied to the standard-basis second derivatives of selectedTest. The broader hemGeneratorSourceActionDef now follows locally; keep hsourceLaplacianFunctional, hsourceLaplacianFieldMeas, hemGeneratorLaplacianEventFieldStdBasisDef, hweakFpSourceFieldPointwiseEqLaplacian, htraceFieldEqLaplacian, state-event, and sibling EM leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula", "hemGeneratorStdBasisDef", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle153.reviewer_laplacian_state_integral_source_functional_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorLaplacianStateIntegral is narrowed first to hemGeneratorSourceActionDef plus hsourceLaplacianFunctional through SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfSourceFunctional and then, for lower_2, hemGeneratorSourceActionDef is narrowed to hemGeneratorStdBasisDef through SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional. Sibling EM/weak-FP leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle153GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralSourceFunctionalLowerObligation", "SALD.cycle153GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralStdBasisSourceFunctionalLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 153 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 154: EM generator Laplacian state integral from trace field -/ /-- Cycle-154 narrowing of the state-integral standard-basis source premise. -/ def cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceFieldSourceFunctionalLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle154_em_generator_laplacian_state_integral_trace_field_source_functional" statement := "Cycle 154 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceFieldSourceFunctional for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: hemGeneratorStdBasisDef under SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional. The state-integral route now takes the smaller source-cited pair hemGeneratorTraceActionDef plus htraceFieldStdBasis, then reuses SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField and SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional. It keeps hsourceLaplacianFunctional, hsourceLaplacianFieldMeas, hhatRhoS, hhatX, htraceFieldStdBasis, hemGeneratorLaplacianEventFieldStdBasisDef, hweakFpSourceFieldPointwiseEqLaplacian, htraceFieldEqLaplacian, state-event, and sibling EM leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceFieldSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "hemGeneratorTraceActionDef", "htraceFieldStdBasis", "hsourceLaplacianFunctional", "hsourceLaplacianFieldMeas", "hhatRhoS", "hhatX", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet inside the EM conditional-law/state-event set-integral illness area. It targets only the supplied hemGeneratorStdBasisDef premise in the accepted cycle-153 state-integral path. It does not prove hemGeneratorTraceActionDef, htraceFieldStdBasis, source-functional construction, source-Laplacian measurability, EM event-field standard-basis definition, weak-FP pointwise source field identity, trace-field Laplacian identity, state-event formula, theorem closure, SLT import, Lake/toolchain change, theorem-status promotion, fake closure, broad audit, downstream total-event wrapper churn, non-EM fallback, or sald_version_2.tex content." /-- Cycle-154 lower_1 scout narrowing of the trace-field state-integral inputs. -/ def cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceLawIntegralLaplacianFieldScoutObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle154_lower1_em_generator_laplacian_state_integral_trace_law_integral_laplacian_field" statement := "Cycle 154 lower_1 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceLawIntegralLaplacianField for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining hemGeneratorTraceActionDef and htraceFieldStdBasis inputs under SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceFieldSourceFunctional are no longer primitive for the active state-integral route; they follow from the smaller source-cited hemGeneratorTraceLawIntegral law-space trace integral and htraceFieldEqLaplacian trace-field/Laplacian identity, while hsourceLaplacianFunctional, hsourceLaplacianFieldMeas, hhatRhoS, and hhatX remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceLawIntegralLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorTraceActionDefOfTraceLawIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceFieldSourceFunctional", "hemGeneratorTraceLawIntegral", "htraceFieldEqLaplacian", "hsourceLaplacianFunctional", "hsourceLaplacianFieldMeas", "hhatRhoS", "hhatX", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 proof-scout packet in the EM conditional-law/state-event set-integral illness area. It composes already compiled local SALD bridges only: the trace-action law-integral bridge and the trace-field/Laplacian standard-basis bridge. It does not prove hemGeneratorTraceLawIntegral, htraceFieldEqLaplacian, hsourceLaplacianFunctional, source-Laplacian measurability, EM event-field standard-basis definition, weak-FP pointwise source field identity, state-event formula, theorem closure, SLT import, Lake/toolchain change, theorem-status promotion, fake closure, broad audit, downstream total-event wrapper churn, non-EM fallback, or sald_version_2.tex content." /-- Cycle-154 lower_2 narrowing of the trace-law state-integral input. -/ def cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceStateIntegralLaplacianFieldLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle154_lower2_em_generator_laplacian_state_integral_trace_state_integral_laplacian_field" statement := "Cycle 154 lower_2 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceStateIntegralLaplacianField for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining hemGeneratorTraceLawIntegral input under SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceLawIntegralLaplacianField is no longer primitive for the active state-integral route; it follows from the smaller sample-space trace integral hemGeneratorTraceStateIntegral, htraceFieldMeas, hhatRhoS, and hhatX through SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral, while htraceFieldEqLaplacian, hsourceLaplacianFunctional, and hsourceLaplacianFieldMeas remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceStateIntegralLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceLawIntegralLaplacianField", "hemGeneratorTraceStateIntegral", "htraceFieldMeas", "hhatRhoS", "hhatX", "htraceFieldEqLaplacian", "hsourceLaplacianFunctional", "hsourceLaplacianFieldMeas", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 worker packet in the EM conditional-law/state-event set-integral illness area. It reuses the existing Mathlib-backed local map-law bridge SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral and does not prove hemGeneratorTraceStateIntegral, htraceFieldMeas, htraceFieldEqLaplacian, source-functional construction, source-Laplacian measurability, EM event-field standard-basis definition, weak-FP pointwise source field identity, state-event formula, theorem closure, SLT import, Lake/toolchain change, theorem-status promotion, fake closure, broad audit, downstream total-event wrapper churn, non-EM fallback, or sald_version_2.tex content." /-- Cycle-154 proof-DAG pane for the trace-field source-functional state-integral split. -/ def cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceFieldSourceFunctionalDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle154.packet.laplacian_state_integral_from_trace_field_source_functional" interface := "Compiled worker theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceFieldSourceFunctional derives hemGeneratorLaplacianStateIntegral from hemGeneratorTraceActionDef plus htraceFieldStdBasis by first recovering hemGeneratorStdBasisDef through SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField, then reusing the accepted cycle-153 state-integral standard-basis source-functional bridge." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceFieldSourceFunctionalLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceFieldSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional", "hemGeneratorTraceActionDef", "htraceFieldStdBasis", "hsourceLaplacianFunctional", "hsourceLaplacianFieldMeas", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndPointwiseEventFormula", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle154.lower_1_packet.laplacian_state_integral_from_trace_law_laplacian_field" interface := "Compiled lower_1 scout theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceLawIntegralLaplacianField derives hemGeneratorLaplacianStateIntegral from hemGeneratorTraceLawIntegral and htraceFieldEqLaplacian by recovering hemGeneratorTraceActionDef through SALD.generalMovingTargetDiscreteEmGeneratorTraceActionDefOfTraceLawIntegral, recovering htraceFieldStdBasis through SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField, and then reusing the cycle-154 trace-field source-functional state-integral bridge." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceLawIntegralLaplacianFieldScoutObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceLawIntegralLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorTraceActionDefOfTraceLawIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceFieldSourceFunctional", "hemGeneratorTraceLawIntegral", "htraceFieldEqLaplacian", "hsourceLaplacianFunctional", "hsourceLaplacianFieldMeas", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceFieldSourceFunctional", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle154.lower_2_packet.laplacian_state_integral_from_trace_state_laplacian_field" interface := "Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceStateIntegralLaplacianField derives hemGeneratorLaplacianStateIntegral from hemGeneratorTraceStateIntegral, htraceFieldMeas, hhatRhoS, hhatX, and htraceFieldEqLaplacian by first recovering hemGeneratorTraceLawIntegral through SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral, then reusing the lower_1 trace-law/Laplacian-field state-integral bridge." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceStateIntegralLaplacianFieldLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceStateIntegralLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceLawIntegralLaplacianField", "hemGeneratorTraceStateIntegral", "htraceFieldMeas", "hhatRhoS", "hhatX", "htraceFieldEqLaplacian", "hsourceLaplacianFunctional", "hsourceLaplacianFieldMeas", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceLawIntegralLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle154.remaining_trace_state_trace_field_source_functional" interface := "Remaining exact theorem boundary after cycle 154 lower_2: prove hemGeneratorTraceStateIntegral for the named frozen EM trace-field action, htraceFieldMeas for map-law transport, and htraceFieldEqLaplacian for the trace-field/Laplacian identity. The direct hemGeneratorStdBasisDef, hemGeneratorTraceActionDef, htraceFieldStdBasis, and hemGeneratorTraceLawIntegral inputs are no longer primitive for this state-integral path; keep hsourceLaplacianFunctional, hsourceLaplacianFieldMeas, hemGeneratorLaplacianEventFieldStdBasisDef, hweakFpSourceFieldPointwiseEqLaplacian, state-event, and sibling EM leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceStateIntegralLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceLawIntegralLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceFieldSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "hemGeneratorTraceStateIntegral", "htraceFieldMeas", "htraceFieldEqLaplacian", "hhatRhoS", "hhatX", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle154.reviewer_trace_field_source_functional_check" interface := "Reviewer check: accept only if the packet classification is discharges-supplied-hypothesis for the first cycle-154 bridge and narrows-source-cited-boundary for the lower_1/lower_2 continuation bridges; hemGeneratorStdBasisDef is discharged under the accepted cycle-153 state-integral bridge, hemGeneratorTraceActionDef and htraceFieldStdBasis are narrowed to hemGeneratorTraceLawIntegral plus htraceFieldEqLaplacian, and hemGeneratorTraceLawIntegral is then narrowed to hemGeneratorTraceStateIntegral plus htraceFieldMeas, hhatRhoS, and hhatX. Sibling EM/weak-FP leaves remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, downstream wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceFieldSourceFunctionalLowerObligation", "SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceLawIntegralLaplacianFieldScoutObligation", "SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceStateIntegralLaplacianFieldLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceFieldSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceLawIntegralLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceStateIntegralLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 154 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 155: trace state integral from trace-field/Laplacian identity -/ /-- Cycle-155 narrowing of the trace-state integral source boundary. -/ def cycle155GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralLaplacianFieldLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle155_em_generator_trace_state_integral_laplacian_field" statement := "Cycle 155 compiles SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegralLaplacianField for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining hemGeneratorTraceStateIntegral premise under SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceStateIntegralLaplacianField is no longer primitive; it follows from the direct selected-test Laplacian state-integral leaf hemGeneratorLaplacianStateIntegral plus the already explicit trace-field/Laplacian identity htraceFieldEqLaplacian by rewriting the sample-space trace integrand along hatXAtS. The packet keeps htraceFieldMeas, hsourceLaplacianFunctional, hsourceLaplacianFieldMeas, hhatRhoS, hhatX, hemGeneratorLaplacianEventFieldStdBasisDef, hweakFpSourceFieldPointwiseEqLaplacian, state-event, and sibling EM leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegralLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceStateIntegralLaplacianField", "hemGeneratorTraceStateIntegral", "hemGeneratorLaplacianStateIntegral", "htraceFieldEqLaplacian", "htraceFieldMeas", "hsourceLaplacianFunctional", "hsourceLaplacianFieldMeas", "hhatRhoS", "hhatX", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf middle packet in the EM conditional-law/state-event set-integral illness area. It implements only the requested trace-state sample-integral narrowing and does not add a downstream total-event/source-functional consumer wrapper. It does not prove hemGeneratorLaplacianStateIntegral, htraceFieldMeas, hsourceLaplacianFunctional, source-Laplacian measurability, EM event-field standard-basis definition, weak-FP pointwise source field identity, state-event formula, theorem closure, SLT import, Lake/toolchain change, theorem-status promotion, fake closure, broad audit, non-EM fallback, or sald_version_2.tex content." /-- Cycle-155 proof-DAG pane for the trace-state sample-integral split. -/ def cycle155GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralLaplacianFieldDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle155.packet.trace_state_integral_from_laplacian_state_integral" interface := "Compiled theorem: SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegralLaplacianField derives hemGeneratorTraceStateIntegral from hemGeneratorLaplacianStateIntegral and htraceFieldEqLaplacian by rewriting the sample-space trace-field integrand along hatXAtS." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle155GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralLaplacianFieldLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegralLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceStateIntegralLaplacianField", "hemGeneratorLaplacianStateIntegral", "htraceFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceStateIntegralLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle155.remaining_trace_state_integral_laplacian_field" interface := "Remaining exact theorem boundary after cycle 155: prove hemGeneratorLaplacianStateIntegral or one of its smaller accepted source-functional leaves, prove htraceFieldMeas for the map-law transport, and prove htraceFieldEqLaplacian for the named trace field. Keep hsourceLaplacianFunctional, hsourceLaplacianFieldMeas, hemGeneratorLaplacianEventFieldStdBasisDef, hweakFpSourceFieldPointwiseEqLaplacian, state-event, and sibling EM leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegralLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceStateIntegralLaplacianField", "hemGeneratorLaplacianStateIntegral", "htraceFieldMeas", "htraceFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle155.reviewer_trace_state_laplacian_field_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorTraceStateIntegral is narrowed to hemGeneratorLaplacianStateIntegral plus htraceFieldEqLaplacian through SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegralLaplacianField, htraceFieldMeas remains explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, downstream wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle155GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralLaplacianFieldLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegralLaplacianField", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 155 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 156: trace-field equality from pointwise trace identity -/ /-- Cycle-156 narrowing of the trace-field/Laplacian source boundary. -/ def cycle156GeneralMovingTargetDiscreteEmGeneratorTraceFieldPointwiseLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle156_em_generator_trace_field_pointwise" statement := "Cycle 156 compiles SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldEqLaplacianOfPointwise, SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseEqLaplacianOfStdBasis, and SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseStdBasisOfEventFieldStdBasis for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundaries narrowed: htraceFieldEqLaplacian for the named frozen EM trace field follows from the smaller pointwise source identity htraceFieldPointwiseEqLaplacian by function extensionality; htraceFieldPointwiseEqLaplacian follows from the smaller pointwise standard-basis Hessian-trace display htraceFieldPointwiseStdBasis plus SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv; htraceFieldPointwiseStdBasis follows from the smaller source-facing pair hemGeneratorLaplacianEventFieldEqTraceField plus hemGeneratorLaplacianEventFieldStdBasisDef. The packet keeps hemGeneratorLaplacianStateIntegral, htraceFieldMeas, hsourceLaplacianFunctional, hsourceLaplacianFieldMeas, hemGeneratorLaplacianEventFieldEqTraceField, hemGeneratorLaplacianEventFieldStdBasisDef, hweakFpSourceFieldPointwiseEqLaplacian, state-event, and sibling EM leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldEqLaplacianOfPointwise", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseEqLaplacianOfStdBasis", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseStdBasisOfEventFieldStdBasis", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "htraceFieldEqLaplacian", "htraceFieldPointwiseEqLaplacian", "htraceFieldPointwiseStdBasis", "hemGeneratorLaplacianEventFieldEqTraceField", "hemGeneratorLaplacianEventFieldStdBasisDef", "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegralLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area refiner packet for the EM conditional-law/state-event backend. It implements only the requested htraceFieldEqLaplacian-to-pointwise narrowing and does not add a stale hemGeneratorTraceLawIntegral/hemGeneratorTraceStateIntegral wrapper, downstream total-event/source-functional consumer, SLT import, Lake/toolchain change, theorem-status promotion, fake closure, broad audit, non-EM fallback, or sald_version_2.tex content." /-- Cycle-156 proof-DAG pane for the trace-field pointwise split. -/ def cycle156GeneralMovingTargetDiscreteEmGeneratorTraceFieldPointwiseDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle156.packet.trace_field_eq_laplacian_from_pointwise" interface := "Compiled theorem: SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldEqLaplacianOfPointwise derives htraceFieldEqLaplacian from htraceFieldPointwiseEqLaplacian by function extensionality over the selected weak-test index." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle156GeneralMovingTargetDiscreteEmGeneratorTraceFieldPointwiseLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldEqLaplacianOfPointwise", "htraceFieldPointwiseEqLaplacian", "htraceFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegralLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle156.lower_1_packet.trace_field_pointwise_laplacian_from_stdbasis" interface := "Compiled lower_1 scout theorem: SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseEqLaplacianOfStdBasis derives htraceFieldPointwiseEqLaplacian from the pointwise standard-basis trace display htraceFieldPointwiseStdBasis plus SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle156GeneralMovingTargetDiscreteEmGeneratorTraceFieldPointwiseLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseEqLaplacianOfStdBasis", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "htraceFieldPointwiseStdBasis", "htraceFieldPointwiseEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldEqLaplacianOfPointwise", "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegralLaplacianField", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle156.lower_2_packet.trace_field_pointwise_stdbasis_from_event_field" interface := "Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseStdBasisOfEventFieldStdBasis derives htraceFieldPointwiseStdBasis from the smaller source-facing pair hemGeneratorLaplacianEventFieldEqTraceField plus hemGeneratorLaplacianEventFieldStdBasisDef." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle156GeneralMovingTargetDiscreteEmGeneratorTraceFieldPointwiseLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseStdBasisOfEventFieldStdBasis", "htraceFieldPointwiseStdBasis", "hemGeneratorLaplacianEventFieldEqTraceField", "hemGeneratorLaplacianEventFieldStdBasisDef", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseEqLaplacianOfStdBasis", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldEqLaplacianOfPointwise", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle156.remaining_trace_field_pointwise_laplacian" interface := "Remaining exact theorem boundary after cycle 156 lower_2: prove the event/trace field identification hemGeneratorLaplacianEventFieldEqTraceField and the event-field standard-basis source definition hemGeneratorLaplacianEventFieldStdBasisDef for the named frozen EM Laplacian contribution. Keep hemGeneratorLaplacianStateIntegral, htraceFieldMeas, hsourceLaplacianFunctional, hsourceLaplacianFieldMeas, hweakFpSourceFieldPointwiseEqLaplacian, state-event, and sibling EM leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldEqLaplacianOfPointwise", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseEqLaplacianOfStdBasis", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseStdBasisOfEventFieldStdBasis", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "htraceFieldPointwiseEqLaplacian", "htraceFieldPointwiseStdBasis", "hemGeneratorLaplacianEventFieldEqTraceField", "hemGeneratorLaplacianEventFieldStdBasisDef", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle156.reviewer_trace_field_pointwise_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, htraceFieldEqLaplacian is narrowed to htraceFieldPointwiseEqLaplacian through SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldEqLaplacianOfPointwise, htraceFieldPointwiseStdBasis is narrowed to hemGeneratorLaplacianEventFieldEqTraceField plus hemGeneratorLaplacianEventFieldStdBasisDef through SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseStdBasisOfEventFieldStdBasis, stale trace-law/trace-state wrappers are rejected, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, downstream wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle156GeneralMovingTargetDiscreteEmGeneratorTraceFieldPointwiseLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldEqLaplacianOfPointwise", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseStdBasisOfEventFieldStdBasis", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := ["cycle 156 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 157: event-field standard-basis definition from pointwise display -/ /-- Cycle-157 narrowing of the EM Laplacian event-field standard-basis boundary. -/ def cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle157_em_generator_laplacian_event_field_pointwise_stdbasis" statement := "Cycle 157 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfPointwise for appendix.tex:984-995, appendix.tex:1368-1387, and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hemGeneratorLaplacianEventFieldStdBasisDef for the named frozen EM Laplacian event field is no longer primitive; it follows from the smaller pointwise source display hEmGeneratorLaplacianEventFieldPointwiseStdBasisDef by function extensionality over the selected weak-test index. The packet keeps hemGeneratorLaplacianEventFieldEqTraceField, hemGeneratorLaplacianStateIntegral, htraceFieldMeas, hsourceLaplacianFunctional, hsourceLaplacianFieldMeas, hweakFpSourceFieldPointwiseEqLaplacian, state-event, and sibling EM leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfPointwise", "hEmGeneratorLaplacianEventFieldPointwiseStdBasisDef", "hemGeneratorLaplacianEventFieldStdBasisDef", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Illness-area refiner packet for the EM conditional-law/state-event backend. It implements only the requested pointwise-to-field narrowing for the named EM Laplacian event field and does not add a stale htraceFieldEqLaplacian wrapper, downstream total-event/source-functional consumer, SLT import, Lake/toolchain change, theorem-status promotion, fake closure, broad audit, non-EM fallback, or sald_version_2.tex content." /-- Cycle-157 lower_1 scout split for the remaining pointwise event-field display. -/ def cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseLaplacianScoutObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle157_em_generator_laplacian_event_field_pointwise_laplacian_scout" statement := "Cycle 157 lower_1 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisDefOfPointwiseLaplacian. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hEmGeneratorLaplacianEventFieldPointwiseStdBasisDef is no longer primitive; it follows from the smaller source-facing pointwise Delta identity hemGeneratorLaplacianEventFieldEqLaplacian together with SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv. Lower_2 should prove hemGeneratorLaplacianEventFieldEqLaplacian directly from the frozen EM interpolation and Fokker--Planck Laplacian display at appendix.tex:984-995 and appendix.tex:1379-1387, not by reusing the cycle-151/152 source-field or trace-field consumers." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisDefOfPointwiseLaplacian", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "hemGeneratorLaplacianEventFieldEqLaplacian", "hEmGeneratorLaplacianEventFieldPointwiseStdBasisDef", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Natural-language proof scout plus compiled local Mathlib bridge. This narrows only the pointwise standard-basis display to the source Delta identity for the named frozen EM event field. It does not prove the conditional-drift Fokker--Planck theorem, weak-FP source-field equality, event/trace equality, state-event integral formula, SLT import, downstream consumer wrapper, broad audit, theorem-status promotion, fake closure, or sald_version_2.tex content." /-- Cycle-157 lower_2 narrowing of the pointwise event-field Laplacian leaf. -/ def cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseScalarLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle157_em_generator_laplacian_event_field_pointwise_scalar" statement := "Cycle 157 lower_2 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfPointwiseScalar. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hemGeneratorLaplacianEventFieldEqLaplacian is no longer primitive; it follows from the smaller statewise source display hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian : testRegular -> forall phi x, emGeneratorLaplacianEventField phi x = Laplacian.laplacian (selectedTest phi) x. The remaining analytic source boundary is precisely that scalar display, cited to the frozen EM interpolation and Fokker--Planck Laplacian lines at appendix.tex:984-995 and appendix.tex:1379-1387." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfPointwiseScalar", "hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian", "hemGeneratorLaplacianEventFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Lower_2 illness-area refiner for the EM conditional-law/state-event backend. It narrows the exact pointwise Delta identity to a statewise scalar display and explicitly rejects the older cycle-151/152 weak-FP source-field route, trace-field route, downstream total-event/source-functional wrapper, SLT import, Lake/toolchain change, theorem-status promotion, fake closure, broad audit, non-EM fallback, or sald_version_2.tex content." /-- Cycle-157 proof-DAG pane for the event-field pointwise standard-basis split. -/ def cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle157.packet.event_field_stdbasis_from_pointwise" interface := "Compiled theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfPointwise derives hemGeneratorLaplacianEventFieldStdBasisDef from the smaller pointwise standard-basis source display hEmGeneratorLaplacianEventFieldPointwiseStdBasisDef by function extensionality over selected weak tests." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfPointwise", "hEmGeneratorLaplacianEventFieldPointwiseStdBasisDef", "hemGeneratorLaplacianEventFieldStdBasisDef", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseStdBasisOfEventFieldStdBasis", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqSourceFieldOfStdBasisFields", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle157.lower_1_packet.event_field_pointwise_stdbasis_from_laplacian" interface := "Compiled lower_1 theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisDefOfPointwiseLaplacian derives hEmGeneratorLaplacianEventFieldPointwiseStdBasisDef from the smaller pointwise source Delta identity hemGeneratorLaplacianEventFieldEqLaplacian plus SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseLaplacianScoutObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisDefOfPointwiseLaplacian", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "hemGeneratorLaplacianEventFieldEqLaplacian", "hEmGeneratorLaplacianEventFieldPointwiseStdBasisDef", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfPointwise", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle157.lower_2_packet.event_field_laplacian_from_pointwise_scalar" interface := "Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfPointwiseScalar derives hemGeneratorLaplacianEventFieldEqLaplacian from the smaller statewise source display hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian by function extensionality over the state variable." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseScalarLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfPointwiseScalar", "hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian", "hemGeneratorLaplacianEventFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisDefOfPointwiseLaplacian", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle157.remaining_event_field_pointwise_scalar" interface := "Remaining exact theorem boundary after the lower_2 split: prove hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian, the scalar source Delta identity for the named frozen EM Laplacian event field at each selected weak test and state point, directly from the frozen EM interpolation and Fokker--Planck Laplacian display. Do not close it via the cycle-151/152 source-field or trace-field consumers. Keep hemGeneratorLaplacianEventFieldEqTraceField, hemGeneratorLaplacianStateIntegral, htraceFieldMeas, hsourceLaplacianFunctional, hsourceLaplacianFieldMeas, hweakFpSourceFieldPointwiseEqLaplacian, state-event, and sibling EM leaves explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfPointwiseScalar", "hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisDefOfPointwiseLaplacian", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle157.reviewer_event_field_pointwise_stdbasis_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hemGeneratorLaplacianEventFieldStdBasisDef is narrowed to hEmGeneratorLaplacianEventFieldPointwiseStdBasisDef through SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfPointwise, hemGeneratorLaplacianEventFieldEqLaplacian is narrowed to hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian through SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfPointwiseScalar, stale htraceFieldEqLaplacian/downstream total-event wrappers are rejected, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, downstream wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisLowerObligation", "SALD.cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseScalarLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfPointwise", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfPointwiseScalar", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := ["cycle 157 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 158: scalar event-field Delta source audit -/ /-- Cycle-158 middle audit for the remaining scalar event-field Delta identity. This is an explicit wrapper-churn rejection for the current illness area. Lean still sees the named frozen EM Laplacian event field as an abstract field, so a theorem that simply assumes `hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian` and returns an equivalent field identity would not reduce the paper boundary. The next lower packet must instead expose the source definition of the Brownian generator/Fokker--Planck Laplacian event field, or narrow exactly that source definition further. -/ def cycle158GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseScalarAuditObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle158_em_generator_laplacian_event_field_pointwise_scalar_audit" statement := "Cycle 158 middle rejects wrapper churn around hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian. Classification: rejected-wrapper-churn. Exact boundary reviewed: hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian remains the live scalar source Delta identity for the named frozen EM Laplacian event field at each selected weak test and state point. Source-dependency audit classification: source-contract-gap/internal-paper-step. Lean still treats emGeneratorLaplacianEventField as an abstract field, while appendix.tex:984-995 gives the frozen EM interpolation and appendix.tex:1379-1387 gives the Fokker--Planck diffusion term that must define the Brownian-generator Laplacian event field as Laplacian.laplacian (selectedTest phi) x. Valid lower work must prove that source definition directly, or narrow it to one smaller named Brownian-generator event-field definition; wrappers that assume the same scalar identity, weak-FP source-field or trace-field detours, downstream total-event/source-functional consumers, broad audits, theorem-status promotion, fake closure, SLT import, Lake/toolchain change, non-EM fallback, and sald_version_2.tex use are rejected." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian", "emGeneratorLaplacianEventField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfPointwiseScalar", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisDefOfPointwiseLaplacian", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfPointwise", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Middle illness-area packet for the active EM conditional-law/state-event backend. Local SLT was not consulted beyond verifying the configured SLT project is absent; the missing ingredient is the paper-internal source definition of the frozen Brownian generator Laplacian event field, not an external SLT theorem." /-- Cycle-158 lower_2 source-definition boundary after the Brownian event-field split. -/ def cycle158GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle158_em_generator_laplacian_event_field_brownian_def_lower" statement := "Cycle 158 lower_2 narrows hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian to hEmGeneratorLaplacianEventFieldBrownianDef, the source-facing definition that the named frozen EM Laplacian event field is SALD.emFrozenBrownianLaplacianEventField selectedTest. The compiled theorem SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseEqLaplacianOfBrownianDef unfolds that definition to recover the scalar statewise Delta display. The remaining source-cited boundary is to prove the Brownian-generator event-field definition from eq:general_moving_target_SALD_frozen_interp and the Fokker--Planck diffusion term at appendix.tex:1379-1387, without weak-FP source-field, trace-field, total-event/source-functional, or non-EM fallbacks." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.emFrozenBrownianLaplacianEventField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseEqLaplacianOfBrownianDef", "hEmGeneratorLaplacianEventFieldBrownianDef", "hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Lower_2 dynamic-leaf/illness-area refiner. Local SLT files were not consulted because the narrowed boundary is the paper-internal Brownian-generator event-field definition, not a probability/concentration reuse theorem." /-- Cycle-158 proof-DAG pane for the scalar event-field Delta blocker. -/ def cycle158GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseScalarAuditDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle158.middle_packet.scalar_event_field_delta_source_gap" interface := "Source-contract gap: prove hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian by defining the named frozen EM Laplacian event field from the Brownian generator/Fokker--Planck diffusion term, so that for every selected weak test phi and state x it equals Laplacian.laplacian (selectedTest phi) x." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle158GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseScalarAuditObligation", "hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian", "emGeneratorLaplacianEventField", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfPointwiseScalar", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle158.lower_2_packet.scalar_delta_from_brownian_event_field_def" interface := "Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseEqLaplacianOfBrownianDef derives hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian from the smaller source-facing Brownian-generator event-field definition hEmGeneratorLaplacianEventFieldBrownianDef by unfolding SALD.emFrozenBrownianLaplacianEventField." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle158GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefLowerObligation", "SALD.emFrozenBrownianLaplacianEventField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseEqLaplacianOfBrownianDef", "hEmGeneratorLaplacianEventFieldBrownianDef", "hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfPointwiseScalar", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisDefOfPointwiseLaplacian", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle158.remaining_brownian_event_field_def" interface := "Remaining exact source boundary after lower_2: prove hEmGeneratorLaplacianEventFieldBrownianDef, namely that the named frozen EM Laplacian event field is SALD.emFrozenBrownianLaplacianEventField selectedTest, directly from the frozen interpolation and the Fokker--Planck diffusion display." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle158GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefLowerObligation", "SALD.emFrozenBrownianLaplacianEventField", "hEmGeneratorLaplacianEventFieldBrownianDef", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseEqLaplacianOfBrownianDef", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle158.middle_packet.rejected_scalar_delta_wrapper_churn" interface := "Rejected wrapper candidates: any theorem that merely takes hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian, a weak-FP source-field equality, a trace-field equality, or a downstream total-event/source-functional consumer and returns the same scalar event-field Delta identity without exposing the Brownian-generator event-field definition." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle158GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseScalarAuditObligation", "ASTIS.SALD.cycle157.remaining_event_field_pointwise_scalar", "hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := ["cycle 158 lower packet selection", "cycle 158 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 159: Brownian event-field pointwise definition boundary -/ /-- Cycle-159 middle narrowing of the Brownian event-field definition. The remaining cycle-158 source boundary was the function equality `hEmGeneratorLaplacianEventFieldBrownianDef`. This packet exposes the smaller pointwise source display that the frozen Brownian-generator Laplacian event field agrees with `SALD.emFrozenBrownianLaplacianEventField selectedTest` at each selected weak test and state. -/ def cycle159GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle159_em_generator_laplacian_event_field_brownian_pointwise" statement := "Cycle 159 middle compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefOfPointwise. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hEmGeneratorLaplacianEventFieldBrownianDef is no longer primitive; it follows from the smaller pointwise Brownian-generator event-field source display hEmGeneratorLaplacianEventFieldBrownianPointwiseDef : testRegular -> forall phi x, emGeneratorLaplacianEventField phi x = SALD.emFrozenBrownianLaplacianEventField selectedTest phi x. The remaining source-cited boundary is to prove that pointwise definition directly from eq:general_moving_target_SALD_frozen_interp and the Fokker--Planck diffusion display at appendix.tex:1379-1387, without weak-FP source-field, trace-field, downstream total-event/source-functional, or non-EM fallbacks." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefOfPointwise", "SALD.emFrozenBrownianLaplacianEventField", "hEmGeneratorLaplacianEventFieldBrownianPointwiseDef", "hEmGeneratorLaplacianEventFieldBrownianDef", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf middle worker packet inside the EM conditional-law/state-event backend. Local SLT files were not consulted because this is a paper-internal Brownian-generator event-field definition; the bridge uses only function extensionality and the local Brownian event-field definition. It rejects wrapper churn through weak-FP source fields, trace fields, total-event/source-functional consumers, theorem-status promotion, fake closure, Lake/toolchain changes, non-EM fallback, or sald_version_2.tex." /-- Cycle-159 lower_2 standard-basis split of the Brownian pointwise boundary. -/ def cycle159GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle159_em_generator_laplacian_event_field_brownian_pointwise_stdbasis" statement := "Cycle 159 lower_2 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseDefOfStdBasis. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hEmGeneratorLaplacianEventFieldBrownianPointwiseDef is no longer primitive; it follows from the smaller coordinate Brownian-generator Hessian-trace display hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef : testRegular -> forall phi x, emGeneratorLaplacianEventField phi x = sum_i iteratedFDeriv Real 2 (selectedTest phi) x ![e_i,e_i], together with SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv and SALD.emFrozenBrownianLaplacianEventField. The remaining source-cited boundary is to prove this coordinate trace display directly from eq:general_moving_target_SALD_frozen_interp and the Fokker--Planck diffusion display at appendix.tex:1379-1387, without weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM, wrapper, broad-audit, fake-closure, theorem-status-promotion, Lake/toolchain, SLT-import, or sald_version_2 routes." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseDefOfStdBasis", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "SALD.emFrozenBrownianLaplacianEventField", "hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef", "hEmGeneratorLaplacianEventFieldBrownianPointwiseDef", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 worker packet inside the EM conditional-law/state-event backend. Consulted local Mathlib Laplacian support through the existing SALD bridge; no local SLT file was consulted or imported because the remaining theorem is the paper-internal Brownian coordinate trace display." /-- Cycle-159 proof-DAG pane for the Brownian event-field pointwise split. -/ def cycle159GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle159.middle_packet.brownian_event_field_def_from_pointwise" interface := "Compiled theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefOfPointwise derives hEmGeneratorLaplacianEventFieldBrownianDef from the smaller pointwise Brownian-generator event-field display hEmGeneratorLaplacianEventFieldBrownianPointwiseDef by function extensionality over selected tests and states." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle159GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefOfPointwise", "SALD.emFrozenBrownianLaplacianEventField", "hEmGeneratorLaplacianEventFieldBrownianPointwiseDef", "hEmGeneratorLaplacianEventFieldBrownianDef", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseEqLaplacianOfBrownianDef", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfPointwiseScalar", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle159.lower_2_packet.brownian_pointwise_def_from_stdbasis" interface := "Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseDefOfStdBasis derives hEmGeneratorLaplacianEventFieldBrownianPointwiseDef from the smaller coordinate Brownian-generator Hessian-trace display hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef, using SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv to rewrite the trace as SALD.emFrozenBrownianLaplacianEventField selectedTest." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle159GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseDefOfStdBasis", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "SALD.emFrozenBrownianLaplacianEventField", "hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef", "hEmGeneratorLaplacianEventFieldBrownianPointwiseDef", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefOfPointwise", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseEqLaplacianOfBrownianDef", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle159.remaining_brownian_event_field_pointwise_stdbasis_def" interface := "Remaining exact source boundary after the lower_2 split: prove hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef, namely that the named frozen EM Brownian-generator Laplacian event field is the coordinate Hessian trace of each selected weak test at each state point, directly from the frozen interpolation and the Fokker--Planck diffusion display." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle159GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisLowerObligation", "hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseDefOfStdBasis", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefOfPointwise", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle159.reviewer_brownian_event_field_pointwise_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hEmGeneratorLaplacianEventFieldBrownianPointwiseDef is narrowed to hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef through SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseDefOfStdBasis, stale weak-FP source-field/trace-field/consumer wrappers are rejected, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, downstream wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle159GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseLowerObligation", "SALD.cycle159GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseDefOfStdBasis", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefOfPointwise", "SALD.cycle158GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefLowerObligation", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := ["cycle 159 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 160: frozen scalar Brownian Ito generator event-field boundary -/ /-- Cycle-160 middle narrowing of the Brownian coordinate-trace boundary. The remaining cycle-159 source boundary was the pointwise coordinate Hessian-trace display `hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef`. This packet exposes one smaller named stochastic-generator interface for the frozen scalar Brownian increment in `eq:general_moving_target_SALD_frozen_interp`: identify the abstract EM Laplacian event field with `SALD.emFrozenScalarBrownianItoGeneratorEventField selectedTest`. -/ def cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle160_em_generator_laplacian_event_field_frozen_scalar_brownian_ito" statement := "Cycle 160 middle compiles SALD.emFrozenScalarBrownianItoGeneratorEventField and SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDefOfFrozenScalarBrownianItoGenerator. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef is no longer primitive; it follows from the named source-cited stochastic-generator interface hFrozenScalarBrownianItoGeneratorEventFieldDef : testRegular -> emGeneratorLaplacianEventField = SALD.emFrozenScalarBrownianItoGeneratorEventField selectedTest. The remaining source-cited boundary is to prove that named frozen scalar Brownian/Ito generator event-field equality directly from eq:general_moving_target_SALD_frozen_interp and appendix.tex:1379-1387, with the sigma_eta^2/2 coefficient kept outside the event field, and without weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM, wrapper, broad-audit, fake-closure, theorem-status-promotion, Lake/toolchain, SLT-import, or sald_version_2 routes." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.emFrozenScalarBrownianItoGeneratorEventField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDefOfFrozenScalarBrownianItoGenerator", "hFrozenScalarBrownianItoGeneratorEventFieldDef", "hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseDefOfStdBasis", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf / illness-area middle worker packet inside the EM conditional-law/state-event backend. The configured local SLT project path was checked and is absent; no SLT theorem was consulted or imported. The only Mathlib/local ingredient already in scope is the finite-dimensional standard-basis calculus vocabulary, but this cycle's remaining work is the paper-internal frozen scalar Brownian/Ito generator identification." /-- Cycle-160 lower_1 pointwise scout split of the frozen scalar Ito generator boundary. -/ def cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseScoutObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle160_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_pointwise_scout" statement := "Cycle 160 lower_1 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoGeneratorDefOfPointwise. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hFrozenScalarBrownianItoGeneratorEventFieldDef is no longer primitive; it follows from the pointwise stochastic-generator display hFrozenScalarBrownianItoGeneratorEventFieldPointwiseDef : testRegular -> forall phi x, emGeneratorLaplacianEventField phi x = SALD.emFrozenScalarBrownianItoGeneratorEventField selectedTest phi x. The remaining source-cited boundary is to prove that pointwise equality directly from the scalar Brownian increment in eq:general_moving_target_SALD_frozen_interp, the Ito/Taylor generator formula, and the positive FP diffusion display at appendix.tex:1379-1387, with sigma_eta^2/2 kept outside the event field and without weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM, wrapper, broad-audit, fake-closure, theorem-status-promotion, Lake/toolchain, SLT-import, or sald_version_2 routes." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoGeneratorDefOfPointwise", "SALD.emFrozenScalarBrownianItoGeneratorEventField", "hFrozenScalarBrownianItoGeneratorEventFieldPointwiseDef", "hFrozenScalarBrownianItoGeneratorEventFieldDef", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 proof scout inside the EM conditional-law/state-event backend. Consulted appendix.tex:984-995 and appendix.tex:1379-1387 plus the existing Mathlib Laplacian bridge and local Gaussian/covariance inventory. The configured local SLT project path is absent, and no SLT theorem was imported; the remaining dependency is an internal-paper stochastic-generator theorem." /-- Cycle-160 lower_2 coordinate-generator split of the pointwise frozen scalar Ito generator boundary. -/ def cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseCoordinateLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle160_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_pointwise_coordinate" statement := "Cycle 160 lower_2 compiles SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hFrozenScalarBrownianItoGeneratorEventFieldPointwiseDef is no longer primitive; it follows from the smaller coordinate-generator source pair hFrozenScalarBrownianItoEventFieldCoordinateSum : testRegular -> forall phi x, emGeneratorLaplacianEventField phi x = sum_i brownianCoordinateGenerator phi x i and hFrozenScalarBrownianItoCoordinateGeneratorDef : testRegular -> forall phi x i, brownianCoordinateGenerator phi x i = iteratedFDeriv Real 2 (selectedTest phi) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i]. The remaining source-cited boundary is to prove those coordinate decomposition and per-coordinate Ito/Taylor generator facts directly from eq:general_moving_target_SALD_frozen_interp and appendix.tex:1379-1387, with sigma_eta^2/2 kept outside the event field and without weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM, wrapper, broad-audit, fake-closure, theorem-status-promotion, Lake/toolchain, SLT-import, or sald_version_2 routes." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator", "SALD.emFrozenScalarBrownianItoGeneratorEventField", "hFrozenScalarBrownianItoGeneratorEventFieldPointwiseDef", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "hFrozenScalarBrownianItoCoordinateGeneratorDef", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 proof-producing packet inside the EM conditional-law/state-event backend. The compiled theorem is only a finite coordinate-sum assembly from two source-cited stochastic-generator facts; it rejects weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM fallback, broad audit, theorem-status promotion, fake closure, Lake/toolchain change, SLT import, and sald_version_2.tex routes." /-- Cycle-160 proof-DAG pane for the frozen scalar Brownian Ito generator split. -/ def cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle160.middle_packet.brownian_stdbasis_from_frozen_scalar_ito_generator" interface := "Compiled middle theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDefOfFrozenScalarBrownianItoGenerator derives hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef from the named source-cited stochastic-generator interface hFrozenScalarBrownianItoGeneratorEventFieldDef by unfolding SALD.emFrozenScalarBrownianItoGeneratorEventField." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLowerObligation", "SALD.emFrozenScalarBrownianItoGeneratorEventField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDefOfFrozenScalarBrownianItoGenerator", "hFrozenScalarBrownianItoGeneratorEventFieldDef", "hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseDefOfStdBasis", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefOfPointwise", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle160.remaining_frozen_scalar_brownian_ito_generator_event_field_def" interface := "Compiled lower_1 theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoGeneratorDefOfPointwise derives hFrozenScalarBrownianItoGeneratorEventFieldDef from the smaller pointwise stochastic-generator display hFrozenScalarBrownianItoGeneratorEventFieldPointwiseDef by function extensionality over selected weak tests and state points." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseScoutObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoGeneratorDefOfPointwise", "hFrozenScalarBrownianItoGeneratorEventFieldDef", "hFrozenScalarBrownianItoGeneratorEventFieldPointwiseDef", "SALD.emFrozenScalarBrownianItoGeneratorEventField", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDefOfFrozenScalarBrownianItoGenerator", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle160.lower_2_packet.frozen_scalar_ito_pointwise_from_coordinate_generator" interface := "Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator derives hFrozenScalarBrownianItoGeneratorEventFieldPointwiseDef from the smaller source-cited coordinate-generator pair hFrozenScalarBrownianItoEventFieldCoordinateSum plus hFrozenScalarBrownianItoCoordinateGeneratorDef by finite-sum congruence and unfolding SALD.emFrozenScalarBrownianItoGeneratorEventField." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseCoordinateLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator", "SALD.emFrozenScalarBrownianItoGeneratorEventField", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "hFrozenScalarBrownianItoCoordinateGeneratorDef", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoGeneratorDefOfPointwise", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDefOfFrozenScalarBrownianItoGenerator", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle160.remaining_frozen_scalar_brownian_ito_coordinate_generator" interface := "Remaining exact source boundary after lower_2: prove hFrozenScalarBrownianItoEventFieldCoordinateSum, identifying the abstract EM Laplacian event field as the sum of frozen Brownian coordinate generator contributions, and hFrozenScalarBrownianItoCoordinateGeneratorDef, identifying each coordinate contribution with the corresponding diagonal second derivative. The proof route is the scalar Brownian/Ito generator for the frozen increment in eq:general_moving_target_SALD_frozen_interp: the drift contributes to the first-order transport field, the Brownian increment has covariance sigma_eta^2 I per unit s-time, and the second-order Taylor/Ito term gives the coordinate Hessian trace while the coefficient sigma_eta^2/2 remains outside this event field in the weak-FP action." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseCoordinateLowerObligation", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "hFrozenScalarBrownianItoCoordinateGeneratorDef", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "Mathlib.Probability.Distributions.Gaussian.Real", "Mathlib.Probability.Moments.CovarianceBilin", "Mathlib.Analysis.InnerProductSpace.Laplacian" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoGeneratorDefOfPointwise", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDefOfFrozenScalarBrownianItoGenerator", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle160.reviewer_frozen_scalar_brownian_ito_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef is narrowed to hFrozenScalarBrownianItoGeneratorEventFieldDef through the compiled middle theorem, then to hFrozenScalarBrownianItoGeneratorEventFieldPointwiseDef through the compiled lower_1 theorem, and then to hFrozenScalarBrownianItoEventFieldCoordinateSum plus hFrozenScalarBrownianItoCoordinateGeneratorDef through the compiled lower_2 coordinate-generator theorem. Stale weak-FP source-field/trace-field/consumer wrappers are rejected, and python3 tools/astis.py check passes without SLT import, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, downstream wrapper churn, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLowerObligation", "SALD.cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseScoutObligation", "SALD.cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseCoordinateLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDefOfFrozenScalarBrownianItoGenerator", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoGeneratorDefOfPointwise", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator", "SALD.cycle159GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisLowerObligation", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "hFrozenScalarBrownianItoCoordinateGeneratorDef", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := ["cycle 160 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 161: per-coordinate scalar Brownian Ito Taylor boundary -/ /-- Cycle-161 narrowing of the per-coordinate frozen scalar Brownian Ito generator. The remaining cycle-160 coordinate-generator pair included the supplied per-coordinate theorem `hFrozenScalarBrownianItoCoordinateGeneratorDef`. This packet exposes the smaller one-dimensional Brownian/Ito Taylor boundary along a fixed standard Brownian coordinate. The sibling coordinate-sum theorem remains explicit. -/ def cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle161_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_coordinate_generator" statement := "Cycle 161 middle compiles SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator and SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hFrozenScalarBrownianItoCoordinateGeneratorDef is no longer primitive; it follows from the smaller source-cited one-dimensional Brownian/Ito Taylor boundary hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor : testRegular -> forall phi x i, brownianCoordinateGenerator phi x i = SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest phi) x ((stdOrthonormalBasis Real E) i). The remaining source work is to prove that one-dimensional coordinate contribution directly from the scalar Brownian increment in eq:general_moving_target_SALD_frozen_interp, the Brownian second moment/covariance, and the Taylor/Ito expansion over appendix.tex:984-995 and appendix.tex:1379-1387. The sibling hFrozenScalarBrownianItoEventFieldCoordinateSum remains explicit; sigma_eta^2/2 stays outside the event field. Weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM, broad-audit, wrapper-churn, fake-closure, theorem-status-promotion, Lake/toolchain, SLT-import, and sald_version_2 routes are rejected." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor", "hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor", "hFrozenScalarBrownianItoCoordinateGeneratorDef", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "SALD.gaussianRealZeroSecondMoment", "Mathlib.Probability.Distributions.Gaussian.Real", "Mathlib.Probability.Moments.CovarianceBilin", "Mathlib.Analysis.InnerProductSpace.Laplacian", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf middle worker packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib Gaussian.Real, CovarianceBilin, and Laplacian files for available Brownian/covariance/calculus vocabulary; the configured local SLT clone is absent, so no SLT theorem was imported or marked formalized. The compiled theorem is a narrow source-facing bridge from the old coordinate-generator supplied hypothesis to a one-dimensional Taylor boundary, not a weak-FP source-field, trace-field, total-event/source-functional, non-EM, or route-audit wrapper." /-- Cycle-161 lower_1 scout route below the one-dimensional Brownian/Ito Taylor boundary. -/ def cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorScoutObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle161_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_one_dim_taylor_scout" statement := "Cycle 161 lower_1 proof-scout packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor should no longer treat the scalar Brownian second moment as open; SALD.gaussianRealZeroSecondMoment compiles the centered real Gaussian second moment needed for one Brownian coordinate. The remaining smaller source-cited theorem is the stochastic Taylor/generator passage hFrozenScalarBrownianItoOneDimTaylorExpansion : testRegular -> forall phi x i, the source-defined brownianCoordinateGenerator for the scalar increment sigma_eta (W_s-W_{s_k}) has derivative-per-unit-time contribution SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest phi) x ((stdOrthonormalBasis Real E) i), using selected-test C^2 regularity, zero first moment for the centered Brownian coordinate, the compiled second moment, and drift separation into the first-order conditional drift field. The coefficient sigma_eta^2/2 remains in the surrounding weak-FP action; hFrozenScalarBrownianItoEventFieldCoordinateSum remains explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.gaussianRealZeroSecondMoment", "hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor", "hFrozenScalarBrownianItoOneDimTaylorExpansion", "SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator", "ProbabilityTheory.integral_id_gaussianReal", "ProbabilityTheory.variance_id_gaussianReal", "ProbabilityTheory.variance_eq_integral", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "Mathlib.Probability.Distributions.Gaussian.Real", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 proof scout inside the scalar Brownian/Ito illness area. Consulted Mathlib Gaussian.Real for mean/variance of real Gaussian coordinates and the existing local Laplacian bridge; CovarianceBilin remains a future vector-covariance option, but the current one-dimensional leaf only needs the scalar centered second moment. The configured local SLT clone is absent, and no SLT theorem was imported or marked formalized." /-- Cycle-161 lower_2 moment-algebra narrowing below the one-dimensional Taylor boundary. -/ def cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorMomentLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle161_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_one_dim_taylor_moment_lower" statement := "Cycle 161 lower_2 packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hFrozenScalarBrownianItoOneDimTaylorExpansion should no longer treat the centered scalar Gaussian first/second moment algebra as open; SALD.gaussianRealZeroOneDimTaylorMomentContribution compiles the local fact that a linear Taylor coefficient times E[Z] plus a quadratic coefficient times E[Z^2] collapses to the quadratic coefficient times the variance for Z ~ gaussianReal 0 v. The remaining smaller source-cited theorem is hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit: under selected-test C^2 regularity, integrability, and dominated Taylor remainder hypotheses, the source-defined brownianCoordinateGenerator for sigma_eta (W_s-W_{s_k}) has derivative-per-unit-time contribution SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator after drift separation. The coefficient sigma_eta^2/2 remains outside this event field, and hFrozenScalarBrownianItoEventFieldCoordinateSum remains explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.gaussianRealZeroOneDimTaylorMomentContribution", "SALD.gaussianRealZeroSecondMoment", "ProbabilityTheory.integral_id_gaussianReal", "hFrozenScalarBrownianItoOneDimTaylorExpansion", "hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit", "SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "Mathlib.Probability.Distributions.Gaussian.Real", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 worker packet inside the scalar Brownian/Ito illness area. It uses only Mathlib Gaussian.Real moment facts already imported in SALD, with no local SLT clone or SLT import. This is not a weak-FP source-field, trace-field, total-event/source-functional, non-EM, route-audit, or wrapper-churn packet; it leaves the analytic dominated Taylor/generator limit explicit." /-- Cycle-161 proof-DAG pane for the one-dimensional coordinate Taylor split. -/ def cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle161.middle_packet.coordinate_generator_from_one_dim_taylor" interface := "Compiled theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor derives hFrozenScalarBrownianItoCoordinateGeneratorDef from the smaller source-cited one-dimensional Brownian/Ito Taylor boundary hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor by unfolding SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator along the standard Brownian coordinate." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorLowerObligation", "SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor", "hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor", "hFrozenScalarBrownianItoCoordinateGeneratorDef", "SALD.gaussianRealZeroSecondMoment", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle161.lower_2_packet.one_dim_taylor_moment_collapse" interface := "Compiled lower_2 theorem: SALD.gaussianRealZeroOneDimTaylorMomentContribution collapses the scalar Taylor linear/quadratic Gaussian moment contribution using ProbabilityTheory.integral_id_gaussianReal and SALD.gaussianRealZeroSecondMoment. The remaining source-cited gap below hFrozenScalarBrownianItoOneDimTaylorExpansion is now the dominated Taylor remainder and generator-limit theorem hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit, not the centered moment algebra." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorMomentLowerObligation", "SALD.gaussianRealZeroOneDimTaylorMomentContribution", "SALD.gaussianRealZeroSecondMoment", "ProbabilityTheory.integral_id_gaussianReal", "hFrozenScalarBrownianItoOneDimTaylorExpansion", "hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle161.remaining_one_dim_brownian_ito_taylor_boundary" interface := "Remaining exact source boundary after the cycle-161 lower_2 moment split: prove hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor, identifying the scalar Brownian coordinate contribution with SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator for the selected weak test and standard Brownian basis direction. Cycle 161 lower_1 compiles SALD.gaussianRealZeroSecondMoment, and lower_2 compiles SALD.gaussianRealZeroOneDimTaylorMomentContribution, so the remaining smaller source-cited gap is the dominated Taylor remainder and generator-limit passage hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit under selected-test C^2/integrability hypotheses and drift separation; sigma_eta^2/2 remains outside this event field." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorScoutObligation", "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorMomentLowerObligation", "hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor", "hFrozenScalarBrownianItoOneDimTaylorExpansion", "hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit", "SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator", "SALD.gaussianRealZeroSecondMoment", "SALD.gaussianRealZeroOneDimTaylorMomentContribution", "Mathlib.Probability.Distributions.Gaussian.Real", "Mathlib.Probability.Moments.CovarianceBilin", "Mathlib.Analysis.InnerProductSpace.Laplacian", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle161.residual_coordinate_sum_boundary" interface := "Residual sibling source boundary: hFrozenScalarBrownianItoEventFieldCoordinateSum remains explicit and should only be discharged by a source proof that the abstract EM Laplacian event field is the sum of scalar Brownian coordinate generator contributions. This cycle does not hide it behind a downstream weak-FP or total-event consumer." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "hFrozenScalarBrownianItoEventFieldCoordinateSum", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle161.reviewer_one_dim_taylor_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hFrozenScalarBrownianItoCoordinateGeneratorDef is narrowed to hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor through the compiled cycle-161 theorem, hFrozenScalarBrownianItoEventFieldCoordinateSum remains explicit, and python3 tools/astis.py check passes without weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, SLT import, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorLowerObligation", "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorScoutObligation", "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorMomentLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor", "SALD.gaussianRealZeroSecondMoment", "SALD.gaussianRealZeroOneDimTaylorMomentContribution", "hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := ["cycle 161 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 162: scalar Brownian Ito Taylor remainder split -/ /-- Cycle-162 narrowing below the one-dimensional Brownian/Ito Taylor boundary. The remaining cycle-161 source theorem `hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit` is no longer treated as one opaque generator equality. The compiled bridge below consumes the Gaussian moment collapse and leaves only the source Taylor moment decomposition, quadratic-variation normalization, and normalized Taylor-remainder vanishing as explicit analytic obligations. -/ def cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorRemainderLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle162_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_taylor_remainder" statement := "Cycle 162 middle packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit should no longer be an opaque one-dimensional Brownian/Ito generator theorem. SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder compiles the algebraic handoff from a source Taylor moment decomposition, Brownian quadratic-variation normalization, and vanishing normalized Taylor remainder to hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor, using SALD.gaussianRealZeroOneDimTaylorMomentContribution. The remaining smaller source-cited theorem is hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes together with its exact Taylor moment decomposition and quadratic-variation normalization hypotheses under selected-test C^2/integrability assumptions. The sibling hFrozenScalarBrownianItoEventFieldCoordinateSum remains explicit; sigma_eta^2/2 stays outside the event field. Weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM fallback, wrapper churn, broad audit, theorem-status promotion, fake closure, Lake/toolchain change, SLT import, and sald_version_2.tex routes are rejected." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder", "SALD.gaussianRealZeroOneDimTaylorMomentContribution", "SALD.gaussianRealZeroSecondMoment", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit", "hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator", "ProbabilityTheory.integral_id_gaussianReal", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "Mathlib.Probability.Distributions.Gaussian.Real", "Mathlib.Analysis.Calculus.Taylor", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet inside the scalar Brownian/Ito illness area. Consulted Mathlib Gaussian.Real for the moment inputs already imported locally; inspected Mathlib Taylor/FDeriv file availability for the remaining analytic remainder boundary. The local SLT clone is absent and no SLT theorem was imported or marked formalized." /-- Cycle-162 lower_1 scout route for the normalized scalar Taylor remainder. -/ def cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderScoutObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle162_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_normalized_remainder_scout" statement := "Cycle 162 lower_1 proof-scout packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes should be implemented as a dominated-convergence theorem for the normalized scalar Taylor remainder integral, separate from hFrozenScalarBrownianItoTaylorMomentDecomposition and hFrozenScalarBrownianItoQuadraticVariationNormalization. Lower_2-ready theorem shape: SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT proves Tendsto (fun h => integral z, normalizedRemainder h z d gaussianReal 0 v) (nhdsWithin 0 (Set.Ioi 0)) (nhds 0) from eventual AEStronglyMeasurable, eventual ae domination by an Integrable bound, and ae pointwise convergence of normalizedRemainder h z to 0. The follow-up source Taylor step should supply that pointwise convergence for r |-> selectedTest phi (x + r • e_i) using Real.taylor_tendsto or taylor_isLittleO under selected-test C^2/ContDiffOn hypotheses. The sibling hFrozenScalarBrownianItoEventFieldCoordinateSum remains explicit; sigma_eta^2/2 stays outside the event field." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.obligation dependsOn := [ "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder", "MeasureTheory.tendsto_integral_filter_of_dominated_convergence", "Real.taylor_tendsto", "taylor_isLittleO", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Analysis.Calculus.Taylor", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 proof scout inside the scalar Brownian/Ito illness area. Consulted Mathlib Taylor for the scalar path little-o theorem and Mathlib dominated convergence for the integral-limit handoff; the configured local SLT clone is absent, and no SLT theorem was imported or marked formalized." /-- Cycle-162 lower_2 compiled DCT theorem for the normalized scalar Taylor remainder. -/ def cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderDctLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle162_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_normalized_remainder_dct_lower" statement := "Cycle 162 lower_2 packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes no longer needs to keep the dominated-convergence integral theorem open. SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT compiles the Mathlib DCT step for the normalized scalar Taylor remainder integral under ProbabilityTheory.gaussianReal 0 v from eventual AEStronglyMeasurable, eventual a.e. domination by an integrable bound, and a.e. pointwise convergence to zero along nhdsWithin 0 (Set.Ioi 0). The remaining smaller source-cited theorem is the source Taylor pointwise limit for the concrete selected-test scalar path, plus the separate hFrozenScalarBrownianItoTaylorMomentDecomposition and hFrozenScalarBrownianItoQuadraticVariationNormalization hypotheses. The sibling hFrozenScalarBrownianItoEventFieldCoordinateSum remains explicit; sigma_eta^2/2 stays outside the event field." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.formalized dependsOn := [ "SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT", "MeasureTheory.tendsto_integral_filter_of_dominated_convergence", "MeasureTheory.AEStronglyMeasurable", "MeasureTheory.Integrable", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "Real.taylor_tendsto", "taylor_isLittleO", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Analysis.Calculus.Taylor", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 proof-producing packet inside the scalar Brownian/Ito illness area. Consulted Mathlib dominated convergence for the compiled integral-limit theorem; Mathlib Taylor remains cited for the next pointwise source step. The configured local SLT clone is absent, and no SLT theorem was imported or marked formalized." /-- Cycle-162 proof-DAG pane for the scalar Taylor remainder split. -/ def cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorRemainderDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle162.middle_packet.one_dim_taylor_from_moment_remainder" interface := "Compiled theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder derives hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor from a coordinate Taylor moment decomposition, quadratic-variation normalization, and vanishing normalized Taylor remainder, using SALD.gaussianRealZeroOneDimTaylorMomentContribution." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorRemainderLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder", "SALD.gaussianRealZeroOneDimTaylorMomentContribution", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle162.remaining_normalized_taylor_remainder_boundary" interface := "Remaining exact source boundary after cycle 162: prove the selected-test second-order Taylor moment decomposition for one scalar Brownian coordinate, the Brownian quadratic-variation normalization after factoring sigma_eta^2/2 outside the event field, and the vanishing normalized Taylor remainder under C^2/integrability/dominated-convergence hypotheses." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "Mathlib.Analysis.Calculus.Taylor", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle162.lower_1_packet.normalized_taylor_remainder_dct_scout" interface := "Lower_1 proof-scout refinement: the next lower_2-ready theorem should target only the normalized Taylor remainder integral, not the whole scalar Brownian/Ito generator theorem. Expected compiled shape: SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT, proving Tendsto (fun h => integral z, normalizedRemainder h z d gaussianReal 0 v) (nhdsWithin 0 (Set.Ioi 0)) (nhds 0) from eventual AEStronglyMeasurable, an eventual ae bound by an Integrable function, and ae pointwise Tendsto of normalizedRemainder h z to 0. The source C^2 Taylor step should then supply that pointwise limit via Mathlib Real.taylor_tendsto/taylor_isLittleO for r |-> selectedTest phi (x + r • e_i), while hFrozenScalarBrownianItoTaylorMomentDecomposition and hFrozenScalarBrownianItoQuadraticVariationNormalization remain separate source bookkeeping hypotheses." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderScoutObligation", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder", "MeasureTheory.tendsto_integral_filter_of_dominated_convergence", "Real.taylor_tendsto", "taylor_isLittleO", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Analysis.Calculus.Taylor", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle162.lower_2_packet.normalized_taylor_remainder_dct" interface := "Compiled lower_2 theorem: SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT proves the normalized scalar Taylor remainder integral tends to zero under gaussianReal 0 v from eventual AEStronglyMeasurable, eventual a.e. domination by an integrable bound, and a.e. pointwise convergence to zero along nhdsWithin 0 (Set.Ioi 0). The remaining source theorem is the concrete selected-test scalar Taylor pointwise limit supplying hPoint, while hFrozenScalarBrownianItoTaylorMomentDecomposition and hFrozenScalarBrownianItoQuadraticVariationNormalization remain separate." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderDctLowerObligation", "SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT", "MeasureTheory.tendsto_integral_filter_of_dominated_convergence", "MeasureTheory.AEStronglyMeasurable", "MeasureTheory.Integrable", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle162.residual_coordinate_sum_boundary" interface := "Residual sibling boundary: hFrozenScalarBrownianItoEventFieldCoordinateSum remains explicit and is not changed by the scalar Taylor-remainder split." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "hFrozenScalarBrownianItoEventFieldCoordinateSum", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle162.reviewer_taylor_remainder_check" interface := "Reviewer check: accept only if the packet is classified narrows-source-cited-boundary, the remaining hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit boundary is split into a smaller normalized Taylor-remainder/decomposition theorem, hFrozenScalarBrownianItoEventFieldCoordinateSum stays explicit, and python3 tools/astis.py check passes." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorRemainderLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder", "hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := ["cycle 162 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 163: source Taylor pointwise limit for the scalar remainder -/ /-- Cycle-163 lower packet for the selected-test scalar Taylor pointwise limit. The refreshed blueprint after cycle 162 names the source-specific `hPoint` input of `gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT` as the next dynamic leaf. This packet compiles the Mathlib Taylor step for the one-dimensional line `r ↦ sourceTest (x + r • e)` and keeps the remaining paper bookkeeping separate. -/ def cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseTaylorLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle163_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_pointwise_taylor" statement := "Cycle 163 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the concrete source Taylor pointwise hPoint input below SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT is no longer open for the selected scalar line r |-> sourceTest (x + r • e). SALD.gaussianRealSelectedTestLineSecondOrderTaylorRemainderPointwiseAE compiles the a.e. Gaussian pointwise Tendsto statement from ContDiffOn Real 2 of that line and Mathlib Real.taylor_tendsto, with scalar coordinate scaling r = h*z. Lower_1 also compiles SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfSelectedTestLineEq, which turns an eventual a.e. source equality between the paper's normalizedRemainder and that scaled line remainder into the DCT integral limit. Remaining source-cited inputs are the source equality hSourceEq, the event/measurable domination hypotheses hMeas/hBound/hBoundInt, hFrozenScalarBrownianItoTaylorMomentDecomposition, hFrozenScalarBrownianItoQuadraticVariationNormalization, and the sibling hFrozenScalarBrownianItoEventFieldCoordinateSum. Weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM fallback, wrapper churn, broad audit, theorem-status promotion, fake closure, Lake/toolchain change, SLT import, and sald_version_2.tex routes are rejected." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.formalized dependsOn := [ "SALD.gaussianRealSelectedTestLineSecondOrderTaylorRemainderPointwiseAE", "SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfSelectedTestLineEq", "SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT", "Real.taylor_tendsto", "Filter.Tendsto.congr'", "taylorWithinEval", "hSourceEq", "hPoint", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "Mathlib.Analysis.Calculus.Taylor", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib Analysis.Calculus.Taylor and the already compiled DCT theorem; the configured local SLT clone is absent, and no SLT theorem was imported, ported, or marked formalized." /-- Cycle-163 lower_2 packet for the normalized-remainder source equality. -/ def cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderSourceEqLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle163_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_normalized_remainder_source_eq" statement := "Cycle 163 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hSourceEq input below SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfSelectedTestLineEq is no longer open for the concrete selected-test scalar Taylor normalizedRemainder. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainder defines the source-shaped expression, SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderSourceEq proves the eventual Gaussian-a.e. source equality by definitional equality, and SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero derives the DCT integral limit without an hSourceEq premise. Remaining source-cited inputs are hMeas, hBound, and hBoundInt for this concrete remainder, plus hFrozenScalarBrownianItoTaylorMomentDecomposition, hFrozenScalarBrownianItoQuadraticVariationNormalization, and the sibling hFrozenScalarBrownianItoEventFieldCoordinateSum. Weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM fallback, wrapper churn, broad audit, theorem-status promotion, fake closure, Lake/toolchain change, SLT import, and sald_version_2.tex routes are rejected." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.formalized dependsOn := [ "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainder", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderSourceEq", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero", "SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfSelectedTestLineEq", "SALD.gaussianRealSelectedTestLineSecondOrderTaylorRemainderPointwiseAE", "SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT", "taylorWithinEval", "Filter.Eventually.of_forall", "hMeas", "hBound", "hBoundInt", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "Mathlib.Analysis.Calculus.Taylor", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 packet inside the scalar Brownian/Ito illness area. It discharges the source-equality bridge for the source-shaped selected-test normalized remainder and leaves only concrete DCT regularity/domination plus the separate moment, quadratic-variation, and coordinate-sum source leaves. Local Mathlib Taylor/DCT APIs were already imported; the configured local SLT clone is absent, and no SLT theorem was imported, ported, or marked formalized." /-- Cycle-163 proof-DAG pane for the source Taylor pointwise limit. -/ def cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseTaylorDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle163.lower_packet.selected_test_line_taylor_hpoint" interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineSecondOrderTaylorRemainderPointwiseAE supplies the a.e. Gaussian pointwise Tendsto hPoint for the scaled selected-test line r |-> sourceTest (x + r • e), using ContDiffOn Real 2 and Mathlib Real.taylor_tendsto." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseTaylorLowerObligation", "SALD.gaussianRealSelectedTestLineSecondOrderTaylorRemainderPointwiseAE", "SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT", "Real.taylor_tendsto", "Mathlib.Analysis.Calculus.Taylor", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle163.lower_1_packet.normalized_remainder_source_eq_to_dct" interface := "Compiled lower_1 scout bridge: SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfSelectedTestLineEq consumes an eventual Gaussian-a.e. source equality hSourceEq between normalizedRemainder and the scaled selected-test line Taylor remainder, then derives the DCT integral limit from the compiled pointwise Taylor theorem and SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfSelectedTestLineEq", "SALD.gaussianRealSelectedTestLineSecondOrderTaylorRemainderPointwiseAE", "SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT", "Filter.Tendsto.congr'", "hSourceEq", "hMeas", "hBound", "hBoundInt", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle163.lower_2_packet.normalized_remainder_source_eq" interface := "Compiled lower_2 bridge: SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderSourceEq supplies the hSourceEq input by definitional equality for the concrete source-shaped selected-test normalized remainder, and SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero derives the DCT integral limit without an hSourceEq premise." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderSourceEqLowerObligation", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainder", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderSourceEq", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero", "SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfSelectedTestLineEq", "taylorWithinEval", "Filter.Eventually.of_forall", "hMeas", "hBound", "hBoundInt", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle163.remaining_normalized_remainder_source_identification" interface := "Remaining exact theorem after the lower_2 source-shaped remainder bridge: provide eventual AEStronglyMeasurable and domination/integrability for the concrete selected-test normalized remainder, and separately prove the Taylor moment decomposition plus Brownian quadratic-variation normalization." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.gaussianRealSelectedTestLineSecondOrderTaylorRemainderPointwiseAE", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero", "SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT", "hMeas", "hBound", "hBoundInt", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle163.reviewer_pointwise_taylor_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, the compiled theorem targets the source Taylor hPoint or the concrete normalized-remainder source equality below the normalized remainder DCT theorem, the Taylor moment and quadratic-variation hypotheses remain explicit, and python3 tools/astis.py check passes." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseTaylorLowerObligation", "SALD.cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderSourceEqLowerObligation", "SALD.gaussianRealSelectedTestLineSecondOrderTaylorRemainderPointwiseAE", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderSourceEq", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero", "SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := ["cycle 163 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 164: measurability for the concrete normalized remainder -/ /-- Cycle-164 lower packet for the concrete selected-test remainder `hMeas`. The refreshed blueprint and upper handoff select only the eventual `AEStronglyMeasurable` input below the concrete DCT theorem. This packet discharges that supplied hypothesis from the selected one-dimensional line regularity; domination and integrability remain separate source-cited inputs. -/ def cycle164GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderMeasLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle164_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_normalized_remainder_meas" statement := "Cycle 164 dynamic-leaf worker packet. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: the hMeas input to SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero for the concrete selected-test scalar normalized Taylor remainder. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable compiles eventual AEStronglyMeasurable under ProbabilityTheory.gaussianReal 0 v from the selected scalar line regularity ContDiffOn Real 2 (fun r => sourceTest (x + r • e)) Set.univ, using ContDiffOn.continuousOn, continuity of taylorWithinEval in the evaluation variable via hasDerivAt_taylorWithinEval_succ, Continuous.comp_aestronglyMeasurable, and AEStronglyMeasurable arithmetic closure. Remaining source-cited inputs are hBound and hBoundInt for the concrete normalized remainder, plus hFrozenScalarBrownianItoTaylorMomentDecomposition, hFrozenScalarBrownianItoQuadraticVariationNormalization, and hFrozenScalarBrownianItoEventFieldCoordinateSum. Weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM fallback, wrapper churn, broad audit, theorem-status promotion, fake closure, Lake/toolchain change, SLT import, and sald_version_2.tex routes are rejected." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.formalized dependsOn := [ "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainder", "ContDiffOn.continuousOn", "hasDerivAt_taylorWithinEval_succ", "Continuous.comp_aestronglyMeasurable", "MeasureTheory.AEStronglyMeasurable.sub", "MeasureTheory.AEStronglyMeasurable.div₀", "MeasureTheory.AEStronglyMeasurable.mul", "MeasureTheory.AEStronglyMeasurable.pow", "hMeas", "hBound", "hBoundInt", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable", "Mathlib.Analysis.Calculus.Taylor", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet inside the scalar Brownian/Ito illness area. Local Mathlib consulted through existing imports: Analysis.Calculus.Taylor for taylorWithinEval/hasDerivAt_taylorWithinEval_succ and MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable for composition/arithmetic closure. The configured local SLT clone is absent, and no SLT theorem was imported, ported, or marked formalized." /-- Cycle-164 lower_2 packet for the concrete selected-test quadratic bound `hBoundInt`. Once the source Taylor domination leaf is stated with the quadratic Gaussian bound `fun z => C * z ^ 2`, this packet discharges the DCT integrability input using Mathlib's Gaussian exponential-moment theorem. The pointwise domination `hBound` remains the next source Taylor estimate. -/ def cycle164GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundIntLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle164_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_normalized_remainder_quadratic_bound_int" statement := "Cycle 164 lower_2 dynamic-leaf worker packet. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: the hBoundInt input to SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero for the quadratic Gaussian domination bound fun z => C * z ^ 2. SALD.gaussianRealSelectedTestLineSecondOrderQuadraticBoundIntegrable compiles MeasureTheory.Integrable (fun z : Real => C * z ^ 2) (ProbabilityTheory.gaussianReal 0 v) from ProbabilityTheory.integrable_exp_mul_gaussianReal and ProbabilityTheory.integrable_pow_of_integrable_exp_mul. The remaining source-cited DCT input is hBound, namely the pointwise domination of the concrete selected-test normalized remainder by that quadratic bound; Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves remain separate. Weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM fallback, wrapper churn, broad audit, theorem-status promotion, fake closure, Lake/toolchain change, SLT import, and sald_version_2.tex routes are rejected." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.formalized dependsOn := [ "SALD.gaussianRealSelectedTestLineSecondOrderQuadraticBoundIntegrable", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero", "ProbabilityTheory.integrable_exp_mul_gaussianReal", "ProbabilityTheory.integrable_pow_of_integrable_exp_mul", "MeasureTheory.Integrable.const_mul", "hBound", "hBoundInt", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "Mathlib.Probability.Distributions.Gaussian.Real", "Mathlib.Probability.Moments.IntegrableExpMul", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet inside the scalar Brownian/Ito illness area. It proves only Gaussian quadratic-bound integrability for the DCT hBoundInt input after the intended bound has the source shape C * z^2. The configured local SLT clone is absent, and no SLT theorem was imported, ported, or marked formalized." /-- Cycle-164 proof-DAG pane for the concrete normalized-remainder measurability leaf. -/ def cycle164GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderMeasDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle164.lower_packet.normalized_remainder_hmeas" interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable supplies the eventual AEStronglyMeasurable hMeas input for the concrete selected-test normalized Taylor remainder from ContDiffOn Real 2 line regularity." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle164GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderMeasLowerObligation", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainder", "ContDiffOn.continuousOn", "hasDerivAt_taylorWithinEval_succ", "Continuous.comp_aestronglyMeasurable", "MeasureTheory.AEStronglyMeasurable.div₀", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle164.lower_2_packet.normalized_remainder_quadratic_bound_integrable" interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineSecondOrderQuadraticBoundIntegrable supplies the DCT hBoundInt input for the quadratic Gaussian bound fun z => C * z ^ 2, using Mathlib Gaussian exponential moments and integrable powers. The remaining domination leaf is hBound for the concrete selected-test normalized remainder against that bound." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle164GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundIntLowerObligation", "SALD.gaussianRealSelectedTestLineSecondOrderQuadraticBoundIntegrable", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero", "ProbabilityTheory.integrable_exp_mul_gaussianReal", "ProbabilityTheory.integrable_pow_of_integrable_exp_mul", "MeasureTheory.Integrable.const_mul", "Mathlib.Probability.Distributions.Gaussian.Real", "Mathlib.Probability.Moments.IntegrableExpMul", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle164.remaining_normalized_remainder_domination" interface := "Remaining exact theorem after the hMeas and quadratic hBoundInt discharge: provide hBound for the concrete selected-test normalized remainder against the quadratic Gaussian bound fun z => C * z ^ 2, and separately prove the Taylor moment decomposition, Brownian quadratic-variation normalization, and coordinate-sum leaves." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable", "SALD.gaussianRealSelectedTestLineSecondOrderQuadraticBoundIntegrable", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero", "hBound", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle164.reviewer_hmeas_check" interface := "Reviewer check: accept only if the packet classification is discharges-supplied-hypothesis, the concrete hMeas premise is eliminated by a compiled theorem rather than renamed, hBound/hBoundInt and the moment/normalization/coordinate-sum leaves remain explicit, and python3 tools/astis.py check passes." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle164GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderMeasLowerObligation", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := ["cycle 164 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 165: quadratic domination for the concrete normalized remainder -/ /-- Cycle-165 lower packet for the concrete selected-test remainder `hBound`. The refreshed blueprint after cycle 164 leaves the pointwise domination input as the next dynamic leaf. This packet compiles the local algebraic bridge from a deterministic scalar Taylor quotient bound to the exact eventual a.e. quadratic domination shape consumed by the DCT theorem. -/ def cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle165_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_normalized_remainder_quadratic_bound" statement := "Cycle 165 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hBound input to SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero for the concrete selected-test scalar normalized Taylor remainder is no longer primitive once a deterministic scalar Taylor quotient estimate is supplied. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound compiles that bridge: from forall r, the norm of (sourceTest (x + r • e) - taylorWithinEval (fun r => sourceTest (x + r • e)) 2 Set.univ 0 r) / r^2 is at most C, it produces the eventual Gaussian-a.e. domination by fun z => C * z^2. Remaining source-cited work is the deterministic Taylor quotient bound itself, plus hFrozenScalarBrownianItoTaylorMomentDecomposition, hFrozenScalarBrownianItoQuadraticVariationNormalization, and hFrozenScalarBrownianItoEventFieldCoordinateSum. hMeas and hBoundInt remain discharged by the cycle-164 compiled theorems. Weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM fallback, wrapper churn, broad audit, theorem-status promotion, fake closure, Lake/toolchain change, SLT import, and sald_version_2.tex routes are rejected." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.formalized dependsOn := [ "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable", "SALD.gaussianRealSelectedTestLineSecondOrderQuadraticBoundIntegrable", "hTaylorQuotientBound", "hBound", "hBoundInt", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "taylorWithinEval", "norm_mul", "mul_le_mul_of_nonneg_right", "Mathlib.Analysis.Calculus.Taylor", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet inside the scalar Brownian/Ito illness area. Local Mathlib use is only arithmetic/norm closure and the already imported Taylor expression vocabulary; the configured local SLT clone is absent, and no SLT theorem was consulted, imported, ported, or marked formalized." /-- Cycle-165 lower_2 packet splitting the deterministic Taylor quotient bound. -/ def cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorQuotientSplitLower2Obligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle165_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_taylor_quotient_split" statement := "Cycle 165 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the deterministic hTaylorQuotientBound input to SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound is no longer primitive once a first-order quadratic Taylor-remainder estimate hFirst, a second Taylor coefficient bound hSecondCoeff, and constant bookkeeping 0 <= C and C1 + C2 <= C are supplied. SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff compiles this deterministic algebraic split using taylorWithinEval_succ, taylorCoeffWithin, division by r^2 off r = 0, and the triangle inequality. Remaining source-cited work is to prove the smaller first-order quadratic remainder/second-coefficient estimates from the selected-test regularity route, plus hFrozenScalarBrownianItoTaylorMomentDecomposition, hFrozenScalarBrownianItoQuadraticVariationNormalization, and hFrozenScalarBrownianItoEventFieldCoordinateSum. Weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM fallback, wrapper churn, broad audit, theorem-status promotion, fake closure, Lake/toolchain change, SLT import, and sald_version_2.tex routes are rejected." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.formalized dependsOn := [ "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound", "hTaylorQuotientBound", "hFirst", "hSecondCoeff", "taylorWithinEval_succ", "taylorCoeffWithin", "taylorWithinEval_self", "norm_sub_le", "field_simp", "Mathlib.Analysis.Calculus.Taylor", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet in the selected scalar Brownian/Ito normalized-remainder branch. Local Mathlib consulted only through Analysis.Calculus.Taylor and basic norm/field arithmetic already available in the project. The configured local SLT clone is absent, and no SLT theorem was consulted, imported, ported, or marked formalized." /-- Cycle-165 proof-DAG pane for the concrete normalized-remainder domination leaf. -/ def cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle165.lower_packet.normalized_remainder_hbound_from_taylor_quotient" interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound supplies the DCT hBound input for the concrete selected-test normalized Taylor remainder from a deterministic scalar Taylor quotient bound along r |-> sourceTest (x + r • e)." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundLowerObligation", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainder", "hTaylorQuotientBound", "hBound", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle165.lower_2_packet.taylor_quotient_split_first_order_second_coeff" interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff supplies the deterministic selected-line second-order Taylor quotient bound from a first-order quadratic remainder estimate, a second Taylor coefficient bound, and constant bookkeeping." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorQuotientSplitLower2Obligation", "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff", "hFirst", "hSecondCoeff", "taylorWithinEval_succ", "taylorCoeffWithin", "norm_sub_le", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "hTaylorQuotientBound", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle165.remaining_taylor_quotient_bound" interface := "Remaining exact theorem after the lower_2 split: prove the smaller source-cited first-order quadratic Taylor remainder estimate hFirst and the selected-line second Taylor coefficient bound hSecondCoeff, with constant bookkeeping C1 + C2 <= C. Keep the Taylor moment decomposition, Brownian quadratic-variation normalization, and coordinate-sum leaves separate." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff", "hFirst", "hSecondCoeff", "selected-test regularity or compact-support/bounded-Hessian source hypothesis", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable", "SALD.gaussianRealSelectedTestLineSecondOrderQuadraticBoundIntegrable", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle165.reviewer_hbound_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, the compiled theorem targets the concrete hBound domination shape rather than a new wrapper, the remaining source theorem is the deterministic Taylor quotient bound, and python3 tools/astis.py check passes." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundLowerObligation", "SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorQuotientSplitLower2Obligation", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound", "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := ["cycle 165 reviewer"] status := ProofStatus.obligation } ] /-! ### Cycle 166: first-order selected-line quadratic remainder quotient -/ /-- Cycle-166 dynamic-leaf packet for the selected-line first-order remainder. The cycle-165 split left `hFirst` as a supplied deterministic quotient bound. This packet narrows that supplied quotient to the source-facing non-quotient quadratic Taylor-remainder estimate for the degree-one selected scalar line. -/ def cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle166_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_first_order_remainder" statement := "Cycle 166 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hFirst input to SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff is no longer primitive once the source-facing non-quotient first-order quadratic Taylor-remainder estimate is supplied. SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder compiles the bookkeeping bridge from forall r, norm (sourceTest (x + r • e) - taylorWithinEval (fun r => sourceTest (x + r • e)) 1 Set.univ 0 r) <= C1 * r^2, plus 0 <= C1, to the exact quotient bound hFirst. Remaining source-cited work is to prove that non-quotient quadratic remainder estimate from the selected-test second-derivative/bounded-Hessian route, including Mathlib Taylor interval-to-Set.univ compatibility and the positive/negative r split. hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves remain separate. Weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM fallback, wrapper churn, broad audit, theorem-status promotion, fake closure, Lake/toolchain change, SLT import, and sald_version_2.tex routes are rejected." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.formalized dependsOn := [ "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder", "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff", "hFirst", "hFirstQuadraticRemainder", "hC1", "taylorWithinEval", "norm_div", "div_le_div_of_nonneg_right", "field_simp", "Mathlib.Analysis.Calculus.Taylor.taylor_mean_remainder_bound", "Mathlib.Analysis.Calculus.MeanValue", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf worker packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib Taylor. The available theorem taylor_mean_remainder_bound is interval-based, so the remaining named theory/source gap is the non-quotient quadratic remainder estimate over Set.univ, including interval-to-Set.univ Taylor compatibility and sign split. The configured local SLT clone is absent, and no SLT theorem was imported, ported, or marked formalized." /-- Cycle-166 lower_1 nonnegative interval Taylor proof-scout packet. -/ def cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoIntervalTaylorLower1Obligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle166_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_interval_taylor_nonneg" statement := "Cycle 166 lower_1 dynamic-leaf proof-scout/worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the non-quotient first-order quadratic Taylor-remainder estimate is no longer wholly primitive on the nonnegative side. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor compiles the r >= 0 case from Mathlib taylor_mean_remainder_bound, assuming ContDiffOn Real 2 on Icc 0 r, a selected-line interval second-derivative bound, and the interval-to-Set.univ Taylor-polynomial compatibility equality. Remaining boundary: prove the negative/reflection side, source the selected-line interval second-derivative bound and compatibility from the paper's bounded-Hessian/selected-test regularity, then combine signed sides to supply hFirstQuadraticRemainder. hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves remain separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.formalized dependsOn := [ "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor", "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder", "Mathlib.Analysis.Calculus.Taylor.taylor_mean_remainder_bound", "ContDiffOn Real 2", "iteratedDerivWithin", "Set.Icc", "Set.univ", "hTaylorCompat", "selected-line interval second-derivative bound", "negative/reflection sign split", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib Taylor; no SLT theorem was needed because this is deterministic one-dimensional calculus. The nonnegative side is now a compiled local theorem; the remaining named theory/source gap is the signed combination plus the source proof of the interval derivative bound and Taylor-polynomial compatibility." /-- Cycle-166 lower_2 signed interval Taylor combination packet. -/ def cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSignedIntervalTaylorLower2Obligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle166_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_signed_interval_taylor" statement := "Cycle 166 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the source-facing all-r non-quotient first-order quadratic Taylor-remainder estimate hFirstQuadraticRemainder is no longer primitive once signed interval Taylor data are supplied. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor compiles the signed combination: for r >= 0 it uses SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor on the selected line q |-> sourceTest (x + q • e), and for r < 0 it applies the same theorem to the reflected line q |-> sourceTest (x + q • (-e)) at -r, with an explicit Set.univ Taylor-polynomial reflection compatibility. Remaining source-cited work is to prove the signed interval ContDiffOn/second-derivative domination, interval-to-Set.univ Taylor compatibility, and reflected Taylor-polynomial compatibility from the paper's selected-test bounded-Hessian/regularity hypothesis. hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves remain separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.formalized dependsOn := [ "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor", "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder", "hFirstQuadraticRemainder", "hNonnegCont", "hNonnegSecond", "hNonnegTaylorCompat", "hNegCont", "hNegSecond", "hNegTaylorCompat", "hNegTaylorReflect", "Mathlib.Analysis.Calculus.Taylor.taylor_mean_remainder_bound", "selected-test second-derivative or bounded-Hessian source hypothesis", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib Taylor through the lower_1 theorem and basic order/reflection arithmetic; no SLT theorem was needed because this is deterministic one-dimensional calculus. The remaining named source/theory gap is now the signed interval regularity/second-derivative domination and Taylor-compatibility data, not a primitive all-r quadratic remainder." /-- Cycle-166 proof-DAG pane for the first-order selected-line remainder split. -/ def cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle166.lower_packet.hfirst_from_quadratic_remainder" interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder supplies hFirst from the smaller source-facing non-quotient quadratic Taylor-remainder estimate and 0 <= C1." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderLowerObligation", "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder", "hFirst", "hFirstQuadraticRemainder", "hC1", "taylorWithinEval", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle166.lower_1_packet.nonnegative_interval_taylor_remainder" interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor proves the r >= 0 non-quotient quadratic remainder bound from Mathlib taylor_mean_remainder_bound, explicit ContDiffOn Real 2 on Icc 0 r, interval second-derivative domination, and interval-to-Set.univ Taylor compatibility." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoIntervalTaylorLower1Obligation", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor", "Mathlib.Analysis.Calculus.Taylor.taylor_mean_remainder_bound", "ContDiffOn Real 2", "iteratedDerivWithin", "hTaylorCompat", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "hFirstQuadraticRemainder", "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle166.lower_2_packet.signed_interval_taylor_remainder" interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor supplies the all-r non-quotient quadratic remainder from signed interval Taylor data: the nonnegative selected-line interval theorem, the reflected -e theorem for r < 0, and explicit Set.univ Taylor reflection compatibility." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSignedIntervalTaylorLower2Obligation", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor", "hFirstQuadraticRemainder", "hNonnegCont", "hNonnegSecond", "hNonnegTaylorCompat", "hNegCont", "hNegSecond", "hNegTaylorCompat", "hNegTaylorReflect", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "hFirst", "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder", "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle166.remaining_nonquotient_quadratic_remainder" interface := "Remaining exact theorem boundary after the signed interval combination: prove the signed interval ContDiffOn/second-derivative domination, interval-to-Set.univ Taylor compatibility, and reflected Set.univ Taylor-polynomial compatibility from the paper's selected-test second-derivative/bounded-Hessian hypothesis; hSecondCoeff remains separate." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor", "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder", "hFirstQuadraticRemainder", "Mathlib.Analysis.Calculus.Taylor.taylor_mean_remainder_bound", "Mathlib.Analysis.Calculus.MeanValue", "selected-test second-derivative or bounded-Hessian source hypothesis", "interval-to-Set.univ taylorWithinEval compatibility", "reflected Set.univ taylorWithinEval compatibility", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "hFirst", "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle166.reviewer_hfirst_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hFirst is narrowed to the non-quotient quadratic remainder estimate rather than restated, hSecondCoeff and the Taylor moment/quadratic-variation/coordinate-sum leaves remain separate, and python3 tools/astis.py check passes." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderLowerObligation", "SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSignedIntervalTaylorLower2Obligation", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor", "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := ["cycle 166 reviewer"] status := ProofStatus.obligation } ] /-- Cycle-167 reflected Taylor compatibility discharge packet. -/ def cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorReflectMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle167_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_taylor_reflect" statement := "Cycle 167 middle dynamic-leaf worker packet. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: hNegTaylorReflect in SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor is no longer primitive. SALD.gaussianRealSelectedTestLineFirstOrderTaylorReflectSetUniv compiles the reflected Set.univ first-order Taylor-polynomial compatibility for q |-> sourceTest (x + q • (-e)) at -r from the domain reflection q |-> -q and Mathlib deriv_comp_neg. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect then supplies the all-r first-order quadratic remainder from signed interval ContDiffOn, signed second-derivative domination, and interval-to-Set.univ Taylor compatibility alone. Remaining source-cited boundary: derive hNonnegCont, hNonnegSecond, hNonnegTaylorCompat, hNegCont, hNegSecond, and hNegTaylorCompat from the paper's selected-test bounded-Hessian/regularity hypotheses. hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves remain separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.formalized dependsOn := [ "SALD.gaussianRealSelectedTestLineFirstOrderTaylorReflectSetUniv", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor", "hNegTaylorReflect", "deriv_comp_neg", "taylorWithinEval_succ", "Set.univ", "hNonnegCont", "hNonnegSecond", "hNonnegTaylorCompat", "hNegCont", "hNegSecond", "hNegTaylorCompat", "selected-test second-derivative or bounded-Hessian source hypothesis", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf middle packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib Taylor and Deriv.Shift through the imported Taylor stack; no local SLT clone is present and no SLT theorem was imported or marked formalized. This discharges one explicit reflected Taylor compatibility hypothesis and leaves the signed interval regularity/second-derivative/interval-compatibility data as the smaller source-facing boundary." /-- Cycle-167 proof-DAG pane for the reflected Taylor compatibility discharge. -/ def cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorReflectDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle167.middle_packet.reflected_set_univ_taylor_compat" interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderTaylorReflectSetUniv proves the reflected Set.univ first-order Taylor-polynomial compatibility for the selected scalar line from the algebraic reflection q |-> -q and Mathlib deriv_comp_neg." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.gaussianRealSelectedTestLineFirstOrderTaylorReflectSetUniv", "deriv_comp_neg", "taylorWithinEval_succ", "Set.univ", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect", "hFirstQuadraticRemainder" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle167.middle_packet.signed_interval_without_reflect_hypothesis" interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect removes hNegTaylorReflect from the signed interval first-order quadratic remainder theorem by supplying it internally from the reflection lemma." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorReflectMiddleObligation", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor", "SALD.gaussianRealSelectedTestLineFirstOrderTaylorReflectSetUniv", "hNonnegCont", "hNonnegSecond", "hNonnegTaylorCompat", "hNegCont", "hNegSecond", "hNegTaylorCompat", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder", "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle167.remaining_signed_interval_regularity_data" interface := "Remaining exact theorem boundary after discharging hNegTaylorReflect: prove signed interval ContDiffOn, signed interval second-derivative domination, and interval-to-Set.univ Taylor compatibility for the selected and reflected lines from the paper's selected-test bounded-Hessian/regularity hypotheses; hSecondCoeff remains separate." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect", "hNonnegCont", "hNonnegSecond", "hNonnegTaylorCompat", "hNegCont", "hNegSecond", "hNegTaylorCompat", "selected-test second-derivative or bounded-Hessian source hypothesis", "interval-to-Set.univ taylorWithinEval compatibility", "appendix.tex:984-995", "appendix.tex:1379-1387" ] reusedBy := [ "hFirst", "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff" ] status := ProofStatus.obligation } ] /-- Cycle-167 lower_1 Taylor-compatibility narrowing packet. -/ def cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorCompatLower1Obligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle167_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_taylor_compat_lower1" statement := "Cycle 167 lower_1 dynamic-leaf worker/scout packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hNonnegTaylorCompat and hNegTaylorCompat below SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect are no longer primitive compatibility assumptions once the selected and reflected scalar lines are differentiable at 0. SALD.gaussianRealSelectedTestLineFirstOrderTaylorCompatOfDifferentiableAt compiles the interval-to-Set.univ first-order Taylor-polynomial compatibility for any r >= 0 from DifferentiableAt at the expansion point, using the r = 0 algebraic case and Mathlib uniqueDiffOn_Icc/DifferentiableAt.derivWithin for 0 < r. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff then supplies the all-r first-order quadratic remainder from signed interval ContDiffOn, signed interval second-derivative domination, and base differentiability of the selected/reflected scalar lines. Remaining source-cited boundary: derive hNonnegCont, hNonnegSecond, hNegCont, hNegSecond, hNonnegBaseDiff, and hNegBaseDiff from the paper's selected-test bounded-Hessian/regularity hypotheses. hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves remain separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.formalized dependsOn := [ "SALD.gaussianRealSelectedTestLineFirstOrderTaylorCompatOfDifferentiableAt", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect", "hNonnegTaylorCompat", "hNegTaylorCompat", "hNonnegBaseDiff", "hNegBaseDiff", "hNonnegCont", "hNonnegSecond", "hNegCont", "hNegSecond", "uniqueDiffOn_Icc", "DifferentiableAt.derivWithin", "derivWithin_univ", "taylorWithinEval_succ", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_1 packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib Taylor plus Deriv/FDeriv endpoint derivative-within APIs and TangentCone.Real uniqueDiffOn_Icc. The configured local SLT clone is absent; no SLT theorem was imported, ported, or marked formalized. This rejects wrapper churn by removing two explicit Taylor-compatibility hypotheses and exposing base differentiability as the smaller regularity leaf." /-- Cycle-167 lower_2 global line regularity narrowing packet. -/ def cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLineContDiffLower2Obligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle167_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_line_contdiff_lower2" statement := "Cycle 167 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hNonnegCont, hNegCont, hNonnegBaseDiff, and hNegBaseDiff below SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff are no longer primitive once the selected scalar line has global ContDiffOn Real 2 regularity on Set.univ. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff compiles the all-r first-order quadratic remainder from one global selected-line ContDiffOn hypothesis plus the two signed interval second-derivative domination hypotheses. The reflected negative-line regularity is supplied by composing the global line with q |-> -q, using Mathlib ContDiff.comp, contDiff_neg, and contDiff_id; base differentiability at 0 is supplied by ContDiffOn.contDiffAt and ContDiffAt.differentiableAt. Remaining source-cited boundary: prove the global selected-line ContDiffOn Real 2 fact and the signed interval second-derivative domination from the paper's selected-test bounded-Hessian/regularity assumptions. hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves remain separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.formalized dependsOn := [ "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff", "hLine", "hNonnegSecond", "hNegSecond", "hNonnegCont", "hNegCont", "hNonnegBaseDiff", "hNegBaseDiff", "ContDiffOn.mono", "contDiffOn_univ", "ContDiff.comp", "contDiff_neg", "contDiff_id", "ContDiffOn.contDiffAt", "ContDiffAt.differentiableAt", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf lower_2 packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib ContDiff composition/basic operations and Taylor-adjacent APIs already imported by this file; the configured local SLT clone is absent, and no SLT theorem was imported, ported, or marked formalized. This rejects wrapper churn by removing four interval/base regularity hypotheses and exposing global line ContDiff plus second-derivative domination as the smaller source-facing boundary." /-- Cycle-168 ambient selected-test regularity narrowing packet. -/ def cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceContDiffMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle168_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_source_contdiff_middle" statement := "Cycle 168 middle dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hLine below SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff is no longer primitive once the selected source test has ambient global ContDiffOn Real 2 regularity on Set.univ. SALD.gaussianRealSelectedTestLineContDiffOnOfSourceContDiffOn compiles the affine-line composition q |-> sourceTest (x + q • e), and SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn feeds that line regularity into the cycle-167 global-line Taylor bridge. Remaining source-cited boundary: prove ambient selected-test ContDiffOn Real 2 and the two signed interval second-derivative domination hypotheses from the paper's selected-test bounded-Hessian/regularity assumptions. hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves remain separate." source := saldGeneralMovingTargetDiscreteWeakFpSource status := ProofStatus.formalized dependsOn := [ "SALD.gaussianRealSelectedTestLineContDiffOnOfSourceContDiffOn", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff", "hSource", "hLine", "hNonnegSecond", "hNegSecond", "contDiffOn_univ", "ContDiff.comp", "ContDiff.add", "ContDiff.smul_const", "contDiff_const", "contDiff_id", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "Dynamic-leaf middle packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib ContDiff Basic/Operations/Comp APIs; the configured local SLT clone is absent, and no SLT theorem was imported, ported, or marked formalized. This rejects wrapper churn by only supplying hLine from ambient selected-test C^2 regularity while preserving the signed second-derivative domination leaves." /-- Cycle-168 lower_2 global line-second-derivative narrowing packet. -/ def cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLineSecondLower2Obligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle168_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_line_second_lower2" statement := "Cycle 168 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hNonnegSecond and hNegSecond below SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn are no longer primitive signed interval iteratedDerivWithin 2 domination hypotheses once global selected/reflected scalar-line iteratedDeriv 2 bounds are supplied. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds compiles the bridge: for r=0 it uses accPt_iff_nhds plus derivWithin_zero_of_not_accPt to close the singleton interval by hC1; for 0 forall phi x i z, sourceSelectedEndpoint phi x i z = x + z smul stdOrthonormalBasis Real E i", "leaf=hSelectedEndpointCoordinateLineDef", "error_class=source_contract_gap_missing_selected_endpoint_coordinate_line_definition", "blocked_by=sourceSelectedEndpoint is an abstract Lean parameter and the original paper source does not provide a Lean-facing endpoint coordinate-line definition", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source_contract_gap_missing_selected_endpoint_coordinate_line_definition", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] note := "typed_verifier_feedback: leaf=hSelectedEndpointCoordinateLineDef; error_class=source_contract_gap_missing_selected_endpoint_coordinate_line_definition; needed_shape=testRegular -> forall phi x i z, sourceSelectedEndpoint phi x i z = x + z smul stdOrthonormalBasis Real E i; source_lines=appendix.tex:958-970;appendix.tex:983-996;appendix.tex:1161-1170;appendix.tex:1379-1387; blocked_by=sourceSelectedEndpoint is an abstract parameter of the compiled cycle-190 bridges and no original-paper Lean-facing definition of that endpoint coordinate line was found; lower_1_handoff=absent_from_dialogue_at_lower_2_start; packet_type=dynamic-leaf worker packet." /-- Cycle-202 proof-DAG pane for the selected endpoint coordinate-line gap. -/ def cycle202GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedEndpointCoordinateLineLower2Dag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle202.lower_2_packet.selected_endpoint_coordinate_line_source_gap" interface := "Typed source-cited boundary: hSelectedEndpointCoordinateLineDef is not closed by another wrapper; the existing cycle-190 bridge already shows that this field is the smaller endpoint-coordinate source contract needed for hSelectedIncrementCoordinateLineDef and hSourceTaylorIntegrandRawDef." source := saldGeneralMovingTargetDiscreteEmSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle202GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedEndpointCoordinateLineLower2Obligation", "SALD.selectedWeakTestSelectedIncrementCoordinateLineDefOfEndpointAndLineDef", "SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementEndpointAndLineDef", "hSelectedEndpointCoordinateLineDef", "hSelectedIncrementEndpointDef", "sourceSelectedEndpoint", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source_contract_gap_missing_selected_endpoint_coordinate_line_definition" ] reusedBy := [ "hSelectedIncrementCoordinateLineDef", "hSourceTaylorIntegrandRawDef", "hBrownianCoordinateGeneratorTaylorIntegralDef", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff" ] status := ProofStatus.obligation } ] /-! ### Cycle 203 lower_2: selected-increment endpoint definition source gap -/ /-- Cycle-203 lower_2 packet for the selected-increment endpoint field. The cycle-190 local bridge already reduces the selected-increment coordinate line and raw source Taylor integrand leaves to two endpoint-facing source fields. Cycle 202 recorded the endpoint coordinate-line field. This packet records the remaining companion field: the source selected-line increment must be the selected weak-test value at the named endpoint minus its base value. Local inspection found `sourceSelectedLineIncrement` only as an abstract source-facing parameter of the compiled bridges, not as a reducible definition, so this remains a precise source contract rather than a theorem claim. -/ def cycle203GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementEndpointLower2Obligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle203_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_selected_increment_endpoint_lower2" statement := "Cycle 203 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact missing theorem boundary narrowed: hSelectedIncrementEndpointDef is the remaining source-facing selected-increment endpoint definition after the compiled cycle-190 bridge SALD.selectedWeakTestSelectedIncrementCoordinateLineDefOfEndpointAndLineDef and raw-integrand bridge SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementEndpointAndLineDef. Lean inspection shows sourceSelectedLineIncrement, like sourceSelectedEndpoint, is an abstract parameter of those bridges rather than a reducible local definition, so the needed source-backed shape is testRegular -> forall phi x i z, sourceSelectedLineIncrement phi x i z = selectedTest phi (sourceSelectedEndpoint phi x i z) - selectedTest phi x. Source anchors appendix.tex:958-970, appendix.tex:983-996, appendix.tex:1161-1170, and appendix.tex:1379-1387 identify the EM increment, frozen interpolation endpoint, normalized Brownian scalar coordinate, and weak-FP diffusion consumer while keeping sigma_eta^2 / 2 outside this scalar selected-test increment field. No endpoint-coordinate replay, normalized-remainder replay, source-Hessian work, testRegular repackaging, VP score-Hessian substitution, direct SLT dependency, broad route audit, or consumer-wrapper churn is introduced." source := saldGeneralMovingTargetDiscreteEmSource status := ProofStatus.obligation dependsOn := [ "hSelectedIncrementEndpointDef", "hSelectedEndpointCoordinateLineDef", "hSelectedIncrementCoordinateLineDef", "hSourceTaylorIntegrandRawDef", "sourceSelectedLineIncrement", "sourceSelectedEndpoint", "sourceTaylorIntegrand", "selectedTest", "stdOrthonormalBasis", "SALD.selectedWeakTestSelectedIncrementCoordinateLineDefOfEndpointAndLineDef", "SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementEndpointAndLineDef", "SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementAndLineDef", "SALD.cycle202GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedEndpointCoordinateLineLower2Obligation", "needed_shape=testRegular -> forall phi x i z, sourceSelectedLineIncrement phi x i z = selectedTest phi (sourceSelectedEndpoint phi x i z) - selectedTest phi x", "leaf=hSelectedIncrementEndpointDef", "error_class=source_contract_gap_missing_selected_increment_endpoint_definition", "blocked_by=sourceSelectedLineIncrement is an abstract source-facing parameter in the compiled cycle-190 bridges", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source_contract_gap_missing_selected_increment_endpoint_definition", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] note := "typed_verifier_feedback: leaf=hSelectedIncrementEndpointDef; error_class=source_contract_gap_missing_selected_increment_endpoint_definition; needed_shape=testRegular -> forall phi x i z, sourceSelectedLineIncrement phi x i z = selectedTest phi (sourceSelectedEndpoint phi x i z) - selectedTest phi x; source_lines=appendix.tex:958-970;appendix.tex:983-996;appendix.tex:1161-1170;appendix.tex:1379-1387; blocked_by=sourceSelectedLineIncrement is an abstract parameter of the compiled cycle-190 bridges and no original-paper Lean-facing definition of the selected increment as an endpoint difference was found; lower_1_handoff=absent_from_dialogue_at_lower_2_start; packet_type=dynamic-leaf worker packet." /-- Cycle-203 proof-DAG pane for the selected-increment endpoint gap. -/ def cycle203GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementEndpointLower2Dag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle203.lower_2_packet.selected_increment_endpoint_source_gap" interface := "Typed source-cited boundary: hSelectedIncrementEndpointDef is not closed by a same-shape wrapper; the existing cycle-190 bridge already shows that this field, together with the cycle-202 hSelectedEndpointCoordinateLineDef boundary, is the smaller source contract needed for hSelectedIncrementCoordinateLineDef and hSourceTaylorIntegrandRawDef." source := saldGeneralMovingTargetDiscreteEmSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle203GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementEndpointLower2Obligation", "SALD.selectedWeakTestSelectedIncrementCoordinateLineDefOfEndpointAndLineDef", "SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementEndpointAndLineDef", "SALD.cycle202GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedEndpointCoordinateLineLower2Obligation", "hSelectedIncrementEndpointDef", "hSelectedEndpointCoordinateLineDef", "sourceSelectedLineIncrement", "sourceSelectedEndpoint", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source_contract_gap_missing_selected_increment_endpoint_definition" ] reusedBy := [ "hSelectedIncrementCoordinateLineDef", "hSourceTaylorIntegrandRawDef", "hBrownianCoordinateGeneratorTaylorIntegralDef", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff" ] status := ProofStatus.obligation } ] /-! ### Cycle 204 lower_2: source Taylor integrand selected-increment source gap -/ /-- Cycle-204 lower_2 packet for the source Taylor integrand naming field. The cycle-189 and cycle-190 local bridges already reduce the raw source Taylor integrand leaf to the source naming field below plus endpoint-facing fields. Local Lean inspection found `sourceTaylorIntegrand` and `sourceSelectedLineIncrement` only as abstract parameters of those bridges, not as reducible definitions, so lower_2 records this exact source contract rather than claiming a definitional proof. -/ def cycle204GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandSelectedIncrementLower2Obligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle204_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_source_taylor_integrand_selected_increment_lower2" statement := "Cycle 204 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact missing theorem boundary narrowed: hSourceTaylorIntegrandSelectedIncrementDef is the source-facing naming field still required after the compiled cycle-189 raw-integrand bridge SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementAndLineDef and the cycle-190 endpoint/raw-integrand bridge SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementEndpointAndLineDef. Lean inspection shows sourceTaylorIntegrand and sourceSelectedLineIncrement are abstract parameters of those bridges rather than reducible local definitions, so the needed source-backed shape is testRegular -> forall phi x i z, sourceTaylorIntegrand phi x i z = sourceSelectedLineIncrement phi x i z. Source anchors appendix.tex:958-970, appendix.tex:983-996, appendix.tex:1161-1170, and appendix.tex:1379-1387 identify the EM increment, frozen interpolation, normalized Brownian coordinate, and weak-FP consumer while keeping sigma_eta^2 / 2 outside this scalar naming field. No endpoint replay, normalized-remainder replay, source-Hessian work, testRegular repackaging, VP score-Hessian substitution, direct SLT dependency, broad route audit, or consumer-wrapper churn is introduced." source := saldGeneralMovingTargetDiscreteEmSource status := ProofStatus.obligation dependsOn := [ "hSourceTaylorIntegrandSelectedIncrementDef", "hSourceTaylorIntegrandRawDef", "hSelectedIncrementCoordinateLineDef", "hSelectedIncrementEndpointDef", "hSelectedEndpointCoordinateLineDef", "sourceTaylorIntegrand", "sourceSelectedLineIncrement", "sourceSelectedEndpoint", "selectedTest", "stdOrthonormalBasis", "SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementAndLineDef", "SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementEndpointAndLineDef", "SALD.selectedWeakTestSelectedIncrementCoordinateLineDefOfEndpointAndLineDef", "SALD.cycle202GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedEndpointCoordinateLineLower2Obligation", "SALD.cycle203GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementEndpointLower2Obligation", "needed_shape=testRegular -> forall phi x i z, sourceTaylorIntegrand phi x i z = sourceSelectedLineIncrement phi x i z", "leaf=hSourceTaylorIntegrandSelectedIncrementDef", "error_class=source_contract_gap_missing_source_taylor_integrand_selected_increment_definition", "blocked_by=sourceTaylorIntegrand and sourceSelectedLineIncrement are abstract parameters in the compiled selected-increment/raw-integrand bridges", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source_contract_gap_missing_source_taylor_integrand_selected_increment_definition", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] note := "typed_verifier_feedback: leaf=hSourceTaylorIntegrandSelectedIncrementDef; error_class=source_contract_gap_missing_source_taylor_integrand_selected_increment_definition; needed_shape=testRegular -> forall phi x i z, sourceTaylorIntegrand phi x i z = sourceSelectedLineIncrement phi x i z; source_lines=appendix.tex:958-970;appendix.tex:983-996;appendix.tex:1161-1170;appendix.tex:1379-1387; blocked_by=sourceTaylorIntegrand and sourceSelectedLineIncrement are abstract parameters of the compiled cycle-189/190 bridges and no original-paper Lean-facing definition identifying the source Taylor integrand with the selected weak-test increment was found; lower_1_handoff=absent_from_dialogue_at_lower_2_start; packet_type=dynamic-leaf worker packet." /-- Cycle-204 proof-DAG pane for the source Taylor integrand naming gap. -/ def cycle204GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandSelectedIncrementLower2Dag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle204.lower_2_packet.source_taylor_integrand_selected_increment_source_gap" interface := "Typed source-cited boundary: hSourceTaylorIntegrandSelectedIncrementDef is not closed by a same-shape wrapper; the existing cycle-189/190 bridges already show that this field is the smaller source contract needed before endpoint-coordinate and endpoint-difference fields can feed hSourceTaylorIntegrandRawDef." source := saldGeneralMovingTargetDiscreteEmSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle204GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandSelectedIncrementLower2Obligation", "SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementAndLineDef", "SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementEndpointAndLineDef", "SALD.selectedWeakTestSelectedIncrementCoordinateLineDefOfEndpointAndLineDef", "SALD.cycle202GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedEndpointCoordinateLineLower2Obligation", "SALD.cycle203GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementEndpointLower2Obligation", "hSourceTaylorIntegrandSelectedIncrementDef", "sourceTaylorIntegrand", "sourceSelectedLineIncrement", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source_contract_gap_missing_source_taylor_integrand_selected_increment_definition" ] reusedBy := [ "hSourceTaylorIntegrandRawDef", "hBrownianCoordinateGeneratorTaylorIntegralDef", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff" ] status := ProofStatus.obligation } ] /-! ### Cycle 205 lower_2: selected-line raw Taylor split source gap -/ /-- Cycle-205 lower_2 packet for the selected-line raw Taylor split. The cycle-188 bridge already reduces `hSelectedLineTaylorSplitDef` to the raw one-dimensional selected-line Taylor identity plus source-term naming fields. Local inspection found `normalizedRemainder` only as an abstract parameter in that interface and did not find an original-paper definition of it as the residual after the first and second scalar Taylor terms, so this packet records that exact source contract instead of promoting the Taylor split. -/ def cycle205GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorRawSplitLower2Obligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle205_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_selected_line_taylor_raw_split_lower2" statement := "Cycle 205 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact missing theorem boundary narrowed: hSelectedLineTaylorRawSplitDef is the raw selected scalar-line Taylor identity still required below the accepted cycle-204 hSourceTaylorIntegrandSelectedIncrementDef naming gap. Existing compiled bridges SALD.selectedWeakTestSelectedLineTaylorSplitDefOfRawTaylorAndTermDefs, SALD.selectedWeakTestSourceTaylorIntegrandDefOfRawAndLineTaylorSplit, SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralRawTaylorAndTermDefs, SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfScalarPushforwardRawTaylorAndTermDefs, and SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfScalarPushforwardRawTaylorAndDominatedRemainder already reuse this raw Taylor field, so lower_2 must not add another consumer wrapper. Lean inspection shows normalizedRemainder is an abstract parameter of those bridges rather than a reducible residual definition, and a targeted original-source search excluding sald_version_2.tex found no displayed selected-line Taylor expansion or definition of normalizedRemainder as that residual. The needed source-backed shape is testRegular -> forall phi x i z, selectedTest phi (x + z smul stdOrthonormalBasis Real E i) - selectedTest phi x = deriv (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 0 * z + ((2 : Real) * taylorCoeffWithin (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 2 Set.univ 0) * z ^ 2 + normalizedRemainder phi x i z. Source anchors appendix.tex:958-970, appendix.tex:983-996, appendix.tex:1161-1176, and appendix.tex:1379-1387 identify the EM increment, frozen interpolation, normalized Gaussian coordinate, and weak-FP consumer while keeping sigma_eta^2 / 2 outside this scalar Taylor residual field. No endpoint/naming replay, normalized-remainder-bound replay, source-Hessian work, testRegular repackaging, VP score-Hessian substitution, direct SLT dependency, broad route audit, or wrapper churn is introduced." source := saldGeneralMovingTargetDiscreteEmSource status := ProofStatus.obligation dependsOn := [ "hSelectedLineTaylorRawSplitDef", "hSelectedLineTaylorSplitDef", "hSourceTaylorIntegrandSelectedIncrementDef", "hSourceTaylorIntegrandRawDef", "hSourceLinearTermTaylorDef", "hScalarLineFirstCoeffDef", "hSourceQuadraticTermTaylorDef", "hScalarLineTaylorCoeffDef", "selectedTest", "normalizedRemainder", "deriv", "taylorCoeffWithin", "Set.univ", "stdOrthonormalBasis", "SALD.selectedWeakTestSelectedLineTaylorSplitDefOfRawTaylorAndTermDefs", "SALD.selectedWeakTestSourceTaylorIntegrandDefOfRawAndLineTaylorSplit", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralRawTaylorAndTermDefs", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfScalarPushforwardRawTaylorAndTermDefs", "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfScalarPushforwardRawTaylorAndDominatedRemainder", "SALD.cycle188GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorSplitLower2Obligation", "SALD.cycle204GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandSelectedIncrementLower2Obligation", "runs/20260613-050236-675526-ASTIS-SALD-001-cycle205/lower_1_selected_line_taylor_raw_split_route.md", "runs/20260613-050236-675526-ASTIS-SALD-001-cycle205/lower_3_selected_line_taylor_api_churn_rejection.md", "needed_shape=testRegular -> forall phi x i z, selectedTest phi (x + z smul stdOrthonormalBasis Real E i) - selectedTest phi x = deriv (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 0 * z + ((2 : Real) * taylorCoeffWithin (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 2 Set.univ 0) * z ^ 2 + normalizedRemainder phi x i z", "leaf=hSelectedLineTaylorRawSplitDef", "error_class=source_contract_gap_missing_selected_line_taylor_raw_split_definition", "blocked_by=normalizedRemainder is an abstract parameter in the compiled Taylor bridges and no original-paper definition identifies it with the selected scalar Taylor residual", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1176", "appendix.tex:1379-1387", "source_contract_gap_missing_selected_line_taylor_raw_split_definition", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] note := "typed_verifier_feedback: leaf=hSelectedLineTaylorRawSplitDef; error_class=source_contract_gap_missing_selected_line_taylor_raw_split_definition; needed_shape=testRegular -> forall phi x i z, selectedTest phi (x + z smul stdOrthonormalBasis Real E i) - selectedTest phi x = deriv (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 0 * z + ((2 : Real) * taylorCoeffWithin (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 2 Set.univ 0) * z ^ 2 + normalizedRemainder phi x i z; source_lines=appendix.tex:958-970;appendix.tex:983-996;appendix.tex:1161-1176;appendix.tex:1379-1387; blocked_by=normalizedRemainder is an abstract parameter of the compiled cycle-188/189/192/193 bridges and no original-paper or local Lean definition identifies it with the residual of the selected scalar Taylor expansion; lower_1_route_artifact=runs/20260613-050236-675526-ASTIS-SALD-001-cycle205/lower_1_selected_line_taylor_raw_split_route.md; lower_3_api_artifact=runs/20260613-050236-675526-ASTIS-SALD-001-cycle205/lower_3_selected_line_taylor_api_churn_rejection.md; packet_type=dynamic-leaf worker packet." /-- Cycle-205 proof-DAG pane for the selected-line raw Taylor split gap. -/ def cycle205GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorRawSplitLower2Dag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle205.lower_2_packet.selected_line_taylor_raw_split_source_gap" interface := "Typed source-cited boundary: hSelectedLineTaylorRawSplitDef is not closed by a same-shape wrapper; the existing cycle-188/189/192/193 bridges already show that this raw scalar Taylor identity is the smaller source contract needed for hSelectedLineTaylorSplitDef, hBrownianCoordinateGeneratorTaylorIntegralDef, and the Brownian/Ito Taylor moment consumer." source := saldGeneralMovingTargetDiscreteEmSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle205GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorRawSplitLower2Obligation", "SALD.selectedWeakTestSelectedLineTaylorSplitDefOfRawTaylorAndTermDefs", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralRawTaylorAndTermDefs", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfScalarPushforwardRawTaylorAndTermDefs", "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfScalarPushforwardRawTaylorAndDominatedRemainder", "runs/20260613-050236-675526-ASTIS-SALD-001-cycle205/lower_1_selected_line_taylor_raw_split_route.md", "runs/20260613-050236-675526-ASTIS-SALD-001-cycle205/lower_3_selected_line_taylor_api_churn_rejection.md", "hSelectedLineTaylorRawSplitDef", "selectedTest", "normalizedRemainder", "deriv", "taylorCoeffWithin", "Set.univ", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1176", "appendix.tex:1379-1387", "source_contract_gap_missing_selected_line_taylor_raw_split_definition" ] reusedBy := [ "hSelectedLineTaylorSplitDef", "hBrownianCoordinateGeneratorTaylorIntegralDef", "hRemainderGeneratorLimitDef", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff" ] status := ProofStatus.obligation } ] /-! ### Cycle 206 middle: normalized-remainder pullback source boundary -/ /-- Cycle-206 middle packet for the normalized-remainder pullback field. The cycle-185 scalar-pushforward bridge already consumes `hRemainderPullbackDef` to derive the normalized-law remainder generator identity. This packet therefore does not add another consumer wrapper; it records the smaller source contract that lower agents must either discharge from a reducible source definition or preserve as a typed source gap. -/ def cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle206_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_remainder_pullback_middle" statement := "Cycle 206 middle_formalizer dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact missing theorem boundary narrowed: hRemainderPullbackDef, the source-facing sample-space pullback definition required by SALD.selectedWeakTestRemainderGeneratorNormalizedLawDefOfScalarPushforward before hRemainderGeneratorLimitDef can be derived from scalar Brownian coordinate law transport. Existing compiled bridges SALD.selectedWeakTestRemainderGeneratorNormalizedLawDefOfScalarPushforward, SALD.selectedWeakTestRemainderGeneratorLimitDefOfStdGaussianVectorLaw, and SALD.selectedWeakTestRemainderGeneratorLimitDefOfScalarPushforwardAndStdGaussianVectorLaw already consume this field, so lower_2 must not add a same-shape consumer wrapper. The lower_2-ready shape is testRegular -> forall phi x i, remainderGeneratorLimit phi x i = integral omega, normalizedRemainder phi x i (scalarBrownianCoordinate phi x i omega) dP. Source anchors appendix.tex:958-970, appendix.tex:983-996, appendix.tex:1161-1170, and appendix.tex:1379-1387 identify the EM increment, frozen interpolation, normalized Brownian coordinate, and weak-FP consumer while keeping sigma_eta^2 / 2 outside the scalar remainder pullback. Targeted source search excluding sald_version_2.tex found no named normalized-remainder or pullback definition in the original source, so if the local definitions do not reduce then lower_2 should record typed feedback leaf=hRemainderPullbackDef error_class=source_contract_gap_missing_remainder_pullback_definition. No source-Hessian work, selected-line raw Taylor replay, endpoint/naming replay, direct SLT dependency, project-article work, or wrapper churn is introduced." source := saldGeneralMovingTargetDiscreteEmSource status := ProofStatus.obligation dependsOn := [ "hRemainderPullbackDef", "hRemainderGeneratorLimitDef", "hRemainderGeneratorNormalizedLawDef", "hScalarMeas", "hNormalizedCoordinateLawDef", "hNormalizedRemainderMeas", "hNormalizedVectorLaw", "hCoordinateLawDef", "hVarianceDef", "scalarBrownianCoordinate", "normalizedCoordinateLaw", "normalizedRemainder", "remainderGeneratorLimit", "MeasureTheory.integral_map", "SALD.selectedWeakTestRemainderGeneratorNormalizedLawDefOfScalarPushforward", "SALD.selectedWeakTestRemainderGeneratorLimitDefOfStdGaussianVectorLaw", "SALD.selectedWeakTestRemainderGeneratorLimitDefOfScalarPushforwardAndStdGaussianVectorLaw", "AutoSamplingTheory.TechnicalLemmas.SALDExtracted.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw", "AutoSamplingTheory.TechnicalLemmas.Gaussian.nnrealVarianceOneOfGaussianRealUnitLaw", "runs/20260613-052444-683667-ASTIS-SALD-001-cycle206/middle_remainder_pullback_packet.md", "needed_shape=testRegular -> forall phi x i, remainderGeneratorLimit phi x i = integral omega, normalizedRemainder phi x i (scalarBrownianCoordinate phi x i omega) dP", "leaf=hRemainderPullbackDef", "error_class=source_contract_gap_missing_remainder_pullback_definition", "blocked_by=remainderGeneratorLimit and normalizedRemainder are source-facing abstract fields in the compiled scalar-pushforward remainder bridge unless lower_2 finds a reducible local definition", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source_contract_gap_missing_remainder_pullback_definition", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] note := "typed_verifier_feedback: leaf=hRemainderPullbackDef; error_class=source_contract_gap_missing_remainder_pullback_definition; needed_shape=testRegular -> forall phi x i, remainderGeneratorLimit phi x i = integral omega, normalizedRemainder phi x i (scalarBrownianCoordinate phi x i omega) dP; source_lines=appendix.tex:958-970;appendix.tex:983-996;appendix.tex:1161-1170;appendix.tex:1379-1387; blocked_by=remainderGeneratorLimit and normalizedRemainder are source-facing abstract fields in the compiled scalar-pushforward remainder bridge unless lower_2 finds a reducible local definition; lower_packet_artifact=runs/20260613-052444-683667-ASTIS-SALD-001-cycle206/middle_remainder_pullback_packet.md; packet_type=dynamic-leaf worker packet." /-- Cycle-206 proof-DAG pane for the normalized-remainder pullback gap. -/ def cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackMiddleDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle206.middle_packet.remainder_pullback_source_gap" interface := "Typed source-cited boundary: hRemainderPullbackDef is the sample-space definition of remainderGeneratorLimit as the expectation of normalizedRemainder evaluated at the normalized scalar Brownian coordinate. This is strictly below hRemainderGeneratorLimitDef because the compiled scalar-pushforward and Gaussian-law bridges already consume hRemainderPullbackDef." source := saldGeneralMovingTargetDiscreteEmSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackMiddleObligation", "SALD.selectedWeakTestRemainderGeneratorNormalizedLawDefOfScalarPushforward", "SALD.selectedWeakTestRemainderGeneratorLimitDefOfScalarPushforwardAndStdGaussianVectorLaw", "hRemainderPullbackDef", "hScalarMeas", "hNormalizedCoordinateLawDef", "hNormalizedRemainderMeas", "scalarBrownianCoordinate", "normalizedRemainder", "remainderGeneratorLimit", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source_contract_gap_missing_remainder_pullback_definition" ] reusedBy := [ "hRemainderGeneratorNormalizedLawDef", "hRemainderGeneratorLimitDef", "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsDominatedRemainderAndRemainderLimitScalarPushforward", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff" ] status := ProofStatus.obligation } ] /-! ### Cycle 206 lower_2: normalized-remainder pullback source gap -/ /-- Cycle-206 lower_2 packet for the normalized-remainder pullback field. Lower_2 inspected the local Lean interfaces and found `remainderGeneratorLimit`, `normalizedRemainder`, and `scalarBrownianCoordinate` only as parameters of the scalar-pushforward remainder bridges, not as reducible definitions. This records the exact source contract still needed below `hRemainderGeneratorLimitDef` without adding another consumer wrapper around the existing compiled bridge. -/ def cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackLower2Obligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle206_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_remainder_pullback_lower2" statement := "Cycle 206 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact missing theorem boundary narrowed: hRemainderPullbackDef, the sample-space normalized-remainder expectation required by SALD.selectedWeakTestRemainderGeneratorNormalizedLawDefOfScalarPushforward before hRemainderGeneratorLimitDef can be derived. Lean inspection shows remainderGeneratorLimit, normalizedRemainder, and scalarBrownianCoordinate are abstract parameters of the compiled scalar-pushforward and Gaussian-law remainder bridges rather than reducible local definitions, so lower_2 cannot close the pullback identity by rfl or simp. The needed source-backed shape is testRegular -> forall phi x i, remainderGeneratorLimit phi x i = integral omega, normalizedRemainder phi x i (scalarBrownianCoordinate phi x i omega) dP. A targeted original-source search excluding sald_version_2.tex found no named normalizedRemainder, remainderGeneratorLimit, or remainder-pullback definition; the source anchors appendix.tex:958-970, appendix.tex:983-996, appendix.tex:1161-1170, and appendix.tex:1379-1387 only identify the EM increment, frozen interpolation, normalized Brownian coordinate representation, and weak-FP consumer. No source-Hessian work, selected-line Taylor replay, endpoint/naming replay, direct SLT dependency, theorem-status promotion, or consumer-wrapper churn is introduced." source := saldGeneralMovingTargetDiscreteEmSource status := ProofStatus.obligation dependsOn := [ "hRemainderPullbackDef", "hRemainderGeneratorLimitDef", "hRemainderGeneratorNormalizedLawDef", "hScalarMeas", "hNormalizedCoordinateLawDef", "hNormalizedRemainderMeas", "hNormalizedVectorLaw", "hCoordinateLawDef", "hVarianceDef", "scalarBrownianCoordinate", "normalizedCoordinateLaw", "normalizedRemainder", "remainderGeneratorLimit", "MeasureTheory.integral_map", "SALD.selectedWeakTestRemainderGeneratorNormalizedLawDefOfScalarPushforward", "SALD.selectedWeakTestRemainderGeneratorLimitDefOfStdGaussianVectorLaw", "SALD.selectedWeakTestRemainderGeneratorLimitDefOfScalarPushforwardAndStdGaussianVectorLaw", "SALD.cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackMiddleObligation", "SALD.cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackMiddleDag", "AutoSamplingTheory.TechnicalLemmas.SALDExtracted.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw", "AutoSamplingTheory.TechnicalLemmas.Gaussian.nnrealVarianceOneOfGaussianRealUnitLaw", "runs/20260613-052444-683667-ASTIS-SALD-001-cycle206/lower_1_remainder_pullback_route.md", "runs/20260613-052444-683667-ASTIS-SALD-001-cycle206/middle_remainder_pullback_packet.md", "runs/20260613-052444-683667-ASTIS-SALD-001-cycle206/lower_2_remainder_pullback_boundary.md", "needed_shape=testRegular -> forall phi x i, remainderGeneratorLimit phi x i = integral omega, normalizedRemainder phi x i (scalarBrownianCoordinate phi x i omega) dP", "leaf=hRemainderPullbackDef", "error_class=source_contract_gap_missing_remainder_pullback_definition", "blocked_by=remainderGeneratorLimit, normalizedRemainder, and scalarBrownianCoordinate are abstract parameters in the compiled scalar-pushforward remainder bridge", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source_contract_gap_missing_remainder_pullback_definition", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] note := "typed_verifier_feedback: leaf=hRemainderPullbackDef; error_class=source_contract_gap_missing_remainder_pullback_definition; needed_shape=testRegular -> forall phi x i, remainderGeneratorLimit phi x i = integral omega, normalizedRemainder phi x i (scalarBrownianCoordinate phi x i omega) dP; source_lines=appendix.tex:958-970;appendix.tex:983-996;appendix.tex:1161-1170;appendix.tex:1379-1387; blocked_by=remainderGeneratorLimit, normalizedRemainder, and scalarBrownianCoordinate are abstract parameters of the compiled scalar-pushforward remainder bridge and no original-paper or local Lean definition identifies remainderGeneratorLimit with the sample-space expectation of normalizedRemainder along the scalar Brownian coordinate; lower_1_route_artifact=runs/20260613-052444-683667-ASTIS-SALD-001-cycle206/lower_1_remainder_pullback_route.md; lower_2_artifact=runs/20260613-052444-683667-ASTIS-SALD-001-cycle206/lower_2_remainder_pullback_boundary.md; packet_type=dynamic-leaf worker packet." /-- Cycle-206 proof-DAG pane for the lower_2 remainder-pullback gap. -/ def cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackLower2Dag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle206.lower_2_packet.remainder_pullback_source_gap" interface := "Typed source-cited boundary: hRemainderPullbackDef remains the exact sample-space definition of remainderGeneratorLimit as the expectation of normalizedRemainder evaluated at scalarBrownianCoordinate. Local Lean inspection found no reducible definition for the three names, and the existing scalar-pushforward bridge already consumes this field to derive hRemainderGeneratorNormalizedLawDef and hRemainderGeneratorLimitDef." source := saldGeneralMovingTargetDiscreteEmSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackLower2Obligation", "SALD.cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackMiddleObligation", "SALD.selectedWeakTestRemainderGeneratorNormalizedLawDefOfScalarPushforward", "SALD.selectedWeakTestRemainderGeneratorLimitDefOfScalarPushforwardAndStdGaussianVectorLaw", "hRemainderPullbackDef", "hScalarMeas", "hNormalizedCoordinateLawDef", "hNormalizedRemainderMeas", "scalarBrownianCoordinate", "normalizedRemainder", "remainderGeneratorLimit", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source_contract_gap_missing_remainder_pullback_definition" ] reusedBy := [ "hRemainderGeneratorNormalizedLawDef", "hRemainderGeneratorLimitDef", "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsDominatedRemainderAndRemainderLimitScalarPushforward", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff" ] status := ProofStatus.obligation } ] /-- Cycle-130 proof-DAG pane for the weak Laplacian integration-by-parts sub-boundary inside the EM diffusion source action. -/ def cycle130GeneralMovingTargetDiscreteEmLaplacianIbPDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle130.lower_packet.weak_laplacian_ibp_boundary" interface := "Compiled lower-ready continuation: SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianActionOfIntegrationByParts derives the old hlaplacianAction from the positive sigmaCoeff-scaled density-Laplacian weak term and the weak Laplacian integration-by-parts identity; SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianIntegrationByParts then reuses the cycle-129 diffusion-source helper while keeping hdiffusionAction separate." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle130GeneralMovingTargetDiscreteEmLaplacianIbPLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianActionOfIntegrationByParts", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianIntegrationByParts", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianAction", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle130.lower_1_packet.green_laplacian_ibp_route" interface := "Compiled lower_1 scout route: SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity derives the old hweakLaplacianIbP shape from first-Green density-Laplacian-to-negative-gradient-pairing, second-Green negative-gradient-pairing-to-test-Laplacian, and test-Laplacian normalization facts. SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfGreenLaplacianIbP feeds that route into the existing cycle-130 diffusion-source helper without reintroducing direct hweakLaplacianIbP." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle130GeneralMovingTargetDiscreteEmGreenLaplacianIbPScoutObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfGreenLaplacianIbP", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianIntegrationByParts", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle130.lower_2_packet.first_green_no_boundary_flux" interface := "Compiled lower_2 continuation: SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfFirstGreenNoBoundaryFlux removes the direct hfirstGreen premise from the lower_1 Green route by deriving the first Green identity from a density-Laplacian residual identity, a divergence-theorem boundary-flux identity, and zero boundary flux. The second Green identity, test-Laplacian normalization, hdiffusionAction, and hdiffusionLaplacianTerm remain explicit." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle130GeneralMovingTargetDiscreteEmFirstGreenNoBoundaryFluxLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfFirstGreenNoBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfGreenLaplacianIbP", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "appendix.tex:1379-1387" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.formalized }, { id := "ASTIS.SALD.cycle130.remaining_laplacian_ibp_analysis" interface := "Remaining exact theorem boundary after cycle 130 lower_2: prove the density-Laplacian weak term for the EM/Brownian diffusion contribution, instantiate the first-Green residual/divergence/zero-flux facts for the source density and admissible tests, then prove the second Green identity and test-Laplacian normalization." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "appendix.tex:1379-1387", "appendix.tex:1368-1377", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfFirstGreenNoBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfGreenLaplacianIbP", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle130.reviewer_laplacian_ibp_check" interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, the new theorem statement lacks the direct hlaplacianAction premise, hdiffusionAction remains separate, the remaining boundary is the density-Laplacian weak term plus weak Laplacian integration by parts, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, or sald_version_2 use." source := saldGeneralMovingTargetDiscreteWeakFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle130GeneralMovingTargetDiscreteEmLaplacianIbPLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianIntegrationByParts", "appendix.tex:1379-1387" ] reusedBy := ["cycle 130 reviewer"] status := ProofStatus.obligation } ] /-- Cycle-70 proof-DAG pane for the EM conditional-law/measurability backfill. -/ def cycle70GeneralMovingTargetDiscreteConditionalLawDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle70.global_phase_judgment" interface := "Cycle 70 judgment: cycle 69 passed and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for one cited-theory/SDE backend backfill; the risk-reducing packet is sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to conditional-law/measurability and named conditional drift interfaces." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle69MainSkeletonAnalyticMiddleObligation", "SALD.cycle69GeneralMovingTargetDiscreteEmFpSourceSignsLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete", "sald.general_moving_target_discrete.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle70.middle_conditional_law_interface" interface := "Middle conversion window for appendix.tex:1368-1377: name the regular conditional kernel, component conditional fields, selected bar b_{k,s}, and regularity side conditions before the weak conditional Fokker-Planck handoff." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawMiddleObligation", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle70.lower_named_conditional_drift" interface := "Lower proof-producing wrappers: from supplied conditional-expectation linearity and named component fields, rewrite bar b_{k,s} into dot t_k*condC_{k,s}+(sigma_eta^2/2)*condScore_{k,s}; derive the combination regularity from component regularity using add/smul closure, then transfer it to bar b_{k,s} by pointwise equality." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawLowerObligation", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] /-- Cycle-69 proof-DAG pane for the post-route analytic-interface ledger. -/ def cycle69MainSkeletonAnalyticInterfaceDag : List ProofDagBlock := [ { id := "ASTIS.SALD.cycle69.global_phase_judgment" interface := "Upper judgment for cycle 69: cycle 68 passed and needs no recovery; Phase 1 is stable enough for the post-route analytic-interface ledger but not broad backfill; select the shared EM interpolation conditional-law/Fokker-Planck backend as the one lower packet." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle69MainSkeletonAnalyticInterfaceLedger", "SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation", "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeLowerObligation" ] reusedBy := ["ASTIS-SALD-001 cycle 69", "sald.general_moving_target_discrete.em_interpolation_fp"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle69.five_backend_check" interface := "Recheck Gronwall, DV, LSI/KL/FI, continuous forward-KL Fokker-Planck/KL derivative, and Euler-Maruyama interpolation Fokker-Planck interfaces after all six theorem-route skeletons are wired." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle69.theorem_route_recheck" interface := "Verify that the five analytic interfaces feed the six theorem skeletons in source order: forward-KL, discrete forward-KL, guided residual, continuous general moving target, unified forward-KL, and discrete general moving target." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle69MainSkeletonAnalyticInterfaceObligation", "SALD.continuousSaldContract", "SALD.discreteSaldContract", "SALD.guidedResidualContract", "SALD.generalVaSaldContract", "SALD.unifiedForwardKlContract", "SALD.generalVaSaldDiscreteContract", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle69.middle_interface_audit" interface := "Middle audit for cycle 69: synchronize the post-route five-backend ledger with the six theorem skeletons, keep all theorem statements and statuses fixed, and hand lower work to the shared EM interpolation conditional-law/Fokker-Planck backend over appendix.tex:1358-1387." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle69MainSkeletonAnalyticMiddleContract", "SALD.cycle69MainSkeletonAnalyticMiddleObligation", "SALD.cycle69MainSkeletonAnalyticInterfaceObligation", "SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation", "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeLowerObligation", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := [ "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "cycle 69 lower handoff" ] status := ProofStatus.obligation }, { id := "ASTIS.SALD.cycle69.lower_packet.em_interpolation_fp" interface := "Selected lower packet: sharpen sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, exposing common-space/density prerequisites, regular conditional drift, and the weak conditional Fokker-Planck equation with source signs while leaving KL differentiation, LSI, DV, and Gronwall separate." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle69MainSkeletonAnalyticInterfaceObligation", "SALD.cycle69MainSkeletonAnalyticMiddleObligation", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination", "SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination", "SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff", "SALD.cycle69GeneralMovingTargetDiscreteEmFpSourceSignsLowerObligation", "sald.general_moving_target_discrete.cycle64_conditional_drift_lower", "sald.general_moving_target_discrete.cycle69_em_fp_source_signs_lower", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] reusedBy := [ "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] status := ProofStatus.obligation } ] def guidedResidualNormalizerObligation : ProofObligation where id := "sald.guided_path_residual.normalizer_derivative" statement := "Formalize appendix lines 630-656: for Z_t=int p_t exp(-f_t), use partial_t p_t=-div(p_t*u_t), differentiation under the integral, and integration by parts to prove dot Z_t/Z_t=-E_{pi_t}[partial_t f_t+nabla f_t^T u_t]." source := saldGuidedResidualProofSource status := ProofStatus.obligation dependsOn := ["TransportVelocityContract", "GuidedTiltContract"] note := "The paper performs this calculation directly; Lean must expose positive finite Z_t, dominated convergence, and boundary decay as obligations." def guidedResidualIdentityObligation : ProofObligation where id := "sald.guided_path_residual.identity" statement := "Formalize appendix lines 658-704: differentiate pi_t=Z_t^(-1)*p_t*exp(-f_t), add div(pi_t*u_t), cancel the divergence terms, substitute dot Z_t/Z_t, and prove the residual identity plus mean-zero property." source := saldGuidedResidualSource status := ProofStatus.obligation dependsOn := ["sald.guided_path_residual.normalizer_derivative", "TransportVelocityContract", "GuidedTiltContract"] note := "This proposition supports thm:unified-forward-KL; do not replace the centered residual with an uncentered guide derivative." def generalMovingTargetDerivativeObligation : ProofObligation where id := "sald.general_moving_target.kl_derivative" statement := "Formalize appendix lines 765-884: differentiate KL(rho_s||pi_{t(s)}), insert the general VA-SALD Fokker--Planck equation, combine c_t with the target transport velocity v_t to expose m_t=v_t-c_t, apply Young's inequality with epsilon=2*dot{t}(s)/sigma_{t(s)}^2, change from s to t, and apply LSI." source := saldGeneralMovingTargetDerivativeSource status := ProofStatus.obligation dependsOn := [ "eq:general_moving_target_SALD", "TransportVelocityContract", "FokkerPlanckContract", "KLContract", "FIContract", "probability.lsi_to_kl_fi", "SALD.generalMovingTargetKlDerivativeResidualSplitScalar", "SALD.generalMovingTargetKlDerivativeScaledResidualDisplayScalar", "SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar", "SALD.generalMovingTargetPostYoungDerivativeBoundScalar", "SALD.generalMovingTargetLsiDerivativeBoundScalar", "SALD.generalMovingTargetTimeChangedDerivativeBoundScalar", "SALD.generalMovingTargetPreDvDerivativeBoundScalar", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change" ] note := "generalMovingTargetDerivativeCandidateContract records the sigma-weighted derivative route and its implicit regularity gaps before proof search. Cycle 47 lower adds compiled scalar/order handoffs for residual Young, LSI substitution, and inverse-schedule coefficient rewriting; cycle 57 lower adds the raw KL-derivative residual split before those handoffs; cycle 62 lower adds the dot{t}(s)-scaled residual sign handoff for the target-transport term. The Fokker-Planck, mass conservation, integration-by-parts, target transport, LSI density-test, and schedule backends remain obligations." def generalMovingTargetDvEnergyObligation : ProofObligation where id := "sald.general_moving_target.dv_m_energy" statement := "Formalize appendix lines 886-907: apply Donsker--Varadhan with Z=alpha*||m_t||^2 to obtain ||m_t||_{L2(rho_{s(t)})}^2 <= alpha^(-1)*K(t)+E_alpha(pi_t,m_t)." source := saldGeneralMovingTargetResidualDvSource status := ProofStatus.obligation dependsOn := [ "probability.dv_variational_formula", "def:alpha-complexity", "KLContract", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling" ] note := "The DV formula remains source-cited; this theorem-specific obligation must expose the finite log-mgf witness from the alpha0 assumption on m_t." def generalMovingTargetDvPositiveAlphaScalingObligation : ProofObligation where id := "sald.general_moving_target.dv_positive_alpha_scaling" statement := "Formalize appendix lines 887-907 after DV has been applied with Z=alpha*||m_t||^2: use alpha>0 to divide alpha*E_{rho_{s(t)}}[||m_t||^2] <= K(t)+log E_{pi_t}[exp(alpha*||m_t||^2)], rewrite the log-mgf quotient as E_alpha(pi_t,m_t), and preserve the downstream coefficient sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1) on K(t)." source := saldGeneralMovingTargetResidualDvSource status := ProofStatus.obligation dependsOn := [ "probability.dv_variational_formula", "def:alpha-complexity", "SALD.saldDvFiniteLogMgfContract", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "KLContract" ] note := "This lower packet isolates only the positive-alpha scalar division and E_alpha rewrite inside the residual m_t DV witness. It adds no hypothesis to thm:general-moving-target-SALD or thm:unified-forward-KL and does not mark DV formalized." def generalMovingTargetDvFiniteLogMgfWitnessObligation : ProofObligation where id := "sald.general_moving_target.dv_finite_log_mgf_witness" statement := "Formalize the residual DV side interface for appendix lines 885-895: with nu=rho_{s(t)}, mu=pi_t, and Z=alpha*||m_t||^2, derive the finite log-mgf condition for 00, and preserve the downstream sigma_t^(-2)*dot{s}(t)^(-1) coefficient." source := saldGeneralMovingTargetResidualDvSource status := ProofStatus.obligation dependsOn := [ "probability.dv_variational_formula", "def:alpha-complexity", "SALD.saldDvFiniteLogMgfContract", "sald.dv_variation.finite_log_mgf_interface", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.kl_derivative" ] note := "This narrows the Cycle 12 upper target to the residual m_t DV witness. The positive-alpha division and E_alpha rewrite are now split into sald.general_moving_target.dv_positive_alpha_scaling. It is not a new theorem assumption and does not mark DV or exponential-moment monotonicity formalized." def generalMovingTargetGronwallApplicationObligation : ProofObligation where id := "sald.general_moving_target.gronwall_application" statement := "Instantiate lem:gronwall on appendix lines 908-934 with a(t)=(sigma_t^2/2)*dot{s}(t)*C_LSI(t)-sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1) and b(t)=sigma_t^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t), preserving the theorem's sigma-weighted exponent factors." source := saldGeneralMovingTargetDvGronwallSource status := ProofStatus.obligation dependsOn := ["sald.gronwall.integrating_factor", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_m_energy"] note := "Do not normalize away sigma_t or replace the source Gronwall signs; generalMovingTargetGronwallInstantiationContract records the exact a(t), b(t), and residual-exponential simplification." def cycle24GeneralVaSaldGronwallMiddleObligation : ProofObligation where id := "sald.general_moving_target.cycle24_gronwall_middle" statement := "Maintain the cycle-24 middle source-to-Lean map for appendix lines 909-945 and theorem display lines 727-743: route the Gronwall output through endpoint K(T)/K(0) rewrites, coefficient regularity for the sigma-weighted a(t), b(t), exponent splitting, residual-exponent monotonicity, and zero-residual alpha-complexity without changing thm:general-moving-target-SALD or thm:unified-forward-KL." source := saldGeneralMovingTargetDvGronwallSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle24GeneralVaSaldUpperPacket", "SALD.cycle24GeneralVaSaldMiddleContract", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.kl_derivative", "sald.forward_kl.schedule_time_change", "sald.gronwall.integrating_factor", "sald.gronwall.exponent_rewrite", "SALD.generalMovingTargetGronwallCoeffAdjacentIntervalIntegrable", "SALD.generalMovingTargetGronwallExpProductRewriteIntegralCongrOfPieces", "probability.lsi_to_kl_fi" ] note := "This is middle-role synchronization data. It selects coefficient regularity and adjacent interval-integrability as the preferred first lower sub-slice; the compiled cycle-24 lower helpers package only the local interval-integrability closure after source hypotheses are supplied and do not prove thm:general-moving-target-SALD, Gronwall, endpoint rewrites, exponent monotonicity, or pure contraction." def generalMovingTargetGronwallSideConditionObligation : ProofObligation where id := "sald.general_moving_target.gronwall_side_conditions" statement := "Formalize appendix lines 908-945: after applying lem:gronwall, identify K(T)=KL(rho_S||pi_T) and K(0)=KL(rho_0||pi_0), prove the sigma-weighted a(t), b(t) regularity needed by Gronwall, split exp(-int a), justify dropping the nonpositive LSI contribution from the residual exponent, and show c_t=v_t makes E_alpha(pi_t,m_t)=0." source := saldGeneralMovingTargetDvGronwallSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle24GeneralVaSaldUpperPacket", "SALD.cycle24GeneralVaSaldMiddleContract", "sald.general_moving_target.cycle24_gronwall_middle", "sald.forward_kl.schedule_time_change", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_application", "sald.gronwall.integrating_factor", "SALD.generalMovingTargetGronwallCoeffAdjacentIntervalIntegrable", "SALD.generalMovingTargetGronwallExpProductRewriteIntegralCongrOfPieces", "def:alpha-complexity" ] note := "generalMovingTargetGronwallSideConditionContract records endpoint rewrites, coefficient regularity, exponent splitting, residual-exponent sign facts, and zero-residual alpha-complexity without changing thm:general-moving-target-SALD. SALD.generalMovingTargetGronwallCoeffAdjacentIntervalIntegrable only assembles a(t) and carries b(t) interval-integrability after the theorem-specific regularity hypotheses are provided." def generalMovingTargetPureContractionObligation : ProofObligation where id := "sald.general_moving_target.pure_contraction" statement := "Formalize appendix lines 936-945: under c_t=v_t, m_t=0 and E_alpha(pi_t,m_t)=0, so the residual integral in eq:general_moving_target_KL_bound vanishes and the pure contraction estimate remains." source := saldGeneralMovingTargetPureContractionSource status := ProofStatus.obligation dependsOn := ["sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.gronwall_application", "def:alpha-complexity"] note := "This is an algebraic specialization of the source theorem, not a separate stronger theorem." def cycle16UnifiedForwardKlTransportBridgeMiddleObligation : ProofObligation where id := "sald.unified_forward_kl.transport_bridge_middle" statement := "Maintain the cycle-16 middle source-to-Lean map for the unified VA-SALD transport bridge: main_body.tex lines 359-368 must route from prop:guided_path_residual and eq:poisson-eq to u_t+w_t transporting pi_t, then appendix.tex lines 949-951 must specialize thm:general-moving-target-SALD with c_t=u_t and m_t=w_t." source := saldUnifiedTransportBridgeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle16GeneralVaSaldUpperPacket", "SALD.cycle16UnifiedForwardKlTransportBridgeMiddleContract", "SALD.unifiedForwardKlSpecializationContract", "sald.guided_path_residual.identity", "eq:poisson-eq", "sald.unified_forward_kl.transport_velocity_bridge" ] note := "This is middle-role synchronization data. It selects the transport-velocity bridge as the lower slice and keeps correction-field existence, divergence regularity, DV, Gronwall, and discrete EM work as separate obligations." def cycle16UnifiedForwardKlTransportBridgeLowerObligation : ProofObligation where id := "sald.unified_forward_kl.transport_bridge_lower" statement := "Refine the lower slice for main_body.tex lines 359-368: add the residual equation and correction equation with their source signs, expose the divergence-linearity backend needed to rewrite div(pi_t*(u_t+w_t)), cancel the centered guide residual, and hand v_t=u_t+w_t, c_t=u_t, m_t=w_t to the unified specialization." source := saldUnifiedTransportBridgeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle16UnifiedForwardKlTransportBridgeMiddleContract", "SALD.cycle16UnifiedForwardKlTransportBridgeLowerContract", "SALD.unifiedForwardKlSpecializationContract", "sald.guided_path_residual.identity", "eq:poisson-eq", "TransportVelocityContract" ] note := "This lower sub-obligation remains inside sald.unified_forward_kl.transport_velocity_bridge. It records only the signed cancellation and divergence-linearity interface; correction-field existence, residual DV, Gronwall, and discrete EM remain separate." def unifiedForwardKlTransportBridgeObligation : ProofObligation where id := "sald.unified_forward_kl.transport_velocity_bridge" statement := "Formalize main_body.tex lines 359-368: combine prop:guided_path_residual, partial_t pi_t+div(pi_t*u_t)=-pi_t*(g_t-E_pi_t[g_t]), with eq:poisson-eq, div(pi_t*w_t)=pi_t*(g_t-E_pi_t[g_t]), to derive partial_t pi_t+div(pi_t*(u_t+w_t))=0; then record the source specialization v_t=u_t+w_t, c_t=u_t, and m_t=w_t." source := saldCorrectionFieldSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle16UnifiedForwardKlTransportBridgeMiddleContract", "SALD.cycle16UnifiedForwardKlTransportBridgeLowerContract", "SALD.unifiedForwardKlSpecializationContract", "sald.guided_path_residual.identity", "eq:poisson-eq", "sald.unified_forward_kl.transport_bridge_lower", "TransportVelocityContract" ] note := "Cycle 16 isolates only the correction-field transport bridge used by thm:unified-forward-KL. Existence and regularity of w_t remain local obligations, and the source proof route through thm:general-moving-target-SALD is unchanged." def unifiedForwardKlSpecializationObligation : ProofObligation where id := "sald.unified_forward_kl.specialization" statement := "Formalize appendix lines 949-951 and main_body.tex lines 359-395: specialize thm:general-moving-target-SALD by setting c_t=u_t, use prop:guided_path_residual and eq:poisson-eq to obtain the transport velocity v_t=u_t+w_t for pi_t, identify m_t=v_t-c_t=w_t, and preserve the sigma-weighted VA-SALD bound." source := saldUnifiedForwardKlSource status := ProofStatus.obligation dependsOn := [ "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_bridge_middle", "sald.unified_forward_kl.transport_velocity_bridge", "sald.guided_path_residual.identity", "eq:poisson-eq", "eq:SALD_Ito", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction" ] note := "unifiedForwardKlSpecializationContract records the source bridge from the guided residual and correction-field equation to the general theorem specialization. The proof is a specialization only; no new path-space or direct SALD argument should be substituted." def generalVaSaldGuidedPathMiddleObligation : ProofObligation where id := "sald.general_va_sald.guided_path_middle" statement := "Maintain the cycle-12 middle source-to-Lean map for the guided/general VA-SALD path: prop:guided_path_residual, the correction-field bridge, thm:general-moving-target-SALD, thm:unified-forward-KL, thm:general-moving-target-SALD-discrete, and the discrete guided specialization must each point to explicit Lean-facing contracts, cited results, or proof obligations without changing theorem statements." source := saldGeneralMovingTargetSource status := ProofStatus.obligation dependsOn := [ "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "sald.unified_forward_kl.transport_bridge_middle", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions", "sald.general_moving_target_discrete.unified_specialization" ] note := "cycle12GeneralVaSaldMiddleContract is a lower-ready map, not a theorem proof. It preserves appendix.tex:619-951, appendix.tex:1354-1603, and main_body.tex:359-395 as obligations until local Lean proofs replace the analytic backends." def cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle64_conditional_drift_lower" statement := "Cycle 64 lower sharpens the regular conditional drift interface for appendix.tex lines 1368-1377: SALD.generalMovingTargetDiscreteConditionalDriftContract records the common-space, regular conditional law/kernel, measurability, integrability, and selected conditional expectation needed to define bar b_{k,s}; SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination and SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination compile only the abstract linearity algebra for the two drift summands after the analytic conditional-expectation backend is supplied." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination", "SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "sald.general_moving_target_discrete.em_interpolation_fp", "FokkerPlanckContract", "KLContract", "FIContract" ] note := "Proof-producing work is limited to conditional-expectation linearity algebra under explicit hypotheses. Regular conditional laws, density/absolute-continuity, weak Fokker-Planck, KL differentiation, integration by parts, LSI, DV, Gronwall, and theorem status remain obligations." def generalMovingTargetDiscreteEmInterpolationObligation : ProofObligation where id := "sald.general_moving_target_discrete.em_interpolation_fp" statement := "Formalize appendix lines 957-996 and 1354-1387: the discrete general VA-SALD EM interpolation has endpoint laws rho_k^eta, rho_{k+1}^eta and satisfies the conditional-drift Fokker--Planck equation with diffusion coefficient sigma_eta^2/2. The named-interpolation endpoint law pair now compiles from pointwise endpoint identities and law representations, cycle 64 records the regular conditional drift interface for bar b_{k,s} and compiles only its abstract linear-combination algebra, cycle 70 compiles only the named component drift and component-regularity handoffs, cycle 71 compiles only the Measure.map second-marginal identification and supplied kernel-compatibility transport from the joint law to the named hat rho_s marginal, cycle 72 records the weak-test FP source-sign interface and compiles only the test-indexed and admissible-test coefficient/sign packaging with -div drift and +(sigma_eta^2/2) Delta diffusion, cycle 73 compiles only the scalar substitution of that supplied weak FP identity into the differentiated KL formula at the log-ratio test, including a direct composition through the cycle-72 admissible source-sign wrapper, cycle 74 records the blocked condExpKernel/conditional-kernel Mathlib measure interface as sourceCited, middle-synchronizes the exact condDistrib/condExpKernel audit map for appendix.tex:1368-1377, and lower compiles only a supplied-kernel compatibility/regularity wrapper, cycle 75 middle sharpens the condDistrib orientation handoff from the existing (X_k^eta,hat X_s) joint-law naming to Mathlib's (hat X_s,X_k^eta) conditioning order, while cycle 75 lower compiles only the first-marginal, swap-orientation, and supplied-kernel regularity wrapper under explicit component-integral hypotheses, cycle 76 compiles only endpoint Measure.map-to-conditional compatibility wrappers that package endpoint laws, the swapped first marginal, the original second marginal, the swap equality, and original-orientation kernel compatibility under supplied hypotheses, cycle 80 lower compiles only the composition of that endpoint/conditional compatibility package with supplied component conditional-integral regularity to obtain measurable/integrable bar b_{k,s}, cycle 77 compiles only generator-level weak-FP source-sign handoffs from supplied generator/time-derivative, generator-expansion, and split drift/diffusion source-action hypotheses to the exact -div drift and +(sigma_eta^2/2) Delta diffusion statement, cycle 78 middle maps those supplied generator pieces to the differentiated KL display and cycle 78 lower compiles only the equality composition at the admissible log-ratio test, cycle 69 compiles only the source-sign/coefficient rewrite for the supplied weak FP identity, and the cycle-54 lower handoff compiles only the sigma-weighted divergence regrouping after the FP and Laplacian identities are supplied; the conditional-drift Fokker--Planck and KL differentiation backends remain open." source := saldGeneralMovingTargetDiscreteEmSource status := ProofStatus.obligation dependsOn := ["sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit", "SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination", "SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap", "SALD.generalMovingTargetDiscreteConditionalKernelCompatibilityOfJointMapMarginal", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityOfJointMap", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "sald.general_moving_target_discrete.cycle64_conditional_drift_lower", "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawMiddleObligation", "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawLowerObligation", "sald.general_moving_target_discrete.cycle70_named_conditional_drift_lower", "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "sald.general_moving_target_discrete.cycle71_endpoint_conditional_lower", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.cycle72GeneralMovingTargetDiscreteWeakFpMiddleObligation", "SALD.cycle72GeneralMovingTargetDiscreteWeakFpLowerObligation", "sald.general_moving_target_discrete.cycle72_weak_fp_source_signs_lower", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns", "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpUpperObligation", "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpMiddleObligation", "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpLowerObligation", "sald.general_moving_target_discrete.cycle73_kl_derivative_weak_fp_lower", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceUpperObligation", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation", "sald.general_moving_target_discrete.cycle74_conditional_kernel_measure_interface", "sald.general_moving_target_discrete.cycle74_measure_interface_middle", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToSwappedConditionalCompatibility", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "AutoSamplingTheory.lawMapProdSwap", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorMiddleObligation", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "sald.general_moving_target_discrete.cycle77_weak_fp_generator_lower", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorUpperObligation", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation", "sald.general_moving_target_discrete.cycle78_kl_derivative_generator_lower", "SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff", "SALD.cycle69GeneralMovingTargetDiscreteEmFpSourceSignsLowerObligation", "sald.general_moving_target_discrete.cycle69_em_fp_source_signs_lower", "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "sald.general_moving_target_discrete.cycle54_em_fp_lower", "EulerMaruyamaContract", "FokkerPlanckContract", "KLContract", "FIContract"] note := "This is the general VA analogue of the SALD EM interpolation backend; endpoint-law handoffs, conditional-drift linearity algebra, named component regularity wrappers, endpoint-to-conditional marginal transport and packaged compatibility, weak-test and admissible-test source-sign packaging, weak-FP-to-KL scalar substitution including the cycle-72 composition wrapper, the cycle-74 source-cited conditional-kernel measure interface, middle source map, lower supplied-kernel regularity wrapper, the cycle-75 condDistrib orientation source map and lower swapped-joint regularity wrapper, cycle-76 endpoint Measure.map-to-conditional compatibility packaging in both swapped and original marginal views, cycle-80 endpoint/conditional drift-regularity composition, cycle-77 generator-level source-sign handoffs including the split drift/diffusion component wrapper, cycle-78 middle source map and lower generator-piece-to-KL derivative equality composition, source-sign coefficient rewriting, and the sigma-weighted FP regrouping compile or are recorded under explicit hypotheses, while the source's common-space construction, regular conditional law, density/absolute-continuity, actual weak Fokker-Planck theorem, log-ratio admissibility, KL differentiation, and integration by parts remain obligations." def generalMovingTargetDiscreteConstantScheduleObligation : ProofObligation where id := "sald.general_moving_target_discrete.constant_schedule_stitching" statement := "Formalize the constant inverse-schedule and stitched-interval interface used in appendix lines 1315 and 1573-1600: dot{t}(s)=dot{s}(t)^(-1) is constant, endpoint laws identify K(T) with KL(rho_K^eta||pi_T), and interval-wise derivative inequalities are admissible for the final Gronwall step." source := saldGeneralMovingTargetDiscreteSource status := ProofStatus.obligation dependsOn := ["sald.general_moving_target_discrete.em_interpolation_fp", "sald.forward_kl.schedule_time_change", "sald.gronwall.integrating_factor"] note := "The paper states the constant-schedule condition but does not spell out the closed-interval regularity needed for a global Gronwall application." def generalMovingTargetDiscreteFrozenDeltaObligation : ProofObligation where id := "sald.general_moving_target_discrete.frozen_delta_cross_lip" statement := "Formalize Lemma lem:frozen_delta_cross_lip, appendix lines 1026-1307: under the source c_t/score space-time Lipschitz assumptions, exponential-complexity assumptions, and step-size condition, bound the VA frozen-field cross term by (sigma_eta^2/8)*FI + 2*eta^2*alpha'^(-1)*Gamma*K + 2*eta*Delta." source := saldFrozenDeltaCrossLipGeneralSource status := ProofStatus.obligation dependsOn := ["eq:SALD_general_EM", "eq:general_discrete_delta_def", "lem:dv_variation", "def:alpha-complexity"] note := "generalFrozenDeltaCrossLipContract records the source proof split through the joint law, increment estimate, DV bounds for c_t/score/1+M, and Gamma/Delta collection." def cycle28GeneralVaSaldDerivativeSideMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle28_derivative_side_middle" statement := "Maintain the cycle-28 middle source-to-Lean map for appendix lines 1469-1511: rewrite the derivative cross field as delta_pi^VA+dot t(s)*m_{t(s)}, split the derivative cross term into frozen and residual parts, preserve the two sigma_eta^2/8 Young shares, and route the remaining analytic work to sald.general_moving_target_discrete.derivative_side_conditions without changing theorem coefficients." source := saldGeneralMovingTargetDiscreteYoungSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle28GeneralVaSaldUpperPacket", "SALD.cycle28GeneralVaSaldMiddleContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteFrozenResidualAlgebraVector", "SALD.generalMovingTargetDiscreteYoungFisherShareScalar", "SALD.generalMovingTargetDiscreteTwoYoungFisherBudgetScalar", "SALD.generalMovingTargetDiscreteResidualYoungCoefficientScalar", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "sald.general_moving_target_discrete.derivative_side_conditions", "probability.lsi_to_kl_fi" ] note := "This is middle-role synchronization data. The compiled module helper closes only algebra after the concrete field identifications are supplied, and the three scalar helpers close only real coefficient bookkeeping for the Young shares. Inner-product Young, frozen-delta, LSI, DV, time change, and the full derivative theorem remain obligations." def cycle28GeneralVaSaldDerivativeSideLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle28_derivative_side_lower" statement := "Record the cycle-28 lower compiled algebra for appendix lines 1469-1478: once eq:general_discrete_delta_def identifies delta_pi^VA as dot t(s)*c_t plus the score term minus the frozen conditional drift, tilde v_s is dot t(s)*v_t, and m_t=v_t-c_t, the cross field rewrites as delta_pi^VA+dot t(s)*m_t. The conditional drift, score, slowed-transport, pointwise vector-field identifications, Young inequalities, frozen-delta lemma, LSI, DV, time change, and theorem statement remain obligations." source := saldGeneralMovingTargetDiscreteYoungSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle28GeneralVaSaldMiddleContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteFrozenResidualAlgebraVector", "eq:general_discrete_delta_def", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.derivative_side_conditions" ] note := "The compiled theorem is only module-level algebra. It does not prove the analytic equality of the paper's concrete vector fields or any Young/FI/frozen-delta side condition." def generalMovingTargetDiscreteDerivativeSideConditionObligation : ProofObligation where id := "sald.general_moving_target_discrete.derivative_side_conditions" statement := "Formalize the side interfaces used in appendix lines 1354-1600 before the discrete general VA-SALD derivative inequality: EM endpoint laws through the compiled named-interpolation handoff, endpoint-to-conditional marginal compatibility through the cycle-71 Measure.map wrappers, regular conditional drift through the cycle-64 drift contract and linearity wrappers, the cycle-74 source-cited condExpKernel/conditional-kernel measure interface plus middle source map and lower supplied-kernel regularity wrapper for the blocked conditional-law theorem, the cycle-75 middle condDistrib orientation handoff and lower swapped-joint supplied-kernel wrapper from the existing (X_k^eta,hat X_s) joint-law name to Mathlib's (hat X_s,X_k^eta) conditioning order, the cycle-76 endpoint Measure.map-to-conditional compatibility wrappers connecting endpoint laws, named hat rho_s marginal in both swapped first-marginal and original second-marginal views, swap equality, and original-orientation kernel compatibility under supplied hypotheses, the cycle-80 endpoint/conditional drift-regularity wrapper that composes those marginal/kernel facts with supplied component conditional-integral regularity to obtain measurable/integrable bar b_{k,s}, the cycle-72 weak-test source-sign interface and wrappers for the conditional Fokker--Planck equation, including the admissible-test predicate variant, the cycle-77 generator-level handoffs splitting the supplied weak FP theorem into generator/time-derivative, generator source-expansion, and component drift/diffusion source-action hypotheses, the cycle-73 weak-FP-to-KL derivative handoff at the log-ratio test and its direct composition through the cycle-72 admissible source-sign wrapper, the cycle-78 generator-piece-to-KL derivative handoff composing the cycle-77 source-action hypotheses directly into the same log-ratio derivative display, the cycle-69 source-sign/coefficient handoff for the supplied weak conditional Fokker--Planck equation, conditional Fokker--Planck split with the cycle-54 sigma-weighted divergence regrouping, slowed transport velocity, frozen/residual algebra delta_pi^VA+dot t*m, the two sigma_eta^2/8 Young splits, DV finite-log-mgf side condition, and stitched time-change to K(t)." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := ["sald.general_moving_target_discrete.cycle28_derivative_side_middle", "sald.general_moving_target_discrete.cycle28_derivative_side_lower", "SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap", "SALD.generalMovingTargetDiscreteConditionalKernelCompatibilityOfJointMapMarginal", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityOfJointMap", "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination", "SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff", "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawMiddleObligation", "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawLowerObligation", "sald.general_moving_target_discrete.cycle64_conditional_drift_lower", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceUpperObligation", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation", "sald.general_moving_target_discrete.cycle74_conditional_kernel_measure_interface", "sald.general_moving_target_discrete.cycle74_measure_interface_middle", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToSwappedConditionalCompatibility", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "AutoSamplingTheory.lawMapProdSwap", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.cycle72GeneralMovingTargetDiscreteWeakFpMiddleObligation", "SALD.cycle72GeneralMovingTargetDiscreteWeakFpLowerObligation", "sald.general_moving_target_discrete.cycle72_weak_fp_source_signs_lower", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorMiddleObligation", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "sald.general_moving_target_discrete.cycle77_weak_fp_generator_lower", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorUpperObligation", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation", "sald.general_moving_target_discrete.cycle78_kl_derivative_generator_lower", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns", "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpUpperObligation", "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpMiddleObligation", "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpLowerObligation", "sald.general_moving_target_discrete.cycle73_kl_derivative_weak_fp_lower", "SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff", "SALD.cycle69GeneralMovingTargetDiscreteEmFpSourceSignsLowerObligation", "sald.general_moving_target_discrete.cycle69_em_fp_source_signs_lower", "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "sald.general_moving_target_discrete.cycle54_em_fp_lower", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "sald.forward_kl.schedule_time_change", "TransportVelocityContract", "FokkerPlanckContract", "KLContract", "FIContract"] note := "generalMovingTargetDiscreteDerivativeSideConditionContract keeps these source steps explicit so the theorem target is not strengthened by hidden regularity or coefficient assumptions. The cycle-71 wrappers prove only Measure.map marginal/equality transport for a supplied conditional-kernel predicate, the cycle-64 lower theorem proves only conditional-expectation linear-combination algebra under supplied linearity hypotheses, cycle 74 records the source-cited Mathlib conditional-kernel interface and middle source map and proves only a supplied-kernel compatibility/regularity wrapper, cycle 75 records the condDistrib orientation map and compiles only the first-marginal/swap supplied-kernel regularity wrapper, cycle 76 compiles only endpoint Measure.map-to-conditional compatibility packaging with both swapped and original marginal views, cycle 80 compiles only the supplied-hypothesis composition from those endpoint/kernel facts and component regularity to measurable/integrable bar b_{k,s}, cycle 77 compiles only generator-level source-sign handoffs after generator/time-derivative, generator-expansion, and split drift/diffusion source-action hypotheses are supplied, the cycle-72 and cycle-69 lower theorems prove only weak-test/source-sign coefficient rewriting after the weak FP identity is supplied, cycle 73 proves only the scalar substitution of that supplied weak FP identity into the KL derivative display at an admissible log-ratio test, including the wrapper that composes through cycle 72, cycle 78 proves only the equality composition from supplied generator pieces into that same KL derivative display, and the cycle-54 lower theorem proves only the additive/sigma algebra after analytic FP identities are supplied." def cycle54GeneralMovingTargetDiscreteEmFpLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle54_em_fp_lower" statement := "Cycle 54 lower compiles the source-shaped sigma-weighted Fokker--Planck regrouping from appendix.tex lines 1380-1387: from partial_s hat rho_s = -div(hat rho_s*bar b_{k,s})+(sigma_eta^2/2)*Delta hat rho_s and Delta hat rho_s = div(hat rho_s*A_s)+div(hat rho_s*nabla log tilde pi_s), SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff derives the regrouped divergence form with (sigma_eta^2/2)*div(hat rho_s*A_s)+div(hat rho_s*((sigma_eta^2/2)*nabla log tilde pi_s-bar b_{k,s}))." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "FokkerPlanckContract" ] note := "Proof-producing algebra only. Common-space construction, regular conditional drift, density/absolute-continuity, weak Fokker-Planck identity, KL differentiation under the integral, mass conservation, boundary integration by parts, and endpoint stitching remain analytic obligations." def generalMovingTargetDiscreteDerivativeObligation : ProofObligation where id := "sald.general_moving_target_discrete.kl_derivative" statement := "Formalize appendix lines 1354-1598: differentiate KL(hat rho_s||tilde pi_s), insert the general EM interpolation Fokker--Planck equation through the cycle-73 weak-FP-to-KL handoff at the log-ratio test and the cycle-78 generator-piece-to-KL composition, keep the cycle-74 source-cited conditional-kernel measure interface, the cycle-75 condDistrib orientation/source-map obligation, the cycle-80 conditional-law/measurability middle source map and lower supplied-hypothesis drift-regularity handoff, the cycle-76 endpoint-to-conditional marginal wrappers, and the cycle-77 generator-level weak-FP source-sign handoffs as blockers for the actual weak FP theorem, decompose the cross term into delta_pi^VA and dot{t}*m_t, use Young, lem:frozen_delta_cross_lip, LSI, DV, and the time change to derive the source differential inequality in t." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit", "SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap", "SALD.generalMovingTargetDiscreteConditionalKernelCompatibilityOfJointMapMarginal", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityOfJointMap", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff", "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawMiddleObligation", "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawLowerObligation", "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle72GeneralMovingTargetDiscreteWeakFpMiddleObligation", "SALD.cycle72GeneralMovingTargetDiscreteWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorMiddleObligation", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces", "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpUpperObligation", "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpMiddleObligation", "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpLowerObligation", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorUpperObligation", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorMiddleObligation", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceUpperObligation", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillUpperObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityMiddleObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "AutoSamplingTheory.lawMapProdSwap", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "sald.general_moving_target_discrete.derivative_side_conditions", "SALD.generalMovingTargetDiscretePostYoungDerivativeBoundScalar", "SALD.generalMovingTargetDiscretePostLsiDerivativeBoundScalar", "SALD.generalMovingTargetDiscretePostDvDerivativeBoundScalar", "SALD.generalMovingTargetDiscreteTimeChangedDerivativeBoundScalar", "SALD.generalMovingTargetDiscreteDerivativeDvTimeChangedScalar", "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "sald.general_moving_target_discrete.cycle54_em_fp_lower", "sald.general_moving_target_discrete.cycle53_derivative_dv_lower", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "sald.general_moving_target_discrete.constant_schedule_stitching", "probability.lsi_to_kl_fi", "probability.dv_variational_formula", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change" ] note := "This is the discrete general analogue of generalMovingTargetDerivativeObligation, with additional frozen-delta and stitched-EM interfaces. The side-condition contract records the analytic setup, cycle 74 records only the source-cited conditional-kernel measure blocker and middle source map, cycle 75 records only the condDistrib orientation and component-integral audit map, cycle 76 records only endpoint-to-conditional marginal packaging under supplied kernel hypotheses, cycle 80 records only the supplied-hypothesis composition from endpoint/kernel compatibility plus component regularity to measurable/integrable bar b_{k,s}, cycle 77 compiles only generator-level weak-FP source-sign handoffs under supplied analytic hypotheses, including the split drift/diffusion component wrapper, cycle 73 compiles only the weak-FP-to-KL scalar substitution wrappers, and cycle 53 lower compiles only the scalar Young/LSI/DV/time-change handoff after those analytic inputs are supplied." def cycle53GeneralMovingTargetDiscreteDerivativeDvLowerObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle53_derivative_dv_lower" statement := "Cycle 53 lower compiles the source-shaped scalar handoff from appendix.tex lines 1469-1583 for sald.general_moving_target_discrete.kl_derivative: after the EM Fokker--Planck/KL derivative identity, frozen/residual decomposition, two Young splits, frozen-delta bound, LSI half-Fisher comparison, residual DV estimate, and constant inverse-schedule identity are supplied explicitly, SALD.generalMovingTargetDiscreteDerivativeDvTimeChangedScalar derives the exact t-time differential inequality with doubled residual coefficient 2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*alpha^(-1) and frozen Gamma/Delta coefficients." source := saldGeneralMovingTargetDiscreteDerivativeSource status := ProofStatus.obligation dependsOn := [ "SALD.generalMovingTargetDiscreteDerivativeDvTimeChangedScalar", "SALD.generalMovingTargetDiscretePostYoungDerivativeBoundScalar", "SALD.generalMovingTargetDiscretePostLsiDerivativeBoundScalar", "SALD.generalMovingTargetDiscretePostDvDerivativeBoundScalar", "SALD.generalMovingTargetDiscreteTimeChangedDerivativeBoundScalar", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "probability.lsi_to_kl_fi", "sald.general_moving_target_discrete.dv_m_energy", "sald.forward_kl.schedule_time_change" ] note := "The scalar lemmas compile, but the conditional-drift Fokker--Planck theorem, KL differentiation under the integral, boundary integration by parts, frozen-delta estimate, LSI density-test theorem, residual finite-log-mgf/common-space witnesses, source-cited DV, stitched regularity, and Gronwall remain obligations. No theorem statement or source constant is changed." def generalMovingTargetDiscreteDvMEnergyObligation : ProofObligation where id := "sald.general_moving_target_discrete.dv_m_energy" statement := "Formalize appendix lines 1544-1552: apply Donsker--Varadhan with Z=alpha*||m_t||^2 under the EM interpolation law to obtain ||m_t||_{L2(hat rho_s)}^2 <= alpha^(-1)*KL(hat rho_s||tilde pi_s)+E_alpha(pi_{t(s)},m_{t(s)})." source := saldGeneralMovingTargetDiscreteResidualDvSource status := ProofStatus.obligation dependsOn := [ "probability.dv_variational_formula", "def:alpha-complexity", "KLContract", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.em_interpolation_fp" ] note := "The DV formula remains source-cited; this obligation exposes the theorem-specific finite log-mgf witness for the residual field m_t along hat rho_s." def generalMovingTargetDiscreteDvFiniteLogMgfWitnessObligation : ProofObligation where id := "sald.general_moving_target_discrete.dv_finite_log_mgf_witness" statement := "Formalize the discrete residual DV side interface for appendix lines 1544-1552: with nu=hat rho_s, mu=tilde pi_s=pi_{t(s)}, and Z=alpha*||m_{t(s)}||^2, reuse the continuous residual alpha0-to-alpha finite-log-mgf witness, expose EM-interpolation common-space and absolute-continuity, divide by alpha>0, and preserve the doubled coefficient 2*sigma_eta^(-2)*dot t(s)^2 before time change." source := saldGeneralMovingTargetDiscreteResidualDvSource status := ProofStatus.obligation dependsOn := [ "probability.dv_variational_formula", "def:alpha-complexity", "SALD.saldDvFiniteLogMgfContract", "sald.dv_variation.finite_log_mgf_interface", "sald.general_moving_target.dv_finite_log_mgf_witness", "SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.derivative_side_conditions" ] note := "This narrows the Cycle 12 upper target to the residual m_t DV witness under the general EM interpolation. It does not add hypotheses to the discrete theorem or mark DV formalized." def generalMovingTargetDiscreteGronwallApplicationObligation : ProofObligation where id := "sald.general_moving_target_discrete.gronwall_application" statement := "Instantiate lem:gronwall for appendix lines 1584-1600 with a(t)=(sigma_eta(t)^2/2)*dot{s}(t)*C_LSI(t)-2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t) and b(t)=2*sigma_eta(t)^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_t)+2*dot{s}(t)*eta*Delta(t), yielding eq:general_moving_target_KL_bound_discrete." source := saldGeneralMovingTargetDiscreteGronwallSource status := ProofStatus.obligation dependsOn := ["sald.gronwall.integrating_factor", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput", "SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.constant_schedule_stitching"] note := "Do not drop or renormalize the doubled sigma_eta^{-2} residual term; generalMovingTargetDiscreteGronwallInstantiationContract records the exact source coefficients." def cycle20GeneralVaSaldDiscreteGronwallMiddleObligation : ProofObligation where id := "sald.general_moving_target_discrete.cycle20_gronwall_middle" statement := "Maintain the cycle-20 middle source-to-Lean map for appendix lines 1573-1600 and theorem display lines 1316-1347: define K(t) from the stitched EM interpolation, apply the constant inverse-schedule rewrite to the residual and frozen-delta coefficients, expose the a(t), b(t) regularity required by lem:gronwall, and route exact display matching to sald.general_moving_target_discrete.gronwall_side_conditions without changing coefficients." source := saldGeneralMovingTargetDiscreteGronwallSideConditionSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle20GeneralVaSaldUpperPacket", "SALD.cycle20GeneralVaSaldMiddleContract", "SALD.generalMovingTargetDiscreteGronwallInstantiationContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar", "SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.constant_schedule_stitching", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "sald.gronwall.integrating_factor" ] note := "This is middle-role synchronization data. It selects the constant-schedule coefficient rewrite as the preferred first lower sub-slice, but does not prove thm:general-moving-target-SALD-discrete, Gronwall, endpoint stitching, or coefficient regularity." def generalMovingTargetDiscreteGronwallSideConditionObligation : ProofObligation where id := "sald.general_moving_target_discrete.gronwall_side_conditions" statement := "Formalize appendix lines 1573-1600 against theorem display lines 1316-1347: stitch the EM interval endpoint laws into K(t), identify K(T)=KL(rho_K^eta||pi_T) and K(0)=KL(rho_0||pi_0), apply the constant inverse-schedule rewrite to the residual and frozen-delta coefficients, prove the a(t), b(t) regularity required by lem:gronwall, and check that the Gronwall output matches eq:general_moving_target_KL_bound_discrete without changing coefficients." source := saldGeneralMovingTargetDiscreteGronwallSideConditionSource status := ProofStatus.obligation dependsOn := [ "SALD.cycle20GeneralVaSaldUpperPacket", "SALD.cycle20GeneralVaSaldMiddleContract", "sald.general_moving_target_discrete.cycle20_gronwall_middle", "sald.gronwall.integrating_factor", "sald.general_moving_target_discrete.constant_schedule_stitching", "SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar", "SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput", "SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar", "SALD.cycle59GeneralMovingTargetDiscreteGronwallLowerObligation", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "sald.general_moving_target_discrete.gronwall_application", "sald.forward_kl.schedule_time_change" ] note := "generalMovingTargetDiscreteGronwallSideConditionContract records endpoint stitching, constant-schedule coefficient rewrites, coefficient regularity, and exact theorem-display matching as obligations, not hidden theorem assumptions." def discreteUnifiedVaSaldSpecializationObligation : ProofObligation where id := "sald.general_moving_target_discrete.unified_specialization" statement := "Formalize appendix line 1603: the discrete-time VA-SALD algorithm is a special case of eq:SALD_general_EM by replacing c with u, so thm:general-moving-target-SALD-discrete yields the discrete guided VA-SALD convergence rate." source := saldGeneralMovingTargetDiscreteSource status := ProofStatus.obligation dependsOn := ["sald.guided_path_residual.identity", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions"] note := "This is a specialization note only; the source does not introduce a separate direct proof for the guided discrete theorem." def gronwallContract : TheoremContract where id := "ASTIS.SALD.gronwall" title := "Gronwall inequality used by SALD KL bounds" mode := "faithfulPaper" source := saldGronwallSource targetLean := "AutoSamplingTheory/SALD.lean" statementSummary := "Differential inequality dK/dt <= -a_t K_t + b_t implies the integrated exponential bound." proofStatus := ProofStatus.obligation obligations := [ gronwallAnalyticObligation, gronwallEndpointCalculusObligation, gronwallExponentRewriteObligation, cycle36GronwallUpperObligation, cycle36GronwallMiddleObligation, cycle41GronwallMiddleObligation, cycle41GronwallLowerObligation, firstAppendixMiddleAuditObligation ] def dvContract : TheoremContract where id := "ASTIS.SALD.dv_variation" title := "Donsker--Varadhan variational formula" mode := "faithfulPaper" source := saldDvVariationSource targetLean := "AutoSamplingTheory/Probability.lean" statementSummary := "KL(nu || mu) is the supremum over random variables Z of E_nu[Z] - log E_mu[exp Z] under finite log-mgf." proofStatus := ProofStatus.sourceCited obligations := [ dvVariationalObligation saldDvVariationSource, cycle32DvVariationInterfaceObligation, cycle32DvVariationMiddleObligation, cycle32DvVariationLowerObligation, cycle37DvVariationUpperObligation, cycle37DvVariationMiddleObligation, cycle37DvVariationLowerObligation, cycle42DvVariationMiddleObligation, cycle42DvVariationLowerObligation, dvFiniteLogMgfInterfaceObligation, firstAppendixMiddleAuditObligation ] def piDefinitionContract : TheoremContract where id := "ASTIS.SALD.def_PI" title := "Poincare inequality definition" mode := "faithfulPaper" source := saldPiSource targetLean := "AutoSamplingTheory/Probability.lean" statementSummary := "PI with constant C_PI > 0 states Var_mu(phi) <= C_PI^{-1} integral ||nabla phi||^2 dmu for all smooth phi." proofStatus := ProofStatus.contractOnly obligations := [ piVelocityNormBackendObligation, cycle25FirstAppendixPiVelocityNormMiddleObligation, cycle25PiVelocityNormLowerObligation, firstAppendixMiddleAuditObligation ] def lsiKlFiVocabularyContract : TheoremContract where id := "ASTIS.SALD.eq_LSI_KL_FI" title := "LSI, KL, and Fisher information vocabulary" mode := "faithfulPaper" source := saldKlFiLsiSource targetLean := "AutoSamplingTheory/Probability.lean" statementSummary := "The paper defines LSI, derives KL(rho||pi) <= FI(rho||pi)/(2 C_LSI), and names KL/FI integrals." proofStatus := ProofStatus.obligation obligations := [ lsiKlFiDensityTestObligation, cycle29LsiKlFiDensityTestMiddleObligation, cycle29LsiKlFiDensityTestLowerObligation, cycle33LsiKlFiDensityTestMiddleObligation, cycle33LsiKlFiDensityTestLowerObligation, cycle38LsiKlFiUpperObligation, cycle38LsiKlFiMiddleObligation, cycle38LsiKlFiLowerObligation, cycle43LsiKlFiUpperObligation, cycle43LsiKlFiMiddleObligation, cycle43LsiKlFiLowerObligation, lsiToKlFiObligation saldKlFiLsiSource, firstAppendixMiddleAuditObligation ] def continuousSaldContract : TheoremContract where id := "ASTIS.SALD.forward_KL" title := "Continuous-time SALD forward KL theorem" mode := "faithfulPaper" source := saldForwardKlSource targetLean := "AutoSamplingTheory/SALD.lean" statementSummary := "For SALD law rho_s tracking pi_t with transport velocity v_t, LSI constants C_LSI(t), and finite alpha-complexity E_alpha(pi_t,v_t), the terminal KL is bounded by the source's two-exponential initial-error term plus the alpha-complexity residual integral." proofStatus := ProofStatus.contractOnly obligations := [ cycle44MainSkeletonAnalyticInterfaceObligation, cycle49MainSkeletonAnalyticReadinessObligation, cycle49MainSkeletonAnalyticMiddleObligation, cycle54MainSkeletonAnalyticInterfaceObligation, cycle54MainSkeletonAnalyticMiddleObligation, cycle59MainSkeletonAnalyticInterfaceObligation, cycle59MainSkeletonAnalyticMiddleObligation, cycle64MainSkeletonAnalyticInterfaceObligation, cycle64MainSkeletonAnalyticMiddleObligation, cycle69MainSkeletonAnalyticInterfaceObligation, cycle69MainSkeletonAnalyticMiddleObligation, cycle65ForwardKlSkeletonObligation, cycle65ForwardKlSkeletonMiddleObligation, cycle65ForwardKlDerivativePointwiseLowerObligation, cycle60ForwardKlSkeletonObligation, cycle60ForwardKlSkeletonMiddleObligation, cycle60ForwardKlDerivativeRawLowerObligation, cycle55ForwardKlSkeletonObligation, cycle55ForwardKlSkeletonMiddleObligation, cycle55ForwardKlDerivativeMassLowerObligation, cycle45ForwardKlSkeletonObligation, cycle45ForwardKlSkeletonMiddleObligation, cycle50ForwardKlSkeletonObligation, cycle50ForwardKlSkeletonMiddleObligation, cycle50ForwardKlDerivativeLowerObligation, lsiToKlFiObligation saldKlFiLsiSource, dvVariationalObligation saldDvVariationSource, gronwallAnalyticObligation, forwardKlMiddleSourceToLeanMapObligation, forwardKlMovingTargetDependencyObligation, forwardKlEndpointScheduleObligation, forwardKlCoefficientChainObligation, forwardKlGronwallSideConditionObligation, cycle30ForwardKlDerivativeSideUpperObligation, cycle30ForwardKlDerivativeSideMiddleObligation, forwardKlDensityBoundaryObligation, cycle30ForwardKlDensityBoundaryLowerObligation, cycle34ForwardKlDerivativeScalarObligation, cycle34ForwardKlTargetYoungLowerObligation, cycle34ForwardKlDerivativeMiddleObligation, cycle39ForwardKlDerivativeUpperObligation, cycle39ForwardKlDerivativeMiddleObligation, forwardKlScheduleTimeChangeObligation, forwardKlDerivativeObligation, forwardKlDvEnergyObligation, forwardKlDvAlphaMonotonicityObligation, forwardKlDvFiniteLogMgfWitnessObligation, cycle26ForwardKlDvWitnessMiddleObligation, cycle26ForwardKlDvPositiveAlphaLowerObligation, forwardKlGronwallApplicationObligation ] def forwardKlProofDag : List ProofDagBlock := cycle49MainSkeletonAnalyticReadinessDag ++ cycle54MainSkeletonAnalyticInterfaceDag ++ cycle59MainSkeletonAnalyticInterfaceDag ++ cycle64MainSkeletonAnalyticInterfaceDag ++ cycle69MainSkeletonAnalyticInterfaceDag ++ cycle50ForwardKlSkeletonDag ++ cycle55ForwardKlSkeletonDag ++ cycle60ForwardKlSkeletonDag ++ cycle65ForwardKlSkeletonDag ++ [ { id := "ASTIS.SALD.forward_KL.middle_source_to_lean_map" interface := "Classify appendix lines 168-252 and main_body lines 218-248 as Lean contracts, cited results, or named obligations, and hand off the current cycle's selected lower slice." source := saldForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle14ForwardKlUpperPacket", "SALD.cycle14ForwardKlMiddleContract", "SALD.cycle18ForwardKlUpperPacket", "SALD.cycle18ForwardKlMiddleContract", "SALD.cycle22ForwardKlUpperPacket", "SALD.cycle22ForwardKlMiddleContract", "SALD.cycle26ForwardKlUpperPacket", "SALD.cycle26ForwardKlMiddleContract", "SALD.cycle30ForwardKlUpperPacket", "SALD.cycle30ForwardKlMiddleContract", "SALD.cycle34ForwardKlDerivativeUpperPacket", "SALD.cycle34ForwardKlDerivativeMiddleContract", "SALD.cycle39ForwardKlDerivativeUpperPacket", "SALD.cycle39ForwardKlDerivativeMiddleContract", "SALD.cycle45ForwardKlSkeletonUpperPacket", "SALD.cycle45ForwardKlSkeletonObligation", "SALD.cycle45ForwardKlSkeletonMiddleContract", "SALD.cycle45ForwardKlSkeletonMiddleObligation", "ASTIS.SALD.forward_KL.cycle45_theorem_skeleton_route", "sald.forward_kl.cycle45_middle_route_audit", "SALD.cycle50ForwardKlSkeletonUpperPacket", "SALD.cycle50ForwardKlSkeletonObligation", "SALD.cycle50ForwardKlSkeletonMiddleContract", "SALD.cycle50ForwardKlSkeletonMiddleObligation", "ASTIS.SALD.forward_KL.cycle50_theorem_skeleton_route", "ASTIS.SALD.forward_KL.cycle50_middle_route_audit", "sald.forward_kl.cycle50_theorem_skeleton_route", "sald.forward_kl.cycle50_middle_route_audit", "SALD.continuousForwardKlStatementContract", "sald.forward_kl.endpoint_schedule_identities", "sald.forward_kl.moving_target_dependency_chain", "sald.forward_kl.cycle30_derivative_side_upper", "sald.forward_kl.cycle30_derivative_side_middle", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.cycle34_derivative_scalar", "sald.forward_kl.cycle34_derivative_middle", "sald.forward_kl.cycle39_derivative_upper", "sald.forward_kl.cycle39_derivative_middle", "SALD.forwardKlInverseScheduleDerivativeScalar", "SALD.forwardKlTimeChangeSquareCoefficientRewriteOfProductScalar", "SALD.forwardKlVelocitySquareScalingScalar", "SALD.forwardKlTimeChangedDerivativeBoundOfProductScalar", "SALD.forwardKlPreDvDerivativeBoundOfProductScalar", "SALD.forwardKlPreDvDerivativeBoundOfVelocityScalingScalar", "SALD.forwardKlLsiDerivativeBoundOfKlFiScalar", "SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "probability.lsi_to_kl_fi", "sald.forward_kl.cycle26_dv_witness_middle", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.coefficient_chain_audit", "sald.forward_kl.gronwall_side_conditions" ] reusedBy := ["thm:forward-KL", "cycle14/cycle18/cycle22/cycle26/cycle30 lower packets"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle45_theorem_skeleton_route" interface := "Wire the five slow analytic interfaces into the continuous forward-KL theorem skeleton: derivative/Fokker-Planck, LSI/KL/FI, DV finite-log-mgf, Gronwall endpoint/exponent side conditions, and downstream EM interpolation visibility." source := saldForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle45ForwardKlSkeletonUpperPacket", "SALD.cycle45ForwardKlSkeletonObligation", "SALD.cycle45ForwardKlSkeletonDag", "SALD.cycle45ForwardKlSkeletonMiddleContract", "SALD.cycle45ForwardKlSkeletonMiddleObligation", "SALD.continuousForwardKlStatementContract", "sald.forward_kl.middle_source_to_lean_map", "sald.forward_kl.cycle45_middle_route_audit", "sald.forward_kl.moving_target_dependency_chain", "sald.forward_kl.kl_derivative", "probability.lsi_to_kl_fi", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "sald.forward_kl.gronwall_application", "sald.forward_kl.gronwall_side_conditions", "sald.discrete_forward_kl.em_interpolation_fp" ] reusedBy := ["thm:forward-KL", "ASTIS-SALD-001 main skeleton sprint 2"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle45_middle_route_audit" interface := "Middle source-to-Lean audit for the continuous forward-KL skeleton: verify the exact main_body.tex:238-247 statement and appendix.tex:168-252 route, then select the theorem-level Gronwall side-condition backend for lower work." source := saldForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle45ForwardKlSkeletonMiddleContract", "SALD.cycle45ForwardKlSkeletonMiddleObligation", "SALD.cycle45ForwardKlSkeletonObligation", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "sald.forward_kl.gronwall_side_conditions" ] reusedBy := ["thm:forward-KL", "cycle 45 lower Gronwall-side-condition packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.moving_target_dependencies" interface := "Audit the theorem-level moving-target assumptions and wire them to the derivative, LSI, DV, and Gronwall proof blocks without changing thm:forward-KL." source := saldForwardKlSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "sald.forward_kl.middle_source_to_lean_map", "sald.forward_kl.endpoint_schedule_identities", "eq:SALD", "eq:FP-eq", "TransportVelocityContract", "eq:LSI-KL-FI", "def:alpha-complexity", "lem:dv_variation", "lem:gronwall", "sald.forward_kl.schedule_time_change" ] reusedBy := ["thm:forward-KL", "thm:forward-KL-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.endpoint_schedule_identities" interface := "Isolate the inverse-schedule endpoint rewrites s(0)=0, S=s(T), t(s(T))=T, tilde_pi_{s(t)}=pi_t, and the K(0)/K(T) identifications used after Gronwall." source := saldForwardKlEndpointScheduleSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.forwardKlEndpointScheduleContract", "sald.forward_kl.moving_target_dependency_chain", "sald.forward_kl.schedule_time_change", "sald.forward_kl.gronwall_side_conditions" ] reusedBy := ["thm:forward-KL", "cycle14 lower packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle30_derivative_side_upper" interface := "Upper-role lower packet for the derivative-side interface: mass conservation, KL differentiation, SALD and target integration by parts, slowed-target transport, Young 1/2 coefficients, and the time-change handoff." source := saldForwardKlDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle30ForwardKlUpperPacket", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlMovingTargetDependencyContract", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "eq:SALD", "eq:FP-eq", "eq:LSI-KL-FI", "TransportVelocityContract" ] reusedBy := ["thm:forward-KL", "cycle30 lower derivative-side packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle30_derivative_side_middle" interface := "Middle source-to-Lean map for appendix.tex:168-208, selecting appendix.tex:168-185 density/boundary/FI identification as the first lower sub-slice and leaving target transport, LSI, time-change, DV, and Gronwall as separate obligations." source := saldForwardKlDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle30ForwardKlUpperPacket", "SALD.cycle30ForwardKlMiddleContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDensityBoundaryObligation", "sald.forward_kl.cycle30_derivative_side_upper", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.kl_derivative", "eq:SALD", "eq:FP-eq", "KLContract", "FIContract", "FokkerPlanckContract" ] reusedBy := ["thm:forward-KL", "cycle30 lower density-boundary packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle30_density_boundary_lower" interface := "Lower scalar handoff for appendix.tex:168-185: substitute the supplied first-term identity firstTerm=-FI into the KL derivative display, while keeping mass conservation, Fokker-Planck, integration by parts, and FI identification as obligations." source := saldForwardKlDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle30ForwardKlMiddleContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlFirstTermFisherSubstitutionScalar", "sald.forward_kl.cycle30_derivative_side_middle", "sald.forward_kl.density_boundary_regular", "eq:FP-eq", "KLContract", "FIContract", "FokkerPlanckContract" ] reusedBy := ["thm:forward-KL", "sald.forward_kl.kl_derivative"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle34_derivative_scalar" interface := "Proof-producing scalar derivative handoff for appendix.tex:168-228: combine the first-term Fisher substitution, target-side Young bound, supplied LSI half-Fisher comparison, and scalar inverse-schedule coefficient handoff while leaving analytic time-change inputs open." source := saldForwardKlDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle34ForwardKlDerivativeUpperPacket", "SALD.cycle34ForwardKlDerivativeMiddleContract", "SALD.forwardKlTargetTransportYoungBoundScalar", "SALD.forwardKlPostYoungDerivativeBoundScalar", "SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar", "SALD.forwardKlLsiDerivativeBoundScalar", "SALD.forwardKlLsiDerivativeBoundOfKlFiScalar", "SALD.forwardKlTimeChangedDerivativeBoundScalar", "SALD.forwardKlFirstTermFisherSubstitutionScalar", "sald.forward_kl.cycle34_target_young_lower", "sald.forward_kl.cycle30_density_boundary_lower", "sald.forward_kl.density_boundary_regular", "probability.lsi_to_kl_fi", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative" ] reusedBy := ["thm:forward-KL", "sald.forward_kl.kl_derivative"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle39_derivative_upper" interface := "Upper proof-closure packet for appendix.tex:168-228: use the compiled pre-DV scalar pipeline and assign lower work to the remaining density/boundary and schedule-time-change inputs before any ledger expansion or EM interpolation work." source := saldForwardKlDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle39ForwardKlDerivativeUpperPacket", "SALD.cycle39ForwardKlDerivativeMiddleContract", "SALD.cycle39ForwardKlDerivativeUpperObligation", "SALD.forwardKlPreDvDerivativeBoundScalar", "SALD.forwardKlPreDvDerivativeBoundOfProductScalar", "SALD.forwardKlPreDvDerivativeBoundOfVelocityScalingScalar", "SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar", "SALD.forwardKlInverseScheduleDerivativeScalar", "SALD.forwardKlTimeChangeSquareCoefficientRewriteOfProductScalar", "SALD.forwardKlVelocitySquareScalingScalar", "SALD.forwardKlTimeChangedDerivativeBoundOfProductScalar", "SALD.forwardKlFirstTermFisherSubstitutionScalar", "SALD.forwardKlTargetTransportYoungBoundScalar", "SALD.forwardKlPostYoungDerivativeBoundScalar", "SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar", "SALD.forwardKlLsiDerivativeBoundScalar", "SALD.forwardKlTimeChangedDerivativeBoundScalar", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "probability.lsi_to_kl_fi", "sald.forward_kl.kl_derivative" ] reusedBy := ["thm:forward-KL", "sald.forward_kl.kl_derivative"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle39_derivative_middle" interface := "Middle source-to-Lean map for appendix.tex:191-228: expose source-shaped inverse-schedule product and slowed-velocity norm-square scalar lemmas feeding the pre-DV derivative pipeline." source := saldForwardKlDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle39ForwardKlDerivativeMiddleContract", "SALD.cycle39ForwardKlDerivativeMiddleObligation", "SALD.forwardKlInverseScheduleDerivativeScalar", "SALD.forwardKlTimeChangeSquareCoefficientRewriteOfProductScalar", "SALD.forwardKlVelocitySquareScalingScalar", "SALD.forwardKlTimeChangedDerivativeBoundOfProductScalar", "SALD.forwardKlPreDvDerivativeBoundOfProductScalar", "SALD.forwardKlPreDvDerivativeBoundOfVelocityScalingScalar", "SALD.forwardKlLsiDerivativeBoundOfKlFiScalar", "SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar", "SALD.forwardKlPreDvDerivativeBoundScalar", "sald.forward_kl.cycle39_derivative_upper", "sald.forward_kl.schedule_time_change", "sald.forward_kl.density_boundary_regular", "probability.lsi_to_kl_fi", "sald.forward_kl.kl_derivative" ] reusedBy := ["thm:forward-KL", "sald.forward_kl.kl_derivative", "sald.forward_kl.schedule_time_change"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.coefficient_chain_audit" interface := "Preserve the source LSI, DV, and Gronwall coefficient flow from appendix lines 210-252, including the source-line ledger, scalar side conditions, and final exponent split in the theorem statement." source := saldForwardKlDependencyChainSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "eq:LSI-KL-FI", "def:alpha-complexity", "lem:dv_variation", "lem:gronwall", "sald.forward_kl.kl_derivative", "sald.forward_kl.dv_energy_bound", "sald.forward_kl.gronwall_application", "sald.forward_kl.gronwall_side_conditions" ] reusedBy := ["thm:forward-KL", "thm:forward-KL-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.gronwall_side_conditions" interface := "Expose endpoint K(0)/K(T) identifications, coefficient regularity, exponent splitting, and residual-exponent sign facts for the final Gronwall display." source := saldForwardKlGronwallSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle18ForwardKlUpperPacket", "SALD.cycle18ForwardKlMiddleContract", "SALD.cycle22ForwardKlUpperPacket", "SALD.cycle22ForwardKlMiddleContract", "sald.forward_kl.moving_target_dependency_chain", "sald.forward_kl.endpoint_schedule_identities", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "sald.forward_kl.dv_energy_bound", "sald.gronwall.integrating_factor", "sald.gronwall.exponent_rewrite", "SALD.gronwallNegIntegralRewriteScalar", "SALD.gronwallExpProductRewriteScalar", "SALD.gronwallExpProductRewriteIntervalIntegral", "SALD.gronwallExpProductRewriteIntegralCongr", "SALD.forwardKlGronwallCoeffIntervalIntegrable", "SALD.forwardKlGronwallCoeffAdjacentIntervalIntegrable", "SALD.forwardKlGronwallExpProductRewriteIntegralCongrOfPieces", "SALD.forwardKlGronwallCoeffIntegralSub", "SALD.forwardKlGronwallInitialExponentSplitScalar", "SALD.forwardKlGronwallInitialExponentSplitOfPieces", "SALD.forwardKlGronwallResidualExponentDropScalar", "SALD.forwardKlGronwallResidualExponentDropIntegral" ] reusedBy := ["thm:forward-KL"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.alpha_complexity" interface := "Define E_alpha(pi_t,v_t) and A_alpha(pi,v); expose finite log-mgf obligations for alpha in (0, alpha0]." source := saldAlphaComplexitySource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := ["exponential moment vocabulary"] reusedBy := ["thm:forward-KL", "thm:forward-KL-discrete", "thm:general-moving-target-SALD"] status := ProofStatus.contractOnly }, { id := "ASTIS.SALD.forward_KL.dv_alpha_mgf_monotonicity" interface := "Derive the theorem-specific finite log-mgf at alpha from the source alpha0-complexity assumption for Z=alpha*||v_t||^2 before applying DV." source := saldForwardKlDvEnergySource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "def:alpha-complexity", "SALD.saldDvFiniteLogMgfContract", "sald.dv_variation.finite_log_mgf_interface" ] reusedBy := ["thm:forward-KL", "sald.forward_kl.dv_finite_log_mgf_witness"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.cycle26_middle_dv_witness" interface := "Middle source-dependency map for the DV witness: common-space/absolute-continuity, measurability of Z=alpha*||v_t||^2, alpha0-to-alpha finite log-mgf, and positive-alpha division." source := saldForwardKlDvEnergySource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle26ForwardKlUpperPacket", "SALD.cycle26ForwardKlMiddleContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "SALD.forwardKlDvAlphaMonotonicityContract", "SALD.saldDvFiniteLogMgfContract", "probability.dv_variational_formula", "sald.dv_variation.finite_log_mgf_interface", "sald.forward_kl.dv_alpha_mgf_monotonicity", "SALD.forwardKlDvPositiveAlphaScalingScalar", "SALD.forwardKlDvPositiveAlphaCoefficientScalar", "sald.forward_kl.moving_target_dependency_chain" ] reusedBy := ["sald.forward_kl.dv_finite_log_mgf_witness", "thm:forward-KL"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.derivative" interface := "Derive the source differential inequality before DV: dK/dt <= -dot{s}*C_LSI*K + (1/2)*dot{s}^(-1)*||v_t||^2_{L2(rho)}." source := saldForwardKlDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle30ForwardKlUpperPacket", "SALD.cycle30ForwardKlMiddleContract", "SALD.cycle39ForwardKlDerivativeUpperPacket", "SALD.cycle39ForwardKlDerivativeMiddleContract", "sald.forward_kl.cycle30_derivative_side_upper", "sald.forward_kl.cycle30_derivative_side_middle", "sald.forward_kl.cycle30_density_boundary_lower", "sald.forward_kl.cycle39_derivative_upper", "sald.forward_kl.cycle39_derivative_middle", "eq:SALD", "eq:FP-eq", "eq:LSI-KL-FI", "TransportVelocityContract", "SALD.forwardKlFirstTermFisherSubstitutionScalar", "SALD.forwardKlPostYoungDerivativeBoundScalar", "SALD.forwardKlLsiDerivativeBoundScalar", "sald.forward_kl.cycle34_derivative_scalar", "SALD.forwardKlPreDvDerivativeBoundScalar", "SALD.forwardKlPreDvDerivativeBoundOfProductScalar", "SALD.forwardKlPreDvDerivativeBoundOfVelocityScalingScalar", "SALD.forwardKlInverseScheduleDerivativeScalar", "SALD.forwardKlVelocitySquareScalingScalar", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change" ] reusedBy := ["thm:forward-KL", "thm:forward-KL-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.dv_finite_log_mgf_witness" interface := "Provide the theorem-specific DV finite-log-mgf and common-space witness for Z=alpha*||v_t||^2 from the alpha0-complexity assumption before the DV-energy inequality is used." source := saldForwardKlDvEnergySource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle26ForwardKlUpperPacket", "SALD.cycle26ForwardKlMiddleContract", "ASTIS.SALD.forward_KL.cycle26_middle_dv_witness", "sald.forward_kl.cycle26_dv_witness_middle", "lem:dv_variation", "def:alpha-complexity", "SALD.saldDvFiniteLogMgfContract", "sald.dv_variation.finite_log_mgf_interface", "sald.forward_kl.dv_alpha_mgf_monotonicity", "SALD.forwardKlDvPositiveAlphaScalingScalar", "SALD.forwardKlDvPositiveAlphaCoefficientScalar", "sald.forward_kl.cycle26_dv_positive_alpha_lower", "sald.forward_kl.moving_target_dependency_chain" ] reusedBy := ["thm:forward-KL", "thm:forward-KL-discrete DV pattern"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.dv_energy" interface := "Instantiate DV with Z=alpha*||v_t||^2 and rewrite the log-mgf term as E_alpha(pi_t,v_t)." source := saldForwardKlDvEnergySource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "lem:dv_variation", "def:alpha-complexity", "SALD.cycle26ForwardKlMiddleContract", "sald.forward_kl.cycle26_dv_witness_middle", "sald.forward_kl.dv_alpha_mgf_monotonicity", "sald.forward_kl.dv_finite_log_mgf_witness", "SALD.forwardKlDvPositiveAlphaScalingScalar", "SALD.forwardKlDvPositiveAlphaCoefficientScalar", "sald.forward_kl.cycle26_dv_positive_alpha_lower" ] reusedBy := ["thm:forward-KL", "thm:forward-KL-discrete", "thm:general-moving-target-SALD"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL.gronwall_application" interface := "Apply lem:gronwall with the paper's a(t), b(t), then split the exponent exactly as in the theorem statement." source := saldForwardKlGronwallSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := ["lem:gronwall", "sald.forward_kl.kl_derivative", "sald.forward_kl.dv_energy_bound"] reusedBy := ["thm:forward-KL", "thm:forward-KL-discrete"] status := ProofStatus.obligation } ] def discreteForwardKlProofDag : List ProofDagBlock := cycle49MainSkeletonAnalyticReadinessDag ++ cycle54MainSkeletonAnalyticInterfaceDag ++ cycle59MainSkeletonAnalyticInterfaceDag ++ cycle64MainSkeletonAnalyticInterfaceDag ++ cycle69MainSkeletonAnalyticInterfaceDag ++ cycle70GeneralMovingTargetDiscreteConditionalLawDag ++ cycle71GeneralMovingTargetDiscreteEndpointConditionalDag ++ cycle72GeneralMovingTargetDiscreteWeakFpDag ++ cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpDag ++ cycle74GeneralMovingTargetDiscreteMeasureInterfaceDag ++ cycle75GeneralMovingTargetDiscreteConditionalLawBackfillDag ++ cycle76GeneralMovingTargetDiscreteEndpointConditionalDag ++ cycle77GeneralMovingTargetDiscreteWeakFpGeneratorDag ++ cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorDag ++ cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureDag ++ cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityDag ++ cycle81GeneralMovingTargetDiscreteEndpointConditionalDag ++ cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsDag ++ cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpDag ++ cycle84GeneralMovingTargetDiscreteActiveEmBackendDag ++ cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryDag ++ cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryDag ++ cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryDag ++ cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityDag ++ cycle89DiscreteForwardKlClosurePressureDag ++ cycle90DiscreteForwardKlMassConservationDag ++ cycle91GeneralMovingTargetDiscreteConditionalKernelDag ++ cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitDag ++ cycle93GeneralMovingTargetDiscreteKlMassDerivativeDag ++ cycle94GeneralMovingTargetDiscreteWeakFpDriftActionDag ++ cycle95DiscreteForwardKlClosurePressureDag ++ cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingDag ++ cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingDag ++ cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryDag ++ cycle99GeneralMovingTargetDiscreteRawKlDerivativeDag ++ cycle100GeneralMovingTargetDiscreteBarBWeakGradDefDag ++ cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundDag ++ cycle101DiscreteForwardKlClosurePressureDag ++ cycle103GeneralMovingTargetDiscreteConditionalKernelVersionDag ++ cycle61DiscreteForwardKlSkeletonDag ++ cycle66DiscreteForwardKlSkeletonDag ++ [ { id := "ASTIS.SALD.forward_KL_discrete.cycle46_theorem_skeleton_route" interface := "Main skeleton sprint 3 route wrapper: compose the source statement, EM endpoint/conditional-FP backend, discrete KL derivative, frozen-defect/LSI handoff, DV velocity witness, Gronwall accumulation, and linear-slowdown accumulated-error bridge without changing the theorem display." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle46DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle46DiscreteForwardKlSkeletonObligation", "SALD.cycle46DiscreteForwardKlSkeletonDag", "SALD.cycle46DiscreteForwardKlSkeletonMiddleContract", "SALD.cycle46DiscreteForwardKlSkeletonMiddleObligation", "sald.discrete_forward_kl.cycle46_middle_route_audit", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.discreteForwardKlDerivativeCandidateContract", "SALD.frozenDeltaCrossLipSaldContract", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.saldGronwallCandidateContract", "dvVariationalFormulaInterface saldDvVariationSource" ] reusedBy := ["thm:forward-KL-discrete", "ASTIS-SALD-001 cycle 46"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle46_middle_route_audit" interface := "Middle source-to-Lean audit for the discrete forward-KL skeleton: verify the exact main_body.tex:299-323 statement and appendix.tex:260-592 route, then select the accumulated-error bridge for lower work." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle46DiscreteForwardKlSkeletonMiddleContract", "SALD.cycle46DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle46DiscreteForwardKlSkeletonObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.discreteForwardKlDerivativeCandidateContract", "SALD.frozenDeltaCrossLipSaldContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "sald.discrete_forward_kl.accumulated_error_bridge" ] reusedBy := ["thm:forward-KL-discrete", "cycle 46 lower accumulated-error packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle51_theorem_interface_route" interface := "Post-cycle-50 discrete forward-KL route: consume the five-backend readiness check and the continuous derivative/DV scalar handoff, then route the discrete theorem through EM endpoint/conditional-FP, KL derivative with frozen defect and LSI, DV velocity, Gronwall, and accumulated-error interfaces without reproving EM/Fokker-Planck." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle51DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle51DiscreteForwardKlSkeletonObligation", "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle46DiscreteForwardKlSkeletonObligation", "SALD.cycle46DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle50ForwardKlDerivativeLowerObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "sald.discrete_forward_kl.kl_derivative", "SALD.frozenDeltaCrossLipSaldContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlAccumulatedErrorBridgeContract" ] reusedBy := ["thm:forward-KL-discrete", "ASTIS-SALD-001 cycle 51"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle51_middle_route_audit" interface := "Middle source-to-Lean audit for the cycle-51 discrete forward-KL route: select appendix.tex:334-491 as the KL derivative lower backend, consuming appendix.tex:334-385 EM endpoint/conditional-FP interfaces explicitly rather than reproving them." source := saldForwardKlDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle51DiscreteForwardKlSkeletonMiddleContract", "SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlSkeletonObligation", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "SALD.discreteForwardKlDerivativeObligation", "SALD.discreteForwardKlPostLsiDerivativeBoundScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfKlFiScalar", "SALD.frozenDeltaCrossLipSaldContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlAccumulatedErrorBridgeContract" ] reusedBy := ["thm:forward-KL-discrete", "cycle 51 lower derivative packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle51_derivative_lower" interface := "Lower scalar handoff for appendix.tex:388-491: consume the supplied EM conditional-FP derivative identity, frozen-defect bound, moving Young estimate, and LSI/KL/FI comparison to obtain the exact source pre-DV inequality before the separate DV velocity witness." source := saldForwardKlDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.discreteForwardKlPostLsiDerivativeBoundScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfKlFiScalar", "SALD.discreteForwardKlDerivativeCandidateContract", "SALD.discreteForwardKlDerivativeObligation", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.frozen_delta_cross_lip", "SALD.frozenDeltaCrossLipSaldContract", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi" ] reusedBy := ["sald.discrete_forward_kl.kl_derivative", "thm:forward-KL-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle56_theorem_interface_route" interface := "Cycle-56 theorem-level route: consume the explicit EM endpoint/conditional-FP interfaces, cycle-51 derivative/LSI scalar handoff, discrete DV velocity witness, and appendix.tex:526-592 Gronwall/accumulated-error interfaces to match main_body.tex:299-323 without proving EM/Fokker-Planck from scratch." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle56DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle56DiscreteForwardKlSkeletonObligation", "SALD.cycle55ForwardKlSkeletonObligation", "SALD.cycle51DiscreteForwardKlSkeletonObligation", "SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "sald.discrete_forward_kl.kl_derivative", "SALD.discreteForwardKlPostLsiDerivativeBoundOfKlFiScalar", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.discreteForwardKlGronwallInstantiationContract", "sald.discrete_forward_kl.gronwall_accumulation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "sald.discrete_forward_kl.accumulated_error_bridge", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI" ] reusedBy := ["thm:forward-KL-discrete", "ASTIS-SALD-001 cycle 56", "cycle56 lower Gronwall packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle56_middle_route_audit" interface := "Cycle-56 middle audit: verify the upper discrete theorem route against main_body.tex:299-323 and appendix.tex:260-592, keep EM/Fokker-Planck as named source-cited interfaces, and select sald.discrete_forward_kl.gronwall_accumulation over appendix.tex:526-592 as the lower packet." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle56DiscreteForwardKlSkeletonMiddleContract", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlSkeletonObligation", "SALD.cycle54MainSkeletonAnalyticMiddleObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "sald.discrete_forward_kl.kl_derivative", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.discreteForwardKlGronwallInstantiationContract", "sald.discrete_forward_kl.gronwall_accumulation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "sald.discrete_forward_kl.accumulated_error_bridge" ] reusedBy := ["thm:forward-KL-discrete", "cycle56 lower Gronwall packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle56_gronwall_lower" interface := "Lower scalar handoff for appendix.tex:526-553: consume the post-DV s-time inequality and inverse-schedule inputs to produce the exact t-time Gronwall coefficient and residual used by lem:gronwall." source := saldForwardKlDiscreteGronwallSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged", "SALD.discreteForwardKlTimeChangeSquareCoefficientRewriteScalar", "SALD.discreteForwardKlGronwallInstantiationContract", "sald.discrete_forward_kl.kl_derivative", "sald.discrete_forward_kl.dv_velocity_bound", "sald.forward_kl.schedule_time_change", "lem:gronwall" ] reusedBy := ["sald.discrete_forward_kl.gronwall_accumulation", "thm:forward-KL-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.euler_maruyama" interface := "Pin the EM update and continuous frozen interpolation hat X_s with endpoint laws rho_k^eta." source := saldForwardKlDiscreteInterpolationSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := ["EulerMaruyamaContract", "sald.discrete_forward_kl.em_endpoint_laws"] reusedBy := ["thm:forward-KL-discrete"] status := ProofStatus.contractOnly }, { id := "ASTIS.SALD.forward_KL_discrete.cycle35_em_fp_upper" interface := "Upper proof-closure packet for appendix.tex:260-385 selecting the EM endpoint laws and conditional-drift Fokker--Planck backend as the lower target after the earlier Gronwall, DV, LSI, and continuous-derivative proof-sprint slices." source := saldForwardKlDiscreteInterpolationSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle35DiscreteForwardKlEmFpUpperPacket", "sald.discrete_forward_kl.cycle35_em_fp_upper", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.em_interpolation_fp" ] reusedBy := ["thm:forward-KL-discrete", "cycle35 lower EM-FP packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle35_em_fp_middle" interface := "Middle source-to-Lean packet for appendix.tex:260-385: compiled endpoint-vector algebra and divergence-regrouping algebra, with stochastic endpoint laws and conditional-drift Fokker--Planck still explicit obligations." source := saldForwardKlDiscreteConditionalFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle35DiscreteForwardKlEmFpMiddleContract", "sald.discrete_forward_kl.cycle35_em_fp_middle", "SALD.discreteForwardKlEmInterpolationLeftEndpointVector", "SALD.discreteForwardKlEmInterpolationRightEndpointVector", "SALD.discreteForwardKlConditionalFpDivergenceDriftSplit", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck" ] reusedBy := ["thm:forward-KL-discrete", "cycle35 lower EM-FP packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle35_em_fp_lower" interface := "Lower proof-producing handoff for appendix.tex:357-385: compose the supplied conditional-drift Fokker--Planck equation and supplied Laplacian split into the source regrouped divergence form, without promoting the analytic FP, density, or integration-by-parts backend." source := saldForwardKlDiscreteConditionalFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle35DiscreteForwardKlEmFpMiddleContract", "SALD.cycle35DiscreteForwardKlEmFpMiddleObligation", "SALD.discreteForwardKlConditionalFpDivergenceDriftSplit", "SALD.discreteForwardKlConditionalFpLaplacianSplitHandoff", "SALD.cycle35DiscreteForwardKlEmFpLowerObligation", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.em_interpolation_fp" ] reusedBy := ["thm:forward-KL-discrete", "sald.discrete_forward_kl.em_conditional_fokker_planck"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle40_em_fp_middle" interface := "Middle proof-closure handoff for appendix.tex:260-385: add abstract endpoint-law equality transport from pointwise EM interpolation identities, while keeping conditional drift density and conditional-drift Fokker--Planck as obligations." source := saldForwardKlDiscreteInterpolationSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle40DiscreteForwardKlEmFpMiddleContract", "SALD.cycle40DiscreteForwardKlEmFpMiddleObligation", "SALD.discreteForwardKlLawEqOfPointwise", "SALD.discreteForwardKlEmInterpolationLeftEndpointLawHandoff", "SALD.discreteForwardKlEmInterpolationRightEndpointLawHandoff", "SALD.discreteForwardKlEmEndpointLawPairHandoff", "SALD.discreteForwardKlConditionalFpLaplacianSplitHandoff", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.em_interpolation_fp" ] reusedBy := ["thm:forward-KL-discrete", "cycle40 lower EM endpoint/conditional-FP packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle40_em_endpoint_lower" interface := "Lower endpoint-law representation handoff for appendix.tex:334-335: from explicit named law representations for hat rho_s and rho_k^eta, prove both endpoint law equalities using the compiled law handoff lemmas." source := saldForwardKlDiscreteInterpolationSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.discreteForwardKlEmEndpointLawPairHandoff", "SALD.discreteForwardKlEmInterpolationLeftEndpointLawHandoff", "SALD.discreteForwardKlEmInterpolationRightEndpointLawHandoff", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck" ] reusedBy := ["thm:forward-KL-discrete", "sald.discrete_forward_kl.em_endpoint_laws"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle15_upper_packet" interface := "Upper packet selecting the EM interpolation side-condition spine as the next lower target while keeping the discrete theorem statement and accumulated-error constants unchanged." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle15DiscreteForwardKlUpperPacket", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.accumulated_error_bridge" ] reusedBy := ["thm:forward-KL-discrete", "cycle15 lower packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle15_middle_em_spine" interface := "Middle packet mapping the cycle-15 EM side-condition spine into ordered lower obligations, with the conditional-drift Fokker--Planck equation as the first lower slice." source := saldForwardKlDiscreteProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle15DiscreteForwardKlMiddleContract", "sald.discrete_forward_kl.cycle15_middle_em_spine", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.accumulated_error_bridge" ] reusedBy := ["thm:forward-KL-discrete", "cycle15 lower packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.conditional_drift_density" interface := "Lower sub-obligation for appendix.tex:347-354: construct the regular conditional-law, density, measurability, and integrability interface needed to treat bar b_{k,s} as a drift field." source := saldForwardKlDiscreteConditionalFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle15DiscreteForwardKlConditionalDriftDensityContract", "sald.discrete_forward_kl.conditional_drift_density", "eq:frozen_interp_terminal_disc_prop_additive_final", "sald.discrete_forward_kl.em_endpoint_laws" ] reusedBy := ["sald.discrete_forward_kl.em_conditional_fokker_planck", "thm:forward-KL-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle15_conditional_fp_lower_packet" interface := "Lower-ready line ledger for appendix.tex:347-385: conditional frozen drift, conditional-drift Fokker--Planck equation, Laplacian split, and KL-derivative handoff." source := saldForwardKlDiscreteConditionalFpSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle15DiscreteForwardKlEmConditionalFpLowerContract", "SALD.cycle15DiscreteForwardKlConditionalDriftDensityContract", "sald.discrete_forward_kl.cycle15_conditional_fp_lower_packet", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.em_endpoint_laws" ] reusedBy := ["thm:forward-KL-discrete", "cycle15 lower packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.em_interpolation_side_conditions" interface := "Separate endpoint law matching, conditional-drift Fokker--Planck, and stitched-interval regularity for the EM interpolation." source := saldForwardKlDiscreteInterpolationSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity" ] reusedBy := ["thm:forward-KL-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.frozen_delta" interface := "Bound the one-step frozen score-defect cross term by (1/4)*FI plus Gamma and Delta contributions." source := saldFrozenDeltaCrossLipSaldSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := ["lem:frozen_delta_cross_lip", "eq:lip_SALD_1", "eq:lip_SALD_2", "lem:dv_variation"] reusedBy := ["thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete specialization"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.derivative" interface := "Differentiate KL(hat rho_s||tilde pi_s) on each EM interval and derive the discrete differential inequality before Gronwall." source := saldForwardKlDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := ["sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.frozen_delta_cross_lip", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "eq:LSI-KL-FI"] reusedBy := ["thm:forward-KL-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.dv_finite_log_mgf_witness" interface := "Expose the EM-interpolation DV witness for nu=hat rho_s, mu=tilde pi_s, and Z=alpha*||v_{t(s)}||^2 before bounding ||tilde v_s||^2." source := saldForwardKlDiscreteDvVelocitySource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "lem:dv_variation", "def:alpha-complexity", "sald.forward_kl.dv_alpha_mgf_monotonicity", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.em_interpolation_fp" ] reusedBy := ["thm:forward-KL-discrete", "sald.discrete_forward_kl.dv_velocity_bound"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.gronwall_accumulation" interface := "Apply lem:gronwall with the added Gamma term to the source general-schedule differential inequality." source := saldForwardKlDiscreteGronwallSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := ["lem:gronwall", "sald.discrete_forward_kl.kl_derivative", "sald.discrete_forward_kl.dv_velocity_bound", "sald.discrete_forward_kl.stitched_interval_regularity"] reusedBy := ["thm:forward-KL-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle19_upper_packet" interface := "Upper packet selecting the accumulated-error bridge from the appendix Gronwall display to the main-body linear-slowdown theorem constants." source := saldForwardKlDiscreteAccumulatedErrorSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle19DiscreteForwardKlUpperPacket", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces", "SALD.discreteForwardKlResidualExponentBoundScalar", "SALD.discreteForwardKlResidualExpBoundScalar", "sald.discrete_forward_kl.accumulated_error_bridge" ] reusedBy := ["thm:forward-KL-discrete", "cycle19 lower packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle19_middle_accumulated_error" interface := "Middle packet translating the accumulated-error bridge into lower-ready endpoint, exponent, residual-exponent, and integral-collection sub-slices." source := saldForwardKlDiscreteAccumulatedErrorSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle19DiscreteForwardKlUpperPacket", "SALD.cycle19DiscreteForwardKlMiddleContract", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces", "SALD.discreteForwardKlResidualExponentBoundScalar", "SALD.discreteForwardKlResidualExpBoundScalar", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.cycle19_accumulated_error_middle" ] reusedBy := ["thm:forward-KL-discrete", "cycle19 lower accumulated-error bridge"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.linear_slowdown_specialization" interface := "Substitute dot{s}=r, collect bar Gamma and bar Delta, and recover the main-body discrete theorem constants." source := saldForwardKlDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := ["sald.discrete_forward_kl.gronwall_accumulation", "def:alpha-complexity"] reusedBy := ["thm:forward-KL-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.residual_exponent_bound" interface := "Bound the residual Gronwall exponent after linear slowdown by the full positive factor T/(r*alpha)+2*r*eta^2*barGamma/alpha'." source := saldForwardKlDiscreteAccumulatedErrorSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.gronwall.integrating_factor", "SALD.discreteForwardKlResidualExponentBoundScalar", "SALD.discreteForwardKlResidualExpBoundScalar" ] reusedBy := ["thm:forward-KL-discrete", "sald.discrete_forward_kl.accumulated_error_bridge"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.accumulated_error_bridge" interface := "Rewrite the appendix Gronwall output into the main-body theorem display by endpoint matching, exponent splitting, and barGamma/barDelta collection." source := saldForwardKlDiscreteAccumulatedErrorSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "SALD.discreteForwardKlGronwallCoeffIntervalIntegrable", "SALD.discreteForwardKlGronwallCoeffIntegralSubSub", "SALD.discreteForwardKlGronwallInitialExponentSplitScalar", "SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces", "SALD.discreteForwardKlResidualExponentBoundScalar", "SALD.discreteForwardKlResidualExpBoundScalar", "SALD.discreteForwardKlAlphaComplexityCollectionScalar", "SALD.discreteForwardKlDeltaAccumulationScalar", "SALD.discreteForwardKlAccumulatedErrorCollectionScalar", "SALD.discreteForwardKlResidualIntegralDisplayBoundScalar", "sald.discrete_forward_kl.accumulated_error_bridge" ] reusedBy := ["thm:forward-KL-discrete", "sald.discrete_forward_kl.coefficient_chain_audit"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle27_upper_accumulated_collection" interface := "Upper packet selecting the accumulated-error collection slice after the coefficient-chain audit: endpoint rewrites, linear-slowdown exponent split, and A_alpha/barGamma/barDelta collection." source := saldForwardKlDiscreteAccumulatedErrorSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle27DiscreteForwardKlUpperPacket", "sald.discrete_forward_kl.cycle27_accumulated_collection_upper", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces", "SALD.discreteForwardKlResidualExponentBoundScalar", "SALD.discreteForwardKlResidualExpBoundScalar", "SALD.discreteForwardKlAlphaComplexityCollectionScalar", "SALD.discreteForwardKlDeltaAccumulationScalar", "SALD.discreteForwardKlAccumulatedErrorCollectionScalar", "sald.discrete_forward_kl.coefficient_chain_audit" ] reusedBy := ["thm:forward-KL-discrete", "cycle27 lower accumulated-error bridge"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle27_middle_accumulated_collection" interface := "Middle packet translating the cycle-27 accumulated-error target into lower-ready endpointBridge, alphaComplexityCollection, and deltaAccumulation sub-slices." source := saldForwardKlDiscreteAccumulatedErrorSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle27DiscreteForwardKlUpperPacket", "SALD.cycle27DiscreteForwardKlMiddleContract", "sald.discrete_forward_kl.cycle27_accumulated_collection_upper", "sald.discrete_forward_kl.cycle27_accumulated_collection_middle", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces", "SALD.discreteForwardKlResidualExponentBoundScalar", "SALD.discreteForwardKlResidualExpBoundScalar", "SALD.discreteForwardKlAlphaComplexityCollectionScalar", "SALD.discreteForwardKlDeltaAccumulationScalar", "SALD.discreteForwardKlAccumulatedErrorCollectionScalar", "sald.discrete_forward_kl.coefficient_chain_audit", "def:alpha-complexity" ] reusedBy := ["thm:forward-KL-discrete", "cycle27 lower accumulated-error bridge"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle27_lower_accumulated_collection" interface := "Lower scalar/integral core for collecting the additive E_alpha and Delta residual integrals into r^(-1)*A_alpha plus 2*r*eta*barDelta after linear slowdown." source := saldForwardKlDiscreteAccumulatedErrorSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle27DiscreteForwardKlMiddleContract", "SALD.discreteForwardKlAlphaComplexityCollectionScalar", "SALD.discreteForwardKlDeltaAccumulationScalar", "SALD.discreteForwardKlAccumulatedErrorCollectionScalar", "sald.discrete_forward_kl.cycle27_accumulated_collection_lower", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.linear_slowdown_specialization", "def:alpha-complexity" ] reusedBy := ["thm:forward-KL-discrete", "sald.discrete_forward_kl.accumulated_error_bridge"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.em_defect_accumulation_middle" interface := "Middle packet mapping the EM interpolation, one-step defects, DV velocity witness, Gronwall accumulation, and accumulated-error collection to explicit lower obligations." source := saldForwardKlDiscreteCoefficientChainSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.frozen_delta_cross_lip", "sald.discrete_forward_kl.kl_derivative", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.discreteForwardKlTimeChangeSquareCoefficientRewriteScalar", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge" ] reusedBy := ["thm:forward-KL-discrete", "cycle11 lower packets"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle23_upper_packet" interface := "Upper packet rebaselining the discrete forward-KL spine and selecting the coefficient-chain audit as the single lower target." source := saldForwardKlDiscreteCoefficientChainSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle23DiscreteForwardKlUpperPacket", "SALD.cycle15DiscreteForwardKlUpperPacket", "SALD.cycle19DiscreteForwardKlUpperPacket", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.frozen_delta_cross_lip", "sald.discrete_forward_kl.dv_velocity_bound", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.coefficient_chain_audit" ] reusedBy := ["thm:forward-KL-discrete", "cycle23 lower coefficient-chain audit"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.cycle23_middle_coefficient_chain" interface := "Middle packet translating the cycle-23 coefficient-chain target into the appendix.tex:454-553 first lower slice, with the accumulated-error bridge kept separate." source := saldForwardKlDiscreteCoefficientChainSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle23DiscreteForwardKlUpperPacket", "SALD.cycle23DiscreteForwardKlMiddleContract", "sald.discrete_forward_kl.cycle23_coefficient_chain_middle", "SALD.discreteForwardKlCoefficientChainAuditContract", "sald.discrete_forward_kl.frozen_delta_cross_lip", "sald.discrete_forward_kl.kl_derivative", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.discreteForwardKlTimeChangeSquareCoefficientRewriteScalar", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.accumulated_error_bridge" ] reusedBy := ["thm:forward-KL-discrete", "sald.discrete_forward_kl.coefficient_chain_audit", "cycle23 lower coefficient-chain audit"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.forward_KL_discrete.coefficient_chain_audit" interface := "Audit the one-step Gamma/Delta coefficients from frozen/moving cross terms through time change, Gronwall, and linear-slowdown accumulation." source := saldForwardKlDiscreteCoefficientChainSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle23DiscreteForwardKlMiddleContract", "sald.discrete_forward_kl.cycle23_coefficient_chain_middle", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.frozen_delta_cross_lip", "sald.discrete_forward_kl.kl_derivative", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.discreteForwardKlTimeChangeSquareCoefficientRewriteScalar", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge" ] reusedBy := ["thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete coefficient pattern"] status := ProofStatus.obligation } ] def discreteSaldContract : TheoremContract where id := "ASTIS.SALD.forward_KL_discrete" title := "Discrete-time SALD forward KL theorem" mode := "faithfulPaper" source := saldForwardKlDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" statementSummary := "For linear slowdown t(s)=s/r, Euler--Maruyama SALD satisfies the source forward-KL bound with the continuous alpha-complexity term plus accumulated discretization terms 2*r*eta^2*barGamma/alpha' and 2*r*eta*barDelta_{alpha'}." proofStatus := ProofStatus.contractOnly obligations := [ cycle44MainSkeletonAnalyticInterfaceObligation, cycle49MainSkeletonAnalyticReadinessObligation, cycle49MainSkeletonAnalyticMiddleObligation, cycle54MainSkeletonAnalyticInterfaceObligation, cycle54MainSkeletonAnalyticMiddleObligation, cycle59MainSkeletonAnalyticInterfaceObligation, cycle59MainSkeletonAnalyticMiddleObligation, cycle64MainSkeletonAnalyticInterfaceObligation, cycle64MainSkeletonAnalyticMiddleObligation, cycle69MainSkeletonAnalyticInterfaceObligation, cycle69MainSkeletonAnalyticMiddleObligation, cycle70GeneralMovingTargetDiscreteConditionalLawMiddleObligation, cycle70GeneralMovingTargetDiscreteConditionalLawLowerObligation, cycle71GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation, cycle71GeneralMovingTargetDiscreteEndpointConditionalLowerObligation, cycle72GeneralMovingTargetDiscreteWeakFpMiddleObligation, cycle72GeneralMovingTargetDiscreteWeakFpLowerObligation, cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpUpperObligation, cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpMiddleObligation, cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpLowerObligation, cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface, cycle74GeneralMovingTargetDiscreteMeasureInterfaceUpperObligation, cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation, cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation, cycle75GeneralMovingTargetDiscreteConditionalLawBackfillUpperObligation, cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation, cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation, cycle76GeneralMovingTargetDiscreteEndpointConditionalUpperObligation, cycle76GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation, cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation, cycle77GeneralMovingTargetDiscreteWeakFpGeneratorMiddleObligation, cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation, cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorUpperObligation, cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorMiddleObligation, cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation, cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface, cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureUpperObligation, cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureMiddleObligation, cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureLowerObligation, cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityUpperObligation, cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityMiddleObligation, cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation, cycle81GeneralMovingTargetDiscreteEndpointConditionalUpperObligation, cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation, cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsUpperObligation, cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation, cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation, cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpUpperObligation, cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpMiddleObligation, cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation, cycle84GeneralMovingTargetDiscreteActiveEmBackendUpperObligation, cycle84GeneralMovingTargetDiscreteActiveEmBackendMiddleObligation, cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation, cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryUpperObligation, cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryMiddleObligation, cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryUpperObligation, cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryMiddleObligation, cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation, cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryUpperObligation, cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryMiddleObligation, cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryLowerObligation, cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityMiddleObligation, cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation, cycle89DiscreteForwardKlClosurePressureUpperObligation, cycle89DiscreteForwardKlClosurePressureMiddleObligation, cycle89DiscreteForwardKlClosurePressureLowerObligation, cycle90DiscreteForwardKlMassConservationUpperObligation, cycle90DiscreteForwardKlMassConservationMiddleObligation, cycle90DiscreteForwardKlMassConservationLowerObligation, cycle91GeneralMovingTargetDiscreteConditionalKernelMiddleObligation, cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation, cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitMiddleObligation, cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation, cycle93GeneralMovingTargetDiscreteKlMassDerivativeMiddleObligation, cycle93GeneralMovingTargetDiscreteKlMassDerivativeLowerObligation, cycle94GeneralMovingTargetDiscreteWeakFpDriftActionMiddleObligation, cycle94GeneralMovingTargetDiscreteWeakFpDriftActionLowerObligation, cycle95DiscreteForwardKlClosurePressureUpperObligation, cycle95DiscreteForwardKlClosurePressureMiddleObligation, cycle95DiscreteForwardKlClosurePressureLowerObligation, cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingMiddleObligation, cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingLowerObligation, cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingMiddleObligation, cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingLowerObligation, cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryMiddleObligation, cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryLowerObligation, cycle99GeneralMovingTargetDiscreteRawKlDerivativeMiddleObligation, cycle99GeneralMovingTargetDiscreteRawKlDerivativeLowerObligation, cycle100GeneralMovingTargetDiscreteBarBWeakGradDefMiddleObligation, cycle100GeneralMovingTargetDiscreteBarBWeakGradDefLowerObligation, cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundLowerObligation, cycle101DiscreteForwardKlClosurePressureMiddleObligation, cycle101DiscreteForwardKlNoBoundaryProductRuleLowerObligation, cycle102DiscreteForwardKlZeroFluxTraceBoundaryMiddleObligation, cycle102DiscreteForwardKlTraceZeroLowerObligation, cycle103GeneralMovingTargetDiscreteConditionalKernelVersionMiddleObligation, cycle66DiscreteForwardKlSkeletonObligation, cycle66DiscreteForwardKlSkeletonMiddleObligation, cycle66DiscreteForwardKlAccumulatedDisplayLowerObligation, cycle61DiscreteForwardKlSkeletonObligation, cycle61DiscreteForwardKlSkeletonMiddleObligation, cycle56DiscreteForwardKlSkeletonObligation, cycle56DiscreteForwardKlSkeletonMiddleObligation, cycle56DiscreteForwardKlGronwallLowerObligation, cycle51DiscreteForwardKlSkeletonObligation, cycle51DiscreteForwardKlSkeletonMiddleObligation, cycle46DiscreteForwardKlSkeletonObligation, cycle46DiscreteForwardKlSkeletonMiddleObligation, lsiToKlFiObligation saldKlFiLsiSource, dvVariationalObligation saldDvVariationSource, gronwallAnalyticObligation, forwardKlDensityBoundaryObligation, forwardKlScheduleTimeChangeObligation, discreteForwardKlEmEndpointObligation, discreteForwardKlEmConditionalFpObligation, cycle15DiscreteForwardKlConditionalDriftDensityObligation, discreteForwardKlStitchedIntervalRegularityObligation, discreteForwardKlEmInterpolationObligation, cycle15DiscreteForwardKlMiddleEmSpineObligation, cycle15DiscreteForwardKlEmConditionalFpLowerObligation, cycle35DiscreteForwardKlEmFpUpperObligation, cycle35DiscreteForwardKlEmFpMiddleObligation, cycle35DiscreteForwardKlEmFpLowerObligation, cycle40DiscreteForwardKlEmFpMiddleObligation, cycle40DiscreteForwardKlEmEndpointLowerObligation, discreteForwardKlFrozenDeltaObligation, discreteForwardKlDerivativeObligation, cycle51DiscreteForwardKlDerivativeLowerObligation, discreteForwardKlDvFiniteLogMgfWitnessObligation, discreteForwardKlDvVelocityObligation, discreteForwardKlGronwallAccumulationObligation, discreteForwardKlLinearSlowdownObligation, discreteForwardKlResidualExponentBoundObligation, discreteForwardKlAccumulatedErrorBridgeObligation, cycle61DiscreteForwardKlAccumulatedErrorLowerObligation, cycle19DiscreteForwardKlAccumulatedErrorMiddleObligation, discreteForwardKlEmDefectAccumulationMiddleObligation, cycle23DiscreteForwardKlCoefficientChainMiddleObligation, discreteForwardKlCoefficientChainObligation, cycle27DiscreteForwardKlAccumulatedCollectionUpperObligation, cycle27DiscreteForwardKlAccumulatedCollectionMiddleObligation, cycle27DiscreteForwardKlAccumulatedCollectionLowerObligation ] def generalVaSaldProofDag : List ProofDagBlock := cycle49MainSkeletonAnalyticReadinessDag ++ cycle54MainSkeletonAnalyticInterfaceDag ++ cycle59MainSkeletonAnalyticInterfaceDag ++ cycle64MainSkeletonAnalyticInterfaceDag ++ cycle69MainSkeletonAnalyticInterfaceDag ++ cycle70GeneralMovingTargetDiscreteConditionalLawDag ++ cycle71GeneralMovingTargetDiscreteEndpointConditionalDag ++ cycle72GeneralMovingTargetDiscreteWeakFpDag ++ cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpDag ++ cycle74GeneralMovingTargetDiscreteMeasureInterfaceDag ++ cycle75GeneralMovingTargetDiscreteConditionalLawBackfillDag ++ cycle76GeneralMovingTargetDiscreteEndpointConditionalDag ++ cycle77GeneralMovingTargetDiscreteWeakFpGeneratorDag ++ cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorDag ++ cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureDag ++ cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityDag ++ cycle81GeneralMovingTargetDiscreteEndpointConditionalDag ++ cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsDag ++ cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpDag ++ cycle84GeneralMovingTargetDiscreteActiveEmBackendDag ++ cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryDag ++ cycle47GuidedGeneralSkeletonDag ++ cycle52GuidedGeneralSkeletonDag ++ cycle57GuidedGeneralSkeletonDag ++ cycle62GuidedGeneralSkeletonDag ++ cycle67GuidedGeneralSkeletonDag ++ cycle68UnifiedDiscreteGeneralDag ++ cycle48UnifiedDiscreteSkeletonDag ++ cycle53UnifiedDiscreteGeneralDag ++ cycle58UnifiedDiscreteGeneralDag ++ cycle63UnifiedDiscreteGeneralDag ++ [ { id := "ASTIS.SALD.guided_path_residual.normalizer" interface := "Differentiate Z_t and prove dot Z_t/Z_t=-E_{pi_t}[g_t] using the p_t transport equation." source := saldGuidedResidualProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := ["TransportVelocityContract", "GuidedTiltContract"] reusedBy := ["prop:guided_path_residual", "thm:unified-forward-KL"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.guided_path_residual.identity" interface := "Differentiate pi_t=Z_t^(-1)*p_t*exp(-f_t), add div(pi_t*u_t), and obtain the centered residual identity." source := saldGuidedResidualSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := ["sald.guided_path_residual.normalizer_derivative"] reusedBy := ["thm:unified-forward-KL"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target.derivative" interface := "Derive the sigma-weighted KL derivative inequality with residual velocity m_t=v_t-c_t before DV." source := saldGeneralMovingTargetDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "eq:general_moving_target_SALD", "TransportVelocityContract", "eq:LSI-KL-FI", "SALD.generalMovingTargetPostYoungDerivativeBoundScalar", "SALD.generalMovingTargetLsiDerivativeBoundScalar", "SALD.generalMovingTargetTimeChangedDerivativeBoundScalar", "SALD.generalMovingTargetPreDvDerivativeBoundScalar", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change" ] reusedBy := ["thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target.dv_finite_log_mgf_witness" interface := "Expose the residual-field DV finite-log-mgf and common-space witness for Z=alpha*||m_t||^2 before bounding ||m_t||^2." source := saldGeneralMovingTargetResidualDvSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "lem:dv_variation", "def:alpha-complexity", "SALD.saldDvFiniteLogMgfContract", "sald.dv_variation.finite_log_mgf_interface", "sald.general_moving_target.kl_derivative" ] reusedBy := ["thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target.dv_positive_alpha_scaling" interface := "After the residual DV instantiation, divide by alpha>0, rewrite the log-mgf quotient as E_alpha(pi_t,m_t), and preserve the sigma-weighted alpha^(-1) coefficient." source := saldGeneralMovingTargetResidualDvSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "lem:dv_variation", "def:alpha-complexity", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract" ] reusedBy := ["thm:general-moving-target-SALD", "thm:unified-forward-KL"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target.dv_m_energy" interface := "Instantiate DV with Z=alpha*||m_t||^2 and rewrite the log-mgf as E_alpha(pi_t,m_t)." source := saldGeneralMovingTargetResidualDvSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := ["lem:dv_variation", "def:alpha-complexity", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling"] reusedBy := ["thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target.gronwall_application" interface := "Apply lem:gronwall with sigma-weighted a(t), b(t), preserving the source exponent factors." source := saldGeneralMovingTargetDvGronwallSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := ["lem:gronwall", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_m_energy"] reusedBy := ["thm:general-moving-target-SALD", "thm:unified-forward-KL"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target.cycle24_upper_packet" interface := "Upper packet selecting the continuous general Gronwall endpoint/exponent/pure-contraction side-condition bridge as the lower target." source := saldGeneralMovingTargetDvGronwallSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle24GeneralVaSaldUpperPacket", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.dv_m_energy", "sald.gronwall.integrating_factor", "def:alpha-complexity" ] reusedBy := ["thm:general-moving-target-SALD", "thm:unified-forward-KL", "cycle24 lower packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target.cycle24_middle_gronwall_bridge" interface := "Middle packet mapping appendix.tex:909-945 to endpoint rewrites, sigma-weighted coefficient regularity, exponent splitting, residual-exponent monotonicity, and pure-contraction alpha-complexity obligations." source := saldGeneralMovingTargetDvGronwallSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle24GeneralVaSaldUpperPacket", "SALD.cycle24GeneralVaSaldMiddleContract", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.kl_derivative", "sald.forward_kl.schedule_time_change", "sald.gronwall.integrating_factor", "SALD.generalMovingTargetGronwallCoeffAdjacentIntervalIntegrable", "SALD.generalMovingTargetGronwallExpProductRewriteIntegralCongrOfPieces" ] reusedBy := ["thm:general-moving-target-SALD", "thm:unified-forward-KL", "cycle24 lower packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target.gronwall_side_conditions" interface := "Expose K(0)/K(T) endpoint rewrites, sigma-weighted coefficient regularity, exponent splitting, residual-exponent sign facts, and zero-residual alpha-complexity for the final theorem display." source := saldGeneralMovingTargetDvGronwallSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle24GeneralVaSaldUpperPacket", "SALD.cycle24GeneralVaSaldMiddleContract", "ASTIS.SALD.general_moving_target.cycle24_upper_packet", "ASTIS.SALD.general_moving_target.cycle24_middle_gronwall_bridge", "sald.general_moving_target.cycle24_gronwall_middle", "sald.forward_kl.schedule_time_change", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_application", "sald.gronwall.integrating_factor", "SALD.generalMovingTargetGronwallCoeffAdjacentIntervalIntegrable", "SALD.generalMovingTargetGronwallExpProductRewriteIntegralCongrOfPieces", "def:alpha-complexity" ] reusedBy := ["thm:general-moving-target-SALD", "thm:unified-forward-KL"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target.pure_contraction" interface := "Specialize c_t=v_t so m_t=0 and the alpha-complexity residual vanishes." source := saldGeneralMovingTargetPureContractionSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := ["sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.gronwall_application", "def:alpha-complexity"] reusedBy := ["thm:general-moving-target-SALD", "thm:unified-forward-KL"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.unified_forward_KL.transport_bridge_middle" interface := "Middle line ledger for main_body.tex:359-368: residual identity plus correction-field divergence cancel to make u_t+w_t transport pi_t, before the appendix specialization." source := saldUnifiedTransportBridgeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle16GeneralVaSaldUpperPacket", "SALD.cycle16UnifiedForwardKlTransportBridgeMiddleContract", "SALD.unifiedForwardKlSpecializationContract", "sald.guided_path_residual.identity", "eq:poisson-eq" ] reusedBy := ["sald.unified_forward_kl.transport_velocity_bridge", "thm:unified-forward-KL", "cycle16 lower packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.unified_forward_KL.transport_bridge_lower" interface := "Lower slice for main_body.tex:359-368: preserve the residual/correction signs, expose divergence linearity, cancel the centered residual, and record v_t=u_t+w_t, c_t=u_t, m_t=w_t." source := saldUnifiedTransportBridgeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle16UnifiedForwardKlTransportBridgeMiddleContract", "SALD.cycle16UnifiedForwardKlTransportBridgeLowerContract", "sald.unified_forward_kl.transport_bridge_middle", "sald.guided_path_residual.identity", "eq:poisson-eq", "TransportVelocityContract" ] reusedBy := ["sald.unified_forward_kl.transport_velocity_bridge", "thm:unified-forward-KL", "cycle16 lower packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.unified_forward_KL.transport_velocity_bridge" interface := "Combine the centered guided residual identity with eq:poisson-eq to show u_t+w_t transports pi_t, then record v_t=u_t+w_t and m_t=w_t." source := saldCorrectionFieldSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "sald.unified_forward_kl.transport_bridge_middle", "sald.unified_forward_kl.transport_bridge_lower", "SALD.cycle16UnifiedForwardKlTransportBridgeLowerContract", "prop:guided_path_residual", "eq:poisson-eq", "sald.guided_path_residual.identity", "TransportVelocityContract" ] reusedBy := ["thm:unified-forward-KL", "cycle16 lower packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.unified_forward_KL.specialization" interface := "Specialize the general moving-target theorem with c_t=u_t, use the correction equation to choose v_t=u_t+w_t, and identify m_t=w_t." source := saldUnifiedForwardKlProofSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "prop:guided_path_residual", "eq:poisson-eq", "eq:SALD_Ito", "thm:general-moving-target-SALD", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization" ] reusedBy := ["thm:unified-forward-KL"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_va_sald.guided_path_middle" interface := "Middle packet mapping the guided residual proposition, continuous general theorem, unified specialization, and discrete general theorem to explicit lower obligations." source := saldGeneralMovingTargetSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "sald.guided_path_residual.identity", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.unified_forward_kl.transport_bridge_middle", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.gronwall_side_conditions", "sald.general_moving_target_discrete.unified_specialization" ] reusedBy := ["thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "cycle12 lower packets"] status := ProofStatus.obligation } ] def generalVaSaldDiscreteProofDag : List ProofDagBlock := cycle49MainSkeletonAnalyticReadinessDag ++ cycle54MainSkeletonAnalyticInterfaceDag ++ cycle59MainSkeletonAnalyticInterfaceDag ++ cycle64MainSkeletonAnalyticInterfaceDag ++ cycle69MainSkeletonAnalyticInterfaceDag ++ cycle70GeneralMovingTargetDiscreteConditionalLawDag ++ cycle71GeneralMovingTargetDiscreteEndpointConditionalDag ++ cycle72GeneralMovingTargetDiscreteWeakFpDag ++ cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpDag ++ cycle74GeneralMovingTargetDiscreteMeasureInterfaceDag ++ cycle75GeneralMovingTargetDiscreteConditionalLawBackfillDag ++ cycle76GeneralMovingTargetDiscreteEndpointConditionalDag ++ cycle77GeneralMovingTargetDiscreteWeakFpGeneratorDag ++ cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorDag ++ cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureDag ++ cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityDag ++ cycle81GeneralMovingTargetDiscreteEndpointConditionalDag ++ cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsDag ++ cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpDag ++ cycle84GeneralMovingTargetDiscreteActiveEmBackendDag ++ cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryDag ++ cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryDag ++ cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryDag ++ cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityDag ++ cycle91GeneralMovingTargetDiscreteConditionalKernelDag ++ cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitDag ++ cycle93GeneralMovingTargetDiscreteKlMassDerivativeDag ++ cycle94GeneralMovingTargetDiscreteWeakFpDriftActionDag ++ cycle95DiscreteForwardKlClosurePressureDag ++ cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingDag ++ cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingDag ++ cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryDag ++ cycle99GeneralMovingTargetDiscreteRawKlDerivativeDag ++ cycle100GeneralMovingTargetDiscreteBarBWeakGradDefDag ++ cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundDag ++ cycle101DiscreteForwardKlClosurePressureDag ++ cycle103GeneralMovingTargetDiscreteConditionalKernelVersionDag ++ cycle48UnifiedDiscreteSkeletonDag ++ cycle53UnifiedDiscreteGeneralDag ++ cycle58UnifiedDiscreteGeneralDag ++ cycle63UnifiedDiscreteGeneralDag ++ cycle68UnifiedDiscreteGeneralDag ++ [ { id := "ASTIS.SALD.general_moving_target_discrete.euler_maruyama" interface := "Pin the general VA-SALD EM update, frozen interpolation, sigma_eta, and endpoint laws." source := saldGeneralMovingTargetDiscreteEmSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := ["EulerMaruyamaContract", "SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit", "sald.general_moving_target_discrete.em_interpolation_fp"] reusedBy := ["thm:general-moving-target-SALD-discrete"] status := ProofStatus.contractOnly }, { id := "ASTIS.SALD.general_moving_target_discrete.frozen_delta" interface := "Bound the VA frozen-field error delta_pi^VA using the source Gamma and Delta definitions." source := saldFrozenDeltaCrossLipGeneralSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := ["eq:general_discrete_delta_def", "lem:dv_variation", "def:alpha-complexity"] reusedBy := ["thm:general-moving-target-SALD-discrete", "lem:frozen_delta_cross_lip_sald"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.cycle28_upper_derivative_side_conditions" interface := "Upper packet selecting the discrete general derivative side-condition bridge, with appendix.tex:1469-1511 frozen/residual algebra and Young coefficient bookkeeping as the preferred lower sub-slice." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle28GeneralVaSaldUpperPacket", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "sald.forward_kl.schedule_time_change", "eq:LSI-KL-FI", "lem:dv_variation" ] reusedBy := ["thm:general-moving-target-SALD-discrete", "cycle28 lower packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.cycle28_middle_derivative_side_conditions" interface := "Middle packet mapping appendix.tex:1469-1511 to the frozen/residual algebra, two sigma_eta^2/8 Young shares, and compiled scalar coefficient helpers before lower proof search." source := saldGeneralMovingTargetDiscreteYoungSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle28GeneralVaSaldUpperPacket", "SALD.cycle28GeneralVaSaldMiddleContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteFrozenResidualAlgebraVector", "SALD.generalMovingTargetDiscreteYoungFisherShareScalar", "SALD.generalMovingTargetDiscreteTwoYoungFisherBudgetScalar", "SALD.generalMovingTargetDiscreteResidualYoungCoefficientScalar", "sald.general_moving_target_discrete.cycle28_derivative_side_middle", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "eq:LSI-KL-FI" ] reusedBy := ["sald.general_moving_target_discrete.derivative_side_conditions", "thm:general-moving-target-SALD-discrete", "cycle28 lower packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.cycle28_lower_derivative_side_conditions" interface := "Lower compiled module algebra for appendix.tex:1469-1478: after delta_pi^VA, tilde v_s, and m_t=v_t-c_t are identified, rewrite the cross field as delta_pi^VA+dot t(s)*m_t." source := saldGeneralMovingTargetDiscreteYoungSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle28GeneralVaSaldMiddleContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteFrozenResidualAlgebraVector", "sald.general_moving_target_discrete.cycle28_derivative_side_lower", "eq:general_discrete_delta_def", "sald.general_moving_target_discrete.em_interpolation_fp" ] reusedBy := ["sald.general_moving_target_discrete.derivative_side_conditions", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.derivative_side_conditions" interface := "Expose endpoint laws, conditional-drift Fokker--Planck, frozen/residual algebra, Young coefficient bookkeeping, DV finite-log-mgf, and stitched time-change interfaces." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle28GeneralVaSaldMiddleContract", "sald.general_moving_target_discrete.cycle28_derivative_side_middle", "sald.general_moving_target_discrete.cycle28_derivative_side_lower", "SALD.generalMovingTargetDiscreteFrozenResidualAlgebraVector", "SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "sald.forward_kl.schedule_time_change" ] reusedBy := ["thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.dv_finite_log_mgf_witness" interface := "Expose the EM-interpolation residual DV witness for nu=hat rho_s, mu=tilde pi_s, and Z=alpha*||m_{t(s)}||^2 before the doubled residual-energy bound." source := saldGeneralMovingTargetDiscreteResidualDvSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "lem:dv_variation", "def:alpha-complexity", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.derivative_side_conditions" ] reusedBy := ["thm:general-moving-target-SALD-discrete", "discrete VA-SALD guided specialization"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.dv_m_energy" interface := "Instantiate DV under the EM interpolation law and preserve the doubled residual-energy coefficient before the s-to-t time change." source := saldGeneralMovingTargetDiscreteResidualDvSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "lem:dv_variation", "def:alpha-complexity", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness" ] reusedBy := ["thm:general-moving-target-SALD-discrete", "discrete VA-SALD guided specialization"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.derivative" interface := "Differentiate KL(hat rho_s||tilde pi_s), split delta_pi^VA and dot{t}*m_t, then derive the source discrete differential inequality." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit", "SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "SALD.generalMovingTargetDiscreteDerivativeDvTimeChangedScalar", "sald.general_moving_target_discrete.cycle53_derivative_dv_lower", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "eq:LSI-KL-FI" ] reusedBy := ["thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.cycle53_derivative_dv_lower" interface := "Lower proof-producing scalar handoff for appendix.tex:1469-1583: compose the supplied EM KL derivative, frozen/residual Young bounds, frozen-delta term, LSI comparison, residual DV estimate, and constant-schedule time change into the exact t-time pre-Gronwall inequality." source := saldGeneralMovingTargetDiscreteDerivativeSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle53GeneralMovingTargetDiscreteDerivativeDvLowerObligation", "SALD.generalMovingTargetDiscretePostYoungDerivativeBoundScalar", "SALD.generalMovingTargetDiscretePostLsiDerivativeBoundScalar", "SALD.generalMovingTargetDiscretePostDvDerivativeBoundScalar", "SALD.generalMovingTargetDiscreteTimeChangedDerivativeBoundScalar", "SALD.generalMovingTargetDiscreteDerivativeDvTimeChangedScalar", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.derivative_side_conditions", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "sald.general_moving_target_discrete.dv_m_energy", "probability.lsi_to_kl_fi", "sald.forward_kl.schedule_time_change" ] reusedBy := ["sald.general_moving_target_discrete.kl_derivative", "thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.constant_schedule" interface := "Expose constant inverse-schedule and stitched-interval regularity for the final Gronwall step." source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := ["sald.forward_kl.schedule_time_change", "sald.general_moving_target_discrete.em_interpolation_fp"] reusedBy := ["thm:general-moving-target-SALD-discrete"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.gronwall_application" interface := "Apply lem:gronwall with the source sigma_eta, Gamma, Delta, and doubled residual-energy coefficients." source := saldGeneralMovingTargetDiscreteGronwallSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "lem:gronwall", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.constant_schedule_stitching" ] reusedBy := ["thm:general-moving-target-SALD-discrete", "discrete VA-SALD guided specialization"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.cycle20_upper_packet" interface := "Upper packet selecting the final discrete general Gronwall side-condition/display bridge as the lower target." source := saldGeneralMovingTargetDiscreteGronwallSideConditionSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle20GeneralVaSaldUpperPacket", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.constant_schedule_stitching", "sald.general_moving_target_discrete.kl_derivative", "sald.gronwall.integrating_factor" ] reusedBy := ["thm:general-moving-target-SALD-discrete", "cycle20 lower packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.cycle20_middle_gronwall_bridge" interface := "Middle source-to-Lean map for appendix.tex:1573-1600: K(t) stitching, constant-schedule coefficient rewrites, coefficient regularity, and exact Gronwall-display matching." source := saldGeneralMovingTargetDiscreteGronwallSideConditionSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle20GeneralVaSaldUpperPacket", "SALD.cycle20GeneralVaSaldMiddleContract", "SALD.generalMovingTargetDiscreteGronwallInstantiationContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar", "SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput", "SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar", "SALD.cycle59GeneralMovingTargetDiscreteGronwallLowerObligation", "sald.general_moving_target_discrete.cycle20_gronwall_middle", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.constant_schedule_stitching", "sald.gronwall.integrating_factor" ] reusedBy := ["thm:general-moving-target-SALD-discrete", "cycle20 lower packet"] status := ProofStatus.obligation }, { id := "ASTIS.SALD.general_moving_target_discrete.gronwall_side_conditions" interface := "Expose stitched endpoint laws, constant-schedule coefficient rewrites, coefficient regularity, and exact matching of the Gronwall output to the theorem display." source := saldGeneralMovingTargetDiscreteGronwallSideConditionSource targetLean := "AutoSamplingTheory/SALD.lean" dependsOn := [ "SALD.cycle20GeneralVaSaldUpperPacket", "SALD.cycle20GeneralVaSaldMiddleContract", "SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar", "SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar", "sald.general_moving_target_discrete.cycle20_gronwall_middle", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.constant_schedule_stitching", "sald.general_moving_target_discrete.kl_derivative", "sald.gronwall.integrating_factor" ] reusedBy := ["thm:general-moving-target-SALD-discrete", "discrete VA-SALD guided specialization"] status := ProofStatus.obligation } ] def guidedResidualContract : TheoremContract where id := "ASTIS.SALD.guided_path_residual" title := "Residual identity for guided path" mode := "faithfulPaper" source := saldGuidedResidualSource targetLean := "AutoSamplingTheory/SALD.lean" statementSummary := "The guided path residual is driven by the centered guide transport derivative." proofStatus := ProofStatus.contractOnly obligations := [ cycle44MainSkeletonAnalyticInterfaceObligation, cycle49MainSkeletonAnalyticReadinessObligation, cycle49MainSkeletonAnalyticMiddleObligation, cycle54MainSkeletonAnalyticInterfaceObligation, cycle54MainSkeletonAnalyticMiddleObligation, cycle59MainSkeletonAnalyticInterfaceObligation, cycle59MainSkeletonAnalyticMiddleObligation, cycle64MainSkeletonAnalyticInterfaceObligation, cycle64MainSkeletonAnalyticMiddleObligation, cycle69MainSkeletonAnalyticInterfaceObligation, cycle69MainSkeletonAnalyticMiddleObligation, cycle47GuidedGeneralSkeletonObligation, cycle47GuidedGeneralSkeletonMiddleObligation, cycle52GuidedGeneralSkeletonObligation, cycle52GuidedGeneralSkeletonMiddleObligation, cycle57GuidedGeneralSkeletonObligation, cycle57GuidedGeneralSkeletonMiddleObligation, cycle62GuidedGeneralSkeletonObligation, cycle62GuidedGeneralSkeletonMiddleObligation, cycle67GuidedGeneralSkeletonObligation, cycle67GuidedGeneralSkeletonMiddleObligation, guidedResidualNormalizerObligation, guidedResidualIdentityObligation ] def generalVaSaldContract : TheoremContract where id := "ASTIS.SALD.general_moving_target" title := "General moving-target VA-SALD" mode := "faithfulPaper" source := saldGeneralMovingTargetSource targetLean := "AutoSamplingTheory/SALD.lean" statementSummary := "VA-SALD uses an implementable velocity c_t and pays only for m_t = v_t - c_t." proofStatus := ProofStatus.contractOnly obligations := [ cycle44MainSkeletonAnalyticInterfaceObligation, cycle49MainSkeletonAnalyticReadinessObligation, cycle49MainSkeletonAnalyticMiddleObligation, cycle54MainSkeletonAnalyticInterfaceObligation, cycle54MainSkeletonAnalyticMiddleObligation, cycle59MainSkeletonAnalyticInterfaceObligation, cycle59MainSkeletonAnalyticMiddleObligation, cycle64MainSkeletonAnalyticInterfaceObligation, cycle64MainSkeletonAnalyticMiddleObligation, cycle69MainSkeletonAnalyticInterfaceObligation, cycle69MainSkeletonAnalyticMiddleObligation, cycle47GuidedGeneralSkeletonObligation, cycle47GuidedGeneralSkeletonMiddleObligation, cycle52GuidedGeneralSkeletonObligation, cycle52GuidedGeneralSkeletonMiddleObligation, cycle52GuidedGeneralDerivativeDvLowerObligation, cycle57GuidedGeneralSkeletonObligation, cycle57GuidedGeneralSkeletonMiddleObligation, cycle57GuidedGeneralDerivativeSplitLowerObligation, cycle62GuidedGeneralSkeletonObligation, cycle62GuidedGeneralSkeletonMiddleObligation, cycle62GuidedGeneralScaledResidualLowerObligation, cycle67GuidedGeneralSkeletonObligation, cycle67GuidedGeneralSkeletonMiddleObligation, cycle67GuidedGeneralResidualGronwallBridgeObligation, cycle67GuidedGeneralResidualGronwallLowerObligation, lsiToKlFiObligation saldKlFiLsiSource, dvVariationalObligation saldDvVariationSource, gronwallAnalyticObligation, forwardKlDensityBoundaryObligation, forwardKlScheduleTimeChangeObligation, generalMovingTargetDerivativeObligation, generalMovingTargetDvPositiveAlphaScalingObligation, generalMovingTargetDvFiniteLogMgfWitnessObligation, generalMovingTargetDvEnergyObligation, generalMovingTargetGronwallApplicationObligation, cycle24GeneralVaSaldGronwallMiddleObligation, generalMovingTargetGronwallSideConditionObligation, generalMovingTargetPureContractionObligation, generalVaSaldGuidedPathMiddleObligation ] def unifiedForwardKlContract : TheoremContract where id := "ASTIS.SALD.unified_forward_KL" title := "Unified VA-SALD forward KL theorem" mode := "faithfulPaper" source := saldUnifiedForwardKlSource targetLean := "AutoSamplingTheory/SALD.lean" statementSummary := "The main-body VA-SALD bound is obtained from the general moving-target theorem with c_t specialized to u_t." proofStatus := ProofStatus.contractOnly obligations := [ cycle44MainSkeletonAnalyticInterfaceObligation, cycle49MainSkeletonAnalyticReadinessObligation, cycle49MainSkeletonAnalyticMiddleObligation, cycle54MainSkeletonAnalyticInterfaceObligation, cycle54MainSkeletonAnalyticMiddleObligation, cycle59MainSkeletonAnalyticInterfaceObligation, cycle59MainSkeletonAnalyticMiddleObligation, cycle64MainSkeletonAnalyticInterfaceObligation, cycle64MainSkeletonAnalyticMiddleObligation, cycle69MainSkeletonAnalyticInterfaceObligation, cycle69MainSkeletonAnalyticMiddleObligation, cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation, cycle48UnifiedDiscreteSkeletonObligation, cycle48UnifiedDiscreteSkeletonMiddleObligation, cycle53UnifiedDiscreteGeneralSkeletonObligation, cycle53UnifiedDiscreteGeneralMiddleObligation, cycle58UnifiedDiscreteGeneralSkeletonObligation, cycle58UnifiedDiscreteGeneralMiddleObligation, cycle63UnifiedDiscreteGeneralSkeletonObligation, cycle63UnifiedDiscreteGeneralMiddleObligation, cycle63UnifiedDiscreteGeneralMeasureBackfillObligation, cycle67GuidedGeneralSkeletonObligation, cycle67GuidedGeneralSkeletonMiddleObligation, cycle67GuidedGeneralResidualGronwallBridgeObligation, cycle67GuidedGeneralResidualGronwallLowerObligation, cycle68UnifiedDiscreteGeneralSkeletonObligation, cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation, guidedResidualIdentityObligation, generalMovingTargetDerivativeObligation, cycle52GuidedGeneralDerivativeDvLowerObligation, cycle57GuidedGeneralDerivativeSplitLowerObligation, generalMovingTargetDvPositiveAlphaScalingObligation, generalMovingTargetDvFiniteLogMgfWitnessObligation, generalMovingTargetDvEnergyObligation, generalMovingTargetGronwallApplicationObligation, cycle24GeneralVaSaldGronwallMiddleObligation, generalMovingTargetGronwallSideConditionObligation, cycle16UnifiedForwardKlTransportBridgeMiddleObligation, cycle16UnifiedForwardKlTransportBridgeLowerObligation, unifiedForwardKlTransportBridgeObligation, unifiedForwardKlSpecializationObligation, generalVaSaldGuidedPathMiddleObligation ] def generalVaSaldDiscreteContract : TheoremContract where id := "ASTIS.SALD.general_moving_target_discrete" title := "Discrete-time general moving-target VA-SALD" mode := "faithfulPaper" source := saldGeneralMovingTargetDiscreteSource targetLean := "AutoSamplingTheory/SALD.lean" statementSummary := "The discrete-time general VA-SALD theorem applies the general EM interpolation, frozen-delta estimate, DV residual-energy bound, and Gronwall to the moving target with residual m_t=v_t-c_t." proofStatus := ProofStatus.contractOnly obligations := [ cycle44MainSkeletonAnalyticInterfaceObligation, cycle49MainSkeletonAnalyticReadinessObligation, cycle49MainSkeletonAnalyticMiddleObligation, cycle54MainSkeletonAnalyticInterfaceObligation, cycle54MainSkeletonAnalyticMiddleObligation, cycle59MainSkeletonAnalyticInterfaceObligation, cycle59MainSkeletonAnalyticMiddleObligation, cycle64MainSkeletonAnalyticInterfaceObligation, cycle64MainSkeletonAnalyticMiddleObligation, cycle69MainSkeletonAnalyticInterfaceObligation, cycle69MainSkeletonAnalyticMiddleObligation, cycle70GeneralMovingTargetDiscreteConditionalLawMiddleObligation, cycle70GeneralMovingTargetDiscreteConditionalLawLowerObligation, cycle71GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation, cycle71GeneralMovingTargetDiscreteEndpointConditionalLowerObligation, cycle72GeneralMovingTargetDiscreteWeakFpMiddleObligation, cycle72GeneralMovingTargetDiscreteWeakFpLowerObligation, cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpUpperObligation, cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpMiddleObligation, cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpLowerObligation, cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface, cycle74GeneralMovingTargetDiscreteMeasureInterfaceUpperObligation, cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation, cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation, cycle75GeneralMovingTargetDiscreteConditionalLawBackfillUpperObligation, cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation, cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation, cycle76GeneralMovingTargetDiscreteEndpointConditionalUpperObligation, cycle76GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation, cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation, cycle77GeneralMovingTargetDiscreteWeakFpGeneratorMiddleObligation, cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation, cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorUpperObligation, cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorMiddleObligation, cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation, cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface, cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureUpperObligation, cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureMiddleObligation, cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureLowerObligation, cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityUpperObligation, cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityMiddleObligation, cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation, cycle81GeneralMovingTargetDiscreteEndpointConditionalUpperObligation, cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation, cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsUpperObligation, cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation, cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation, cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpUpperObligation, cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpMiddleObligation, cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation, cycle84GeneralMovingTargetDiscreteActiveEmBackendUpperObligation, cycle84GeneralMovingTargetDiscreteActiveEmBackendMiddleObligation, cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation, cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryUpperObligation, cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryMiddleObligation, cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryUpperObligation, cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryMiddleObligation, cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation, cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryUpperObligation, cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryMiddleObligation, cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryLowerObligation, cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityMiddleObligation, cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation, cycle91GeneralMovingTargetDiscreteConditionalKernelMiddleObligation, cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation, cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitMiddleObligation, cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation, cycle93GeneralMovingTargetDiscreteKlMassDerivativeMiddleObligation, cycle93GeneralMovingTargetDiscreteKlMassDerivativeLowerObligation, cycle94GeneralMovingTargetDiscreteWeakFpDriftActionMiddleObligation, cycle94GeneralMovingTargetDiscreteWeakFpDriftActionLowerObligation, cycle95DiscreteForwardKlClosurePressureMiddleObligation, cycle95DiscreteForwardKlClosurePressureLowerObligation, cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingMiddleObligation, cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingLowerObligation, cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingMiddleObligation, cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingLowerObligation, cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryMiddleObligation, cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryLowerObligation, cycle99GeneralMovingTargetDiscreteRawKlDerivativeMiddleObligation, cycle99GeneralMovingTargetDiscreteRawKlDerivativeLowerObligation, cycle100GeneralMovingTargetDiscreteBarBWeakGradDefMiddleObligation, cycle100GeneralMovingTargetDiscreteBarBWeakGradDefLowerObligation, cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundLowerObligation, cycle101DiscreteForwardKlClosurePressureMiddleObligation, cycle101DiscreteForwardKlNoBoundaryProductRuleLowerObligation, cycle102DiscreteForwardKlZeroFluxTraceBoundaryMiddleObligation, cycle102DiscreteForwardKlTraceZeroLowerObligation, cycle103GeneralMovingTargetDiscreteConditionalKernelVersionMiddleObligation, cycle48UnifiedDiscreteSkeletonObligation, cycle48UnifiedDiscreteSkeletonMiddleObligation, cycle53UnifiedDiscreteGeneralSkeletonObligation, cycle53UnifiedDiscreteGeneralMiddleObligation, cycle58UnifiedDiscreteGeneralSkeletonObligation, cycle58UnifiedDiscreteGeneralMiddleObligation, cycle63UnifiedDiscreteGeneralSkeletonObligation, cycle63UnifiedDiscreteGeneralMiddleObligation, cycle63UnifiedDiscreteGeneralMeasureBackfillObligation, cycle68UnifiedDiscreteGeneralSkeletonObligation, cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation, cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation, cycle68UnifiedDiscreteGeneralDiscreteBridgeLowerObligation, lsiToKlFiObligation saldKlFiLsiSource, dvVariationalObligation saldDvVariationSource, gronwallAnalyticObligation, forwardKlDensityBoundaryObligation, forwardKlScheduleTimeChangeObligation, cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation, generalMovingTargetDiscreteEmInterpolationObligation, generalMovingTargetDiscreteConstantScheduleObligation, generalMovingTargetDiscreteFrozenDeltaObligation, cycle28GeneralVaSaldDerivativeSideMiddleObligation, cycle28GeneralVaSaldDerivativeSideLowerObligation, generalMovingTargetDiscreteDerivativeSideConditionObligation, cycle69GeneralMovingTargetDiscreteEmFpSourceSignsLowerObligation, cycle54GeneralMovingTargetDiscreteEmFpLowerObligation, generalMovingTargetDiscreteDerivativeObligation, cycle53GeneralMovingTargetDiscreteDerivativeDvLowerObligation, generalMovingTargetDiscreteDvFiniteLogMgfWitnessObligation, generalMovingTargetDiscreteDvMEnergyObligation, generalMovingTargetDiscreteGronwallApplicationObligation, cycle20GeneralVaSaldDiscreteGronwallMiddleObligation, generalMovingTargetDiscreteGronwallSideConditionObligation, cycle59GeneralMovingTargetDiscreteGronwallLowerObligation, discreteUnifiedVaSaldSpecializationObligation, generalVaSaldGuidedPathMiddleObligation ] def saldTheoremContracts : List TheoremContract := [ gronwallContract, dvContract, piDefinitionContract, lsiKlFiVocabularyContract, continuousSaldContract, discreteSaldContract, guidedResidualContract, generalVaSaldContract, unifiedForwardKlContract, generalVaSaldDiscreteContract ] def saldSourceForLabel (label : String) : SourceAnchor := if label = "lem:gronwall" then saldGronwallSource else if label = "lem:dv_variation" then saldDvVariationSource else if label = "def:PI" then saldPiSource else if label = "eq:LSI-KL-FI" then saldKlFiLsiSource else if label = "thm:forward-KL" then saldForwardKlSource else if label = "thm:forward-KL-discrete" then saldForwardKlDiscreteSource else if label = "prop:guided_path_residual" then saldGuidedResidualSource else if label = "thm:general-moving-target-SALD" then saldGeneralMovingTargetSource else if label = "thm:unified-forward-KL" then saldUnifiedForwardKlSource else if label = "thm:general-moving-target-SALD-discrete" then saldGeneralMovingTargetDiscreteSource else saldAppendixSource def saldLeanTargetForLabel (label : String) : String := if label = "lem:dv_variation" || label = "def:PI" || label = "eq:LSI-KL-FI" then "AutoSamplingTheory/Probability.lean" else "AutoSamplingTheory/SALD.lean" def cycle49MainSkeletonDependencyNames : List String := [ "SALD.cycle49MainSkeletonAnalyticReadinessLedger", "SALD.cycle49MainSkeletonAnalyticReadinessObligation", "SALD.cycle49MainSkeletonAnalyticMiddleContract", "SALD.cycle49MainSkeletonAnalyticMiddleObligation", "SALD.cycle49MainSkeletonAnalyticReadinessDag", "ASTIS.SALD.cycle49.analytic_readiness", "ASTIS.SALD.cycle49.middle_route_audit", "sald.main_skeleton.cycle49_analytic_readiness", "sald.main_skeleton.cycle49_middle_route_audit", "ASTIS.SALD.cycle49.lower_packet.general_discrete_em_fp" ] def cycle50ForwardKlDependencyNames : List String := [ "SALD.cycle50ForwardKlSkeletonUpperPacket", "SALD.cycle50ForwardKlSkeletonObligation", "SALD.cycle50ForwardKlSkeletonMiddleContract", "SALD.cycle50ForwardKlSkeletonMiddleObligation", "SALD.cycle50ForwardKlDerivativeLowerObligation", "SALD.forwardKlDerivativeDvGronwallCoefficientOfKlFiVelocityScalingScalar", "SALD.cycle50ForwardKlSkeletonDag", "ASTIS.SALD.forward_KL.cycle50_theorem_skeleton_route", "ASTIS.SALD.forward_KL.cycle50_middle_route_audit", "ASTIS.SALD.forward_KL.cycle50_derivative_dv_lower", "sald.forward_kl.cycle50_derivative_dv_lower", "sald.forward_kl.cycle50_middle_route_audit", "sald.forward_kl.cycle50_theorem_skeleton_route" ] def cycle51DiscreteForwardKlDependencyNames : List String := [ "SALD.cycle51DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle51DiscreteForwardKlSkeletonObligation", "SALD.cycle51DiscreteForwardKlSkeletonMiddleContract", "SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation", "ASTIS.SALD.forward_KL_discrete.cycle51_theorem_interface_route", "ASTIS.SALD.forward_KL_discrete.cycle51_middle_route_audit", "sald.discrete_forward_kl.cycle51_theorem_interface_route", "sald.discrete_forward_kl.cycle51_middle_route_audit", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.discreteForwardKlPostLsiDerivativeBoundScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfKlFiScalar", "ASTIS.SALD.forward_KL_discrete.cycle51_derivative_lower", "sald.discrete_forward_kl.cycle51_derivative_lower", "SALD.cycle50ForwardKlDerivativeLowerObligation", "SALD.forwardKlDerivativeDvGronwallCoefficientOfKlFiVelocityScalingScalar", "ASTIS.SALD.forward_KL.cycle50_derivative_dv_lower", "sald.forward_kl.cycle50_derivative_dv_lower" ] def cycle52GuidedGeneralDependencyNames : List String := [ "SALD.cycle52GuidedGeneralSkeletonUpperPacket", "SALD.cycle52GuidedGeneralSkeletonObligation", "SALD.cycle52GuidedGeneralSkeletonMiddleContract", "SALD.cycle52GuidedGeneralSkeletonMiddleObligation", "SALD.cycle52GuidedGeneralDerivativeDvLowerObligation", "SALD.generalMovingTargetPostDvGronwallCoefficientScalar", "SALD.generalMovingTargetPostDvGronwallCoefficientOfSigmaScheduleScalar", "SALD.generalMovingTargetDerivativeDvGronwallCoefficientScalar", "SALD.cycle52GuidedGeneralSkeletonDag", "ASTIS.SALD.guided_general.cycle52_upper_route", "ASTIS.SALD.guided_general.cycle52_middle_route_audit", "ASTIS.SALD.general_moving_target.cycle52_derivative_dv_lower", "sald.guided_general.cycle52_upper_route", "sald.guided_general.cycle52_middle_route_audit", "sald.general_moving_target.cycle52_derivative_dv_lower" ] def cycle53UnifiedDiscreteGeneralDependencyNames : List String := [ "SALD.cycle53UnifiedDiscreteGeneralUpperPacket", "SALD.cycle53UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle53UnifiedDiscreteGeneralMiddleContract", "SALD.cycle53UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle53UnifiedDiscreteGeneralDag", "ASTIS.SALD.unified_discrete_general.cycle53_upper_route", "ASTIS.SALD.unified_discrete_general.cycle53_middle_route_audit", "ASTIS.SALD.general_moving_target_discrete.cycle53_measure_map_endpoint_backfill", "AutoSamplingTheory.lawMapEqOfAEEq", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "SALD.cycle53GeneralMovingTargetDiscreteDerivativeDvLowerObligation", "SALD.generalMovingTargetDiscretePostYoungDerivativeBoundScalar", "SALD.generalMovingTargetDiscretePostLsiDerivativeBoundScalar", "SALD.generalMovingTargetDiscretePostDvDerivativeBoundScalar", "SALD.generalMovingTargetDiscreteTimeChangedDerivativeBoundScalar", "SALD.generalMovingTargetDiscreteDerivativeDvTimeChangedScalar", "ASTIS.SALD.general_moving_target_discrete.cycle53_derivative_dv_lower", "sald.general_moving_target_discrete.cycle53_derivative_dv_lower", "sald.unified_discrete_general.cycle53_upper_route", "sald.unified_discrete_general.cycle53_middle_route_audit", "sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] def cycle54MainSkeletonDependencyNames : List String := [ "SALD.cycle54MainSkeletonAnalyticInterfaceLedger", "SALD.cycle54MainSkeletonAnalyticInterfaceObligation", "SALD.cycle54MainSkeletonAnalyticMiddleContract", "SALD.cycle54MainSkeletonAnalyticMiddleObligation", "SALD.cycle54MainSkeletonAnalyticInterfaceDag", "ASTIS.SALD.cycle54.analytic_interface_recheck", "ASTIS.SALD.cycle54.middle_interface_audit", "ASTIS.SALD.cycle54.lower_packet.general_discrete_em_fp", "sald.main_skeleton.cycle54_analytic_interface_ledger", "sald.main_skeleton.cycle54_middle_interface_audit", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "SALD.cycle54GeneralMovingTargetDiscreteEmFpLowerObligation", "ASTIS.SALD.general_moving_target_discrete.cycle54_em_fp_sigma_split", "sald.general_moving_target_discrete.cycle54_em_fp_lower", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] def cycle55ForwardKlDependencyNames : List String := [ "SALD.cycle55ForwardKlSkeletonUpperPacket", "SALD.cycle55ForwardKlSkeletonObligation", "SALD.cycle55ForwardKlSkeletonMiddleContract", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle55ForwardKlDerivativeMassLowerObligation", "SALD.forwardKlMassConservationDropScalar", "SALD.forwardKlMassConservationFirstTermFisherScalar", "SALD.cycle55ForwardKlSkeletonDag", "ASTIS.SALD.forward_KL.cycle55_continuous_skeleton_route", "ASTIS.SALD.forward_KL.cycle55_middle_route_audit", "ASTIS.SALD.forward_KL.cycle55_lower_packet.kl_derivative", "ASTIS.SALD.forward_KL.cycle55_derivative_mass_lower", "sald.forward_kl.cycle55_continuous_skeleton_route", "sald.forward_kl.cycle55_middle_route_audit", "sald.forward_kl.cycle55_derivative_mass_lower", "SALD.cycle54MainSkeletonAnalyticInterfaceLedger", "SALD.cycle54MainSkeletonAnalyticInterfaceObligation", "SALD.cycle54MainSkeletonAnalyticMiddleObligation", "SALD.cycle50ForwardKlSkeletonObligation", "SALD.cycle50ForwardKlSkeletonMiddleObligation", "SALD.cycle50ForwardKlDerivativeLowerObligation", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "sald.forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "sald.forward_kl.gronwall_application", "sald.forward_kl.gronwall_side_conditions", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle56DiscreteForwardKlDependencyNames : List String := [ "SALD.cycle56DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle56DiscreteForwardKlSkeletonObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleContract", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged", "ASTIS.SALD.forward_KL_discrete.cycle56_theorem_interface_route", "ASTIS.SALD.forward_KL_discrete.cycle56_middle_route_audit", "ASTIS.SALD.forward_KL_discrete.cycle56_gronwall_lower", "sald.discrete_forward_kl.cycle56_theorem_interface_route", "sald.discrete_forward_kl.cycle56_middle_route_audit", "sald.discrete_forward_kl.cycle56_gronwall_lower", "SALD.cycle55ForwardKlSkeletonObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlSkeletonObligation", "SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "sald.discrete_forward_kl.kl_derivative", "SALD.discreteForwardKlPostLsiDerivativeBoundScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfKlFiScalar", "SALD.frozenDeltaCrossLipSaldContract", "sald.discrete_forward_kl.frozen_delta_cross_lip", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.discreteForwardKlGronwallInstantiationContract", "sald.discrete_forward_kl.gronwall_accumulation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge" ] def cycle57GuidedGeneralDependencyNames : List String := [ "SALD.cycle57GuidedGeneralSkeletonUpperPacket", "SALD.cycle57GuidedGeneralSkeletonObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleContract", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralDerivativeSplitLowerObligation", "SALD.cycle57GuidedGeneralSkeletonDag", "ASTIS.SALD.guided_general.cycle57_upper_route", "ASTIS.SALD.guided_general.cycle57_middle_route_audit", "ASTIS.SALD.guided_general.cycle57_lower_packet.general_derivative", "ASTIS.SALD.general_moving_target.cycle57_derivative_split_lower", "sald.guided_general.cycle57_upper_route", "sald.guided_general.cycle57_middle_route_audit", "sald.general_moving_target.cycle57_derivative_split_lower", "SALD.cycle56DiscreteForwardKlSkeletonObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle52GuidedGeneralSkeletonObligation", "SALD.cycle52GuidedGeneralSkeletonMiddleObligation", "SALD.cycle52GuidedGeneralDerivativeDvLowerObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "SALD.generalMovingTargetKlDerivativeResidualSplitScalar", "SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar", "SALD.generalMovingTargetPostYoungDerivativeBoundScalar", "SALD.generalMovingTargetLsiDerivativeBoundScalar", "SALD.generalMovingTargetTimeChangedDerivativeBoundScalar", "SALD.generalMovingTargetPreDvDerivativeBoundScalar", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction" ] def cycle58UnifiedDiscreteGeneralDependencyNames : List String := [ "SALD.cycle58UnifiedDiscreteGeneralUpperPacket", "SALD.cycle58UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle58UnifiedDiscreteGeneralMiddleContract", "SALD.cycle58UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle58UnifiedDiscreteGeneralDag", "ASTIS.SALD.unified_discrete_general.cycle58_upper_route", "ASTIS.SALD.unified_discrete_general.cycle58_middle_route_audit", "sald.unified_discrete_general.cycle58_middle_route_audit", "ASTIS.SALD.general_moving_target_discrete.cycle58_lower_packet.gronwall_display", "SALD.cycle57GuidedGeneralSkeletonObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralDerivativeSplitLowerObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle53UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle53UnifiedDiscreteGeneralMiddleObligation", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "sald.general_moving_target.kl_derivative", "sald.general_moving_target.gronwall_side_conditions", "SALD.generalMovingTargetDiscreteGronwallInstantiationContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionObligation", "SALD.cycle20GeneralVaSaldDiscreteGronwallMiddleObligation", "SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar", "SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.constant_schedule_stitching", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_side_conditions" ] def cycle59MainSkeletonDependencyNames : List String := [ "SALD.cycle59MainSkeletonAnalyticInterfaceLedger", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation", "SALD.cycle59MainSkeletonAnalyticMiddleContract", "SALD.cycle59MainSkeletonAnalyticMiddleObligation", "SALD.cycle59MainSkeletonAnalyticInterfaceDag", "ASTIS.SALD.cycle59.analytic_interface_ledger", "ASTIS.SALD.cycle59.theorem_route_rewire", "ASTIS.SALD.cycle59.middle_interface_audit", "ASTIS.SALD.cycle59.lower_packet.general_discrete_gronwall_side_conditions", "sald.main_skeleton.cycle59_analytic_interface_ledger", "sald.main_skeleton.cycle59_middle_interface_audit", "SALD.cycle58UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput", "SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar", "SALD.cycle59GeneralMovingTargetDiscreteGronwallLowerObligation", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "sald.general_moving_target_discrete.gronwall_side_conditions" ] def cycle60ForwardKlDependencyNames : List String := [ "SALD.cycle60ForwardKlSkeletonUpperPacket", "SALD.cycle60ForwardKlSkeletonObligation", "SALD.cycle60ForwardKlSkeletonMiddleContract", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle60ForwardKlDerivativeRawLowerObligation", "SALD.forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar", "SALD.cycle60ForwardKlSkeletonDag", "ASTIS.SALD.forward_KL.cycle60_post_cycle59_route", "ASTIS.SALD.forward_KL.cycle60_middle_route_audit", "ASTIS.SALD.forward_KL.cycle60_five_backend_check", "ASTIS.SALD.forward_KL.cycle60_lower_packet.kl_derivative", "ASTIS.SALD.forward_KL.cycle60_derivative_raw_lower", "sald.forward_kl.cycle60_post_cycle59_route", "sald.forward_kl.cycle60_middle_route_audit", "sald.forward_kl.cycle60_derivative_raw_lower", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation", "SALD.cycle59MainSkeletonAnalyticMiddleObligation", "SALD.cycle55ForwardKlSkeletonObligation", "SALD.cycle55ForwardKlSkeletonMiddleObligation", "SALD.cycle50ForwardKlDerivativeLowerObligation", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "sald.forward_kl.gronwall_side_conditions", "sald.forward_kl.gronwall_application", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle65ForwardKlDependencyNames : List String := [ "SALD.cycle65ForwardKlSkeletonUpperPacket", "SALD.cycle65ForwardKlSkeletonObligation", "SALD.cycle65ForwardKlSkeletonMiddleContract", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.cycle65ForwardKlSkeletonDag", "ASTIS.SALD.forward_KL.cycle65_global_phase_judgment", "ASTIS.SALD.forward_KL.cycle65_five_backend_check", "ASTIS.SALD.forward_KL.cycle65_continuous_route", "ASTIS.SALD.forward_KL.cycle65_middle_route_audit", "ASTIS.SALD.forward_KL.cycle65_lower_packet.kl_derivative", "ASTIS.SALD.forward_KL.cycle65_derivative_pointwise_lower", "sald.forward_kl.cycle65_continuous_route", "sald.forward_kl.cycle65_middle_route_audit", "sald.forward_kl.cycle65_derivative_pointwise_lower", "SALD.cycle64MainSkeletonAnalyticInterfaceObligation", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation", "SALD.cycle60ForwardKlSkeletonObligation", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle60ForwardKlDerivativeRawLowerObligation", "SALD.continuousForwardKlStatementContract", "SALD.forwardKlDerivativeCandidateContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.forwardKlDerivativeObligation", "SALD.forwardKlDensityBoundaryObligation", "SALD.forwardKlScheduleTimeChangeObligation", "SALD.cycle65ForwardKlDerivativePointwiseLowerObligation", "SALD.forwardKlPointwisePreDvDerivativeBoundOfRawKlFiVelocityScaling", "SALD.forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.forwardKlDvFiniteLogMgfWitnessContract", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.forwardKlGronwallInstantiationContract", "SALD.forwardKlGronwallSideConditionContract", "sald.forward_kl.endpoint_schedule_identities", "sald.forward_kl.gronwall_side_conditions", "sald.forward_kl.gronwall_application", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] def cycle61DiscreteForwardKlDependencyNames : List String := [ "SALD.cycle61DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle61DiscreteForwardKlSkeletonObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleContract", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation", "SALD.cycle61DiscreteForwardKlSkeletonDag", "ASTIS.SALD.forward_KL_discrete.cycle61_global_phase_judgment", "ASTIS.SALD.forward_KL_discrete.cycle61_five_backend_check", "ASTIS.SALD.forward_KL_discrete.cycle61_recovered_theorem_route", "ASTIS.SALD.forward_KL_discrete.cycle61_middle_route_audit", "ASTIS.SALD.forward_KL_discrete.cycle61_lower_packet.gronwall_accumulated", "sald.discrete_forward_kl.cycle61_recovered_theorem_route", "sald.discrete_forward_kl.cycle61_middle_route_audit", "sald.discrete_forward_kl.cycle61_accumulated_error_lower", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle60ForwardKlDerivativeRawLowerObligation", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation", "SALD.cycle59MainSkeletonAnalyticMiddleObligation", "SALD.cycle56DiscreteForwardKlSkeletonObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle51DiscreteForwardKlSkeletonObligation", "SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "SALD.discreteForwardKlDerivativeObligation", "sald.discrete_forward_kl.kl_derivative", "SALD.frozenDeltaCrossLipSaldContract", "sald.discrete_forward_kl.frozen_delta_cross_lip", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged", "sald.discrete_forward_kl.gronwall_accumulation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "SALD.discreteForwardKlAccumulatedErrorBridgeObligation", "SALD.discreteForwardKlGronwallCoeffIntervalIntegrable", "SALD.discreteForwardKlGronwallCoeffIntegralSubSub", "SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces", "SALD.discreteForwardKlResidualExponentBoundScalar", "SALD.discreteForwardKlResidualExpBoundScalar", "SALD.discreteForwardKlAlphaComplexityCollectionScalar", "SALD.discreteForwardKlDeltaAccumulationScalar", "SALD.discreteForwardKlAccumulatedErrorCollectionScalar", "SALD.discreteForwardKlResidualIntegralDisplayBoundScalar", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.coefficient_chain_audit", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity", "thm:forward-KL" ] def cycle66DiscreteForwardKlDependencyNames : List String := [ "SALD.cycle66DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle66DiscreteForwardKlSkeletonObligation", "SALD.cycle66DiscreteForwardKlSkeletonMiddleContract", "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle66DiscreteForwardKlAccumulatedDisplayLowerObligation", "SALD.cycle66DiscreteForwardKlSkeletonDag", "ASTIS.SALD.forward_KL_discrete.cycle66_global_phase_judgment", "ASTIS.SALD.forward_KL_discrete.cycle66_five_backend_check", "ASTIS.SALD.forward_KL_discrete.cycle66_discrete_route", "ASTIS.SALD.forward_KL_discrete.cycle66_middle_route_audit", "ASTIS.SALD.forward_KL_discrete.cycle66_lower_packet.accumulated_error", "sald.discrete_forward_kl.cycle66_discrete_route", "sald.discrete_forward_kl.cycle66_middle_route_audit", "sald.discrete_forward_kl.cycle66_accumulated_display_lower", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.cycle65ForwardKlDerivativePointwiseLowerObligation", "SALD.cycle64MainSkeletonAnalyticInterfaceObligation", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.cycle61DiscreteForwardKlSkeletonObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation", "SALD.cycle56DiscreteForwardKlSkeletonObligation", "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.em_interpolation_fp", "SALD.discreteForwardKlDerivativeCandidateContract", "SALD.discreteForwardKlDerivativeObligation", "sald.discrete_forward_kl.kl_derivative", "SALD.frozenDeltaCrossLipSaldContract", "sald.discrete_forward_kl.frozen_delta_cross_lip", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldDvFiniteLogMgfContract", "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.saldGronwallEndpointCalculusContract", "SALD.discreteForwardKlGronwallInstantiationContract", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged", "sald.discrete_forward_kl.gronwall_accumulation", "SALD.discreteForwardKlAccumulatedErrorBridgeContract", "SALD.discreteForwardKlAccumulatedErrorBridgeObligation", "SALD.discreteForwardKlGronwallCoeffIntervalIntegrable", "SALD.discreteForwardKlGronwallCoeffIntegralSubSub", "SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces", "SALD.discreteForwardKlResidualExponentBoundScalar", "SALD.discreteForwardKlResidualExpBoundScalar", "SALD.discreteForwardKlAlphaComplexityCollectionScalar", "SALD.discreteForwardKlDeltaAccumulationScalar", "SALD.discreteForwardKlAccumulatedErrorCollectionScalar", "SALD.discreteForwardKlResidualIntegralDisplayBoundScalar", "SALD.discreteForwardKlMainDisplayBoundScalar", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.coefficient_chain_audit", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity", "thm:forward-KL" ] def cycle62GuidedGeneralDependencyNames : List String := [ "SALD.cycle62GuidedGeneralSkeletonUpperPacket", "SALD.cycle62GuidedGeneralSkeletonObligation", "SALD.cycle62GuidedGeneralSkeletonMiddleContract", "SALD.cycle62GuidedGeneralSkeletonMiddleObligation", "SALD.cycle62GuidedGeneralScaledResidualLowerObligation", "SALD.cycle62GuidedGeneralSkeletonDag", "ASTIS.SALD.guided_general.cycle62_global_phase_judgment", "ASTIS.SALD.guided_general.cycle62_five_backend_check", "ASTIS.SALD.guided_general.cycle62_upper_route", "ASTIS.SALD.guided_general.cycle62_middle_route_audit", "ASTIS.SALD.guided_general.cycle62_lower_packet.route_audit", "ASTIS.SALD.general_moving_target.cycle62_scaled_residual_lower", "sald.guided_general.cycle62_upper_route", "sald.guided_general.cycle62_middle_route_audit", "sald.general_moving_target.cycle62_scaled_residual_lower", "SALD.cycle61DiscreteForwardKlSkeletonObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation", "SALD.cycle57GuidedGeneralSkeletonObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralDerivativeSplitLowerObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "SALD.generalMovingTargetKlDerivativeResidualSplitScalar", "SALD.generalMovingTargetKlDerivativeScaledResidualDisplayScalar", "SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "SALD.unifiedForwardKlSpecializationContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL", "thm:forward-KL-discrete", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] def cycle67GuidedGeneralDependencyNames : List String := [ "SALD.cycle67GuidedGeneralSkeletonUpperPacket", "SALD.cycle67GuidedGeneralSkeletonObligation", "SALD.cycle67GuidedGeneralSkeletonMiddleContract", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation", "SALD.cycle67GuidedGeneralResidualGronwallLowerObligation", "SALD.generalMovingTargetResidualToGronwallBridgeScalar", "SALD.cycle67GuidedGeneralSkeletonDag", "ASTIS.SALD.guided_general.cycle67_global_phase_judgment", "ASTIS.SALD.guided_general.cycle67_five_backend_check", "ASTIS.SALD.guided_general.cycle67_guided_residual_route", "ASTIS.SALD.guided_general.cycle67_general_moving_target_route", "ASTIS.SALD.guided_general.cycle67_middle_route_audit", "ASTIS.SALD.guided_general.cycle67_lower_packet.residual_to_gronwall_bridge", "sald.guided_general.cycle67_upper_route", "sald.guided_general.cycle67_middle_route_audit", "sald.general_moving_target.cycle67_residual_to_gronwall_lower", "sald.general_moving_target.cycle67_residual_to_gronwall_bridge", "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle66DiscreteForwardKlAccumulatedDisplayLowerObligation", "SALD.cycle62GuidedGeneralSkeletonMiddleObligation", "SALD.cycle62GuidedGeneralScaledResidualLowerObligation", "SALD.cycle57GuidedGeneralSkeletonMiddleObligation", "SALD.cycle57GuidedGeneralDerivativeSplitLowerObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.generalMovingTargetDerivativeObligation", "sald.general_moving_target.kl_derivative", "SALD.generalMovingTargetKlDerivativeResidualSplitScalar", "SALD.generalMovingTargetKlDerivativeScaledResidualDisplayScalar", "SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract", "SALD.generalMovingTargetDvPositiveAlphaScalingContract", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "SALD.generalMovingTargetGronwallInstantiationContract", "SALD.generalMovingTargetGronwallSideConditionContract", "sald.general_moving_target.gronwall_application", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.specialization", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "thm:forward-KL", "thm:forward-KL-discrete", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] def cycle68UnifiedDiscreteGeneralDependencyNames : List String := [ "SALD.cycle68UnifiedDiscreteGeneralSkeletonUpperPacket", "SALD.cycle68UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleContract", "SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation", "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation", "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeLowerObligation", "SALD.generalMovingTargetDiscreteGronwallDisplayBridgeScalar", "SALD.cycle68UnifiedDiscreteGeneralDag", "ASTIS.SALD.unified_discrete_general.cycle68_global_phase_judgment", "ASTIS.SALD.unified_discrete_general.cycle68_five_backend_check", "ASTIS.SALD.unified_discrete_general.cycle68_unified_route", "ASTIS.SALD.general_moving_target_discrete.cycle68_theorem_route", "ASTIS.SALD.unified_discrete_general.cycle68_middle_route_audit", "ASTIS.SALD.unified_discrete_general.cycle68_lower_packet.discrete_general_bridge", "sald.unified_discrete_general.cycle68_upper_route", "sald.unified_discrete_general.cycle68_middle_route_audit", "sald.unified_discrete_general.cycle68_discrete_general_bridge", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation", "SALD.cycle67GuidedGeneralResidualGronwallLowerObligation", "SALD.cycle63UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation", "SALD.cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.identity", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDiscreteStatementContract", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "sald.general_moving_target_discrete.kl_derivative", "SALD.saldLsiKlFiDensityTestContract", "probability.lsi_to_kl_fi", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract", "sald.general_moving_target_discrete.dv_m_energy", "SALD.generalMovingTargetDiscreteDerivativeDvTimeChangedScalar", "SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged", "SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput", "SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar", "SALD.generalMovingTargetDiscreteGronwallInstantiationContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.gronwall_side_conditions", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] def cycle69MainSkeletonDependencyNames : List String := [ "SALD.cycle69MainSkeletonAnalyticInterfaceLedger", "SALD.cycle69MainSkeletonAnalyticInterfaceObligation", "SALD.cycle69MainSkeletonAnalyticMiddleContract", "SALD.cycle69MainSkeletonAnalyticMiddleObligation", "SALD.cycle69GeneralMovingTargetDiscreteEmFpSourceSignsLowerObligation", "SALD.cycle69MainSkeletonAnalyticInterfaceDag", "ASTIS.SALD.cycle69.global_phase_judgment", "ASTIS.SALD.cycle69.five_backend_check", "ASTIS.SALD.cycle69.theorem_route_recheck", "ASTIS.SALD.cycle69.middle_interface_audit", "ASTIS.SALD.cycle69.lower_packet.em_interpolation_fp", "sald.main_skeleton.cycle69_analytic_interface_ledger", "sald.main_skeleton.cycle69_middle_interface_audit", "sald.general_moving_target_discrete.cycle69_em_fp_source_signs_lower", "SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation", "SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeLowerObligation", "SALD.cycle67GuidedGeneralSkeletonMiddleObligation", "SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle65ForwardKlSkeletonMiddleObligation", "SALD.cycle64MainSkeletonAnalyticInterfaceObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination", "SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination", "SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] def cycle70EmConditionalLawDependencyNames : List String := [ "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawMiddleObligation", "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawLowerObligation", "SALD.cycle70GeneralMovingTargetDiscreteConditionalLawDag", "ASTIS.SALD.cycle70.global_phase_judgment", "ASTIS.SALD.cycle70.middle_conditional_law_interface", "ASTIS.SALD.cycle70.lower_named_conditional_drift", "sald.general_moving_target_discrete.cycle70_conditional_law_middle", "sald.general_moving_target_discrete.cycle70_named_conditional_drift_lower", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle71EmEndpointConditionalDependencyNames : List String := [ "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap", "SALD.generalMovingTargetDiscreteConditionalKernelCompatibilityOfJointMapMarginal", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityOfJointMap", "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalDag", "ASTIS.SALD.cycle71.global_phase_judgment", "ASTIS.SALD.cycle71.middle_endpoint_conditional_compatibility", "ASTIS.SALD.cycle71.lower_endpoint_conditional_wrapper", "sald.general_moving_target_discrete.cycle71_endpoint_conditional_middle", "sald.general_moving_target_discrete.cycle71_endpoint_conditional_lower", "AutoSamplingTheory.lawMapProdSnd", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle72EmWeakFpDependencyNames : List String := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.cycle72GeneralMovingTargetDiscreteWeakFpMiddleObligation", "SALD.cycle72GeneralMovingTargetDiscreteWeakFpLowerObligation", "SALD.cycle72GeneralMovingTargetDiscreteWeakFpDag", "ASTIS.SALD.cycle72.global_phase_judgment", "ASTIS.SALD.cycle72.middle_weak_fp_source_signs", "ASTIS.SALD.cycle72.lower_weak_fp_source_signs", "sald.general_moving_target_discrete.cycle72_weak_fp_middle", "sald.general_moving_target_discrete.cycle72_weak_fp_source_signs_lower", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle73EmKlDerivativeWeakFpDependencyNames : List String := [ "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpUpperPacket", "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpUpperObligation", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns", "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpMiddleObligation", "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpLowerObligation", "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpDag", "ASTIS.SALD.cycle73.global_phase_judgment", "ASTIS.SALD.cycle73.middle_kl_derivative_weak_fp_handoff", "ASTIS.SALD.cycle73.lower_kl_derivative_weak_fp_substitution", "sald.general_moving_target_discrete.cycle73_kl_derivative_weak_fp_upper", "sald.general_moving_target_discrete.cycle73_kl_derivative_weak_fp_middle", "sald.general_moving_target_discrete.cycle73_kl_derivative_weak_fp_lower", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle74EmConditionalKernelMeasureDependencyNames : List String := [ "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceUpperPacket", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceUpperObligation", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceDag", "ASTIS.SALD.cycle74.global_phase_judgment", "ASTIS.SALD.cycle74.middle_conditional_kernel_source_map", "ASTIS.SALD.cycle74.lower_packet.conditional_kernel_measure_interface", "ASTIS.SALD.cycle74.lower_conditional_kernel_regularity_handoff", "sald.general_moving_target_discrete.cycle74_conditional_kernel_measure_interface", "sald.general_moving_target_discrete.cycle74_measure_interface_upper", "sald.general_moving_target_discrete.cycle74_measure_interface_middle", "sald.general_moving_target_discrete.cycle74_conditional_kernel_lower", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "Mathlib.Probability.Kernel.Condexp", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Disintegration.StandardBorel", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle75EmConditionalLawBackfillDependencyNames : List String := [ "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillUpperPacket", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillUpperObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillDag", "ASTIS.SALD.cycle75.global_phase_judgment", "ASTIS.SALD.cycle75.middle_conditional_law_source_map", "ASTIS.SALD.cycle75.lower_packet.conditional_law_measurability", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation", "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents", "AutoSamplingTheory.lawMapProdSwap", "ProbabilityTheory.compProd_map_condDistrib", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "ProbabilityTheory.condExp_ae_eq_integral_condExpKernel", "Mathlib.Probability.Kernel.Condexp", "Mathlib.Probability.Kernel.CondDistrib", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle76EmEndpointConditionalDependencyNames : List String := [ "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalUpperPacket", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalUpperObligation", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalDag", "ASTIS.SALD.cycle76.global_phase_judgment", "ASTIS.SALD.cycle76.middle_endpoint_conditional_map", "ASTIS.SALD.cycle76.lower_endpoint_to_swapped_conditional", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToSwappedConditionalCompatibility", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap", "SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "AutoSamplingTheory.lawMapProdSwap", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle77EmWeakFpSourceSignsDependencyNames : List String := [ "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorMiddleObligation", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorDag", "ASTIS.SALD.cycle77.global_phase_judgment", "ASTIS.SALD.cycle77.middle_weak_fp_generator_source_signs", "ASTIS.SALD.cycle77.lower_weak_fp_generator_handoff", "sald.general_moving_target_discrete.cycle77_weak_fp_generator_middle", "sald.general_moving_target_discrete.cycle77_weak_fp_generator_lower", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle78EmKlDerivativeGeneratorDependencyNames : List String := [ "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorUpperPacket", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorUpperObligation", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorMiddleObligation", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorDag", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces", "ASTIS.SALD.cycle78.global_phase_judgment", "ASTIS.SALD.cycle78.middle_kl_derivative_generator_source_map", "ASTIS.SALD.cycle78.lower_kl_derivative_generator_handoff", "sald.general_moving_target_discrete.cycle78_kl_derivative_generator_upper", "sald.general_moving_target_discrete.cycle78_kl_derivative_generator_middle", "sald.general_moving_target_discrete.cycle78_kl_derivative_generator_lower", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpLowerObligation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle79EmWeakFpGeneratorMeasureDependencyNames : List String := [ "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureUpperPacket", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureUpperObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureMiddleObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureLowerObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureDag", "ASTIS.SALD.cycle79.global_phase_judgment", "ASTIS.SALD.cycle79.middle_weak_fp_generator_measure_source_map", "ASTIS.SALD.cycle79.lower_packet.weak_fp_generator_measure_interface", "ASTIS.SALD.cycle79.lower_measure_map_integral_handoff", "sald.general_moving_target_discrete.cycle79_weak_fp_generator_measure_interface", "sald.general_moving_target_discrete.cycle79_weak_fp_generator_measure_upper", "sald.general_moving_target_discrete.cycle79_weak_fp_generator_measure_middle", "sald.general_moving_target_discrete.cycle79_weak_fp_generator_measure_lower", "AutoSamplingTheory.lawMapIntegral", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Measure.Map", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle80EmConditionalLawMeasurabilityDependencyNames : List String := [ "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityUpperPacket", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityUpperObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityMiddleObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityDag", "ASTIS.SALD.cycle80.global_phase_judgment", "ASTIS.SALD.cycle80.middle_conditional_law_source_map", "ASTIS.SALD.cycle80.lower_packet.conditional_law_measurability", "ASTIS.SALD.cycle80.lower_endpoint_conditional_drift_regularity", "sald.general_moving_target_discrete.cycle80_conditional_law_upper", "sald.general_moving_target_discrete.cycle80_conditional_law_middle", "sald.general_moving_target_discrete.cycle80_conditional_law_lower", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "Mathlib.Probability.Kernel.Condexp", "Mathlib.Probability.Kernel.CondDistrib", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle81EmEndpointConditionalDependencyNames : List String := [ "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalUpperPacket", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalUpperObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalDag", "ASTIS.SALD.cycle81.global_phase_judgment", "ASTIS.SALD.cycle81.lower_packet.endpoint_conditional_compatibility", "ASTIS.SALD.cycle81.lower_endpoint_measure_map_weak_fp_prereq", "ASTIS.SALD.cycle81.middle_endpoint_conditional_weak_fp_readiness", "sald.general_moving_target_discrete.cycle81_endpoint_conditional_upper", "sald.general_moving_target_discrete.cycle81_endpoint_conditional_middle", "sald.general_moving_target_discrete.cycle81_endpoint_conditional_lower", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff", "SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract", "SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap", "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility", "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "Mathlib.Probability.Kernel.Condexp", "Mathlib.Probability.Kernel.CondDistrib", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle82EmWeakFpSourceSignsDependencyNames : List String := [ "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsUpperPacket", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsUpperObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsDag", "ASTIS.SALD.cycle82.global_phase_judgment", "ASTIS.SALD.cycle82.middle_readiness_to_source_signs", "ASTIS.SALD.cycle82.lower_packet.weak_fp_source_signs", "ASTIS.SALD.cycle82.reviewer_weak_fp_source_signs_check", "sald.general_moving_target_discrete.cycle82_weak_fp_source_signs_upper", "sald.general_moving_target_discrete.cycle82_weak_fp_source_signs_middle", "sald.general_moving_target_discrete.cycle82_weak_fp_source_signs_lower", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff", "SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfReadinessAndGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "SALD.cycle72GeneralMovingTargetDiscreteWeakFpLowerObligation", "SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle83EmKlDerivativeEndpointWeakFpDependencyNames : List String := [ "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpUpperPacket", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpUpperObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpMiddleObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpDag", "ASTIS.SALD.cycle83.global_phase_judgment", "ASTIS.SALD.cycle83.middle_endpoint_source_signs_to_kl", "ASTIS.SALD.cycle83.lower_packet.kl_derivative_endpoint_handoff", "ASTIS.SALD.cycle83.reviewer_kl_derivative_handoff_check", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "sald.general_moving_target_discrete.cycle83_kl_derivative_endpoint_weak_fp_upper", "sald.general_moving_target_discrete.cycle83_kl_derivative_endpoint_weak_fp_middle", "sald.general_moving_target_discrete.cycle83_kl_derivative_endpoint_weak_fp_lower", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle84ActiveEmBackendDependencyNames : List String := [ "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendUpperPacket", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendUpperObligation", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendMiddleObligation", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendDag", "ASTIS.SALD.cycle84.global_phase_judgment", "ASTIS.SALD.cycle84.middle_active_em_backend_source_map", "ASTIS.SALD.cycle84.lower_packet.active_em_backend", "ASTIS.SALD.cycle84.blocked_measure_interface_escape_hatch", "ASTIS.SALD.cycle84.reviewer_active_em_backend_check", "SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoff", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "sald.general_moving_target_discrete.cycle84_active_em_backend_upper", "sald.general_moving_target_discrete.cycle84_active_em_backend_middle", "sald.general_moving_target_discrete.cycle84_active_em_backend_lower", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle85EmConditionalKernelBoundaryDependencyNames : List String := [ "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryUpperPacket", "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryUpperObligation", "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryMiddleObligation", "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryLowerObligation", "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryDag", "ASTIS.SALD.cycle85.global_phase_judgment", "ASTIS.SALD.cycle85.active_em_backend_check", "ASTIS.SALD.cycle85.lower_packet.conditional_kernel_theorem_boundary", "ASTIS.SALD.cycle85.middle_condDistrib_condExpKernel_boundary", "ASTIS.SALD.cycle85.lower_conditional_integral_named_field_regularity", "ASTIS.SALD.cycle85.missing_mathlib_theorem_record", "ASTIS.SALD.cycle85.reviewer_conditional_kernel_boundary_check", "sald.general_moving_target_discrete.cycle85_conditional_kernel_boundary_upper", "sald.general_moving_target_discrete.cycle85_conditional_kernel_boundary_middle", "sald.general_moving_target_discrete.cycle85_conditional_kernel_boundary_lower", "AutoSamplingTheory.condDistribAeEqCondExpKernelMap", "AutoSamplingTheory.condDistribIntegralAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralIntegrable", "AutoSamplingTheory.condDistribIntegralMapAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralMapIntegrable", "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "proof:thm:general-moving-target-SALD-discrete:conditional-drift", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityMiddleObligation", "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation", "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation", "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface", "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents", "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "Mathlib.Probability.Kernel.Condexp", "Mathlib.Probability.Kernel.CondDistrib", "ProbabilityTheory.condExpKernel", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle86EmWeakFpGeneratorBoundaryDependencyNames : List String := [ "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryUpperPacket", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryUpperObligation", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryMiddleObligation", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryDag", "ASTIS.SALD.cycle86.global_phase_judgment", "ASTIS.SALD.cycle86.active_em_backend_check", "ASTIS.SALD.cycle86.middle_weak_fp_generator_to_law_source_map", "ASTIS.SALD.cycle86.lower_packet.weak_fp_generator_to_law_boundary", "ASTIS.SALD.cycle86.reviewer_weak_fp_generator_boundary_check", "AutoSamplingTheory.lawMapIntegral", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface", "SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfReadinessAndGeneratorPiecesHandoff", "SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation", "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryLowerObligation", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle87EmKlLogRatioBoundaryDependencyNames : List String := [ "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryUpperPacket", "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryUpperObligation", "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryMiddleObligation", "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryLowerObligation", "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryDag", "ASTIS.SALD.cycle87.global_phase_judgment", "ASTIS.SALD.cycle87.active_em_backend_check", "ASTIS.SALD.cycle87.lower_packet.kl_log_ratio_boundary", "ASTIS.SALD.cycle87.middle_kl_log_ratio_source_map", "ASTIS.SALD.cycle87.lower_kl_mass_conservation_handoff", "ASTIS.SALD.cycle87.reviewer_kl_log_ratio_boundary_check", "sald.general_moving_target_discrete.cycle87_kl_log_ratio_boundary_upper", "sald.general_moving_target_discrete.cycle87_kl_log_ratio_boundary_middle", "sald.general_moving_target_discrete.cycle87_kl_log_ratio_boundary_lower", "eq:general_KL_derivative_0_discrete", "proof:thm:general-moving-target-SALD-discrete:weak-fp-to-kl-derivative", "SALD.generalMovingTargetDiscreteKlDerivativeMassConservationDropScalar", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlLogRatioLlrDef", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction", "SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation", "SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation", "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "Mathlib.Analysis.Calculus.ParametricIntegral", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] def cycle88EmKlLogRatioAdmissibilityDependencyNames : List String := [ "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityMiddlePacket", "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityMiddleObligation", "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation", "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityDag", "ASTIS.SALD.cycle88.global_phase_judgment", "ASTIS.SALD.cycle88.middle_log_ratio_admissibility_boundary", "ASTIS.SALD.cycle88.lower_packet.log_ratio_admissibility", "ASTIS.SALD.cycle88.reviewer_log_ratio_admissibility_check", "sald.general_moving_target_discrete.cycle88_kl_log_ratio_admissibility_middle", "sald.general_moving_target_discrete.cycle88_kl_log_ratio_admissibility_lower", "SALD.GeneralMovingTargetDiscreteKlLogRatioAdmissibilityClosure", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.generalMovingTargetDiscreteKlLogRatioLlrDef", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction", "SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryLowerObligation", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "Mathlib.Analysis.Calculus.ParametricIntegral", "eq:general_KL_derivative_0_discrete", "proof:thm:general-moving-target-SALD-discrete:weak-fp-to-kl-derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative" ] def cycle89DiscreteForwardKlPressureDependencyNames : List String := [ "SALD.cycle89DiscreteForwardKlClosurePressureUpperPacket", "SALD.cycle89DiscreteForwardKlClosurePressureUpperObligation", "SALD.cycle89DiscreteForwardKlClosurePressureMiddleObligation", "SALD.cycle89DiscreteForwardKlClosurePressureLowerObligation", "SALD.cycle89DiscreteForwardKlClosurePressureDag", "ASTIS.SALD.cycle89.global_phase_judgment", "ASTIS.SALD.cycle89.active_em_backend_check", "ASTIS.SALD.forward_KL_discrete.cycle89_pressure_route", "ASTIS.SALD.forward_KL_discrete.cycle89_middle_route_audit", "ASTIS.SALD.forward_KL_discrete.cycle89_next_blocker", "ASTIS.SALD.forward_KL_discrete.cycle89_lower_derivative_ibp_split", "ASTIS.SALD.cycle89.reviewer_pressure_check", "sald.discrete_forward_kl.cycle89_closure_pressure_middle", "sald.discrete_forward_kl.cycle89_derivative_ibp_lower", "SALD.discreteForwardKlDerivativeSplitOfRawIbpsScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar", "SALD.discreteForwardKlStatementContract", "SALD.discreteForwardKlDerivativeObligation", "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation", "SALD.cycle56DiscreteForwardKlGronwallLowerObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation", "SALD.cycle66DiscreteForwardKlAccumulatedDisplayLowerObligation", "SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.kl_derivative", "sald.discrete_forward_kl.dv_velocity_bound", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "appendix.tex:388-413", "appendix.tex:1358-1387", "eq:KL-derivative-1-discrete", "eq:KL-derivative-2-discrete", "eq:general_KL_derivative_0_discrete" ] def cycle90DiscreteForwardKlMassConservationDependencyNames : List String := [ "SALD.cycle90DiscreteForwardKlMassConservationUpperPacket", "SALD.cycle90DiscreteForwardKlMassConservationUpperObligation", "SALD.cycle90DiscreteForwardKlMassConservationMiddleObligation", "SALD.cycle90DiscreteForwardKlMassConservationLowerObligation", "SALD.cycle90DiscreteForwardKlMassConservationDag", "ASTIS.SALD.cycle90.global_phase_judgment", "ASTIS.SALD.cycle90.active_em_backend_exception_check", "ASTIS.SALD.forward_KL_discrete.cycle90_middle_mass_derivative_route", "ASTIS.SALD.forward_KL_discrete.cycle90_lower_law_constant_test_mass", "ASTIS.SALD.forward_KL_discrete.cycle90_mass_conservation_lower_packet", "ASTIS.SALD.cycle90.reviewer_mass_conservation_check", "sald.discrete_forward_kl.cycle90_mass_conservation_upper", "sald.discrete_forward_kl.cycle90_mass_conservation_middle", "sald.discrete_forward_kl.cycle90_mass_conservation_lower", "SALD.discreteForwardKlMassTermZeroOfTotalMassDerivative", "SALD.discreteForwardKlDerivativeSplitOfMassDerivativeScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfMassDerivativeScalar", "SALD.discreteForwardKlLawConstantTestTotalMassOne", "SALD.discreteForwardKlLawConstantTestHasDerivAtZero", "SALD.discreteForwardKlMassTermZeroOfLawConstantTestDerivative", "SALD.discreteForwardKlDerivativeSplitOfLawConstantTestMassScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfLawConstantTestMassScalar", "AutoSamplingTheory.lawMapIntegral", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.cycle89DiscreteForwardKlClosurePressureLowerObligation", "SALD.discreteForwardKlDerivativeSplitOfRawIbpsScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar", "SALD.discreteForwardKlDerivativeObligation", "sald.discrete_forward_kl.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "eq:KL-derivative-0-discrete", "eq:KL-derivative-1-discrete", "eq:KL-derivative-2-discrete", "eq:general_KL_derivative_0_discrete", "appendix.tex:338-388", "appendix.tex:1358-1387", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "Mathlib.Analysis.Calculus.ParametricIntegral", "mass conservation hmass", "derivative under integral constant weak test" ] def cycle91EmConditionalKernelDependencyNames : List String := [ "SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelMiddleObligation", "SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation", "SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelDag", "ASTIS.SALD.cycle91.global_phase_judgment", "ASTIS.SALD.cycle91.middle_conditional_kernel_source_map", "ASTIS.SALD.cycle91.lower_condDistrib_named_drift_regular", "ASTIS.SALD.cycle91.remaining_conditional_kernel_boundary", "sald.general_moving_target_discrete.cycle91_conditional_kernel_middle", "sald.general_moving_target_discrete.cycle91_conditional_kernel_lower", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularity", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable", "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents", "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "ProbabilityTheory.condDistrib", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle92EmWeakFpGeneratorSplitDependencyNames : List String := [ "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitMiddleObligation", "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation", "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitDag", "ASTIS.SALD.cycle92.middle_generator_to_law_source_map", "ASTIS.SALD.cycle92.lower_sample_split_generator_handoff", "ASTIS.SALD.cycle92.lower_law_derivative_of_sample_split_generator", "ASTIS.SALD.cycle92.remaining_generator_to_law_boundary", "sald.general_moving_target_discrete.cycle92_weak_fp_split_generator_middle", "sald.general_moving_target_discrete.cycle92_weak_fp_split_generator_lower", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff", "AutoSamplingTheory.lawMapIntegral", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle93EmKlMassDerivativeDependencyNames : List String := [ "SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeMiddleObligation", "SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeLowerObligation", "SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeDag", "ASTIS.SALD.cycle93.middle_kl_mass_derivative_source_map", "ASTIS.SALD.cycle93.lower_law_constant_test_mass", "ASTIS.SALD.cycle93.lower_finite_kl_llr_law_mass_handoff", "ASTIS.SALD.cycle93.remaining_raw_kl_target_time_boundary", "sald.general_moving_target_discrete.cycle93_kl_mass_derivative_middle", "sald.general_moving_target_discrete.cycle93_kl_mass_derivative_lower", "SALD.generalMovingTargetDiscreteKlMassTermZeroOfLawConstantTestDerivative", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfLawConstantTestMassAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfFiniteKlLlrLawConstantTestMassWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.Analysis.Calculus.ParametricIntegral", "eq:general_KL_derivative_0_discrete", "appendix.tex:1358-1366", "appendix.tex:1358-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle94EmWeakFpDriftActionDependencyNames : List String := [ "SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionMiddleObligation", "SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionLowerObligation", "SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionDag", "ASTIS.SALD.cycle94.middle_weak_fp_drift_action_source_map", "ASTIS.SALD.cycle94.lower_barB_drift_action_handoff", "ASTIS.SALD.cycle94.remaining_barB_divergence_boundary", "sald.general_moving_target_discrete.cycle94_weak_fp_drift_action_middle", "sald.general_moving_target_discrete.cycle94_weak_fp_drift_action_lower", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation", "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle95DiscreteForwardKlPressureDependencyNames : List String := [ "SALD.cycle95DiscreteForwardKlClosurePressureUpperPacket", "SALD.cycle95DiscreteForwardKlClosurePressureUpperObligation", "SALD.cycle95DiscreteForwardKlClosurePressureMiddleObligation", "SALD.cycle95DiscreteForwardKlClosurePressureLowerObligation", "SALD.cycle95DiscreteForwardKlClosurePressureDag", "ASTIS.SALD.cycle95.global_phase_judgment", "ASTIS.SALD.forward_KL_discrete.cycle95_pressure_route", "ASTIS.SALD.forward_KL_discrete.cycle95_middle_route_audit", "ASTIS.SALD.forward_KL_discrete.cycle95_lower_barB_component_pairing", "ASTIS.SALD.forward_KL_discrete.cycle95_next_blocker", "ASTIS.SALD.cycle95.reviewer_pressure_check", "sald.discrete_forward_kl.cycle95_closure_pressure_upper", "sald.discrete_forward_kl.cycle95_closure_pressure_middle", "sald.discrete_forward_kl.cycle95_closure_pressure_lower", "SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfFiniteKlLlrLawConstantTestMassWithLogAction", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.discreteForwardKlPostLsiDerivativeBoundOfLawConstantTestMassScalar", "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar", "SALD.discreteForwardKlMainDisplayBoundScalar", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "appendix.tex:1358-1387", "main_body.tex:299-323", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.kl_derivative" ] def cycle96EmCondexpGeneratorPairingDependencyNames : List String := [ "SALD.cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingMiddleObligation", "SALD.cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingLowerObligation", "SALD.cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingDag", "ASTIS.SALD.cycle96.middle_non_em_fallback_rejected", "ASTIS.SALD.cycle96.middle_condexp_generator_pairing_boundary", "ASTIS.SALD.cycle96.lower_condDistrib_component_pairing_handoff", "ASTIS.SALD.cycle96.lower_packet.condexp_component_generator_pairing", "SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion", "SALD.cycle95DiscreteForwardKlClosurePressureLowerObligation", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "ProbabilityTheory.condExp_ae_eq_integral_condExpKernel", "lean-stat-learning-theory/SLT/EfronStein.lean", "lean-stat-learning-theory/SLT/GaussianMeasure.lean", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "appendix.tex:1358-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle97EmCanonicalCondDistribPairingDependencyNames : List String := [ "SALD.cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingMiddleObligation", "SALD.cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingLowerObligation", "SALD.cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingDag", "ASTIS.SALD.cycle97.global_phase_judgment", "ASTIS.SALD.cycle97.compiled_condDistrib_disintegration_pairing", "ASTIS.SALD.cycle97.lower_packet.canonical_condDistrib_component_pairing", "ASTIS.SALD.cycle97.rejected_wrapper_churn_guard", "AutoSamplingTheory.condDistribIntegralMapIntegral", "AutoSamplingTheory.condDistribIntegralNamedLawIntegral", "SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfIntegralAction", "SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBComponentPairings", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "ProbabilityTheory.compProd_map_condDistrib", "MeasureTheory.Measure.integral_compProd", "MeasureTheory.integral_map", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "appendix.tex:1358-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle98EmBarBDivergenceNoBoundaryDependencyNames : List String := [ "SALD.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryMiddleObligation", "SALD.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryLowerObligation", "SALD.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryDag", "ASTIS.SALD.cycle98.middle_barB_divergence_no_boundary_source_map", "ASTIS.SALD.cycle98.lower_packet.barB_no_boundary_integral", "ASTIS.SALD.cycle98.remaining_exact_no_boundary_theorem", "SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "appendix.tex:1358-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle99EmRawKlDerivativeDependencyNames : List String := [ "SALD.cycle99GeneralMovingTargetDiscreteRawKlDerivativeMiddleObligation", "SALD.cycle99GeneralMovingTargetDiscreteRawKlDerivativeLowerObligation", "SALD.cycle99GeneralMovingTargetDiscreteRawKlDerivativeDag", "ASTIS.SALD.cycle99.middle_raw_kl_derivative_source_map", "ASTIS.SALD.cycle99.raw_kl_derivative_at_finite_kl_llr", "ASTIS.SALD.cycle99.lower_packet.raw_kl_llr_handoff", "SALD.GeneralMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlr", "SALD.generalMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlrHklRaw", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlBoundaryAtFiniteKlLlrWithLogAction", "SALD.GeneralMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlr", "SALD.generalMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlrHkl", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfNoMassRawKlBoundaryAtFiniteKlLlrWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfFiniteKlLlrLawConstantTestMassWithLogAction", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "eq:general_KL_derivative_0_discrete", "appendix.tex:1358-1366", "appendix.tex:1358-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle100EmBarBWeakGradDefDependencyNames : List String := [ "SALD.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefMiddleObligation", "SALD.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefLowerObligation", "SALD.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefDag", "SALD.cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundLowerObligation", "SALD.cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundDag", "ASTIS.SALD.cycle100.middle_barB_weakGrad_def_source_map", "ASTIS.SALD.cycle100.lower_packet.barB_weakGrad_definition_alignment", "ASTIS.SALD.cycle100.remaining_no_boundary_after_weakGrad_def", "ASTIS.SALD.cycle100.lower_packet.barB_inner_gradient_contraction", "ASTIS.SALD.cycle100.remaining_no_boundary_after_inner_gradient", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryWeakGradDef", "SALD.generalMovingTargetDiscreteBarBPairNormBoundOfInnerGradientBound", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryInnerGradientBound", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral", "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing", "SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle101DiscreteForwardKlPressureDependencyNames : List String := [ "SALD.cycle101DiscreteForwardKlClosurePressureMiddleObligation", "SALD.cycle101DiscreteForwardKlNoBoundaryProductRuleLowerObligation", "SALD.cycle101DiscreteForwardKlClosurePressureDag", "ASTIS.SALD.forward_KL_discrete.cycle101_middle_pressure_sync", "ASTIS.SALD.forward_KL_discrete.cycle101_lower_product_rule_handoff", "ASTIS.SALD.forward_KL_discrete.cycle101_next_non_wrapper_blocker", "SALD.generalMovingTargetDiscreteDriftDivNoBoundaryOfProductRule", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientProductRuleBoundary", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound", "ASTIS.SALD.cycle100.remaining_no_boundary_after_inner_gradient", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "main_body.tex:301-323", "appendix.tex:260-592", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle102DiscreteForwardKlZeroFluxTraceBoundaryDependencyNames : List String := [ "SALD.cycle102DiscreteForwardKlZeroFluxTraceBoundaryMiddleObligation", "SALD.cycle102DiscreteForwardKlTraceZeroLowerObligation", "SALD.cycle102DiscreteForwardKlZeroFluxTraceBoundaryDag", "ASTIS.SALD.forward_KL_discrete.cycle102_zero_flux_trace_boundary", "ASTIS.SALD.forward_KL_discrete.cycle102_test_trace_zero_lower", "ASTIS.SALD.forward_KL_discrete.cycle102_next_trace_boundary_blocker", "SALD.generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero", "SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundary", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientProductRuleBoundary", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound", "MeasureTheory.integral_congr_ae", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle103EmConditionalKernelComponentVersionDependencyNames : List String := [ "SALD.cycle103GeneralMovingTargetDiscreteConditionalKernelVersionMiddleObligation", "SALD.cycle103GeneralMovingTargetDiscreteConditionalKernelVersionLowerObligation", "SALD.cycle103GeneralMovingTargetDiscreteConditionalKernelVersionDag", "ASTIS.SALD.cycle103.middle_conditional_kernel_component_version", "ASTIS.SALD.cycle103.lower_condExpKernel_map_version_bridge", "ASTIS.SALD.cycle103.condC_condDistrib_condExpKernel_sample_version", "ASTIS.SALD.cycle103.rejected_wrapper_churn_guard", "ASTIS.SALD.cycle91.remaining_conditional_kernel_boundary", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfCondExpKernelMap", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularity", "AutoSamplingTheory.condDistribAeEqCondExpKernelMap", "AutoSamplingTheory.condDistribIntegralSampleAeEqOfCondExpKernelMap", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "ProbabilityTheory.condDistrib_apply_ae_eq_condExpKernel_map", "ProbabilityTheory.condExp_ae_eq_integral_condExpKernel", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle104EmWeakFpNamedLawTransportDependencyNames : List String := [ "SALD.cycle104GeneralMovingTargetDiscreteWeakFpNamedLawTransportUpperObligation", "SALD.cycle104GeneralMovingTargetDiscreteWeakFpNamedLawTransportLowerObligation", "SALD.cycle104GeneralMovingTargetDiscreteWeakFpNamedLawTransportDag", "ASTIS.SALD.cycle104.global_phase_judgment", "ASTIS.SALD.cycle104.lower_packet.named_law_generator_to_law_transport", "ASTIS.SALD.cycle104.remaining_generator_to_law_after_named_transport", "ASTIS.SALD.cycle104.reviewer_named_transport_check", "AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndSample", "AutoSamplingTheory.lawMapIntegral", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfSampleSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff", "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation", "ASTIS.SALD.cycle92.remaining_generator_to_law_boundary", "ASTIS.SALD.cycle100.remaining_no_boundary_after_inner_gradient", "ASTIS.SALD.cycle103.condC_condDistrib_condExpKernel_sample_version", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle105EmPureRawKlDerivativeDependencyNames : List String := [ "SALD.cycle105GeneralMovingTargetDiscretePureRawKlDerivativeMiddleObligation", "SALD.cycle105GeneralMovingTargetDiscretePureRawKlDerivativeLowerObligation", "SALD.cycle105GeneralMovingTargetDiscretePureRawKlDerivativeDag", "ASTIS.SALD.cycle105.middle_pure_raw_kl_derivative_source_map", "ASTIS.SALD.cycle105.lower_packet.pure_raw_kl_llr_handoff", "ASTIS.SALD.cycle105.remaining_pure_raw_kl_boundary", "SALD.GeneralMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlr", "SALD.generalMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlrHkl", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfPureNoMassRawKlBoundaryAtFiniteKlLlrWithLogAction", "SALD.GeneralMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlr", "SALD.generalMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlrHkl", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfNoMassRawKlBoundaryAtFiniteKlLlrWithLogAction", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "eq:general_KL_derivative_0_discrete", "appendix.tex:1358-1366", "appendix.tex:1358-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle106EmCanonicalCondDistribDriftDependencyNames : List String := [ "SALD.cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftMiddleObligation", "SALD.cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftLowerObligation", "SALD.cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftDag", "ASTIS.SALD.cycle106.middle_canonical_condDistrib_drift_source_map", "ASTIS.SALD.cycle106.lower_packet.canonical_condDistrib_drift_regularity", "ASTIS.SALD.cycle106.lower_packet.named_barB_canonical_ae_regular", "ASTIS.SALD.cycle106.remaining_named_barB_version_boundary", "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularity", "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample", "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable", "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "lean-stat-learning-theory/SLT/EfronStein.lean", "appendix.tex:1368-1377", "appendix.tex:1358-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle107DiscreteForwardKlBoundaryFluxDependencyNames : List String := [ "SALD.cycle107DiscreteForwardKlBoundaryFluxIntegralLowerObligation", "SALD.cycle107DiscreteForwardKlBoundaryFluxIntegralDag", "ASTIS.SALD.forward_KL_discrete.cycle107.lower_packet.boundary_flux_integral_box", "ASTIS.SALD.forward_KL_discrete.cycle107.remaining_box_trace_instantiation_boundary", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero", "SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero", "SALD.generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle108DiscreteForwardKlHatRhoBarBFluxContinuityDependencyNames : List String := [ "SALD.cycle108DiscreteForwardKlHatRhoBarBFluxContinuityLowerObligation", "SALD.cycle108DiscreteForwardKlHatRhoBarBFluxContinuityDag", "ASTIS.SALD.forward_KL_discrete.cycle108.lower_packet.hatRho_barB_flux_continuity", "ASTIS.SALD.forward_KL_discrete.cycle108.remaining_hatRho_barB_box_trace_boundary", "SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldContinuousOnBox", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle108DiscreteForwardKlHatRhoBarBFluxDerivativeDependencyNames : List String := [ "SALD.cycle108DiscreteForwardKlHatRhoBarBFluxDerivativeLowerObligation", "SALD.cycle108DiscreteForwardKlHatRhoBarBFluxDerivativeDag", "ASTIS.SALD.forward_KL_discrete.cycle108.lower_packet.hatRho_barB_flux_product_derivative", "ASTIS.SALD.forward_KL_discrete.cycle108.remaining_hatRho_barB_box_trace_after_product_derivative", "SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAt", "SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAtOffUnion", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBoxProductDeriv", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero", "Mathlib.Analysis.Calculus.FDeriv.Mul", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle109EmNamedBarBSourceDefDependencyNames : List String := [ "SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefBoundary", "SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefMiddleObligation", "SALD.cycle109GeneralMovingTargetDiscreteNamedBarBCondExpSourceLowerObligation", "SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefDag", "ASTIS.SALD.cycle109.middle_named_barB_source_def_boundary", "ASTIS.SALD.cycle109.compiled_named_barB_source_def_bridge", "ASTIS.SALD.cycle109.lower_named_barB_condExp_source_bridge", "ASTIS.SALD.cycle109.remaining_condExp_source_representative_for_named_barB", "ASTIS.SALD.cycle109.remaining_condExpKernel_source_def_and_kernel_alignment", "ASTIS.SALD.cycle109.reviewer_named_barB_source_def_check", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpKernelSourceDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity", "AutoSamplingTheory.condDistribAeEqCondExpKernelMap", "AutoSamplingTheory.condDistribIntegralSampleAeEqOfCondExpKernelMap", "ProbabilityTheory.condDistrib_apply_ae_eq_condExpKernel_map", "ProbabilityTheory.condExp_ae_eq_integral_condExpKernel", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "lean-stat-learning-theory/SLT/EfronStein.lean", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle110EmNamedBarBEqMeasDependencyNames : List String := [ "SALD.cycle110GeneralMovingTargetDiscreteNamedBarBEqMeasMiddleObligation", "SALD.cycle110GeneralMovingTargetDiscreteNamedBarBEqMeasLowerObligation", "SALD.cycle110GeneralMovingTargetDiscreteNamedBarBEqMeasDag", "ASTIS.SALD.cycle110.middle_named_barB_eq_meas_boundary", "ASTIS.SALD.cycle110.lower_named_barB_eq_meas", "ASTIS.SALD.cycle110.remaining_condExp_source_representative_after_eq_meas", "ASTIS.SALD.cycle110.reviewer_named_barB_eq_meas_check", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpKernelSourceDef", "SALD.cycle109GeneralMovingTargetDiscreteNamedBarBCondExpSourceLowerObligation", "ASTIS.SALD.cycle109.remaining_condExp_source_representative_for_named_barB", "MeasureTheory.StronglyMeasurable.measurableSet_eq_fun", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle110EmWeakFpDominatedGeneratorDependencyNames : List String := [ "SALD.cycle110GeneralMovingTargetDiscreteWeakFpDominatedGeneratorLowerObligation", "SALD.cycle110GeneralMovingTargetDiscreteWeakFpDominatedGeneratorDag", "ASTIS.SALD.cycle110.lower_packet.dominated_generator_to_law_transport", "ASTIS.SALD.cycle110.remaining_parametric_generator_boundary_after_dominated_transport", "ASTIS.SALD.cycle110.reviewer_dominated_generator_to_law_check", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfSampleSplitGeneratorHandoff", "AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndDominated", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfDominated", "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle111EmKlTargetTimeDerivativeDependencyNames : List String := [ "SALD.cycle111GeneralMovingTargetDiscreteTargetTimeDerivativeMiddleObligation", "SALD.cycle111GeneralMovingTargetDiscreteTargetTimeDerivativeLowerObligation", "SALD.cycle111GeneralMovingTargetDiscreteTargetTimeDerivativeDag", "ASTIS.SALD.cycle111.middle_target_time_derivative_boundary", "ASTIS.SALD.cycle111.lower_packet.target_time_dominated_derivative", "ASTIS.SALD.cycle111.lower_packet.target_time_source_ratio_congr", "ASTIS.SALD.cycle111.lower_packet.target_time_fields_for_pure_raw_kl", "ASTIS.SALD.cycle111.remaining_pure_raw_kl_after_target_time", "SALD.generalMovingTargetDiscreteTargetTimeDerivativeOfDominated", "SALD.generalMovingTargetDiscreteTargetTimeDerivativeSourceRatioCongr", "SALD.generalMovingTargetDiscretePureRawKlTargetTimeFieldsOfDominated", "SALD.GeneralMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlr", "SALD.generalMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlrHkl", "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfPureNoMassRawKlBoundaryAtFiniteKlLlrWithLogAction", "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "MeasureTheory.integral_congr_ae", "MeasureTheory.Integrable.congr", "Mathlib.InformationTheory.KullbackLeibler.Basic", "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "eq:general_KL_derivative_0_discrete", "appendix.tex:1358-1366", "appendix.tex:1358-1387", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle112EmNamedBarBCondExpRepresentativeDependencyNames : List String := [ "SALD.cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeMiddleObligation", "SALD.cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeLowerObligation", "SALD.cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeDag", "ASTIS.SALD.cycle112.middle_named_barB_condExp_representative", "ASTIS.SALD.cycle112.lower_packet.named_barB_condExp_uniqueness", "ASTIS.SALD.cycle112.lower_packet.named_barB_source_bridge_without_hbarBCondExp", "ASTIS.SALD.cycle112.lower_packet.named_barB_state_event_set_integral", "ASTIS.SALD.cycle112.remaining_named_barB_set_integral_characterization", "SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef", "SALD.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable", "SALD.cycle109GeneralMovingTargetDiscreteNamedBarBCondExpSourceLowerObligation", "SALD.cycle110GeneralMovingTargetDiscreteNamedBarBEqMeasLowerObligation", "ASTIS.SALD.cycle110.remaining_condExp_source_representative_after_eq_meas", "MeasureTheory.ae_eq_condExp_of_forall_setIntegral_eq", "MeasurableSpace.measurableSet_comap", "MeasureTheory.Integrable.comp_aemeasurable", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.Probability.Kernel.Condexp", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "lean-stat-learning-theory/SLT/EfronStein.lean", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp" ] def cycle113EmNamedBarBStateFieldRegularityDependencyNames : List String := [ "SALD.cycle113GeneralMovingTargetDiscreteNamedBarBStateFieldRegularityLowerObligation", "SALD.cycle113GeneralMovingTargetDiscreteNamedBarBStateFieldRegularityDag", "ASTIS.SALD.cycle113.lower_packet.named_barB_state_field_regularity", "ASTIS.SALD.cycle113.lower_packet.named_barB_state_field_set_integral_bridge", "ASTIS.SALD.cycle113.remaining_named_barB_state_event_set_integral", "SALD.generalMovingTargetDiscreteNamedBarBComapRegularityOfStateField", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateFieldSetIntegralDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef", "SALD.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents", "MeasureTheory.Integrable.comp_measurable", "Measurable.of_comap_le", "ASTIS.SALD.cycle112.remaining_named_barB_set_integral_characterization", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] def cycle114EmCanonicalBarBStateEventSetIntegralDependencyNames : List String := [ "SALD.cycle114GeneralMovingTargetDiscreteCanonicalBarBStateEventSetIntegralLowerObligation", "SALD.cycle114GeneralMovingTargetDiscreteCanonicalBarBStateEventSetIntegralDag", "ASTIS.SALD.cycle114.lower_packet.canonical_barB_state_event_set_integral", "ASTIS.SALD.cycle114.remaining_named_barB_version_after_canonical_state_event_integral", "SALD.generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib", "SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib", "ASTIS.SALD.cycle113.remaining_named_barB_state_event_set_integral", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateFieldSetIntegralDef", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib'", "MeasureTheory.setIntegral_condExp", "MeasureTheory.setIntegral_congr_ae", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] def cycle115EmNamedBarBSelectedVersionDependencyNames : List String := [ "SALD.cycle115GeneralMovingTargetDiscreteNamedBarBSelectedVersionMiddleObligation", "SALD.cycle115GeneralMovingTargetDiscreteNamedBarBCondExpSourceLowerObligation", "SALD.cycle115GeneralMovingTargetDiscreteNamedBarBSelectedVersionDag", "ASTIS.SALD.cycle115.middle_packet.named_barB_selected_version_to_state_events", "ASTIS.SALD.cycle115.lower_packet.named_barB_condExp_source_to_state_events", "ASTIS.SALD.cycle115.remaining_named_barB_condExp_source_representative", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCondExpSourceDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef", "SALD.generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable", "ASTIS.SALD.cycle115.remaining_named_barB_selected_canonical_ae_eq", "ASTIS.SALD.cycle114.remaining_named_barB_version_after_canonical_state_event_integral", "MeasureTheory.ae_eq_comp", "MeasureTheory.setIntegral_congr_ae", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib'", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] def cycle116EmCanonicalBarBCondExpDependencyNames : List String := [ "SALD.cycle116GeneralMovingTargetDiscreteCanonicalBarBCondExpLowerObligation", "SALD.cycle116GeneralMovingTargetDiscreteCanonicalBarBCondExpDag", "ASTIS.SALD.cycle116.lower_packet.canonical_barB_condExp", "ASTIS.SALD.cycle116.lower_packet.canonical_barB_condExp_to_state_events", "ASTIS.SALD.cycle116.remaining_named_barB_selected_version_after_canonical_condExp", "SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib", "SALD.generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib", "SALD.generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCondExpSourceDef", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable", "ASTIS.SALD.cycle115.remaining_named_barB_condExp_source_representative", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib'", "MeasureTheory.integral_add", "MeasureTheory.integral_smul", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete" ] def cycle117EmNamedBarBVersionSelectionDependencyNames : List String := [ "SALD.cycle117GeneralMovingTargetDiscreteNamedBarBVersionSelectionMiddleObligation", "SALD.cycle117GeneralMovingTargetDiscreteNamedBarBVersionSelectionLowerObligation", "SALD.cycle117GeneralMovingTargetDiscreteNamedBarBVersionSelectionDag", "ASTIS.SALD.cycle117.middle_packet.named_barB_version_selection", "ASTIS.SALD.cycle117.lower_packet.selected_barB_pointwise_canonical", "ASTIS.SALD.cycle117.lower_ready.selected_named_barB_canonical_ae_eq", "ASTIS.SALD.cycle116.remaining_named_barB_selected_version_after_canonical_condExp", "ASTIS.SALD.cycle115.remaining_named_barB_condExp_source_representative", "SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib", "SALD.generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalPointwiseEq", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCondExpSourceDef", "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq", "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib'", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle118EmCanonicalBarBDownstreamDependencyNames : List String := [ "SALD.cycle118GeneralMovingTargetDiscreteCanonicalBarBDownstreamMiddleObligation", "SALD.cycle118GeneralMovingTargetDiscreteCanonicalBarBDownstreamLowerObligation", "SALD.cycle118GeneralMovingTargetDiscreteCanonicalBarBDownstreamDag", "ASTIS.SALD.cycle118.middle_packet.direct_canonical_barB_downstream", "ASTIS.SALD.cycle118.lower_ready.direct_canonical_barB_em_state_event_interface", "ASTIS.SALD.cycle117.lower_ready.selected_named_barB_canonical_ae_eq", "ASTIS.SALD.cycle116.remaining_named_barB_selected_version_after_canonical_condExp", "SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface", "SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib", "SALD.generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalPointwiseEq", "SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq", "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib'", "MeasureTheory.setIntegral_condExp", "Mathlib.Probability.Kernel.CondDistrib", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle119EmCanonicalBarBWeakFpConsumerDependencyNames : List String := [ "SALD.cycle119GeneralMovingTargetDiscreteCanonicalBarBWeakFpConsumerMiddleObligation", "SALD.cycle119GeneralMovingTargetDiscreteCanonicalBarBWeakFpConsumerLowerObligation", "SALD.cycle119GeneralMovingTargetDiscreteCanonicalBarBWeakFpConsumerDag", "ASTIS.SALD.cycle119.middle_packet.canonical_barB_weak_fp_consumer", "ASTIS.SALD.cycle119.lower_ready.canonical_barB_weak_fp_consumer_boundary", "ASTIS.SALD.cycle119.remaining_em_path_derivative_domination_and_source_actions", "ASTIS.SALD.cycle119.reviewer_canonical_barB_weak_fp_consumer_check", "ASTIS.SALD.cycle110.remaining_parametric_generator_boundary_after_dominated_transport", "ASTIS.SALD.cycle118.lower_ready.direct_canonical_barB_em_state_event_interface", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction", "AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndDominated", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "Mathlib.Probability.Kernel.CondDistrib", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle120EmPathDerivativeDominationDependencyNames : List String := [ "SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationMiddleObligation", "SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationLowerObligation", "SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationDag", "ASTIS.SALD.cycle120.middle_em_path_derivative_domination", "ASTIS.SALD.cycle120.lower_ready.em_sample_path_derivative_domination", "ASTIS.SALD.cycle120.lower_packet.em_interval_neighborhood", "ASTIS.SALD.cycle120.remaining_source_actions_after_path_derivative", "ASTIS.SALD.cycle120.reviewer_em_path_derivative_domination_check", "ASTIS.SALD.cycle119.remaining_em_path_derivative_domination_and_source_actions", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface", "SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff", "AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndDominated", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle121EmSampleMeasDependencyNames : List String := [ "SALD.cycle121GeneralMovingTargetDiscreteEmSampleMeasLowerObligation", "SALD.cycle121GeneralMovingTargetDiscreteEmSampleMeasDag", "ASTIS.SALD.cycle121.lower_packet.em_sample_meas_from_law_test_meas", "ASTIS.SALD.cycle121.remaining_em_path_derivative_domination_after_sample_meas", "ASTIS.SALD.cycle121.reviewer_em_sample_meas_check", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated", "MeasureTheory.AEStronglyMeasurable.comp_aemeasurable", "MeasureTheory.Measure.map", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle122EmSampleIntDependencyNames : List String := [ "SALD.cycle122GeneralMovingTargetDiscreteEmSampleIntLowerObligation", "SALD.cycle122GeneralMovingTargetDiscreteEmSampleIntDag", "ASTIS.SALD.cycle122.lower_packet.em_sample_int_from_law_test_int", "ASTIS.SALD.cycle122.remaining_em_path_derivative_domination_after_sample_int", "ASTIS.SALD.cycle122.reviewer_em_sample_int_check", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated", "MeasureTheory.Integrable.comp_aemeasurable", "MeasureTheory.integrable_map_measure", "MeasureTheory.Measure.map", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle123EmSampleDerivMeasDependencyNames : List String := [ "SALD.cycle123GeneralMovingTargetDiscreteEmSampleDerivMeasLowerObligation", "SALD.cycle123GeneralMovingTargetDiscreteEmSampleDerivMeasDag", "ASTIS.SALD.cycle123.lower_packet.em_sample_deriv_meas_from_concrete_deriv", "ASTIS.SALD.cycle123.remaining_em_path_derivative_domination_after_deriv_meas", "ASTIS.SALD.cycle123.reviewer_em_sample_deriv_meas_check", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated", "MeasureTheory.AEStronglyMeasurable.congr", "Mathlib.Analysis.Calculus.FDeriv.Measurable.aestronglyMeasurable_deriv_with_param", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle124EmSampleDerivBoundDependencyNames : List String := [ "SALD.cycle124GeneralMovingTargetDiscreteEmSampleDerivBoundLowerObligation", "SALD.cycle124GeneralMovingTargetDiscreteEmSampleDerivBoundDag", "ASTIS.SALD.cycle124.lower_packet.em_sample_deriv_bound_from_concrete_deriv", "ASTIS.SALD.cycle124.remaining_em_path_derivative_domination_after_deriv_bound", "ASTIS.SALD.cycle124.reviewer_em_sample_deriv_bound_check", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDominated", "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle124EmBoundIntDependencyNames : List String := [ "SALD.cycle124GeneralMovingTargetDiscreteEmBoundIntLowerObligation", "SALD.cycle124GeneralMovingTargetDiscreteEmBoundIntDag", "ASTIS.SALD.cycle124.lower_packet.em_bound_int_from_joint_law", "ASTIS.SALD.cycle124.remaining_em_path_derivative_domination_after_bound_int", "ASTIS.SALD.cycle124.reviewer_em_bound_int_check", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundDominated", "MeasureTheory.AEMeasurable.prodMk", "MeasureTheory.Integrable.comp_aemeasurable", "MeasureTheory.Integrable.congr", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle125EmPathDerivDependencyNames : List String := [ "SALD.cycle125GeneralMovingTargetDiscreteEmPathDerivLowerObligation", "SALD.cycle125GeneralMovingTargetDiscreteEmPathDerivDag", "ASTIS.SALD.cycle125.lower_packet.em_path_deriv_from_concrete_differentiability", "ASTIS.SALD.cycle125.remaining_source_actions_after_path_deriv", "ASTIS.SALD.cycle125.reviewer_em_path_deriv_check", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntDominated", "Mathlib.Analysis.Calculus.Deriv.Basic.DifferentiableAt.hasDerivAt", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle126EmDerivValueDependencyNames : List String := [ "SALD.cycle126GeneralMovingTargetDiscreteEmDerivValueLowerObligation", "SALD.cycle126GeneralMovingTargetDiscreteEmDerivValueDag", "ASTIS.SALD.cycle126.lower_packet.em_deriv_value_from_concrete_integral_split", "ASTIS.SALD.cycle126.remaining_canonical_source_actions_after_deriv_value", "ASTIS.SALD.cycle126.reviewer_em_deriv_value_check", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated", "MeasureTheory.integral_congr_ae", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle127EmDriftActionDependencyNames : List String := [ "SALD.cycle127GeneralMovingTargetDiscreteEmDriftActionLowerObligation", "SALD.cycle127GeneralMovingTargetDiscreteEmPairMeasLowerObligation", "SALD.cycle127GeneralMovingTargetDiscreteEmDriftActionDag", "ASTIS.SALD.cycle127.lower_packet.canonical_barB_drift_action_from_components", "ASTIS.SALD.cycle127.lower_packet.canonical_barB_pair_meas_from_fields", "ASTIS.SALD.cycle127.remaining_canonical_source_actions_after_drift_action", "ASTIS.SALD.cycle127.reviewer_em_drift_action_check", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDominated", "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings", "SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfIntegralAction", "MeasureTheory.AEStronglyMeasurable.inner", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle128EmNoBoundaryTraceDependencyNames : List String := [ "SALD.cycle128GeneralMovingTargetDiscreteEmNoBoundaryTraceLowerObligation", "SALD.cycle128GeneralMovingTargetDiscreteEmCanonicalBarBMeasLowerObligation", "SALD.cycle128GeneralMovingTargetDiscreteEmNoBoundaryTraceDag", "ASTIS.SALD.cycle128.lower_packet.canonical_barB_no_boundary_trace", "ASTIS.SALD.cycle128.lower_packet.canonical_barB_meas_from_condDistrib_regular", "ASTIS.SALD.cycle128.remaining_canonical_source_actions_after_no_boundary_trace", "ASTIS.SALD.cycle128.reviewer_em_no_boundary_trace_check", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasDominated", "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity", "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable", "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable", "SALD.generalMovingTargetDiscreteDriftDivNoBoundaryOfProductRule", "SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero", "SALD.generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero", "MeasureTheory.integral_congr_ae", "appendix.tex:1368-1377", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle129EmDiffusionSourceDependencyNames : List String := [ "SALD.cycle129GeneralMovingTargetDiscreteEmDiffusionSourceLowerObligation", "SALD.cycle129GeneralMovingTargetDiscreteEmDiffusionSourceDag", "ASTIS.SALD.cycle129.lower_packet.diffusion_source_boundary", "ASTIS.SALD.cycle129.retired_stale_hpathDeriv_leaf", "ASTIS.SALD.cycle129.reviewer_diffusion_source_check", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianAction", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDiffusionSourceDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated", "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDominated", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle130EmLaplacianIbPDependencyNames : List String := [ "SALD.cycle130GeneralMovingTargetDiscreteEmLaplacianIbPLowerObligation", "SALD.cycle130GeneralMovingTargetDiscreteEmGreenLaplacianIbPScoutObligation", "SALD.cycle130GeneralMovingTargetDiscreteEmFirstGreenNoBoundaryFluxLowerObligation", "SALD.cycle130GeneralMovingTargetDiscreteEmLaplacianIbPDag", "ASTIS.SALD.cycle130.lower_packet.weak_laplacian_ibp_boundary", "ASTIS.SALD.cycle130.lower_1_packet.green_laplacian_ibp_route", "ASTIS.SALD.cycle130.lower_2_packet.first_green_no_boundary_flux", "ASTIS.SALD.cycle130.remaining_laplacian_ibp_analysis", "ASTIS.SALD.cycle130.reviewer_laplacian_ibp_check", "SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianActionOfIntegrationByParts", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianIntegrationByParts", "SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfGreenLaplacianIbP", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfFirstGreenNoBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianAction", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "appendix.tex:1379-1387", "appendix.tex:1368-1377", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle131EmSecondGreenNoBoundaryFluxDependencyNames : List String := [ "SALD.cycle131GeneralMovingTargetDiscreteEmSecondGreenNoBoundaryFluxLowerObligation", "SALD.cycle131GeneralMovingTargetDiscreteEmSecondGreenTraceBoundaryLowerObligation", "SALD.cycle131GeneralMovingTargetDiscreteEmSecondGreenBoxBoundaryFluxLowerObligation", "SALD.cycle131GeneralMovingTargetDiscreteEmSecondGreenNoBoundaryFluxDag", "ASTIS.SALD.cycle131.middle_packet.second_green_no_boundary_flux", "ASTIS.SALD.cycle131.lower_1_packet.second_green_trace_boundary", "ASTIS.SALD.cycle131.lower_2_packet.second_green_box_boundary_flux", "ASTIS.SALD.cycle131.remaining_laplacian_ibp_analysis", "ASTIS.SALD.cycle131.reviewer_second_green_check", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenTraceBoundary", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenNoBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfFirstGreenNoBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfGreenLaplacianIbP", "MeasureTheory.integral_congr_ae", "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable", "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle132EmSecondGreenTestTraceZeroDependencyNames : List String := [ "SALD.cycle132GeneralMovingTargetDiscreteEmSecondGreenTestTraceZeroLowerObligation", "SALD.cycle132GeneralMovingTargetDiscreteEmSecondGreenTraceEqTestTraceZeroScoutObligation", "SALD.cycle132GeneralMovingTargetDiscreteEmSecondGreenPointwiseTraceEqLowerObligation", "SALD.cycle132GeneralMovingTargetDiscreteEmSecondGreenTestTraceZeroDag", "ASTIS.SALD.cycle132.lower_packet.second_green_test_trace_zero", "ASTIS.SALD.cycle132.lower_1_packet.second_green_trace_eq_test_trace_zero", "ASTIS.SALD.cycle132.lower_2_packet.second_green_pointwise_trace_eq", "ASTIS.SALD.cycle132.remaining_laplacian_ibp_analysis", "ASTIS.SALD.cycle132.reviewer_second_green_trace_check", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTraceEqTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFlux", "SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero", "Filter.Eventually.of_forall", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle133EmSecondGreenPointwiseTestTraceZeroDependencyNames : List String := [ "SALD.cycle133GeneralMovingTargetDiscreteEmSecondGreenPointwiseTestTraceZeroLowerObligation", "SALD.cycle133GeneralMovingTargetDiscreteEmTestLaplacianNormalizationScoutObligation", "SALD.cycle133GeneralMovingTargetDiscreteEmTestLaplacianOperatorNormalizationLowerObligation", "SALD.cycle133GeneralMovingTargetDiscreteEmSecondGreenPointwiseTestTraceZeroDag", "ASTIS.SALD.cycle133.middle_packet.second_green_pointwise_test_trace_zero", "ASTIS.SALD.cycle133.lower_1_packet.test_laplacian_normalization", "ASTIS.SALD.cycle133.lower_2_packet.test_laplacian_operator_normalization", "ASTIS.SALD.cycle133.remaining_laplacian_ibp_analysis", "ASTIS.SALD.cycle133.reviewer_pointwise_test_trace_zero_check", "SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfTestLaplacianNormalization", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqTestTraceZero", "Filter.Eventually.of_forall", "appendix.tex:1392-1427", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle134EmTestLaplacianSourcePullbackDependencyNames : List String := [ "SALD.cycle134GeneralMovingTargetDiscreteEmTestLaplacianSourcePullbackLowerObligation", "SALD.cycle134GeneralMovingTargetDiscreteEmSourceLaplacianStdBasisScoutObligation", "SALD.cycle134GeneralMovingTargetDiscreteEmWeakFpLaplacianStdBasisLowerObligation", "SALD.cycle134GeneralMovingTargetDiscreteEmTestLaplacianSourcePullbackDag", "ASTIS.SALD.cycle134.middle_packet.test_laplacian_source_pullback", "ASTIS.SALD.cycle134.lower_1_packet.source_laplacian_std_basis_formula", "ASTIS.SALD.cycle134.lower_2_packet.weak_fp_laplacian_std_basis_definition", "ASTIS.SALD.cycle134.remaining_source_laplacian_definitions", "ASTIS.SALD.cycle134.reviewer_test_laplacian_source_pullback_check", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfSourceLaplacianPullback", "SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfTestLaplacianNormalization", "InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle135EmTestLaplacianStdBasisDependencyNames : List String := [ "SALD.cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisLowerObligation", "SALD.cycle135GeneralMovingTargetDiscreteEmSecondGreenStdBasisConsumerLowerObligation", "SALD.cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisDag", "ASTIS.SALD.cycle135.lower_packet.test_laplacian_std_basis_definition", "ASTIS.SALD.cycle135.lower_2_packet.second_green_std_basis_consumer", "ASTIS.SALD.cycle135.remaining_source_laplacian_definitions", "ASTIS.SALD.cycle135.reviewer_test_laplacian_std_basis_check", "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback", "InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle136EmWeakFpStdBasisSourceDensityDependencyNames : List String := [ "SALD.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation", "SALD.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityDag", "ASTIS.SALD.cycle136.middle_packet.weak_fp_std_basis_source_density", "ASTIS.SALD.cycle136.lower_1_packet.density_laplacian_pointwise_source_formula", "ASTIS.SALD.cycle136.lower_2_packet.source_density_laplacian_laplacian_source_field", "ASTIS.SALD.cycle136.remaining_weak_fp_source_density_formula", "ASTIS.SALD.cycle136.reviewer_weak_fp_std_basis_source_density_check", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula", "SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula", "SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField", "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle137EmWeakFpDensityLaplacianActionDependencyNames : List String := [ "SALD.cycle137GeneralMovingTargetDiscreteEmWeakFpDensityLaplacianActionLowerObligation", "SALD.cycle137GeneralMovingTargetDiscreteEmPointwiseGreenIbPScoutObligation", "SALD.cycle137GeneralMovingTargetDiscreteEmSecondGreenPointwiseBoxBoundaryFluxLowerObligation", "SALD.cycle137GeneralMovingTargetDiscreteEmWeakFpDensityLaplacianActionDag", "ASTIS.SALD.cycle137.middle_packet.weak_fp_density_laplacian_pointwise_ibp", "ASTIS.SALD.cycle137.lower_1_packet.pointwise_green_ibp_scout", "ASTIS.SALD.cycle137.lower_2_packet.second_green_pointwise_box_boundary_flux", "ASTIS.SALD.cycle137.remaining_density_laplacian_source_action", "ASTIS.SALD.cycle137.reviewer_weak_fp_density_laplacian_check", "SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP", "SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity", "SALD.generalMovingTargetDiscreteSecondGreenPointwiseOfBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula", "SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity", "SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula", "SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle138EmFirstGreenPointwiseBoundaryFluxDependencyNames : List String := [ "SALD.cycle138GeneralMovingTargetDiscreteEmFirstGreenPointwiseBoundaryFluxLowerObligation", "SALD.cycle138GeneralMovingTargetDiscreteEmTestLaplacianPointwiseSourcePullbackScoutObligation", "SALD.cycle138GeneralMovingTargetDiscreteEmTestLaplacianPointwiseStdBasisLowerObligation", "SALD.cycle138GeneralMovingTargetDiscreteEmFirstGreenPointwiseBoundaryFluxDag", "ASTIS.SALD.cycle138.middle_packet.first_green_pointwise_boundary_flux", "ASTIS.SALD.cycle138.lower_1_packet.test_laplacian_pointwise_source_pullback", "ASTIS.SALD.cycle138.lower_2_packet.test_laplacian_pointwise_std_basis", "ASTIS.SALD.cycle138.remaining_pointwise_green_after_first_green", "ASTIS.SALD.cycle138.reviewer_first_green_pointwise_boundary_flux_check", "SALD.generalMovingTargetDiscreteFirstGreenPointwiseOfBoundaryFluxZero", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfSourcePullback", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfFirstGreenBoundaryFluxAndSecondGreenBoxBoundaryFlux", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux", "SALD.generalMovingTargetDiscreteSecondGreenPointwiseOfBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero", "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity", "SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle139EmWeakFpSourceLaplacianDependencyNames : List String := [ "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianLowerObligation", "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceActionIntegralScoutObligation", "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianStateIntegralLowerObligation", "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianDag", "ASTIS.SALD.cycle139.middle_packet.weak_fp_source_laplacian_field", "ASTIS.SALD.cycle139.middle_packet.test_laplacian_pointwise_weak_fp_source_field", "ASTIS.SALD.cycle139.lower_1_packet.weak_fp_source_action_state_integral", "ASTIS.SALD.cycle139.lower_2_packet.source_laplacian_state_integral", "ASTIS.SALD.cycle139.remaining_weak_fp_source_laplacian_field", "ASTIS.SALD.cycle139.reviewer_weak_fp_source_laplacian_check", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField", "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula", "MeasureTheory.integral_map", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:1379-1387", "appendix.tex:1392-1427", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle140EmLaplacianSourceStateIntegralDependencyNames : List String := [ "SALD.cycle140GeneralMovingTargetDiscreteEmLaplacianSourceStateIntegralLowerObligation", "SALD.cycle140GeneralMovingTargetDiscreteEmGeneratorLawIntegralScoutObligation", "SALD.cycle140GeneralMovingTargetDiscreteEmGeneratorSourceFunctionalLowerObligation", "SALD.cycle140GeneralMovingTargetDiscreteEmLaplacianSourceStateIntegralDag", "ASTIS.SALD.cycle140.middle_packet.laplacian_source_state_integral_em_generator", "ASTIS.SALD.cycle140.middle_packet.weak_fp_source_action_em_generator_state_integral", "ASTIS.SALD.cycle140.lower_1_packet.em_generator_laplacian_law_integral", "ASTIS.SALD.cycle140.lower_2_packet.em_generator_source_functional", "ASTIS.SALD.cycle140.remaining_em_generator_laplacian_state_integral", "ASTIS.SALD.cycle140.reviewer_em_generator_state_integral_check", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorLawIntegral", "SALD.generalMovingTargetDiscreteLaplacianSourceStateIntegralOfEmGeneratorStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle141EmGeneratorStdBasisSourceDependencyNames : List String := [ "SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorStdBasisSourceLowerObligation", "SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorTraceFieldSourceScoutObligation", "SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorTraceLawIntegralLowerObligation", "SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorStdBasisSourceDag", "ASTIS.SALD.cycle141.middle_packet.em_generator_source_action_std_basis", "ASTIS.SALD.cycle141.middle_packet.weak_fp_source_action_em_generator_std_basis", "ASTIS.SALD.cycle141.lower_1_packet.em_generator_trace_field_source", "ASTIS.SALD.cycle141.lower_1_packet.weak_fp_source_action_em_generator_trace_field", "ASTIS.SALD.cycle141.lower_2_packet.em_generator_trace_law_integral", "ASTIS.SALD.cycle141.remaining_em_generator_trace_field_source", "ASTIS.SALD.cycle141.reviewer_em_generator_std_basis_source_check", "SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceFieldSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle142EmGeneratorTraceStateIntegralDependencyNames : List String := [ "SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralMiddleObligation", "SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianStateIntegralScoutObligation", "SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianLawIntegralLowerObligation", "SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralDag", "ASTIS.SALD.cycle142.middle_packet.em_generator_trace_law_integral_state_transport", "ASTIS.SALD.cycle142.middle_packet.weak_fp_source_action_em_generator_trace_state_integral", "ASTIS.SALD.cycle142.lower_1_packet.trace_field_meas_from_source_laplacian", "ASTIS.SALD.cycle142.lower_1_packet.em_generator_trace_state_from_laplacian_state", "ASTIS.SALD.cycle142.lower_1_packet.weak_fp_source_action_em_generator_trace_laplacian_state", "ASTIS.SALD.cycle142.lower_2_packet.em_generator_laplacian_law_integral", "ASTIS.SALD.cycle142.remaining_em_generator_trace_state_integral", "ASTIS.SALD.cycle142.reviewer_em_generator_trace_state_integral_check", "SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceStateIntegralSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldMeasOfSourceLaplacianFieldMeas", "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceFieldSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "MeasureTheory.integral_map", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle143EmGeneratorLaplacianStateEventDependencyNames : List String := [ "SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianStateEventLowerObligation", "SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianStateEventDag", "SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianPointwiseEventLowerObligation", "SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianPointwiseEventDag", "SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianActionDefLowerObligation", "SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianActionDefDag", "ASTIS.SALD.cycle143.middle_packet.em_generator_laplacian_law_integral_state_event", "ASTIS.SALD.cycle143.middle_packet.weak_fp_source_action_em_generator_trace_laplacian_state_event", "ASTIS.SALD.cycle143.remaining_em_generator_laplacian_state_event_formula", "ASTIS.SALD.cycle143.reviewer_em_generator_laplacian_state_event_check", "ASTIS.SALD.cycle143.lower_1_packet.em_generator_laplacian_state_event_pointwise", "ASTIS.SALD.cycle143.lower_1_packet.weak_fp_source_action_em_generator_trace_laplacian_pointwise_event", "ASTIS.SALD.cycle143.lower_1_remaining_em_generator_laplacian_pointwise_event_formula", "ASTIS.SALD.cycle143.lower_1_reviewer_em_generator_laplacian_pointwise_event_check", "ASTIS.SALD.cycle143.lower_2_packet.em_generator_laplacian_total_event_action_def", "ASTIS.SALD.cycle143.lower_2_packet.weak_fp_source_action_em_generator_trace_laplacian_action_def", "ASTIS.SALD.cycle143.lower_2_remaining_em_generator_laplacian_action_def_pointwise_event_formula", "ASTIS.SALD.cycle143.lower_2_reviewer_em_generator_laplacian_action_def_check", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "MeasureTheory.integral_map", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle144EmGeneratorLaplacianStdBasisEventDependencyNames : List String := [ "SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisEventLowerObligation", "SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisEventDag", "SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionLowerObligation", "SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDag", "ASTIS.SALD.cycle144.middle_packet.em_generator_laplacian_event_field_std_basis", "ASTIS.SALD.cycle144.middle_packet.weak_fp_source_action_em_generator_trace_laplacian_std_basis_event", "ASTIS.SALD.cycle144.remaining_em_generator_laplacian_std_basis_event_and_action_def", "ASTIS.SALD.cycle144.reviewer_em_generator_laplacian_std_basis_event_check", "ASTIS.SALD.cycle144.lower_2_packet.em_generator_laplacian_action_std_basis_action", "ASTIS.SALD.cycle144.lower_2_packet.weak_fp_source_action_em_generator_trace_laplacian_std_basis_action", "ASTIS.SALD.cycle144.lower_2_remaining_em_generator_laplacian_std_basis_action_and_event_defs", "ASTIS.SALD.cycle144.lower_2_reviewer_em_generator_laplacian_std_basis_action_check", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle145EmGeneratorLaplacianStdBasisLawDependencyNames : List String := [ "SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisLawLowerObligation", "SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisLawDag", "SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianTraceEventLowerObligation", "SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianTraceEventDag", "ASTIS.SALD.cycle145.middle_packet.em_generator_laplacian_std_basis_action_from_law_integral", "ASTIS.SALD.cycle145.middle_packet.weak_fp_source_action_em_generator_trace_laplacian_law_integral", "ASTIS.SALD.cycle145.remaining_em_generator_law_integral_and_std_basis_event_defs", "ASTIS.SALD.cycle145.reviewer_em_generator_laplacian_law_integral_check", "ASTIS.SALD.cycle145.lower_2_packet.em_generator_laplacian_event_field_std_basis_from_trace_field", "ASTIS.SALD.cycle145.lower_2_packet.weak_fp_source_action_em_generator_trace_event_law_integral", "ASTIS.SALD.cycle145.lower_2_remaining_em_generator_law_integral_and_trace_event_equality", "ASTIS.SALD.cycle145.lower_2_reviewer_trace_event_check", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle146EmGeneratorTraceEventTotalEventDependencyNames : List String := [ "SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorTraceEventTotalEventLowerObligation", "SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorTraceEventTotalEventDag", "SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorTraceFieldLaplacianLowerObligation", "SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorTraceFieldLaplacianDag", "SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorPointwiseTraceEventLowerObligation", "SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorPointwiseTraceEventDag", "ASTIS.SALD.cycle146.middle_packet.em_generator_trace_event_total_event", "ASTIS.SALD.cycle146.remaining_em_generator_total_event_and_trace_event", "ASTIS.SALD.cycle146.reviewer_trace_event_total_event_check", "ASTIS.SALD.cycle146.lower_1_packet.trace_field_std_basis_from_laplacian", "ASTIS.SALD.cycle146.lower_1_packet.weak_fp_source_action_em_generator_trace_laplacian_total_event_trace_laplacian", "ASTIS.SALD.cycle146.lower_1_remaining_total_event_trace_event_and_trace_laplacian", "ASTIS.SALD.cycle146.lower_1_reviewer_trace_field_laplacian_check", "ASTIS.SALD.cycle146.lower_2_packet.em_generator_laplacian_event_field_trace_from_laplacian_fields", "ASTIS.SALD.cycle146.lower_2_packet.weak_fp_source_action_em_generator_pointwise_event_total_event_trace_laplacian", "ASTIS.SALD.cycle146.lower_2_remaining_total_event_pointwise_event_and_trace_laplacian", "ASTIS.SALD.cycle146.lower_2_reviewer_pointwise_trace_event_check", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventLawIntegralFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle147EmGeneratorPointwiseActionDefTraceLaplacianDependencyNames : List String := [ "SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseActionDefTraceLaplacianLowerObligation", "SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseActionDefTraceLaplacianDag", "SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseStdBasisActionTraceLaplacianLowerObligation", "SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseStdBasisActionTraceLaplacianDag", "SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseLawIntegralTraceLaplacianLowerObligation", "SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseLawIntegralTraceLaplacianDag", "ASTIS.SALD.cycle147.middle_packet.em_generator_laplacian_total_event_from_action_def_trace_laplacian", "ASTIS.SALD.cycle147.middle_packet.weak_fp_source_action_em_generator_pointwise_event_action_def_trace_laplacian", "ASTIS.SALD.cycle147.remaining_action_def_pointwise_event_and_trace_laplacian", "ASTIS.SALD.cycle147.reviewer_pointwise_action_def_trace_laplacian_check", "ASTIS.SALD.cycle147.lower_1_packet.em_generator_laplacian_action_def_from_stdbasis_action_pointwise_event", "ASTIS.SALD.cycle147.lower_1_packet.weak_fp_source_action_em_generator_pointwise_event_stdbasis_action_trace_laplacian", "ASTIS.SALD.cycle147.lower_1_remaining_stdbasis_action_pointwise_event_and_trace_laplacian", "ASTIS.SALD.cycle147.lower_1_reviewer_stdbasis_action_trace_laplacian_check", "ASTIS.SALD.cycle147.lower_2_packet.em_generator_laplacian_stdbasis_action_from_law_integral_pointwise_event", "ASTIS.SALD.cycle147.lower_2_packet.weak_fp_source_action_em_generator_pointwise_event_law_integral_trace_laplacian", "ASTIS.SALD.cycle147.lower_2_remaining_law_integral_pointwise_event_and_trace_laplacian", "ASTIS.SALD.cycle147.lower_2_reviewer_law_integral_trace_laplacian_check", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventLawIntegralTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionPointwiseEventFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStdBasisActionTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle148EmGeneratorPointwiseStateEventTraceLaplacianDependencyNames : List String := [ "SALD.cycle148GeneralMovingTargetDiscreteEmGeneratorPointwiseStateEventTraceLaplacianMiddleObligation", "SALD.cycle148GeneralMovingTargetDiscreteEmGeneratorPointwiseStateEventTraceLaplacianDag", "SALD.cycle148GeneralMovingTargetDiscreteEmGeneratorStateEventPointwiseScoutObligation", "SALD.cycle148GeneralMovingTargetDiscreteEmGeneratorStateEventPointwiseScoutDag", "SALD.cycle148GeneralMovingTargetDiscreteEmGeneratorTotalEventSourceFunctionalLowerObligation", "SALD.cycle148GeneralMovingTargetDiscreteEmGeneratorTotalEventSourceFunctionalDag", "ASTIS.SALD.cycle148.middle_packet.em_generator_laplacian_law_integral_from_state_event_current_pointwise", "ASTIS.SALD.cycle148.middle_packet.weak_fp_source_action_em_generator_pointwise_event_state_event_trace_laplacian", "ASTIS.SALD.cycle148.remaining_state_event_pointwise_event_and_trace_laplacian", "ASTIS.SALD.cycle148.reviewer_pointwise_state_event_trace_laplacian_check", "ASTIS.SALD.cycle148.lower_1_packet.state_event_eq_from_pointwise", "ASTIS.SALD.cycle148.lower_1_reject_duplicate_state_event_wrapper", "ASTIS.SALD.cycle148.lower_2_packet.total_event_from_source_functional", "ASTIS.SALD.cycle148.lower_2_remaining_source_functional_state_event_trace", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfSourceFunctional", "SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceFunctionalLowerObligation", "SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceFunctionalDag", "ASTIS.SALD.cycle149.packet.total_event_from_stdbasis_source_functional", "ASTIS.SALD.cycle149.remaining_stdbasis_source_event_trace", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceFunctional", "SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceEventLowerObligation", "SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceEventDag", "ASTIS.SALD.cycle149.lower_1_packet.total_event_from_stdbasis_source_and_event", "ASTIS.SALD.cycle149.lower_1_remaining_stdbasis_source_event_trace", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceAndEventFormula", "SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceFieldSourceEventLowerObligation", "SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceFieldSourceEventDag", "ASTIS.SALD.cycle149.lower_2_packet.total_event_from_trace_field_source_and_event", "ASTIS.SALD.cycle149.lower_2_remaining_trace_action_event_trace_laplacian", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceFieldSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula", "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventLawIntegralTraceLaplacianFormula", "hemGeneratorTraceActionDef", "hemGeneratorLaplacianEventFieldEqTraceField", "htraceFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle149EmGeneratorTotalEventStdBasisSourceFunctionalDependencyNames : List String := [ "SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceFunctionalLowerObligation", "SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceFunctionalDag", "ASTIS.SALD.cycle149.packet.total_event_from_stdbasis_source_functional", "ASTIS.SALD.cycle149.remaining_stdbasis_source_event_trace", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceFunctional", "SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceEventLowerObligation", "SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceEventDag", "ASTIS.SALD.cycle149.lower_1_packet.total_event_from_stdbasis_source_and_event", "ASTIS.SALD.cycle149.lower_1_remaining_stdbasis_source_event_trace", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceAndEventFormula", "SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceFieldSourceEventLowerObligation", "SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceFieldSourceEventDag", "ASTIS.SALD.cycle149.lower_2_packet.total_event_from_trace_field_source_and_event", "ASTIS.SALD.cycle149.lower_2_remaining_trace_action_event_trace_laplacian", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceFieldSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "hemGeneratorStdBasisDef", "hsourceLaplacianFunctional", "hemGeneratorLaplacianEventFieldEqLaplacian", "hemGeneratorLaplacianEventFieldStdBasisDef", "hemGeneratorTraceActionDef", "hemGeneratorLaplacianEventFieldEqTraceField", "htraceFieldEqLaplacian", "Mathlib.Analysis.InnerProductSpace.Laplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle150EmGeneratorTotalEventTraceLawIntegralDependencyNames : List String := [ "SALD.cycle150GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLawIntegralSourceEventMiddleObligation", "SALD.cycle150GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceStateIntegralSourceEventLowerObligation", "SALD.cycle150GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLaplacianStateIntegralSourceEventLowerObligation", "SALD.cycle150GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLawIntegralSourceEventDag", "ASTIS.SALD.cycle150.middle_packet.trace_action_from_trace_law_integral", "ASTIS.SALD.cycle150.middle_packet.total_event_from_trace_law_integral_source_and_event", "ASTIS.SALD.cycle150.lower_1_packet.total_event_from_trace_state_integral_source_and_event", "ASTIS.SALD.cycle150.lower_2_packet.total_event_from_trace_laplacian_state_integral_source_and_event", "ASTIS.SALD.cycle150.lower_2_remaining_trace_laplacian_state_event_trace_laplacian", "ASTIS.SALD.cycle150.lower_1_remaining_trace_state_event_trace_laplacian", "ASTIS.SALD.cycle150.remaining_trace_law_event_trace_laplacian", "ASTIS.SALD.cycle150.reviewer_trace_law_total_event_check", "SALD.generalMovingTargetDiscreteEmGeneratorTraceActionDefOfTraceLawIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLawIntegralSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceStateIntegralSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldMeasOfSourceLaplacianFieldMeas", "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceFieldSourceAndEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceAndEventFormula", "hemGeneratorTraceLawIntegral", "hemGeneratorTraceStateIntegral", "hemGeneratorLaplacianStateIntegral", "hhatRhoS", "hhatX", "htraceFieldMeas", "hsourceLaplacianFieldMeas", "hsourceLaplacianFunctional", "hemGeneratorLaplacianEventFieldEqTraceField", "htraceFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle151EmGeneratorTotalEventTraceLaplacianStateIntegralPointwiseEventDependencyNames : List String := [ "SALD.cycle151GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLaplacianStateIntegralPointwiseEventObligation", "SALD.cycle151GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLaplacianStateIntegralPointwiseEventDag", "SALD.cycle151GeneralMovingTargetDiscreteEmGeneratorPointwiseEventSourceFieldLowerObligation", "SALD.cycle151GeneralMovingTargetDiscreteEmGeneratorPointwiseEventSourceFieldDag", "ASTIS.SALD.cycle151.packet.total_event_from_pointwise_event_trace_laplacian", "ASTIS.SALD.cycle151.lower_2_packet.pointwise_event_from_source_field", "ASTIS.SALD.cycle151.lower_2_remaining_pointwise_event_source_field", "ASTIS.SALD.cycle151.remaining_trace_laplacian_state_pointwise_event", "ASTIS.SALD.cycle151.reviewer_pointwise_event_trace_laplacian_check", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndPointwiseEventFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfWeakFpSourceField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndEventFormula", "hemGeneratorLaplacianEventFieldEqSourceField", "hweakFpSourceFieldEqLaplacian", "hemGeneratorLaplacianEventFieldEqLaplacian", "htraceFieldEqLaplacian", "hemGeneratorLaplacianStateIntegral", "hsourceLaplacianFieldMeas", "hsourceLaplacianFunctional", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle152EmGeneratorEventSourceFieldStdBasisDependencyNames : List String := [ "SALD.cycle152GeneralMovingTargetDiscreteEmGeneratorEventSourceFieldStdBasisLowerObligation", "SALD.cycle152GeneralMovingTargetDiscreteWeakFpSourceFieldStdBasisScoutObligation", "SALD.cycle152GeneralMovingTargetDiscreteWeakFpSourceFieldPointwiseLowerObligation", "SALD.cycle152GeneralMovingTargetDiscreteEmGeneratorEventSourceFieldStdBasisDag", "ASTIS.SALD.cycle152.packet.event_source_field_from_stdbasis_fields", "ASTIS.SALD.cycle152.lower_1_packet.weak_fp_source_field_stdbasis_from_laplacian", "ASTIS.SALD.cycle152.lower_2_packet.weak_fp_source_field_eq_laplacian_pointwise", "ASTIS.SALD.cycle152.remaining_event_source_field_stdbasis_defs", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqSourceFieldOfStdBasisFields", "SALD.generalMovingTargetDiscreteWeakFpSourceFieldStdBasisDefOfLaplacianField", "SALD.generalMovingTargetDiscreteWeakFpSourceFieldEqLaplacianOfPointwise", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfWeakFpSourceField", "hemGeneratorLaplacianEventFieldEqSourceField", "hemGeneratorLaplacianEventFieldStdBasisDef", "hweakFpSourceFieldStdBasisDef", "hweakFpSourceFieldEqLaplacian", "hweakFpSourceFieldPointwiseEqLaplacian", "htraceFieldEqLaplacian", "hemGeneratorLaplacianStateIntegral", "hsourceLaplacianFieldMeas", "hsourceLaplacianFunctional", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle153EmGeneratorLaplacianStateIntegralSourceFunctionalDependencyNames : List String := [ "SALD.cycle153GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralSourceFunctionalLowerObligation", "SALD.cycle153GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralStdBasisSourceFunctionalLowerObligation", "SALD.cycle153GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralSourceFunctionalDag", "ASTIS.SALD.cycle153.packet.laplacian_state_integral_from_source_functional", "ASTIS.SALD.cycle153.lower_2_packet.laplacian_state_integral_from_stdbasis_source_functional", "ASTIS.SALD.cycle153.lower_2_remaining_laplacian_state_integral_stdbasis_source_functional", "ASTIS.SALD.cycle153.reviewer_laplacian_state_integral_source_functional_check", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "hemGeneratorSourceActionDef", "hemGeneratorStdBasisDef", "hsourceLaplacianFunctional", "hsourceLaplacianFieldMeas", "hhatRhoS", "hhatX", "hemGeneratorLaplacianStateIntegral", "hemGeneratorLaplacianEventFieldStdBasisDef", "hweakFpSourceFieldPointwiseEqLaplacian", "htraceFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle154EmGeneratorLaplacianStateIntegralTraceFieldSourceFunctionalDependencyNames : List String := [ "SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceFieldSourceFunctionalLowerObligation", "SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceLawIntegralLaplacianFieldScoutObligation", "SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceStateIntegralLaplacianFieldLowerObligation", "SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceFieldSourceFunctionalDag", "ASTIS.SALD.cycle154.packet.laplacian_state_integral_from_trace_field_source_functional", "ASTIS.SALD.cycle154.lower_1_packet.laplacian_state_integral_from_trace_law_laplacian_field", "ASTIS.SALD.cycle154.lower_2_packet.laplacian_state_integral_from_trace_state_laplacian_field", "ASTIS.SALD.cycle154.remaining_trace_state_trace_field_source_functional", "ASTIS.SALD.cycle154.reviewer_trace_field_source_functional_check", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceFieldSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceLawIntegralLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceStateIntegralLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorTraceActionDefOfTraceLawIntegral", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfSourceFunctional", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral", "hemGeneratorTraceActionDef", "hemGeneratorTraceLawIntegral", "hemGeneratorTraceStateIntegral", "htraceFieldMeas", "htraceFieldStdBasis", "hemGeneratorStdBasisDef", "hsourceLaplacianFunctional", "hsourceLaplacianFieldMeas", "hhatRhoS", "hhatX", "hemGeneratorLaplacianStateIntegral", "hemGeneratorLaplacianEventFieldStdBasisDef", "hweakFpSourceFieldPointwiseEqLaplacian", "htraceFieldEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle155EmGeneratorTraceStateIntegralLaplacianFieldDependencyNames : List String := [ "SALD.cycle155GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralLaplacianFieldLowerObligation", "SALD.cycle155GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralLaplacianFieldDag", "ASTIS.SALD.cycle155.packet.trace_state_integral_from_laplacian_state_integral", "ASTIS.SALD.cycle155.remaining_trace_state_integral_laplacian_field", "ASTIS.SALD.cycle155.reviewer_trace_state_laplacian_field_check", "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegralLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceStateIntegralLaplacianField", "hemGeneratorTraceStateIntegral", "hemGeneratorLaplacianStateIntegral", "htraceFieldMeas", "htraceFieldEqLaplacian", "hsourceLaplacianFunctional", "hsourceLaplacianFieldMeas", "hhatRhoS", "hhatX", "hemGeneratorLaplacianEventFieldStdBasisDef", "hweakFpSourceFieldPointwiseEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle156EmGeneratorTraceFieldPointwiseDependencyNames : List String := [ "SALD.cycle156GeneralMovingTargetDiscreteEmGeneratorTraceFieldPointwiseLowerObligation", "SALD.cycle156GeneralMovingTargetDiscreteEmGeneratorTraceFieldPointwiseDag", "ASTIS.SALD.cycle156.packet.trace_field_eq_laplacian_from_pointwise", "ASTIS.SALD.cycle156.lower_2_packet.trace_field_pointwise_stdbasis_from_event_field", "ASTIS.SALD.cycle156.remaining_trace_field_pointwise_laplacian", "ASTIS.SALD.cycle156.reviewer_trace_field_pointwise_check", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldEqLaplacianOfPointwise", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseEqLaplacianOfStdBasis", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseStdBasisOfEventFieldStdBasis", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegralLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields", "htraceFieldEqLaplacian", "htraceFieldPointwiseEqLaplacian", "htraceFieldPointwiseStdBasis", "hemGeneratorLaplacianEventFieldEqTraceField", "hemGeneratorLaplacianStateIntegral", "htraceFieldMeas", "hsourceLaplacianFunctional", "hsourceLaplacianFieldMeas", "hemGeneratorLaplacianEventFieldStdBasisDef", "hweakFpSourceFieldPointwiseEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:forward-KL-discrete", "thm:general-moving-target-SALD-discrete" ] def cycle157EmGeneratorLaplacianEventFieldPointwiseStdBasisDependencyNames : List String := [ "SALD.cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisLowerObligation", "SALD.cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisDag", "ASTIS.SALD.cycle157.packet.event_field_stdbasis_from_pointwise", "ASTIS.SALD.cycle157.lower_1_packet.event_field_pointwise_stdbasis_from_laplacian", "ASTIS.SALD.cycle157.lower_2_packet.event_field_laplacian_from_pointwise_scalar", "ASTIS.SALD.cycle157.remaining_event_field_pointwise_scalar", "ASTIS.SALD.cycle157.reviewer_event_field_pointwise_stdbasis_check", "SALD.cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseLaplacianScoutObligation", "SALD.cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseScalarLowerObligation", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfPointwise", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisDefOfPointwiseLaplacian", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfPointwiseScalar", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "hEmGeneratorLaplacianEventFieldPointwiseStdBasisDef", "hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian", "hemGeneratorLaplacianEventFieldEqLaplacian", "hemGeneratorLaplacianEventFieldStdBasisDef", "SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseStdBasisOfEventFieldStdBasis", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqSourceFieldOfStdBasisFields", "hemGeneratorLaplacianEventFieldEqTraceField", "hemGeneratorLaplacianStateIntegral", "htraceFieldMeas", "hsourceLaplacianFunctional", "hsourceLaplacianFieldMeas", "hweakFpSourceFieldPointwiseEqLaplacian", "appendix.tex:984-995", "appendix.tex:1368-1387", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle158EmGeneratorLaplacianEventFieldPointwiseScalarAuditDependencyNames : List String := [ "SALD.cycle158GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseScalarAuditObligation", "SALD.cycle158GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefLowerObligation", "SALD.cycle158GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseScalarAuditDag", "ASTIS.SALD.cycle158.middle_packet.scalar_event_field_delta_source_gap", "ASTIS.SALD.cycle158.lower_2_packet.scalar_delta_from_brownian_event_field_def", "ASTIS.SALD.cycle158.remaining_brownian_event_field_def", "ASTIS.SALD.cycle158.middle_packet.rejected_scalar_delta_wrapper_churn", "SALD.emFrozenBrownianLaplacianEventField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseEqLaplacianOfBrownianDef", "hEmGeneratorLaplacianEventFieldBrownianDef", "hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian", "emGeneratorLaplacianEventField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfPointwiseScalar", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisDefOfPointwiseLaplacian", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfPointwise", "eq:general_moving_target_SALD_frozen_interp", "eq:SALD_general_EM", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle159EmGeneratorLaplacianEventFieldBrownianPointwiseDependencyNames : List String := [ "SALD.cycle159GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseLowerObligation", "SALD.cycle159GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisLowerObligation", "SALD.cycle159GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseDag", "ASTIS.SALD.cycle159.middle_packet.brownian_event_field_def_from_pointwise", "ASTIS.SALD.cycle159.lower_2_packet.brownian_pointwise_def_from_stdbasis", "ASTIS.SALD.cycle159.remaining_brownian_event_field_pointwise_stdbasis_def", "ASTIS.SALD.cycle159.reviewer_brownian_event_field_pointwise_check", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefOfPointwise", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseDefOfStdBasis", "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv", "SALD.emFrozenBrownianLaplacianEventField", "hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef", "hEmGeneratorLaplacianEventFieldBrownianPointwiseDef", "hEmGeneratorLaplacianEventFieldBrownianDef", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseEqLaplacianOfBrownianDef", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfPointwiseScalar", "InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis", "eq:general_moving_target_SALD_frozen_interp", "eq:SALD_general_EM", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle160EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDependencyNames : List String := [ "SALD.cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLowerObligation", "SALD.cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseScoutObligation", "SALD.cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseCoordinateLowerObligation", "SALD.cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDag", "ASTIS.SALD.cycle160.middle_packet.brownian_stdbasis_from_frozen_scalar_ito_generator", "ASTIS.SALD.cycle160.remaining_frozen_scalar_brownian_ito_generator_event_field_def", "ASTIS.SALD.cycle160.lower_2_packet.frozen_scalar_ito_pointwise_from_coordinate_generator", "ASTIS.SALD.cycle160.remaining_frozen_scalar_brownian_ito_coordinate_generator", "ASTIS.SALD.cycle160.reviewer_frozen_scalar_brownian_ito_check", "SALD.emFrozenScalarBrownianItoGeneratorEventField", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDefOfFrozenScalarBrownianItoGenerator", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoGeneratorDefOfPointwise", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator", "hFrozenScalarBrownianItoGeneratorEventFieldDef", "hFrozenScalarBrownianItoGeneratorEventFieldPointwiseDef", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "hFrozenScalarBrownianItoCoordinateGeneratorDef", "hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseDefOfStdBasis", "SALD.emFrozenBrownianLaplacianEventField", "Mathlib.Probability.Distributions.Gaussian.Real", "Mathlib.Probability.Moments.CovarianceBilin", "Mathlib.Analysis.InnerProductSpace.Laplacian", "eq:general_moving_target_SALD_frozen_interp", "eq:SALD_general_EM", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle161EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDependencyNames : List String := [ "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorLowerObligation", "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorScoutObligation", "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorMomentLowerObligation", "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDag", "ASTIS.SALD.cycle161.middle_packet.coordinate_generator_from_one_dim_taylor", "ASTIS.SALD.cycle161.lower_2_packet.one_dim_taylor_moment_collapse", "ASTIS.SALD.cycle161.remaining_one_dim_brownian_ito_taylor_boundary", "ASTIS.SALD.cycle161.residual_coordinate_sum_boundary", "ASTIS.SALD.cycle161.reviewer_one_dim_taylor_check", "SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator", "SALD.gaussianRealZeroSecondMoment", "SALD.gaussianRealZeroOneDimTaylorMomentContribution", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator", "hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor", "hFrozenScalarBrownianItoOneDimTaylorExpansion", "hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit", "hFrozenScalarBrownianItoCoordinateGeneratorDef", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "SALD.emFrozenScalarBrownianItoGeneratorEventField", "Mathlib.Probability.Distributions.Gaussian.Real", "Mathlib.Probability.Moments.CovarianceBilin", "Mathlib.Analysis.InnerProductSpace.Laplacian", "eq:general_moving_target_SALD_frozen_interp", "eq:SALD_general_EM", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle162EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorRemainderDependencyNames : List String := [ "SALD.cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorRemainderLowerObligation", "SALD.cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderScoutObligation", "SALD.cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderDctLowerObligation", "SALD.cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorRemainderDag", "ASTIS.SALD.cycle162.middle_packet.one_dim_taylor_from_moment_remainder", "ASTIS.SALD.cycle162.remaining_normalized_taylor_remainder_boundary", "ASTIS.SALD.cycle162.lower_1_packet.normalized_taylor_remainder_dct_scout", "ASTIS.SALD.cycle162.lower_2_packet.normalized_taylor_remainder_dct", "ASTIS.SALD.cycle162.residual_coordinate_sum_boundary", "ASTIS.SALD.cycle162.reviewer_taylor_remainder_check", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder", "SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT", "SALD.gaussianRealZeroOneDimTaylorMomentContribution", "SALD.gaussianRealZeroSecondMoment", "SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit", "hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "ProbabilityTheory.integral_id_gaussianReal", "Mathlib.Probability.Distributions.Gaussian.Real", "Mathlib.Analysis.Calculus.Taylor", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.Bochner.Basic", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle163EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseTaylorDependencyNames : List String := [ "SALD.cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseTaylorLowerObligation", "SALD.cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderSourceEqLowerObligation", "SALD.cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseTaylorDag", "ASTIS.SALD.cycle163.lower_packet.selected_test_line_taylor_hpoint", "ASTIS.SALD.cycle163.lower_1_packet.normalized_remainder_source_eq_to_dct", "ASTIS.SALD.cycle163.lower_2_packet.normalized_remainder_source_eq", "ASTIS.SALD.cycle163.remaining_normalized_remainder_source_identification", "ASTIS.SALD.cycle163.reviewer_pointwise_taylor_check", "SALD.gaussianRealSelectedTestLineSecondOrderTaylorRemainderPointwiseAE", "SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfSelectedTestLineEq", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainder", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderSourceEq", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero", "SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder", "Real.taylor_tendsto", "Filter.Tendsto.congr'", "Filter.Eventually.of_forall", "taylorWithinEval", "hPoint", "hMeas", "hBound", "hBoundInt", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "Mathlib.Analysis.Calculus.Taylor", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle164EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoMeasDependencyNames : List String := [ "SALD.cycle164GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderMeasLowerObligation", "SALD.cycle164GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundIntLowerObligation", "SALD.cycle164GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderMeasDag", "ASTIS.SALD.cycle164.lower_packet.normalized_remainder_hmeas", "ASTIS.SALD.cycle164.lower_2_packet.normalized_remainder_quadratic_bound_integrable", "ASTIS.SALD.cycle164.remaining_normalized_remainder_domination", "ASTIS.SALD.cycle164.reviewer_hmeas_check", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable", "SALD.gaussianRealSelectedTestLineSecondOrderQuadraticBoundIntegrable", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainder", "ContDiffOn.continuousOn", "hasDerivAt_taylorWithinEval_succ", "Continuous.comp_aestronglyMeasurable", "MeasureTheory.AEStronglyMeasurable.sub", "MeasureTheory.AEStronglyMeasurable.div₀", "MeasureTheory.AEStronglyMeasurable.mul", "MeasureTheory.AEStronglyMeasurable.pow", "hMeas", "hBound", "hBoundInt", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "ProbabilityTheory.integrable_exp_mul_gaussianReal", "ProbabilityTheory.integrable_pow_of_integrable_exp_mul", "MeasureTheory.Integrable.const_mul", "Mathlib.Probability.Moments.IntegrableExpMul", "Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable", "Mathlib.Analysis.Calculus.Taylor", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle165EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHBoundDependencyNames : List String := [ "SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundLowerObligation", "SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorQuotientSplitLower2Obligation", "SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundDag", "ASTIS.SALD.cycle165.lower_packet.normalized_remainder_hbound_from_taylor_quotient", "ASTIS.SALD.cycle165.lower_2_packet.taylor_quotient_split_first_order_second_coeff", "ASTIS.SALD.cycle165.remaining_taylor_quotient_bound", "ASTIS.SALD.cycle165.reviewer_hbound_check", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound", "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable", "SALD.gaussianRealSelectedTestLineSecondOrderQuadraticBoundIntegrable", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainder", "hTaylorQuotientBound", "hFirst", "hSecondCoeff", "hBound", "hBoundInt", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "taylorWithinEval", "taylorWithinEval_succ", "taylorCoeffWithin", "norm_mul", "norm_sub_le", "mul_le_mul_of_nonneg_right", "Mathlib.Analysis.Calculus.Taylor", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle166EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHFirstDependencyNames : List String := [ "SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderLowerObligation", "SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoIntervalTaylorLower1Obligation", "SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSignedIntervalTaylorLower2Obligation", "SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderDag", "ASTIS.SALD.cycle166.lower_packet.hfirst_from_quadratic_remainder", "ASTIS.SALD.cycle166.lower_1_packet.nonnegative_interval_taylor_remainder", "ASTIS.SALD.cycle166.lower_2_packet.signed_interval_taylor_remainder", "ASTIS.SALD.cycle166.remaining_nonquotient_quadratic_remainder", "ASTIS.SALD.cycle166.reviewer_hfirst_check", "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor", "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff", "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound", "hFirst", "hFirstQuadraticRemainder", "hC1", "hSecondCoeff", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "taylorWithinEval", "norm_div", "div_le_div_of_nonneg_right", "field_simp", "Mathlib.Analysis.Calculus.Taylor.taylor_mean_remainder_bound", "Mathlib.Analysis.Calculus.MeanValue", "ContDiffOn Real 2", "iteratedDerivWithin", "Set.Icc", "hTaylorCompat", "hNonnegCont", "hNonnegSecond", "hNonnegTaylorCompat", "hNegCont", "hNegSecond", "hNegTaylorCompat", "hNegTaylorReflect", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle167EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorReflectDependencyNames : List String := [ "SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorReflectMiddleObligation", "SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorReflectDag", "SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorCompatLower1Obligation", "SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLineContDiffLower2Obligation", "SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorCompatLower1Dag", "SALD.cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceContDiffMiddleObligation", "SALD.cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLineSecondLower2Obligation", "SALD.cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceContDiffDag", "ASTIS.SALD.cycle167.middle_packet.reflected_set_univ_taylor_compat", "ASTIS.SALD.cycle167.middle_packet.signed_interval_without_reflect_hypothesis", "ASTIS.SALD.cycle167.remaining_signed_interval_regularity_data", "ASTIS.SALD.cycle167.lower_1_packet.interval_set_univ_taylor_compat_from_base_diff", "ASTIS.SALD.cycle167.lower_1_packet.signed_interval_remainder_from_base_diff", "ASTIS.SALD.cycle167.lower_2_packet.signed_interval_remainder_from_global_line_contdiff", "ASTIS.SALD.cycle167.remaining_signed_interval_data_after_taylor_compat", "ASTIS.SALD.cycle168.middle_packet.ambient_source_contdiff_to_selected_line", "ASTIS.SALD.cycle168.middle_packet.hline_consumer_source_contdiff", "ASTIS.SALD.cycle168.lower_2_packet.signed_interval_second_bounds_from_global_line_second", "ASTIS.SALD.cycle168.remaining_global_line_second_derivative_bounds", "SALD.gaussianRealSelectedTestLineFirstOrderTaylorReflectSetUniv", "SALD.gaussianRealSelectedTestLineFirstOrderTaylorCompatOfDifferentiableAt", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff", "SALD.gaussianRealSelectedTestLineContDiffOnOfSourceContDiffOn", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor", "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder", "hNegTaylorReflect", "hSource", "hLine", "hNonnegCont", "hNonnegSecond", "hNonnegTaylorCompat", "hNegCont", "hNegSecond", "hNegTaylorCompat", "hNonnegBaseDiff", "hNegBaseDiff", "hC1", "hLineSecond", "hNegLineSecond", "deriv_comp_neg", "DifferentiableAt.derivWithin", "uniqueDiffOn_Icc", "accPt_iff_nhds", "derivWithin_zero_of_not_accPt", "iteratedDerivWithin_succ", "iteratedDerivWithin_eq_iteratedDeriv", "derivWithin_univ", "taylorWithinEval_succ", "ContDiffOn.mono", "contDiffOn_univ", "ContDiff.comp", "ContDiff.add", "ContDiff.smul_const", "contDiff_const", "contDiff_neg", "contDiff_id", "ContDiffOn.contDiffAt", "ContDiffAt.differentiableAt", "Mathlib.Analysis.Calculus.TangentCone.Real", "Mathlib.Analysis.Calculus.ContDiff.Comp", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.Taylor", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle168EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceContDiffDependencyNames : List String := [ "SALD.cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceContDiffMiddleObligation", "SALD.cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLineSecondLower2Obligation", "SALD.cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceContDiffDag", "ASTIS.SALD.cycle168.middle_packet.ambient_source_contdiff_to_selected_line", "ASTIS.SALD.cycle168.middle_packet.hline_consumer_source_contdiff", "ASTIS.SALD.cycle168.lower_2_packet.signed_interval_second_bounds_from_global_line_second", "ASTIS.SALD.cycle168.remaining_global_line_second_derivative_bounds", "SALD.gaussianRealSelectedTestLineContDiffOnOfSourceContDiffOn", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff", "hSource", "hC1", "hLine", "hLineSecond", "hNegLineSecond", "hNonnegSecond", "hNegSecond", "accPt_iff_nhds", "derivWithin_zero_of_not_accPt", "iteratedDerivWithin_succ", "uniqueDiffOn_Icc", "iteratedDerivWithin_eq_iteratedDeriv", "contDiffOn_univ", "ContDiff.comp", "ContDiff.add", "ContDiff.smul_const", "contDiff_const", "contDiff_id", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.TangentCone.Real", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.ContDiff.Operations", "Mathlib.Analysis.Calculus.ContDiff.Comp", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle169EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondDependencyNames : List String := [ "SALD.cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondMiddleObligation", "SALD.cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondLower2Obligation", "SALD.cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondDag", "ASTIS.SALD.cycle169.middle_packet.line_second_from_ambient_directional_hessian", "ASTIS.SALD.cycle169.middle_packet.reflected_line_second_from_same_bound", "ASTIS.SALD.cycle169.lower_2_packet.directional_hessian_from_second_fderiv_op_norm", "ASTIS.SALD.cycle169.middle_packet.directional_hessian_consumer", "ASTIS.SALD.cycle169.remaining_second_fderiv_operator_norm_bound", "SALD.gaussianRealSelectedTestLineSecondDerivEqDirectionalFDeriv", "SALD.gaussianRealSelectedTestLineSecondBoundsOfDirectionalSecondBound", "SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds", "hDirectionalSecond", "hSecondFDerivOpNorm", "heUnit", "hSource", "hC1", "hLineSecond", "hNegLineSecond", "iteratedDeriv_vcomp_two", "iteratedDeriv_const_add", "iteratedDeriv_smul_const", "iteratedDeriv_fun_id", "deriv_smul_const", "ContinuousMultilinearMap.map_smul_univ", "ContinuousMultilinearMap.le_mul_prod_of_opNorm_le_of_le", "Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Normed.Module.Multilinear.Basic", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle170EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianOpNormDependencyNames : List String := [ "SALD.cycle170GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianOpNormLower2Obligation", "SALD.cycle170GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianOpNormDag", "ASTIS.SALD.cycle170.lower_2_packet.hsecond_fderiv_from_hessian_op_norm", "ASTIS.SALD.cycle170.remaining_selected_test_hessian_operator_norm_source_contract", "SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm", "SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis", "SALD.gaussianRealStdOrthonormalBasisUnit", "hHessianOpNorm", "hSecondFDerivOpNorm", "norm_iteratedFDeriv_fderiv", "norm_iteratedFDeriv_one", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle171EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDependencyNames : List String := [ "SALD.cycle171GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceMiddleObligation", "SALD.cycle171GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower1Obligation", "SALD.cycle171GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation", "SALD.cycle171GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDag", "ASTIS.SALD.cycle171.middle_refiner.reject_opaque_testregular_hessian_wrapper", "ASTIS.SALD.cycle171.lower_1_refiner.hessian_source_contract_audit", "ASTIS.SALD.cycle171.lower_2_refiner.reject_unsourced_hessian_projection", "SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm", "SALD.gaussianRealStdOrthonormalBasisUnit", "SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis", "hHessianOpNorm", "selected weak-test C2_b/bounded-Hessian source interface", "SelectedWeakTestC2bBoundedHessian projection rejected without source backing", "appendix.tex:984-995", "appendix.tex:1026-1072", "appendix.tex:1379-1387", "main_body.tex:273-305", "paper-wide original-source search excluding sald_version_2.tex", "source-contract-gap", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle172EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDependencyNames : List String := [ "SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceMiddleObligation", "SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower1Obligation", "SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation", "SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDag", "ASTIS.SALD.cycle172.middle_refiner.blueprint_refreshed_hessian_source_contract", "ASTIS.SALD.cycle172.middle_refiner.reject_same_field_projection", "ASTIS.SALD.cycle172.lower_1_refiner.hessian_source_contract_proof_route", "ASTIS.SALD.cycle172.lower_2_refiner.reject_score_or_unsourced_test_hessian_projection", "SALD.cycle171GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation", "SALD.cycle171GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDag", "SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm", "SALD.gaussianRealStdOrthonormalBasisUnit", "SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis", "hHessianOpNorm", "hSecondFDerivOpNorm", "sourceTest", "testRegular", "SourceSelectedWeakTestC2bBoundedHessian proposed only as source-backed field", "selected weak-test C2_b/bounded-Hessian source interface", "SelectedWeakTestC2bBoundedHessian projection rejected without source backing", "appendix.tex:984-995", "appendix.tex:1026-1072", "appendix.tex:1379-1387", "main_body.tex:273-305", "paper-wide original-source search excluding sald_version_2.tex", "source-contract-gap", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle173EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDependencyNames : List String := [ "SALD.cycle173GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceMiddleObligation", "SALD.cycle173GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower1Obligation", "SALD.cycle173GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation", "SALD.cycle173GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDag", "ASTIS.SALD.cycle173.middle_refiner.hessian_source_contract_recheck", "ASTIS.SALD.cycle173.middle_refiner.reject_unsourced_projection_and_score_hessian", "ASTIS.SALD.cycle173.lower_1_refiner.source_hessian_field_proof_route", "ASTIS.SALD.cycle173.lower_2_packet.source_hessian_field_bridge", "SALD.selectedWeakTestHessianOpNormOfSourceHessianField", "SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation", "SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDag", "SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm", "SALD.gaussianRealStdOrthonormalBasisUnit", "SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis", "hHessianOpNorm", "hSecondFDerivOpNorm", "sourceTest", "sourceHessian", "testRegular", "hSourceHasHessian", "hSourceHessianBound", "SourceSelectedWeakTestC2bBoundedHessian proposed only as source-backed field", "SelectedWeakTestC2bBoundedHessian projection rejected without source backing", "iteration_complexity.tex:309-321 score Hessian bound rejected for sourceTest", "appendix.tex:984-995", "appendix.tex:1026-1072", "appendix.tex:1379-1387", "main_body.tex:273-305", "paper-wide original-source search excluding sald_version_2.tex", "source-contract-gap", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle174EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticVariationDependencyNames : List String := [ "SALD.cycle174GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticVariationMiddleObligation", "SALD.cycle174GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticVariationLower1Obligation", "SALD.cycle174GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticVariationLower2Obligation", "SALD.cycle174GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticVariationDag", "ASTIS.SALD.cycle174.middle_refiner.source_hessian_gap_decision", "ASTIS.SALD.cycle174.middle_packet.quadratic_variation_normalization_lower_route", "ASTIS.SALD.cycle174.lower_1_packet.quadratic_variation_coeff_variance_route", "ASTIS.SALD.cycle174.lower_2_packet.quadratic_variation_coeff_variance_bridge", "ASTIS.SALD.cycle174.reviewer_quadratic_variation_check", "SALD.cycle173GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation", "SALD.selectedWeakTestHessianOpNormOfSourceHessianField", "SALD.selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne", "hSourceHasHessian", "hSourceHessianBound", "source-contract-gap for selected weak-test Hessian fields", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder", "SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator", "hFrozenScalarBrownianItoQuadraticVariationNormalization", "hQuadraticCoeffDef", "hVarianceOne", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle175EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationDependencyNames : List String := [ "SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationMiddleObligation", "SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticCoeffLower1Obligation", "SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticCoeffLower2Obligation", "SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationDag", "ASTIS.SALD.cycle175.middle_packet.std_basis_first_order_remainder_from_source_hessian", "ASTIS.SALD.cycle175.remaining_selected_line_taylor_boundary", "ASTIS.SALD.cycle175.lower_1_packet.quadratic_coeff_second_taylor_route", "ASTIS.SALD.cycle175.lower_2_packet.quadratic_coeff_second_taylor_bridge", "ASTIS.SALD.cycle175.reviewer_taylor_domination_check", "SALD.gaussianRealSelectedTestStdOrthonormalFirstOrderQuadraticRemainderBoundOfSourceHessianField", "SALD.selectedWeakTestHessianOpNormOfSourceHessianField", "SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm", "SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis", "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound", "SALD.selectedWeakTestQuadraticCoeffDefOfSecondTaylorCoeffDef", "hSource", "hSourceHasHessian", "hSourceHessianBound", "hSecondCoeff", "hSecondTaylorCoeffDef", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hQuadraticCoeffDef", "hVarianceOne", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1026-1072", "appendix.tex:1379-1387", "main_body.tex:273-305", "source-contract-gap for selected weak-test Hessian fields", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle176EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneDependencyNames : List String := [ "SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneMiddleObligation", "SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneLower1Obligation", "SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneLower2Obligation", "SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneDag", "ASTIS.SALD.cycle176.lower_1_packet.variance_one_normalized_source_route", "ASTIS.SALD.cycle176.middle_packet.variance_one_from_normalized_brownian_field", "ASTIS.SALD.cycle176.lower_2_packet.quadratic_variation_from_second_taylor_and_normalized_variance", "ASTIS.SALD.cycle176.remaining_variance_source_boundary", "ASTIS.SALD.cycle176.reviewer_variance_one_check", "SALD.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef", "SALD.selectedWeakTestQuadraticVariationNormalizationOfSecondTaylorCoeffAndNormalizedVarianceDef", "SALD.selectedWeakTestQuadraticCoeffDefOfSecondTaylorCoeffDef", "SALD.selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne", "hVarianceOne", "hNormalizedVarianceDef", "hSecondTaylorCoeffDef", "hSourceHasHessian", "hSourceHessianBound", "hSecondCoeff", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "eq:increment_basic_bound_revised", "appendix.tex:958-970", "appendix.tex:984-995", "appendix.tex:1170-1176", "appendix.tex:1379-1387", "source-contract-gap for selected weak-test Hessian fields", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle177EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSecondTaylorCoeffDependencyNames : List String := [ "SALD.cycle177GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSecondTaylorCoeffMiddleObligation", "SALD.cycle177GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSecondTaylorCoeffDag", "ASTIS.SALD.cycle177.middle_packet.source_hessian_gap_decision", "ASTIS.SALD.cycle177.middle_packet.scalar_line_second_coeff_bridge", "ASTIS.SALD.cycle177.remaining_scalar_line_second_coeff_boundary", "ASTIS.SALD.cycle177.reviewer_second_taylor_coeff_check", "SALD.selectedWeakTestSecondTaylorCoeffDefOfScalarLineSecondCoeffDef", "SALD.gaussianRealSelectedTestLineSecondDerivEqDirectionalFDeriv", "SALD.selectedWeakTestQuadraticVariationNormalizationOfSecondTaylorCoeffAndNormalizedVarianceDef", "SALD.selectedWeakTestQuadraticCoeffDefOfSecondTaylorCoeffDef", "hSecondTaylorCoeffDef", "hScalarLineSecondCoeffDef", "hSource", "hNormalizedVarianceDef", "hSourceHasHessian", "hSourceHessianBound", "hSecondCoeff", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "eq:increment_basic_bound_revised", "appendix.tex:958-970", "appendix.tex:984-995", "appendix.tex:1026-1072", "appendix.tex:1170-1176", "appendix.tex:1379-1387", "main_body.tex:273-305", "iteration_complexity.tex:309-321 score Hessian bound rejected for sourceTest", "source-contract-gap for selected weak-test Hessian fields", "paper-wide original-source search excluding sald_version_2.tex", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle178EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceDependencyNames : List String := [ "SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceMiddleObligation", "SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower1Obligation", "SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower2Obligation", "SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceDag", "ASTIS.SALD.cycle178.middle_packet.normalized_variance_from_gaussian_law", "ASTIS.SALD.cycle178.lower_1_packet.normalized_coordinate_law_projection_route", "ASTIS.SALD.cycle178.lower_2_packet.normalized_coordinate_law_from_vector_gaussian", "ASTIS.SALD.cycle178.remaining_normalized_coordinate_law_boundary", "ASTIS.SALD.cycle178.reviewer_normalized_variance_check", "SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw", "SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw", "ProbabilityTheory.stdGaussian", "ProbabilityTheory.IsGaussian.map_eq_gaussianReal", "ProbabilityTheory.integral_strongDual_stdGaussian", "ProbabilityTheory.variance_dual_stdGaussian", "ProbabilityTheory.variance_id_gaussianReal", "NNReal.coe_injective", "InnerProductSpace.toDual", "stdOrthonormalBasis", "hNormalizedVarianceDef", "hNormalizedCoordinateLaw", "hNormalizedVectorLaw", "hCoordinateLawDef", "hVarianceDef", "hScalarLineSecondCoeffDef", "hSourceHasHessian", "hSourceHessianBound", "hSecondCoeff", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "eq:increment_basic_bound_revised", "appendix.tex:958-970", "appendix.tex:984-995", "appendix.tex:1170-1176", "appendix.tex:1379-1387", "source-contract-gap for selected weak-test Hessian fields", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle179EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianAuditScalarLineCoeffDependencyNames : List String := [ "SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianAuditScalarLineCoeffMiddleObligation", "SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoScalarLineCoeffLower1Obligation", "SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoScalarLineCoeffLower2Obligation", "SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianAuditScalarLineCoeffDag", "ASTIS.SALD.cycle179.middle_packet.source_hessian_audit_closed", "ASTIS.SALD.cycle179.middle_packet.scalar_line_second_coeff_lower_route", "ASTIS.SALD.cycle179.middle_packet.lower_2_exact_target", "ASTIS.SALD.cycle179.lower_1_packet.scalar_line_taylor_coeff_route", "ASTIS.SALD.cycle179.lower_2_packet.scalar_line_second_coeff_from_taylor_coeff", "ASTIS.SALD.cycle179.reviewer_hessian_audit_scalar_coeff_check", "SALD.selectedWeakTestHessianOpNormOfSourceHessianField", "SALD.selectedWeakTestScalarLineSecondCoeffDefOfTaylorCoeffWithin", "SALD.selectedWeakTestSecondTaylorCoeffDefOfScalarLineSecondCoeffDef", "SALD.selectedWeakTestQuadraticVariationNormalizationOfScalarLineSecondCoeffAndNormalizedVarianceDef", "SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw", "SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw", "hSourceHasHessian", "hSourceHessianBound", "sourceHessian", "hScalarLineSecondCoeffDef", "hScalarLineTaylorCoeffDef", "taylorCoeffWithin", "iteratedDerivWithin_univ", "hNormalizedVectorLaw", "hCoordinateLawDef", "hVarianceDef", "hSecondCoeff", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "eq:increment_basic_bound_revised", "appendix.tex:958-970", "appendix.tex:984-995", "appendix.tex:1026-1072", "appendix.tex:1170-1176", "appendix.tex:1379-1387", "main_body.tex:273-305", "iteration_complexity.tex:309-321 score Hessian bound rejected for sourceTest", "paper-wide original-source search excluding sald_version_2.tex", "source-contract-gap for selected weak-test Hessian fields", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle180EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentDependencyNames : List String := [ "SALD.cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentMiddleObligation", "SALD.cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPolynomialIntegrabilityLower1Obligation", "SALD.cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDominatedRemainderLower2Obligation", "SALD.cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentDag", "ASTIS.SALD.cycle180.middle_packet.source_hessian_gap_preserved", "ASTIS.SALD.cycle180.middle_packet.taylor_moment_integral_split", "ASTIS.SALD.cycle180.lower_1_packet.polynomial_summand_integrability", "ASTIS.SALD.cycle180.lower_2_packet.dominated_remainder_integrability", "ASTIS.SALD.cycle180.remaining_taylor_integral_source_boundary", "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefs", "SALD.gaussianRealLinearQuadraticTaylorSummandsIntegrable", "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndGaussianPolynomialIntegrability", "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndDominatedRemainder", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder", "SALD.gaussianRealZeroOneDimTaylorMomentContribution", "hFrozenScalarBrownianItoTaylorMomentDecomposition", "hBrownianCoordinateGeneratorTaylorIntegralDef", "hLinearInt", "hQuadraticInt", "hLinearInt discharged by Gaussian polynomial integrability", "hQuadraticInt discharged by Gaussian polynomial integrability", "hRemainderInt", "hRemainderInt discharged by dominated remainder integrability", "hRemainderMeas", "hRemainderBound", "hRemainderBoundInt", "hRemainderGeneratorLimitDef", "normalizedRemainder", "remainderBound", "brownianCoordinateGenerator", "linearCoeff", "quadraticCoeff", "remainderGeneratorLimit", "MeasureTheory.integral_add", "MeasureTheory.integral_const_mul", "MeasureTheory.Integrable.mono'", "MeasureTheory.Integrable.const_mul", "ProbabilityTheory.integrable_exp_mul_gaussianReal", "ProbabilityTheory.integrable_pow_of_integrable_exp_mul", "hScalarLineTaylorCoeffDef", "hNormalizedVectorLaw", "hCoordinateLawDef", "hVarianceDef", "hSecondCoeff", "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "hSourceHasHessian", "hSourceHessianBound", "source-contract-gap for selected weak-test Hessian fields", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1170-1176", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "thm:general-moving-target-SALD-discrete" ] def cycle183EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralDependencyNames : List String := [ "SALD.cycle183GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralMiddleObligation", "SALD.cycle183GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegrandLower2Obligation", "SALD.cycle183GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralDag", "ASTIS.SALD.cycle183.middle_packet.source_integral_ae_bridge", "ASTIS.SALD.cycle183.lower_1_packet.taylor_integrand_ae_route", "ASTIS.SALD.cycle183.lower_2_packet.pointwise_source_taylor_identity", "ASTIS.SALD.cycle183.remaining_brownian_taylor_backend", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralAndAE", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegrandAEOfPointwise", "MeasureTheory.integral_congr_ae", "Filter.Eventually.of_forall", "hBrownianCoordinateGeneratorTaylorIntegralDef", "hBrownianCoordinateGeneratorSourceIntegralDef", "hBrownianCoordinateGeneratorTaylorIntegrandAE", "hSourceTaylorIntegrandPointwise", "sourceTaylorIntegrand", "brownianCoordinateGenerator", "linearCoeff", "quadraticCoeff", "normalizedRemainder", "variance", "ProbabilityTheory.gaussianReal", "hRemainderGeneratorLimitDef", "hRemainderMeas", "hRemainderBound", "hRemainderBoundInt", "hScalarLineTaylorCoeffDef", "hNormalizedVectorLaw", "hCoordinateLawDef", "hVarianceDef", "hSourceHasHessian", "hSourceHessianBound", "source-contract-gap for selected weak-test Hessian fields", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1170-1176", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle184EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLawDependencyNames : List String := [ "SALD.cycle184GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLawMiddleObligation", "SALD.cycle184GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedLawLower2Obligation", "SALD.cycle184GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLawDag", "ASTIS.SALD.cycle184.middle_packet.source_integral_law_bridge", "ASTIS.SALD.cycle184.lower_1_packet.normalized_law_route", "ASTIS.SALD.cycle184.lower_2_packet.normalized_law_scalar_pushforward", "ASTIS.SALD.cycle184.remaining_brownian_taylor_backend", "SALD.selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfStdGaussianVectorLaw", "SALD.selectedWeakTestBrownianCoordinateGeneratorNormalizedLawDefOfScalarPushforward", "SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw", "SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw", "MeasureTheory.integral_map", "hBrownianCoordinateGeneratorSourceIntegralDef", "hBrownianCoordinateGeneratorNormalizedLawDef", "hScalarMeas", "hNormalizedCoordinateLawDef", "hSourceTaylorIntegrandMeas", "hGeneratorPullbackDef", "hNormalizedVectorLaw", "hCoordinateLawDef", "hVarianceDef", "normalizedVectorLaw", "normalizedCoordinateLaw", "scalarBrownianCoordinate", "sourceTaylorIntegrand", "brownianCoordinateGenerator", "variance", "ProbabilityTheory.stdGaussian", "ProbabilityTheory.gaussianReal", "ProbabilityTheory.IsGaussian.map_eq_gaussianReal", "ProbabilityTheory.integral_strongDual_stdGaussian", "ProbabilityTheory.variance_dual_stdGaussian", "ProbabilityTheory.variance_id_gaussianReal", "hSourceTaylorIntegrandPointwise", "hRemainderGeneratorLimitDef", "hRemainderMeas", "hRemainderBound", "hRemainderBoundInt", "hSourceHasHessian", "hSourceHessianBound", "source-contract-gap for selected weak-test Hessian fields", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-996", "appendix.tex:1170-1176", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle185EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderLimitDependencyNames : List String := [ "SALD.cycle185GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderLimitLower2Obligation", "SALD.cycle185GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderLimitDag", "ASTIS.SALD.cycle185.lower_1_packet.remainder_generator_limit_route", "ASTIS.SALD.cycle185.lower_2_packet.remainder_limit_normalized_law_bridge", "ASTIS.SALD.cycle185.remaining_brownian_taylor_backend", "SALD.selectedWeakTestRemainderGeneratorLimitDefOfStdGaussianVectorLaw", "SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw", "SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw", "hRemainderGeneratorLimitDef", "hRemainderGeneratorNormalizedLawDef", "hNormalizedVectorLaw", "hCoordinateLawDef", "hVarianceDef", "hScalarMeas", "hNormalizedCoordinateLawDef", "hSourceTaylorIntegrandMeas", "hGeneratorPullbackDef", "hSourceTaylorIntegrandPointwise", "hRemainderMeas", "hRemainderBound", "hRemainderBoundInt", "normalizedVectorLaw", "normalizedCoordinateLaw", "normalizedRemainder", "remainderGeneratorLimit", "variance", "ProbabilityTheory.stdGaussian", "ProbabilityTheory.gaussianReal", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:984-995", "appendix.tex:1170-1176", "appendix.tex:1379-1387", "source-contract-gap for selected weak-test Hessian fields", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete", "SALD.cycle189GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralRawTermsMiddleObligation", "SALD.cycle189GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralRawTermsDag", "ASTIS.SALD.cycle189.middle_packet.coordinate_taylor_integral_raw_terms_bridge", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralRawTaylorAndTermDefs", "hBrownianCoordinateGeneratorSourceIntegralDef", "hSourceTaylorIntegrandRawDef", "hSelectedLineTaylorRawSplitDef", "hSourceLinearTermTaylorDef", "hScalarLineFirstCoeffDef", "hSourceQuadraticTermTaylorDef", "hScalarLineTaylorCoeffDef" ] def cycle186EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandDependencyNames : List String := [ "SALD.cycle186GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandPointwiseMiddleObligation", "SALD.cycle186GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandPointwiseDag", "SALD.cycle186GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceLinearTermLower2Obligation", "SALD.cycle186GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceLinearTermLower2Dag", "ASTIS.SALD.cycle186.lower_1_packet.source_taylor_integrand_pointwise_route", "ASTIS.SALD.cycle186.lower_2_packet.source_taylor_integrand_line_term_bridge", "ASTIS.SALD.cycle186.lower_1_packet.source_linear_term_route", "ASTIS.SALD.cycle186.lower_2_packet.source_linear_term_first_coeff_bridge", "ASTIS.SALD.cycle186.remaining_source_taylor_integrand_backend", "ASTIS.SALD.cycle186.remaining_source_linear_term_backend", "SALD.selectedWeakTestSourceTaylorIntegrandPointwiseOfLineTermDefs", "SALD.selectedWeakTestSourceLinearTermDefOfScalarLineFirstCoeffDef", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegrandAEOfPointwise", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralAndAE", "hSourceTaylorIntegrandPointwise", "hSourceTaylorIntegrandDef", "hSourceLinearTermDef", "hSourceLinearTermTaylorDef", "hScalarLineFirstCoeffDef", "hSourceQuadraticTermDef", "sourceTaylorIntegrand", "sourceLinearTerm", "sourceQuadraticTerm", "selectedTest", "linearCoeff", "quadraticCoeff", "normalizedRemainder", "deriv", "stdOrthonormalBasis", "hScalarMeas", "hNormalizedCoordinateLawDef", "hSourceTaylorIntegrandMeas", "hGeneratorPullbackDef", "hNormalizedRemainderMeas", "hRemainderPullbackDef", "hRemainderMeas", "hRemainderBound", "hRemainderBoundInt", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:984-995", "appendix.tex:1170-1176", "appendix.tex:1379-1387", "source-contract-gap for selected weak-test Hessian fields", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle187EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceQuadraticTermDependencyNames : List String := [ "SALD.cycle187GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceQuadraticTermMiddleObligation", "SALD.cycle187GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceQuadraticTermDag", "ASTIS.SALD.cycle187.lower_1_packet.source_quadratic_term_route", "ASTIS.SALD.cycle187.lower_2_packet.source_quadratic_term_taylor_coeff_bridge", "ASTIS.SALD.cycle187.remaining_source_quadratic_term_backend", "SALD.selectedWeakTestSourceQuadraticTermDefOfScalarLineTaylorCoeffDef", "SALD.selectedWeakTestSourceTaylorIntegrandPointwiseOfLineTermDefs", "SALD.selectedWeakTestSourceLinearTermDefOfScalarLineFirstCoeffDef", "SALD.selectedWeakTestScalarLineSecondCoeffDefOfTaylorCoeffWithin", "hSourceQuadraticTermDef", "hSourceQuadraticTermTaylorDef", "hScalarLineTaylorCoeffDef", "hSourceTaylorIntegrandDef", "hSourceLinearTermTaylorDef", "hScalarLineFirstCoeffDef", "sourceQuadraticTerm", "selectedTest", "quadraticCoeff", "taylorCoeffWithin", "Set.univ", "stdOrthonormalBasis", "hScalarMeas", "hNormalizedCoordinateLawDef", "hSourceTaylorIntegrandMeas", "hGeneratorPullbackDef", "hNormalizedRemainderMeas", "hRemainderPullbackDef", "hRemainderMeas", "hRemainderBound", "hRemainderBoundInt", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:984-995", "appendix.tex:1170-1176", "appendix.tex:1379-1387", "source-contract-gap for selected weak-test Hessian fields", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle188EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandDefDependencyNames : List String := [ "SALD.cycle188GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandDefMiddleObligation", "SALD.cycle188GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandDefDag", "SALD.cycle188GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorSplitLower2Obligation", "SALD.cycle188GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorSplitLower2Dag", "ASTIS.SALD.cycle188.lower_1_packet.source_taylor_integrand_def_route", "ASTIS.SALD.cycle188.lower_2_packet.source_taylor_integrand_raw_split_bridge", "ASTIS.SALD.cycle188.lower_2_packet.selected_line_taylor_raw_term_bridge", "ASTIS.SALD.cycle188.remaining_source_taylor_integrand_def_backend", "ASTIS.SALD.cycle188.remaining_selected_line_taylor_split_backend", "SALD.selectedWeakTestSourceTaylorIntegrandDefOfRawAndLineTaylorSplit", "SALD.selectedWeakTestSelectedLineTaylorSplitDefOfRawTaylorAndTermDefs", "SALD.selectedWeakTestSourceTaylorIntegrandPointwiseOfLineTermDefs", "SALD.selectedWeakTestSourceLinearTermDefOfScalarLineFirstCoeffDef", "SALD.selectedWeakTestSourceQuadraticTermDefOfScalarLineTaylorCoeffDef", "hSourceTaylorIntegrandDef", "hSourceTaylorIntegrandRawDef", "hSelectedLineTaylorSplitDef", "hSelectedLineTaylorRawSplitDef", "hSourceLinearTermTaylorDef", "hScalarLineFirstCoeffDef", "hSourceQuadraticTermTaylorDef", "hScalarLineTaylorCoeffDef", "sourceTaylorIntegrand", "sourceLinearTerm", "sourceQuadraticTerm", "normalizedRemainder", "selectedTest", "stdOrthonormalBasis", "hScalarMeas", "hNormalizedCoordinateLawDef", "hSourceTaylorIntegrandMeas", "hGeneratorPullbackDef", "hNormalizedRemainderMeas", "hRemainderPullbackDef", "hRemainderMeas", "hRemainderBound", "hRemainderBoundInt", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:984-995", "appendix.tex:1170-1176", "appendix.tex:1379-1387", "source-contract-gap for selected weak-test Hessian fields", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle189EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralRawTermsDependencyNames : List String := [ "SALD.cycle189GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralRawTermsMiddleObligation", "SALD.cycle189GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralRawTermsDag", "SALD.cycle189GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandRawLower2Obligation", "SALD.cycle189GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandRawLower2Dag", "ASTIS.SALD.cycle189.middle_packet.coordinate_taylor_integral_raw_terms_bridge", "ASTIS.SALD.cycle189.remaining_coordinate_taylor_integral_raw_terms_backend", "ASTIS.SALD.cycle189.lower_2_packet.source_taylor_integrand_raw_selected_increment_bridge", "ASTIS.SALD.cycle189.remaining_source_taylor_integrand_raw_backend", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralRawTaylorAndTermDefs", "SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementAndLineDef", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralAndAE", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegrandAEOfPointwise", "SALD.selectedWeakTestSourceTaylorIntegrandPointwiseOfLineTermDefs", "SALD.selectedWeakTestSourceTaylorIntegrandDefOfRawAndLineTaylorSplit", "SALD.selectedWeakTestSelectedLineTaylorSplitDefOfRawTaylorAndTermDefs", "SALD.selectedWeakTestSourceLinearTermDefOfScalarLineFirstCoeffDef", "SALD.selectedWeakTestSourceQuadraticTermDefOfScalarLineTaylorCoeffDef", "hBrownianCoordinateGeneratorTaylorIntegralDef", "hBrownianCoordinateGeneratorSourceIntegralDef", "hSourceTaylorIntegrandRawDef", "hSourceTaylorIntegrandSelectedIncrementDef", "hSelectedIncrementCoordinateLineDef", "hSelectedLineTaylorRawSplitDef", "hSourceLinearTermTaylorDef", "hScalarLineFirstCoeffDef", "hSourceQuadraticTermTaylorDef", "hScalarLineTaylorCoeffDef", "sourceTaylorIntegrand", "sourceSelectedLineIncrement", "sourceLinearTerm", "sourceQuadraticTerm", "normalizedRemainder", "selectedTest", "deriv", "taylorCoeffWithin", "Set.univ", "stdOrthonormalBasis", "ProbabilityTheory.gaussianReal", "hScalarMeas", "hNormalizedCoordinateLawDef", "hSourceTaylorIntegrandMeas", "hGeneratorPullbackDef", "hNormalizedRemainderMeas", "hRemainderPullbackDef", "hRemainderGeneratorLimitDef", "hRemainderMeas", "hRemainderBound", "hRemainderBoundInt", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:984-995", "appendix.tex:1161-1170", "appendix.tex:1170-1176", "appendix.tex:1379-1387", "source-contract-gap for selected weak-test Hessian fields", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle190EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementCoordinateLineDependencyNames : List String := [ "SALD.cycle190GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementCoordinateLineMiddleObligation", "SALD.cycle190GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementCoordinateLineDag", "ASTIS.SALD.cycle190.middle_packet.selected_increment_endpoint_coordinate_line_bridge", "ASTIS.SALD.cycle190.middle_packet.source_taylor_integrand_raw_endpoint_bridge", "ASTIS.SALD.cycle190.remaining_selected_increment_coordinate_line_backend", "SALD.selectedWeakTestSelectedIncrementCoordinateLineDefOfEndpointAndLineDef", "SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementEndpointAndLineDef", "SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementAndLineDef", "hSelectedIncrementCoordinateLineDef", "hSourceTaylorIntegrandSelectedIncrementDef", "hSelectedIncrementEndpointDef", "hSelectedEndpointCoordinateLineDef", "hSelectedLineTaylorRawSplitDef", "hSourceLinearTermTaylorDef", "hScalarLineFirstCoeffDef", "hSourceQuadraticTermTaylorDef", "hScalarLineTaylorCoeffDef", "sourceTaylorIntegrand", "sourceSelectedLineIncrement", "sourceSelectedEndpoint", "selectedTest", "stdOrthonormalBasis", "hScalarMeas", "hNormalizedCoordinateLawDef", "hSourceTaylorIntegrandMeas", "hGeneratorPullbackDef", "hNormalizedRemainderMeas", "hRemainderPullbackDef", "hRemainderGeneratorLimitDef", "hRemainderMeas", "hRemainderBound", "hRemainderBoundInt", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source-contract-gap for selected weak-test Hessian fields", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle191EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralScalarPushforwardDependencyNames : List String := [ "SALD.cycle191GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLower2Obligation", "SALD.cycle191GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLower2Dag", "ASTIS.SALD.cycle191.lower_1_packet.source_integral_scalar_pushforward_route", "ASTIS.SALD.cycle191.lower_2_packet.source_integral_scalar_pushforward_bridge", "ASTIS.SALD.cycle191.remaining_source_integral_scalar_pushforward_backend", "SALD.selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfScalarPushforwardAndStdGaussianVectorLaw", "SALD.selectedWeakTestBrownianCoordinateGeneratorNormalizedLawDefOfScalarPushforward", "SALD.selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfStdGaussianVectorLaw", "hBrownianCoordinateGeneratorSourceIntegralDef", "hBrownianCoordinateGeneratorNormalizedLawDef", "hScalarMeas", "hNormalizedCoordinateLawDef", "hSourceTaylorIntegrandMeas", "hGeneratorPullbackDef", "hNormalizedVectorLaw", "hCoordinateLawDef", "hVarianceDef", "hSourceTaylorIntegrandSelectedIncrementDef", "hSelectedIncrementEndpointDef", "hSelectedEndpointCoordinateLineDef", "hSelectedLineTaylorRawSplitDef", "hSourceLinearTermTaylorDef", "hScalarLineFirstCoeffDef", "hSourceQuadraticTermTaylorDef", "hScalarLineTaylorCoeffDef", "hRemainderGeneratorLimitDef", "hRemainderMeas", "hRemainderBound", "hRemainderBoundInt", "MeasureTheory.integral_map", "ProbabilityTheory.stdGaussian", "ProbabilityTheory.gaussianReal", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source-contract-gap for selected weak-test Hessian fields", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle192EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralScalarPushforwardDependencyNames : List String := [ "SALD.cycle192GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralLower2Obligation", "SALD.cycle192GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralLower2Dag", "ASTIS.SALD.cycle192.lower_1_packet.taylor_integral_source_integral_discharge_route", "ASTIS.SALD.cycle192.lower_2_packet.taylor_integral_scalar_pushforward_raw_terms_bridge", "ASTIS.SALD.cycle192.remaining_taylor_integral_scalar_pushforward_backend", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfScalarPushforwardRawTaylorAndTermDefs", "SALD.selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfScalarPushforwardAndStdGaussianVectorLaw", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralRawTaylorAndTermDefs", "SALD.selectedWeakTestBrownianCoordinateGeneratorNormalizedLawDefOfScalarPushforward", "SALD.selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfStdGaussianVectorLaw", "hBrownianCoordinateGeneratorTaylorIntegralDef", "hBrownianCoordinateGeneratorSourceIntegralDef", "hScalarMeas", "hNormalizedCoordinateLawDef", "hSourceTaylorIntegrandMeas", "hGeneratorPullbackDef", "hNormalizedVectorLaw", "hCoordinateLawDef", "hVarianceDef", "hSourceTaylorIntegrandRawDef", "hSelectedLineTaylorRawSplitDef", "hSourceLinearTermTaylorDef", "hScalarLineFirstCoeffDef", "hSourceQuadraticTermTaylorDef", "hScalarLineTaylorCoeffDef", "hRemainderGeneratorLimitDef", "hRemainderMeas", "hRemainderBound", "hRemainderBoundInt", "MeasureTheory.integral_map", "ProbabilityTheory.stdGaussian", "ProbabilityTheory.gaussianReal", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source-contract-gap for selected weak-test Hessian fields", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle193EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentScalarPushforwardDependencyNames : List String := [ "SALD.cycle193GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentMiddleObligation", "SALD.cycle193GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentMiddleDag", "SALD.cycle193GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentRemainderLimitLower2Obligation", "SALD.cycle193GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentRemainderLimitLower2Dag", "ASTIS.SALD.cycle193.middle_packet.taylor_moment_scalar_pushforward_remainder_discharge", "ASTIS.SALD.cycle193.lower_1_packet.remainder_limit_consumer_discharge_route", "ASTIS.SALD.cycle193.lower_2_packet.remainder_limit_consumer_scalar_pushforward_bridge", "ASTIS.SALD.cycle193.remaining_taylor_moment_backend", "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfScalarPushforwardRawTaylorAndDominatedRemainder", "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsDominatedRemainderAndRemainderLimitScalarPushforward", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfScalarPushforwardRawTaylorAndTermDefs", "SALD.selectedWeakTestRemainderGeneratorLimitDefOfScalarPushforwardAndStdGaussianVectorLaw", "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndDominatedRemainder", "SALD.selectedWeakTestRemainderGeneratorNormalizedLawDefOfScalarPushforward", "SALD.selectedWeakTestRemainderGeneratorLimitDefOfStdGaussianVectorLaw", "hBrownianCoordinateGeneratorTaylorIntegralDef", "hRemainderGeneratorLimitDef", "hScalarMeas", "hNormalizedCoordinateLawDef", "hSourceTaylorIntegrandMeas", "hGeneratorPullbackDef", "hNormalizedRemainderMeas", "hRemainderPullbackDef", "hNormalizedVectorLaw", "hCoordinateLawDef", "hVarianceDef", "hSourceTaylorIntegrandRawDef", "hSelectedLineTaylorRawSplitDef", "hSourceLinearTermTaylorDef", "hScalarLineFirstCoeffDef", "hSourceQuadraticTermTaylorDef", "hScalarLineTaylorCoeffDef", "hRemainderMeas", "hRemainderBound", "hRemainderBoundInt", "brownianCoordinateGenerator", "remainderGeneratorLimit", "normalizedRemainder", "remainderBound", "MeasureTheory.integral_map", "MeasureTheory.Integrable.mono'", "ProbabilityTheory.stdGaussian", "ProbabilityTheory.gaussianReal", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source-contract-gap for selected weak-test Hessian fields", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle197EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderBoundDefDependencyNames : List String := [ "SALD.cycle197GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderBoundDefLower2Obligation", "SALD.cycle197GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderBoundDefLower2Dag", "ASTIS.SALD.cycle197.lower_2_packet.normalized_remainder_bound_def_source_gap", "hNormalizedRemainderBoundDef", "remainderBound", "remainderBoundC", "SALD.selectedWeakTestNormalizedRemainderBoundIntOfQuadraticBound", "AutoSamplingTheory.TechnicalLemmas.SALDExtracted.selectedWeakTestNormalizedRemainderBoundIntOfQuadraticBound", "AutoSamplingTheory.TechnicalLemmas.Gaussian.integrable_const_mul_sq_gaussianReal_zero", "gaussian.quadratic-bound-integrable", "sald.normalized-remainder-bound-int-quadratic", "needed_shape=remainderBound phi x i z = remainderBoundC phi x i * z ^ 2", "leaf=hNormalizedRemainderBoundDef", "error_class=source_contract_gap_missing_remainder_bound_definition", "blocked_by=no original-paper definition of remainderBound/remainderBoundC outside sald_version_2.tex", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-996", "appendix.tex:1161-1170", "appendix.tex:1358-1387", "appendix.tex:1422-1434", "paper-wide original-source search excluding sald_version_2.tex", "source_contract_gap_missing_remainder_bound_definition", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle198EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateSumDependencyNames : List String := [ "SALD.cycle198GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateSumLower2Obligation", "SALD.cycle198GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateSumLower2Dag", "ASTIS.SALD.cycle198.lower_2_packet.event_field_coordinate_sum_source_gap", "hFrozenScalarBrownianItoEventFieldCoordinateSum", "emGeneratorLaplacianEventField", "brownianCoordinateGenerator", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator", "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor", "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfScalarPushforwardRawTaylorAndDominatedRemainder", "AutoSamplingTheory.TechnicalLemmas.SALDExtracted.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw", "SALD.emFrozenScalarBrownianItoGeneratorEventField", "needed_shape=emGeneratorLaplacianEventField phi x = Finset.univ.sum (fun i : Fin (Module.finrank Real E) => brownianCoordinateGenerator phi x i)", "leaf=hFrozenScalarBrownianItoEventFieldCoordinateSum", "error_class=source_contract_gap_missing_event_field_coordinate_sum_definition", "blocked_by=no original-paper/Lean definition connecting emGeneratorLaplacianEventField to the finite sum of brownianCoordinateGenerator", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:983-996", "appendix.tex:1379-1387", "source_contract_gap_missing_event_field_coordinate_sum_definition", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle199EmInterpolationWeakFpSourceLaplacianFieldMeasDependencyNames : List String := [ "ASTIS.SALD.cycle199.middle_packet.source_laplacian_field_meas", "ASTIS.SALD.cycle199.lower_1_packet.source_laplacian_meas_route", "ASTIS.SALD.cycle199.lower_2_packet.source_laplacian_measurable_bridge", "ASTIS.SALD.cycle199.lower_3_packet.source_laplacian_meas_retrieval", "SALD.generalMovingTargetDiscreteSourceLaplacianFieldMeasOfSelectedTestLaplacianMeasurable", "SALD.generalMovingTargetDiscreteSelectedTestLaplacianMeasurableOfContinuous", "hsourceLaplacianFieldMeas", "hSelectedTestLaplacianMeasurable", "hSelectedTestLaplacianContinuous", "Laplacian.laplacian", "Measurable.aestronglyMeasurable", "Continuous.measurable", "appendix.tex:983-996", "appendix.tex:1379-1387", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle201EmInterpolationSelectedTestLaplacianContinuityDependencyNames : List String := [ "SALD.cycle201GeneralMovingTargetDiscreteEmInterpolationSelectedTestLaplacianContinuityLower2Obligation", "SALD.cycle201GeneralMovingTargetDiscreteEmInterpolationSelectedTestLaplacianContinuityLower2Dag", "ASTIS.SALD.cycle201.lower_2_packet.selected_test_laplacian_continuity_source_gap", "SALD.generalMovingTargetDiscreteSelectedTestLaplacianMeasurableOfContinuous", "SALD.generalMovingTargetDiscreteSourceLaplacianFieldMeasOfSelectedTestLaplacianMeasurable", "hSelectedTestLaplacianContinuous", "hSelectedTestLaplacianMeasurable", "hsourceLaplacianFieldMeas", "selectedTest", "testRegular", "Laplacian.laplacian", "Continuous.measurable", "Measurable.aestronglyMeasurable", "needed_shape=testRegular -> forall phi, Continuous (Laplacian.laplacian (selectedTest phi))", "leaf=hSelectedTestLaplacianContinuous", "error_class=source_contract_gap_missing_selected_test_laplacian_continuity", "blocked_by=no original-paper line found that states selected-test Laplacian continuity/measurability", "appendix.tex:724-727", "appendix.tex:1028-1070", "appendix.tex:1313-1316", "appendix.tex:983-996", "appendix.tex:1379-1387", "source_contract_gap_missing_selected_test_laplacian_continuity", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle202EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedEndpointCoordinateLineDependencyNames : List String := [ "SALD.cycle202GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedEndpointCoordinateLineLower2Obligation", "SALD.cycle202GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedEndpointCoordinateLineLower2Dag", "ASTIS.SALD.cycle202.lower_2_packet.selected_endpoint_coordinate_line_source_gap", "SALD.selectedWeakTestSelectedIncrementCoordinateLineDefOfEndpointAndLineDef", "SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementEndpointAndLineDef", "SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementAndLineDef", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfScalarPushforwardRawTaylorAndTermDefs", "AutoSamplingTheory.TechnicalLemmas.SALDExtracted.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw", "hSelectedEndpointCoordinateLineDef", "hSelectedIncrementEndpointDef", "hSelectedIncrementCoordinateLineDef", "hSourceTaylorIntegrandRawDef", "sourceSelectedEndpoint", "sourceSelectedLineIncrement", "sourceTaylorIntegrand", "selectedTest", "stdOrthonormalBasis", "needed_shape=testRegular -> forall phi x i z, sourceSelectedEndpoint phi x i z = x + z smul stdOrthonormalBasis Real E i", "leaf=hSelectedEndpointCoordinateLineDef", "error_class=source_contract_gap_missing_selected_endpoint_coordinate_line_definition", "blocked_by=sourceSelectedEndpoint is an abstract Lean parameter and the original paper source does not provide a Lean-facing endpoint coordinate-line definition", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source_contract_gap_missing_selected_endpoint_coordinate_line_definition", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle203EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementEndpointDependencyNames : List String := [ "SALD.cycle203GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementEndpointLower2Obligation", "SALD.cycle203GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementEndpointLower2Dag", "ASTIS.SALD.cycle203.lower_2_packet.selected_increment_endpoint_source_gap", "SALD.selectedWeakTestSelectedIncrementCoordinateLineDefOfEndpointAndLineDef", "SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementEndpointAndLineDef", "SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementAndLineDef", "SALD.cycle202GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedEndpointCoordinateLineLower2Obligation", "hSelectedIncrementEndpointDef", "hSelectedEndpointCoordinateLineDef", "hSelectedIncrementCoordinateLineDef", "hSourceTaylorIntegrandRawDef", "sourceSelectedLineIncrement", "sourceSelectedEndpoint", "sourceTaylorIntegrand", "selectedTest", "stdOrthonormalBasis", "needed_shape=testRegular -> forall phi x i z, sourceSelectedLineIncrement phi x i z = selectedTest phi (sourceSelectedEndpoint phi x i z) - selectedTest phi x", "leaf=hSelectedIncrementEndpointDef", "error_class=source_contract_gap_missing_selected_increment_endpoint_definition", "blocked_by=sourceSelectedLineIncrement is an abstract source-facing parameter in the compiled cycle-190 bridges", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source_contract_gap_missing_selected_increment_endpoint_definition", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle204EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandSelectedIncrementDependencyNames : List String := [ "SALD.cycle204GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandSelectedIncrementLower2Obligation", "SALD.cycle204GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandSelectedIncrementLower2Dag", "ASTIS.SALD.cycle204.lower_2_packet.source_taylor_integrand_selected_increment_source_gap", "SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementAndLineDef", "SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementEndpointAndLineDef", "SALD.selectedWeakTestSelectedIncrementCoordinateLineDefOfEndpointAndLineDef", "SALD.cycle202GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedEndpointCoordinateLineLower2Obligation", "SALD.cycle203GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementEndpointLower2Obligation", "hSourceTaylorIntegrandSelectedIncrementDef", "hSourceTaylorIntegrandRawDef", "hSelectedIncrementCoordinateLineDef", "hSelectedIncrementEndpointDef", "hSelectedEndpointCoordinateLineDef", "sourceTaylorIntegrand", "sourceSelectedLineIncrement", "sourceSelectedEndpoint", "selectedTest", "stdOrthonormalBasis", "needed_shape=testRegular -> forall phi x i z, sourceTaylorIntegrand phi x i z = sourceSelectedLineIncrement phi x i z", "leaf=hSourceTaylorIntegrandSelectedIncrementDef", "error_class=source_contract_gap_missing_source_taylor_integrand_selected_increment_definition", "blocked_by=sourceTaylorIntegrand and sourceSelectedLineIncrement are abstract parameters in the compiled selected-increment/raw-integrand bridges", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source_contract_gap_missing_source_taylor_integrand_selected_increment_definition", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle205EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorRawSplitDependencyNames : List String := [ "SALD.cycle205GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorRawSplitLower2Obligation", "SALD.cycle205GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorRawSplitLower2Dag", "ASTIS.SALD.cycle205.lower_2_packet.selected_line_taylor_raw_split_source_gap", "SALD.selectedWeakTestSelectedLineTaylorSplitDefOfRawTaylorAndTermDefs", "SALD.selectedWeakTestSourceTaylorIntegrandDefOfRawAndLineTaylorSplit", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralRawTaylorAndTermDefs", "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfScalarPushforwardRawTaylorAndTermDefs", "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfScalarPushforwardRawTaylorAndDominatedRemainder", "SALD.cycle188GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorSplitLower2Obligation", "SALD.cycle204GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandSelectedIncrementLower2Obligation", "runs/20260613-050236-675526-ASTIS-SALD-001-cycle205/lower_1_selected_line_taylor_raw_split_route.md", "runs/20260613-050236-675526-ASTIS-SALD-001-cycle205/lower_3_selected_line_taylor_api_churn_rejection.md", "hSelectedLineTaylorRawSplitDef", "hSelectedLineTaylorSplitDef", "hSourceTaylorIntegrandSelectedIncrementDef", "hBrownianCoordinateGeneratorTaylorIntegralDef", "hRemainderGeneratorLimitDef", "selectedTest", "normalizedRemainder", "deriv", "taylorCoeffWithin", "Set.univ", "stdOrthonormalBasis", "needed_shape=testRegular -> forall phi x i z, selectedTest phi (x + z smul stdOrthonormalBasis Real E i) - selectedTest phi x = deriv (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 0 * z + ((2 : Real) * taylorCoeffWithin (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 2 Set.univ 0) * z ^ 2 + normalizedRemainder phi x i z", "leaf=hSelectedLineTaylorRawSplitDef", "error_class=source_contract_gap_missing_selected_line_taylor_raw_split_definition", "blocked_by=normalizedRemainder is an abstract parameter in the compiled Taylor bridges and no original-paper definition identifies it with the selected scalar Taylor residual", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1176", "appendix.tex:1379-1387", "source_contract_gap_missing_selected_line_taylor_raw_split_definition", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle206EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackDependencyNames : List String := [ "SALD.cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackMiddleObligation", "SALD.cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackMiddleDag", "SALD.cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackLower2Obligation", "SALD.cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackLower2Dag", "ASTIS.SALD.cycle206.middle_packet.remainder_pullback_source_gap", "ASTIS.SALD.cycle206.lower_2_packet.remainder_pullback_source_gap", "SALD.selectedWeakTestRemainderGeneratorNormalizedLawDefOfScalarPushforward", "SALD.selectedWeakTestRemainderGeneratorLimitDefOfStdGaussianVectorLaw", "SALD.selectedWeakTestRemainderGeneratorLimitDefOfScalarPushforwardAndStdGaussianVectorLaw", "AutoSamplingTheory.TechnicalLemmas.SALDExtracted.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw", "AutoSamplingTheory.TechnicalLemmas.Gaussian.nnrealVarianceOneOfGaussianRealUnitLaw", "hRemainderPullbackDef", "hRemainderGeneratorLimitDef", "hRemainderGeneratorNormalizedLawDef", "hScalarMeas", "hNormalizedCoordinateLawDef", "hNormalizedRemainderMeas", "hNormalizedVectorLaw", "hCoordinateLawDef", "hVarianceDef", "scalarBrownianCoordinate", "normalizedCoordinateLaw", "normalizedRemainder", "remainderGeneratorLimit", "MeasureTheory.integral_map", "runs/20260613-052444-683667-ASTIS-SALD-001-cycle206/lower_1_remainder_pullback_route.md", "runs/20260613-052444-683667-ASTIS-SALD-001-cycle206/middle_remainder_pullback_packet.md", "runs/20260613-052444-683667-ASTIS-SALD-001-cycle206/lower_2_remainder_pullback_boundary.md", "needed_shape=testRegular -> forall phi x i, remainderGeneratorLimit phi x i = integral omega, normalizedRemainder phi x i (scalarBrownianCoordinate phi x i omega) dP", "leaf=hRemainderPullbackDef", "error_class=source_contract_gap_missing_remainder_pullback_definition", "blocked_by=remainderGeneratorLimit, normalizedRemainder, and scalarBrownianCoordinate are abstract parameters in the compiled scalar-pushforward remainder bridge", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "appendix.tex:958-970", "appendix.tex:983-996", "appendix.tex:1161-1170", "appendix.tex:1379-1387", "source_contract_gap_missing_remainder_pullback_definition", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "divergence/FI/IBP handoff", "thm:general-moving-target-SALD-discrete" ] def cycle63UnifiedDiscreteGeneralDependencyNames : List String := [ "SALD.cycle63UnifiedDiscreteGeneralSkeletonUpperPacket", "SALD.cycle63UnifiedDiscreteGeneralSkeletonObligation", "SALD.cycle63UnifiedDiscreteGeneralMiddleContract", "SALD.cycle63UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation", "SALD.cycle63UnifiedDiscreteGeneralDag", "ASTIS.SALD.unified_discrete_general.cycle63_global_phase_judgment", "ASTIS.SALD.unified_discrete_general.cycle63_five_backend_check", "ASTIS.SALD.unified_discrete_general.cycle63_upper_route", "ASTIS.SALD.unified_discrete_general.cycle63_middle_route_audit", "ASTIS.SALD.general_moving_target_discrete.cycle63_joint_endpoint_law_backfill", "sald.unified_discrete_general.cycle63_upper_route", "sald.unified_discrete_general.cycle63_middle_route_audit", "sald.general_moving_target_discrete.cycle63_joint_endpoint_law_backfill", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "AutoSamplingTheory.lawMapProdEqOfAEEq", "AutoSamplingTheory.lawMapProdFst", "AutoSamplingTheory.lawMapProdSnd", "AutoSamplingTheory.lawMapEqOfAEEq", "SALD.cycle62GuidedGeneralSkeletonMiddleObligation", "SALD.cycle62GuidedGeneralScaledResidualLowerObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation", "SALD.cycle58UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle53UnifiedDiscreteGeneralMiddleObligation", "SALD.guidedResidualIdentityContract", "sald.guided_path_residual.identity", "SALD.generalMovingTargetStatementContract", "SALD.generalMovingTargetDerivativeCandidateContract", "sald.general_moving_target.kl_derivative", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_velocity_bridge", "sald.unified_forward_kl.specialization", "SALD.generalMovingTargetDiscreteStatementContract", "SALD.generalMovingTargetDiscreteDerivativeCandidateContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_side_conditions", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "def:alpha-complexity" ] def cycle64MainSkeletonDependencyNames : List String := [ "SALD.cycle64MainSkeletonAnalyticInterfaceLedger", "SALD.cycle64MainSkeletonAnalyticInterfaceObligation", "SALD.cycle64MainSkeletonAnalyticMiddleContract", "SALD.cycle64MainSkeletonAnalyticMiddleObligation", "SALD.cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation", "SALD.cycle64MainSkeletonAnalyticInterfaceDag", "ASTIS.SALD.cycle64.global_phase_judgment", "ASTIS.SALD.cycle64.five_backend_check", "ASTIS.SALD.cycle64.theorem_route_rewire", "ASTIS.SALD.cycle64.middle_interface_audit", "ASTIS.SALD.cycle64.lower_packet.general_discrete_em_conditional_fp", "sald.main_skeleton.cycle64_analytic_interface_ledger", "sald.main_skeleton.cycle64_middle_interface_audit", "SALD.cycle63UnifiedDiscreteGeneralMiddleObligation", "SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation", "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation", "SALD.cycle60ForwardKlSkeletonMiddleObligation", "SALD.cycle59MainSkeletonAnalyticInterfaceObligation", "SALD.saldGronwallEndpointCalculusContract", "dvVariationalFormulaInterface saldDvVariationSource", "SALD.saldLsiKlFiDensityTestContract", "SALD.forwardKlDerivativeSideConditionContract", "SALD.generalMovingTargetDerivativeCandidateContract", "SALD.discreteForwardKlEmInterpolationSideConditionContract", "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract", "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation", "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation", "SALD.generalMovingTargetDiscreteConditionalDriftContract", "SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination", "SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination", "sald.general_moving_target_discrete.cycle64_conditional_drift_lower", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.discrete_forward_kl.em_interpolation_fp", "lem:gronwall", "lem:dv_variation", "eq:LSI-KL-FI", "thm:forward-KL", "thm:forward-KL-discrete", "prop:guided_path_residual", "thm:general-moving-target-SALD", "thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete" ] def saldDependenciesForLabel (label : String) : List String := if label = "lem:gronwall" then cycle49MainSkeletonDependencyNames ++ cycle54MainSkeletonDependencyNames ++ cycle59MainSkeletonDependencyNames ++ cycle64MainSkeletonDependencyNames ++ cycle69MainSkeletonDependencyNames ++ [ "SALD.cycle44MainSkeletonAnalyticInterfaceLedger", "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle44MainSkeletonAnalyticInterfaceDag", "SALD.cycle13FirstAppendixSourceIndexAuditContract", "SALD.cycle13FirstAppendixMiddleAuditContract", "SALD.cycle17FirstAppendixVocabularyPacket", "SALD.cycle17FirstAppendixMiddleAuditContract", "SALD.cycle21FirstAppendixVocabularyPacket", "SALD.cycle21FirstAppendixMiddleAuditContract", "SALD.cycle25FirstAppendixVocabularyPacket", "SALD.cycle25FirstAppendixMiddleAuditContract", "SALD.cycle29FirstAppendixVocabularyPacket", "SALD.cycle29FirstAppendixMiddleAuditContract", "SALD.cycle36GronwallUpperPacket", "SALD.cycle36GronwallUpperObligation", "SALD.cycle36GronwallMiddleObligation", "SALD.cycle41GronwallMiddleObligation", "SALD.cycle41GronwallLowerObligation", "sald.gronwall.cycle36_upper_packet", "sald.gronwall.cycle36_middle_assembly", "sald.gronwall.cycle41_deriv_wrapper", "sald.gronwall.cycle41_interior_endpoint_bridge", "sald.first_appendix.source_index_audit", "sald.first_appendix.middle_source_to_lean_map", "SALD.saldGronwallCandidateContract", "SALD.saldGronwallEndpointCalculusContract", "SALD.saldGronwallExponentRewriteContract", "SALD.gronwallNegIntegralRewriteScalar", "SALD.gronwallExpProductRewriteScalar", "SALD.gronwallIntervalIntegralAdditivityScalar", "SALD.gronwallExpProductRewriteIntervalIntegral", "SALD.gronwallExpProductRewriteIntegralCongr", "SALD.gronwallIntegratingFactorProductDerivative", "SALD.gronwallIntegratingFactorDerivativeInequalityScalar", "SALD.gronwallIntegratingFactorDerivativeLe", "SALD.gronwallIntegratingFactorDerivativeLeOfIntegral", "SALD.gronwallOrderIntegrationOfHasDerivAt", "SALD.gronwallOrderIntegrationOfHasDerivRight", "SALD.gronwallEndpointEvaluationScalar", "SALD.gronwallEndpointMultiplyByExpNegScalar", "SALD.gronwallEndpointIntegralRewrite", "SALD.gronwallIntegratingFactorBoundOfDerivatives", "SALD.gronwallIntegratingFactorBoundOfIntegral", "SALD.gronwallIntegratingFactorBoundOfInteriorDerivatives", "SALD.gronwallCoefficientSideConditionsOfContinuous", "SALD.gronwallIntegratingFactorBoundOfContinuousData", "SALD.gronwallIntegratingFactorBoundOfDifferentiable", "SALD.gronwallIntegratingFactorBoundOfC1", "SALD.gronwallIntegratingFactorBoundOfInteriorContinuousData", "SALD.gronwallIntegratingFactorBoundOfInteriorC1", "sald.gronwall.integrating_factor", "sald.gronwall.endpoint_calculus", "sald.gronwall.exponent_rewrite", "Real intervalIntegral backend", "endpoint-safe differentiability on [0,t1]" ] else if label = "lem:dv_variation" then cycle49MainSkeletonDependencyNames ++ cycle54MainSkeletonDependencyNames ++ cycle59MainSkeletonDependencyNames ++ cycle64MainSkeletonDependencyNames ++ cycle69MainSkeletonDependencyNames ++ [ "SALD.cycle44MainSkeletonAnalyticInterfaceLedger", "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle44MainSkeletonAnalyticInterfaceDag", "SALD.cycle13FirstAppendixSourceIndexAuditContract", "SALD.cycle13FirstAppendixMiddleAuditContract", "SALD.cycle17FirstAppendixVocabularyPacket", "SALD.cycle17FirstAppendixMiddleAuditContract", "SALD.cycle21FirstAppendixVocabularyPacket", "SALD.cycle21FirstAppendixMiddleAuditContract", "SALD.cycle25FirstAppendixVocabularyPacket", "SALD.cycle25FirstAppendixMiddleAuditContract", "SALD.cycle29FirstAppendixVocabularyPacket", "SALD.cycle29FirstAppendixMiddleAuditContract", "SALD.cycle32DvVariationUpperPacket", "SALD.cycle32DvVariationInterfaceObligation", "SALD.cycle32DvVariationMiddleAuditContract", "SALD.cycle32DvVariationMiddleObligation", "SALD.cycle32DvVariationLowerObligation", "SALD.cycle37DvVariationUpperPacket", "SALD.cycle37DvVariationUpperObligation", "SALD.cycle37DvVariationMiddleAuditContract", "SALD.cycle37DvVariationMiddleObligation", "SALD.cycle37DvVariationLowerObligation", "SALD.cycle42DvVariationMiddleAuditContract", "SALD.cycle42DvVariationMiddleObligation", "SALD.cycle42DvVariationLowerObligation", "AutoSamplingTheory.dvVariationalOneSidedConsequenceScalar", "AutoSamplingTheory.dvVariationalOneSidedFromSupremumScalar", "AutoSamplingTheory.dvVariationalOneSidedOfTiltedRight", "AutoSamplingTheory.dvVariationalTiltedRightOneSidedConsequence", "AutoSamplingTheory.dvFiniteLogMgfOfLeAlpha", "AutoSamplingTheory.dvVariationalOneSidedOfScaledTest", "AutoSamplingTheory.dvVariationalScaledTestEnergyBound", "AutoSamplingTheory.dvVariationalScaledTestEnergyBoundWithCoeff", "sald.first_appendix.source_index_audit", "sald.first_appendix.middle_source_to_lean_map", "probability.dv_variational_formula", "dvVariationalFormulaInterface", "SALD.saldDvFiniteLogMgfContract", "sald.dv_variation.finite_log_mgf_interface", "KLContract", "common probability space" ] else if label = "def:PI" then ["SALD.cycle13FirstAppendixSourceIndexAuditContract", "SALD.cycle13FirstAppendixMiddleAuditContract", "SALD.cycle17FirstAppendixVocabularyPacket", "SALD.cycle17FirstAppendixMiddleAuditContract", "SALD.cycle21FirstAppendixVocabularyPacket", "SALD.cycle21FirstAppendixMiddleAuditContract", "SALD.cycle25FirstAppendixVocabularyPacket", "SALD.cycle25FirstAppendixMiddleAuditContract", "SALD.cycle29FirstAppendixVocabularyPacket", "SALD.cycle29FirstAppendixMiddleAuditContract", "sald.first_appendix.source_index_audit", "sald.first_appendix.middle_source_to_lean_map", "PIContract", "SALD.saldPiVelocityNormDependencyContract", "SALD.piVelocityNormBackendObligation", "SALD.cycle25FirstAppendixPiVelocityNormMiddleObligation", "SALD.cycle25PiVelocityNormLowerObligation", "SALD.piVelocityNormMeanZeroH1UpperScalar", "SALD.piVelocityNormBoundedFunctionalScalar", "sald.pi.velocity_norm_backend", "variance vocabulary", "weighted Sobolev vocabulary", "lem:velocity-norm-bound"] else if label = "eq:LSI-KL-FI" then cycle49MainSkeletonDependencyNames ++ cycle54MainSkeletonDependencyNames ++ cycle59MainSkeletonDependencyNames ++ cycle64MainSkeletonDependencyNames ++ cycle69MainSkeletonDependencyNames ++ [ "SALD.cycle44MainSkeletonAnalyticInterfaceLedger", "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle44MainSkeletonAnalyticInterfaceDag", "SALD.cycle13FirstAppendixSourceIndexAuditContract", "SALD.cycle13FirstAppendixMiddleAuditContract", "SALD.cycle17FirstAppendixVocabularyPacket", "SALD.cycle17FirstAppendixMiddleAuditContract", "SALD.cycle21FirstAppendixVocabularyPacket", "SALD.cycle21FirstAppendixMiddleAuditContract", "SALD.cycle25FirstAppendixVocabularyPacket", "SALD.cycle25FirstAppendixMiddleAuditContract", "SALD.cycle29FirstAppendixVocabularyPacket", "SALD.cycle29FirstAppendixMiddleAuditContract", "sald.first_appendix.source_index_audit", "sald.first_appendix.middle_source_to_lean_map", "KLContract", "FIContract", "LSIContract", "SALD.saldLsiKlFiDensityTestContract", "SALD.lsiKlFiDensityTestObligation", "SALD.cycle29LsiKlFiDensityTestMiddleObligation", "SALD.cycle29LsiKlFiDensityTestLowerObligation", "SALD.cycle33LsiKlFiDensityTestMiddleObligation", "SALD.cycle33LsiKlFiDensityTestLowerObligation", "SALD.cycle38LsiKlFiUpperPacket", "SALD.cycle38LsiKlFiUpperObligation", "SALD.cycle38LsiKlFiMiddleObligation", "SALD.cycle38LsiKlFiLowerObligation", "SALD.cycle43LsiKlFiUpperPacket", "SALD.cycle43LsiKlFiUpperObligation", "SALD.cycle43LsiKlFiMiddleObligation", "SALD.cycle43LsiKlFiLowerObligation", "SALD.lsiKlFiCoefficientAuditScalar", "SALD.lsiKlFiDensityTestBridgeScalar", "SALD.lsiKlFiHalfFisherScalar", "SALD.lsiKlFiDensityTestHalfFisherScalar", "AutoSamplingTheory.lsiKlFiSqrtDensitySquareScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityEntropyIntegrandScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityNormalizationScalar", "AutoSamplingTheory.lsiKlFiRnDerivLIntegralMassOne", "AutoSamplingTheory.lsiKlFiRnDerivDensityMassOne", "AutoSamplingTheory.lsiKlFiSqrtRnDerivTestMassOne", "AutoSamplingTheory.lsiKlFiRnDerivEntropyIntegral", "AutoSamplingTheory.lsiKlFiSqrtRnDerivEntropyIntegral", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainOfDerivativesScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumHandoffScalar", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralFiniteSum", "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralHandoffScalar", "sald.lsi_kl_fi.density_test_interface", "sald.lsi_kl_fi.cycle29_density_test_middle", "sald.lsi_kl_fi.cycle29_density_test_lower", "sald.lsi_kl_fi.cycle33_density_test_middle", "sald.lsi_kl_fi.cycle33_density_test_lower", "sald.lsi_kl_fi.cycle38_upper_packet", "sald.lsi_kl_fi.cycle38_middle_fisher_chain", "sald.lsi_kl_fi.cycle38_lower_finite_sum_fisher_chain", "sald.lsi_kl_fi.cycle43_upper_packet", "sald.lsi_kl_fi.cycle43_middle_density_entropy", "sald.lsi_kl_fi.cycle43_lower_integral_fisher_chain", "probability.lsi_to_kl_fi" ] else if label = "thm:forward-KL" then cycle49MainSkeletonDependencyNames ++ cycle54MainSkeletonDependencyNames ++ cycle59MainSkeletonDependencyNames ++ cycle64MainSkeletonDependencyNames ++ cycle69MainSkeletonDependencyNames ++ cycle50ForwardKlDependencyNames ++ cycle55ForwardKlDependencyNames ++ cycle60ForwardKlDependencyNames ++ cycle65ForwardKlDependencyNames ++ ["SALD.cycle44MainSkeletonAnalyticInterfaceLedger", "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle44MainSkeletonAnalyticInterfaceDag", "SALD.cycle45ForwardKlSkeletonUpperPacket", "SALD.cycle45ForwardKlSkeletonObligation", "SALD.cycle45ForwardKlSkeletonMiddleContract", "SALD.cycle45ForwardKlSkeletonMiddleObligation", "SALD.cycle45ForwardKlSkeletonDag", "ASTIS.SALD.forward_KL.cycle45_theorem_skeleton_route", "sald.forward_kl.cycle45_middle_route_audit", "SALD.cycle14ForwardKlUpperPacket", "SALD.cycle14ForwardKlMiddleContract", "SALD.cycle18ForwardKlUpperPacket", "SALD.cycle18ForwardKlMiddleContract", "SALD.cycle22ForwardKlUpperPacket", "SALD.cycle22ForwardKlMiddleContract", "SALD.cycle26ForwardKlUpperPacket", "SALD.cycle26ForwardKlMiddleContract", "SALD.cycle30ForwardKlUpperPacket", "SALD.cycle30ForwardKlMiddleContract", "SALD.cycle34ForwardKlDerivativeUpperPacket", "SALD.cycle34ForwardKlDerivativeMiddleContract", "SALD.cycle39ForwardKlDerivativeUpperPacket", "SALD.cycle39ForwardKlDerivativeMiddleContract", "sald.forward_kl.cycle39_derivative_upper", "sald.forward_kl.cycle39_derivative_middle", "sald.forward_kl.middle_source_to_lean_map", "SALD.forwardKlEndpointScheduleContract", "sald.forward_kl.endpoint_schedule_identities", "eq:SALD", "eq:FP-eq", "eq:LSI-KL-FI", "def:alpha-complexity", "lem:dv_variation", "lem:gronwall", "sald.forward_kl.moving_target_dependency_chain", "sald.forward_kl.cycle30_derivative_side_upper", "sald.forward_kl.cycle30_derivative_side_middle", "SALD.forwardKlFirstTermFisherSubstitutionScalar", "SALD.cycle30ForwardKlDensityBoundaryLowerObligation", "sald.forward_kl.cycle30_density_boundary_lower", "SALD.forwardKlTargetTransportYoungBoundScalar", "SALD.forwardKlPostYoungDerivativeBoundScalar", "SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar", "SALD.forwardKlLsiDerivativeBoundScalar", "SALD.forwardKlLsiDerivativeBoundOfKlFiScalar", "SALD.forwardKlTimeChangedDerivativeBoundScalar", "SALD.forwardKlInverseScheduleDerivativeScalar", "SALD.forwardKlTimeChangeSquareCoefficientRewriteOfProductScalar", "SALD.forwardKlVelocitySquareScalingScalar", "SALD.forwardKlTimeChangedDerivativeBoundOfProductScalar", "SALD.forwardKlPreDvDerivativeBoundScalar", "SALD.forwardKlPreDvDerivativeBoundOfProductScalar", "SALD.forwardKlPreDvDerivativeBoundOfVelocityScalingScalar", "SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar", "SALD.cycle34ForwardKlDerivativeScalarObligation", "SALD.cycle34ForwardKlTargetYoungLowerObligation", "SALD.cycle34ForwardKlDerivativeMiddleObligation", "SALD.cycle39ForwardKlDerivativeUpperObligation", "SALD.cycle39ForwardKlDerivativeMiddleObligation", "sald.forward_kl.cycle34_target_young_lower", "sald.forward_kl.cycle34_derivative_scalar", "sald.forward_kl.cycle34_derivative_middle", "sald.forward_kl.coefficient_chain_audit", "sald.forward_kl.gronwall_side_conditions", "SALD.gronwallNegIntegralRewriteScalar", "SALD.gronwallExpProductRewriteScalar", "SALD.gronwallExpProductRewriteIntervalIntegral", "SALD.gronwallExpProductRewriteIntegralCongr", "SALD.forwardKlGronwallCoeffIntervalIntegrable", "SALD.forwardKlGronwallCoeffAdjacentIntervalIntegrable", "SALD.forwardKlGronwallExpProductRewriteIntegralCongrOfPieces", "SALD.forwardKlGronwallCoeffIntegralSub", "SALD.forwardKlGronwallInitialExponentSplitScalar", "SALD.forwardKlGronwallInitialExponentSplitOfPieces", "SALD.forwardKlGronwallResidualExponentDropScalar", "SALD.forwardKlGronwallResidualExponentDropIntegral", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "sald.forward_kl.kl_derivative", "sald.forward_kl.cycle26_dv_witness_middle", "sald.forward_kl.dv_alpha_mgf_monotonicity", "SALD.forwardKlDvPositiveAlphaScalingScalar", "SALD.forwardKlDvPositiveAlphaCoefficientScalar", "SALD.forwardKlPostDvGronwallCoefficientScalar", "SALD.forwardKlPostDvGronwallCoefficientOfScheduleScalar", "sald.forward_kl.cycle26_dv_positive_alpha_lower", "sald.forward_kl.dv_finite_log_mgf_witness", "sald.forward_kl.dv_energy_bound", "sald.forward_kl.gronwall_application"] else if label = "thm:forward-KL-discrete" then cycle49MainSkeletonDependencyNames ++ cycle54MainSkeletonDependencyNames ++ cycle59MainSkeletonDependencyNames ++ cycle64MainSkeletonDependencyNames ++ cycle69MainSkeletonDependencyNames ++ cycle70EmConditionalLawDependencyNames ++ cycle71EmEndpointConditionalDependencyNames ++ cycle72EmWeakFpDependencyNames ++ cycle73EmKlDerivativeWeakFpDependencyNames ++ cycle74EmConditionalKernelMeasureDependencyNames ++ cycle75EmConditionalLawBackfillDependencyNames ++ cycle76EmEndpointConditionalDependencyNames ++ cycle77EmWeakFpSourceSignsDependencyNames ++ cycle78EmKlDerivativeGeneratorDependencyNames ++ cycle79EmWeakFpGeneratorMeasureDependencyNames ++ cycle80EmConditionalLawMeasurabilityDependencyNames ++ cycle81EmEndpointConditionalDependencyNames ++ cycle82EmWeakFpSourceSignsDependencyNames ++ cycle83EmKlDerivativeEndpointWeakFpDependencyNames ++ cycle84ActiveEmBackendDependencyNames ++ cycle85EmConditionalKernelBoundaryDependencyNames ++ cycle86EmWeakFpGeneratorBoundaryDependencyNames ++ cycle87EmKlLogRatioBoundaryDependencyNames ++ cycle88EmKlLogRatioAdmissibilityDependencyNames ++ cycle89DiscreteForwardKlPressureDependencyNames ++ cycle90DiscreteForwardKlMassConservationDependencyNames ++ cycle91EmConditionalKernelDependencyNames ++ cycle92EmWeakFpGeneratorSplitDependencyNames ++ cycle93EmKlMassDerivativeDependencyNames ++ cycle94EmWeakFpDriftActionDependencyNames ++ cycle95DiscreteForwardKlPressureDependencyNames ++ cycle96EmCondexpGeneratorPairingDependencyNames ++ cycle97EmCanonicalCondDistribPairingDependencyNames ++ cycle98EmBarBDivergenceNoBoundaryDependencyNames ++ cycle99EmRawKlDerivativeDependencyNames ++ cycle100EmBarBWeakGradDefDependencyNames ++ cycle101DiscreteForwardKlPressureDependencyNames ++ cycle102DiscreteForwardKlZeroFluxTraceBoundaryDependencyNames ++ cycle103EmConditionalKernelComponentVersionDependencyNames ++ cycle104EmWeakFpNamedLawTransportDependencyNames ++ cycle105EmPureRawKlDerivativeDependencyNames ++ cycle106EmCanonicalCondDistribDriftDependencyNames ++ cycle107DiscreteForwardKlBoundaryFluxDependencyNames ++ cycle108DiscreteForwardKlHatRhoBarBFluxContinuityDependencyNames ++ cycle108DiscreteForwardKlHatRhoBarBFluxDerivativeDependencyNames ++ cycle109EmNamedBarBSourceDefDependencyNames ++ cycle110EmNamedBarBEqMeasDependencyNames ++ cycle110EmWeakFpDominatedGeneratorDependencyNames ++ cycle111EmKlTargetTimeDerivativeDependencyNames ++ cycle112EmNamedBarBCondExpRepresentativeDependencyNames ++ cycle113EmNamedBarBStateFieldRegularityDependencyNames ++ cycle114EmCanonicalBarBStateEventSetIntegralDependencyNames ++ cycle115EmNamedBarBSelectedVersionDependencyNames ++ cycle116EmCanonicalBarBCondExpDependencyNames ++ cycle117EmNamedBarBVersionSelectionDependencyNames ++ cycle118EmCanonicalBarBDownstreamDependencyNames ++ cycle119EmCanonicalBarBWeakFpConsumerDependencyNames ++ cycle120EmPathDerivativeDominationDependencyNames ++ cycle121EmSampleMeasDependencyNames ++ cycle122EmSampleIntDependencyNames ++ cycle123EmSampleDerivMeasDependencyNames ++ cycle124EmSampleDerivBoundDependencyNames ++ cycle124EmBoundIntDependencyNames ++ cycle125EmPathDerivDependencyNames ++ cycle126EmDerivValueDependencyNames ++ cycle127EmDriftActionDependencyNames ++ cycle128EmNoBoundaryTraceDependencyNames ++ cycle129EmDiffusionSourceDependencyNames ++ cycle130EmLaplacianIbPDependencyNames ++ cycle131EmSecondGreenNoBoundaryFluxDependencyNames ++ cycle132EmSecondGreenTestTraceZeroDependencyNames ++ cycle133EmSecondGreenPointwiseTestTraceZeroDependencyNames ++ cycle134EmTestLaplacianSourcePullbackDependencyNames ++ cycle135EmTestLaplacianStdBasisDependencyNames ++ cycle136EmWeakFpStdBasisSourceDensityDependencyNames ++ cycle137EmWeakFpDensityLaplacianActionDependencyNames ++ cycle138EmFirstGreenPointwiseBoundaryFluxDependencyNames ++ cycle139EmWeakFpSourceLaplacianDependencyNames ++ cycle140EmLaplacianSourceStateIntegralDependencyNames ++ cycle141EmGeneratorStdBasisSourceDependencyNames ++ cycle142EmGeneratorTraceStateIntegralDependencyNames ++ cycle143EmGeneratorLaplacianStateEventDependencyNames ++ cycle144EmGeneratorLaplacianStdBasisEventDependencyNames ++ cycle145EmGeneratorLaplacianStdBasisLawDependencyNames ++ cycle146EmGeneratorTraceEventTotalEventDependencyNames ++ cycle147EmGeneratorPointwiseActionDefTraceLaplacianDependencyNames ++ cycle148EmGeneratorPointwiseStateEventTraceLaplacianDependencyNames ++ cycle150EmGeneratorTotalEventTraceLawIntegralDependencyNames ++ cycle151EmGeneratorTotalEventTraceLaplacianStateIntegralPointwiseEventDependencyNames ++ cycle152EmGeneratorEventSourceFieldStdBasisDependencyNames ++ cycle153EmGeneratorLaplacianStateIntegralSourceFunctionalDependencyNames ++ cycle154EmGeneratorLaplacianStateIntegralTraceFieldSourceFunctionalDependencyNames ++ cycle155EmGeneratorTraceStateIntegralLaplacianFieldDependencyNames ++ cycle156EmGeneratorTraceFieldPointwiseDependencyNames ++ cycle51DiscreteForwardKlDependencyNames ++ cycle56DiscreteForwardKlDependencyNames ++ cycle61DiscreteForwardKlDependencyNames ++ cycle66DiscreteForwardKlDependencyNames ++ [ "SALD.cycle44MainSkeletonAnalyticInterfaceLedger", "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle44MainSkeletonAnalyticInterfaceDag", "SALD.cycle46DiscreteForwardKlSkeletonUpperPacket", "SALD.cycle46DiscreteForwardKlSkeletonObligation", "SALD.cycle46DiscreteForwardKlSkeletonDag", "ASTIS.SALD.forward_KL_discrete.cycle46_theorem_skeleton_route", "SALD.cycle46DiscreteForwardKlSkeletonMiddleContract", "SALD.cycle46DiscreteForwardKlSkeletonMiddleObligation", "sald.discrete_forward_kl.cycle46_middle_route_audit", "ASTIS.SALD.forward_KL_discrete.cycle46_middle_route_audit", "SALD.cycle15DiscreteForwardKlUpperPacket", "SALD.cycle15DiscreteForwardKlMiddleContract", "SALD.cycle15DiscreteForwardKlEmConditionalFpLowerContract", "SALD.cycle15DiscreteForwardKlConditionalDriftDensityContract", "SALD.cycle35DiscreteForwardKlEmFpUpperPacket", "SALD.cycle35DiscreteForwardKlEmFpUpperObligation", "SALD.cycle35DiscreteForwardKlEmFpMiddleContract", "SALD.cycle35DiscreteForwardKlEmFpMiddleObligation", "SALD.cycle35DiscreteForwardKlEmFpLowerObligation", "SALD.discreteForwardKlEmInterpolationLeftEndpointVector", "SALD.discreteForwardKlEmInterpolationRightEndpointVector", "SALD.cycle40DiscreteForwardKlEmFpMiddleContract", "SALD.cycle40DiscreteForwardKlEmFpMiddleObligation", "SALD.discreteForwardKlLawEqOfPointwise", "SALD.discreteForwardKlEmInterpolationLeftEndpointLawHandoff", "SALD.discreteForwardKlEmInterpolationRightEndpointLawHandoff", "SALD.discreteForwardKlEmEndpointLawPairHandoff", "SALD.discreteForwardKlConditionalFpDivergenceDriftSplit", "SALD.discreteForwardKlConditionalFpLaplacianSplitHandoff", "SALD.cycle19DiscreteForwardKlUpperPacket", "SALD.cycle19DiscreteForwardKlMiddleContract", "SALD.cycle23DiscreteForwardKlUpperPacket", "SALD.cycle23DiscreteForwardKlMiddleContract", "SALD.cycle27DiscreteForwardKlUpperPacket", "SALD.cycle27DiscreteForwardKlMiddleContract", "thm:forward-KL", "eq:frozen_interp_terminal_disc_prop_additive_final", "lem:frozen_delta_cross_lip_sald", "lem:dv_variation", "lem:gronwall", "sald.discrete_forward_kl.cycle15_middle_em_spine", "sald.discrete_forward_kl.cycle15_conditional_fp_lower_packet", "sald.discrete_forward_kl.cycle35_em_fp_upper", "sald.discrete_forward_kl.cycle35_em_fp_middle", "sald.discrete_forward_kl.cycle35_em_fp_lower", "sald.discrete_forward_kl.cycle40_em_fp_middle", "sald.discrete_forward_kl.cycle40_em_endpoint_lower", "sald.discrete_forward_kl.conditional_drift_density", "sald.discrete_forward_kl.em_endpoint_laws", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.stitched_interval_regularity", "sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.frozen_delta_cross_lip", "sald.discrete_forward_kl.kl_derivative", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "sald.discrete_forward_kl.dv_velocity_bound", "SALD.discreteForwardKlTimeChangeSquareCoefficientRewriteScalar", "sald.discrete_forward_kl.gronwall_accumulation", "sald.discrete_forward_kl.linear_slowdown_specialization", "sald.discrete_forward_kl.residual_exponent_bound", "SALD.discreteForwardKlGronwallCoeffIntervalIntegrable", "SALD.discreteForwardKlGronwallCoeffIntegralSubSub", "SALD.discreteForwardKlGronwallInitialExponentSplitScalar", "SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces", "SALD.discreteForwardKlResidualExponentBoundScalar", "SALD.discreteForwardKlResidualExpBoundScalar", "SALD.discreteForwardKlAlphaComplexityCollectionScalar", "SALD.discreteForwardKlDeltaAccumulationScalar", "SALD.discreteForwardKlAccumulatedErrorCollectionScalar", "SALD.discreteForwardKlResidualIntegralDisplayBoundScalar", "sald.discrete_forward_kl.accumulated_error_bridge", "sald.discrete_forward_kl.cycle19_accumulated_error_middle", "sald.discrete_forward_kl.em_defect_accumulation_middle", "sald.discrete_forward_kl.cycle23_coefficient_chain_middle", "sald.discrete_forward_kl.cycle27_accumulated_collection_upper", "sald.discrete_forward_kl.cycle27_accumulated_collection_middle", "sald.discrete_forward_kl.cycle27_accumulated_collection_lower", "ASTIS.SALD.forward_KL_discrete.cycle23_upper_packet", "ASTIS.SALD.forward_KL_discrete.cycle23_middle_coefficient_chain", "ASTIS.SALD.forward_KL_discrete.cycle35_em_fp_upper", "ASTIS.SALD.forward_KL_discrete.cycle35_em_fp_middle", "ASTIS.SALD.forward_KL_discrete.cycle35_em_fp_lower", "ASTIS.SALD.forward_KL_discrete.cycle40_em_fp_middle", "ASTIS.SALD.forward_KL_discrete.cycle40_em_endpoint_lower", "ASTIS.SALD.forward_KL_discrete.cycle27_upper_accumulated_collection", "ASTIS.SALD.forward_KL_discrete.cycle27_middle_accumulated_collection", "ASTIS.SALD.forward_KL_discrete.cycle27_lower_accumulated_collection", "sald.discrete_forward_kl.coefficient_chain_audit" ] else if label = "prop:guided_path_residual" then cycle49MainSkeletonDependencyNames ++ cycle54MainSkeletonDependencyNames ++ cycle59MainSkeletonDependencyNames ++ cycle64MainSkeletonDependencyNames ++ cycle69MainSkeletonDependencyNames ++ cycle52GuidedGeneralDependencyNames ++ cycle57GuidedGeneralDependencyNames ++ cycle62GuidedGeneralDependencyNames ++ cycle67GuidedGeneralDependencyNames ++ [ "SALD.cycle44MainSkeletonAnalyticInterfaceLedger", "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle44MainSkeletonAnalyticInterfaceDag", "SALD.cycle47GuidedGeneralSkeletonUpperPacket", "SALD.cycle47GuidedGeneralSkeletonObligation", "SALD.cycle47GuidedGeneralSkeletonMiddleContract", "SALD.cycle47GuidedGeneralSkeletonMiddleObligation", "SALD.cycle47GuidedGeneralSkeletonDag", "ASTIS.SALD.guided_general.cycle47_theorem_skeleton_route", "ASTIS.SALD.guided_general.cycle47_middle_route_audit", "sald.guided_general.cycle47_middle_route_audit", "TransportVelocityContract", "GuidedTiltContract", "sald.guided_path_residual.normalizer_derivative", "sald.guided_path_residual.identity", "sald.unified_forward_kl.transport_bridge_middle", "SALD.cycle16UnifiedForwardKlTransportBridgeLowerContract", "sald.unified_forward_kl.transport_bridge_lower", "sald.unified_forward_kl.transport_velocity_bridge" ] else if label = "thm:general-moving-target-SALD" then cycle49MainSkeletonDependencyNames ++ cycle54MainSkeletonDependencyNames ++ cycle59MainSkeletonDependencyNames ++ cycle64MainSkeletonDependencyNames ++ cycle69MainSkeletonDependencyNames ++ cycle52GuidedGeneralDependencyNames ++ cycle57GuidedGeneralDependencyNames ++ cycle62GuidedGeneralDependencyNames ++ cycle67GuidedGeneralDependencyNames ++ [ "SALD.cycle44MainSkeletonAnalyticInterfaceLedger", "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle44MainSkeletonAnalyticInterfaceDag", "SALD.cycle47GuidedGeneralSkeletonUpperPacket", "SALD.cycle47GuidedGeneralSkeletonObligation", "SALD.cycle47GuidedGeneralSkeletonMiddleContract", "SALD.cycle47GuidedGeneralSkeletonMiddleObligation", "SALD.cycle47GuidedGeneralSkeletonDag", "ASTIS.SALD.guided_general.cycle47_theorem_skeleton_route", "ASTIS.SALD.guided_general.cycle47_middle_route_audit", "sald.guided_general.cycle47_theorem_skeleton_route", "sald.guided_general.cycle47_middle_route_audit", "SALD.cycle16GeneralVaSaldUpperPacket", "SALD.cycle20GeneralVaSaldUpperPacket", "SALD.cycle24GeneralVaSaldUpperPacket", "SALD.cycle24GeneralVaSaldMiddleContract", "eq:general_moving_target_SALD", "eq:LSI-KL-FI", "def:alpha-complexity", "lem:dv_variation", "lem:gronwall", "sald.general_moving_target.kl_derivative", "SALD.generalMovingTargetPostYoungDerivativeBoundScalar", "SALD.generalMovingTargetLsiDerivativeBoundScalar", "SALD.generalMovingTargetTimeChangedDerivativeBoundScalar", "SALD.generalMovingTargetPreDvDerivativeBoundScalar", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "sald.general_moving_target.dv_m_energy", "sald.general_moving_target.gronwall_application", "ASTIS.SALD.general_moving_target.cycle24_upper_packet", "ASTIS.SALD.general_moving_target.cycle24_middle_gronwall_bridge", "sald.general_moving_target.cycle24_gronwall_middle", "SALD.generalMovingTargetGronwallCoeffAdjacentIntervalIntegrable", "SALD.generalMovingTargetGronwallExpProductRewriteIntegralCongrOfPieces", "sald.general_moving_target.gronwall_side_conditions", "sald.general_moving_target.pure_contraction", "sald.general_va_sald.guided_path_middle" ] else if label = "thm:unified-forward-KL" then cycle49MainSkeletonDependencyNames ++ cycle54MainSkeletonDependencyNames ++ cycle59MainSkeletonDependencyNames ++ cycle64MainSkeletonDependencyNames ++ cycle69MainSkeletonDependencyNames ++ cycle52GuidedGeneralDependencyNames ++ cycle67GuidedGeneralDependencyNames ++ cycle53UnifiedDiscreteGeneralDependencyNames ++ cycle58UnifiedDiscreteGeneralDependencyNames ++ cycle63UnifiedDiscreteGeneralDependencyNames ++ cycle68UnifiedDiscreteGeneralDependencyNames ++ [ "SALD.cycle44MainSkeletonAnalyticInterfaceLedger", "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle44MainSkeletonAnalyticInterfaceDag", "SALD.cycle48UnifiedDiscreteSkeletonUpperPacket", "SALD.cycle48UnifiedDiscreteSkeletonObligation", "SALD.cycle48UnifiedDiscreteSkeletonMiddleContract", "SALD.cycle48UnifiedDiscreteSkeletonMiddleObligation", "SALD.cycle48UnifiedDiscreteSkeletonDag", "ASTIS.SALD.unified_forward_KL.cycle48_theorem_skeleton_route", "ASTIS.SALD.unified_discrete_general.cycle48_middle_route_audit", "sald.unified_discrete_general.cycle48_theorem_skeleton_route", "sald.unified_discrete_general.cycle48_middle_route_audit", "SALD.cycle47GuidedGeneralSkeletonObligation", "SALD.cycle47GuidedGeneralSkeletonMiddleObligation", "SALD.cycle16GeneralVaSaldUpperPacket", "SALD.cycle20GeneralVaSaldUpperPacket", "SALD.cycle24GeneralVaSaldUpperPacket", "SALD.cycle24GeneralVaSaldMiddleContract", "SALD.cycle16UnifiedForwardKlTransportBridgeMiddleContract", "sald.unified_forward_kl.transport_bridge_middle", "SALD.cycle16UnifiedForwardKlTransportBridgeLowerContract", "sald.unified_forward_kl.transport_bridge_lower", "prop:guided_path_residual", "eq:poisson-eq", "eq:SALD_Ito", "thm:general-moving-target-SALD", "SALD.unifiedForwardKlSpecializationContract", "sald.unified_forward_kl.transport_velocity_bridge", "sald.general_moving_target.dv_finite_log_mgf_witness", "sald.general_moving_target.dv_positive_alpha_scaling", "ASTIS.SALD.general_moving_target.cycle24_upper_packet", "ASTIS.SALD.general_moving_target.cycle24_middle_gronwall_bridge", "sald.general_moving_target.cycle24_gronwall_middle", "SALD.generalMovingTargetGronwallCoeffAdjacentIntervalIntegrable", "SALD.generalMovingTargetGronwallExpProductRewriteIntegralCongrOfPieces", "sald.general_moving_target.gronwall_side_conditions", "sald.unified_forward_kl.specialization", "sald.general_va_sald.guided_path_middle" ] else if label = "thm:general-moving-target-SALD-discrete" then cycle49MainSkeletonDependencyNames ++ cycle54MainSkeletonDependencyNames ++ cycle59MainSkeletonDependencyNames ++ cycle64MainSkeletonDependencyNames ++ cycle69MainSkeletonDependencyNames ++ cycle70EmConditionalLawDependencyNames ++ cycle71EmEndpointConditionalDependencyNames ++ cycle72EmWeakFpDependencyNames ++ cycle73EmKlDerivativeWeakFpDependencyNames ++ cycle74EmConditionalKernelMeasureDependencyNames ++ cycle75EmConditionalLawBackfillDependencyNames ++ cycle76EmEndpointConditionalDependencyNames ++ cycle77EmWeakFpSourceSignsDependencyNames ++ cycle78EmKlDerivativeGeneratorDependencyNames ++ cycle79EmWeakFpGeneratorMeasureDependencyNames ++ cycle80EmConditionalLawMeasurabilityDependencyNames ++ cycle81EmEndpointConditionalDependencyNames ++ cycle82EmWeakFpSourceSignsDependencyNames ++ cycle83EmKlDerivativeEndpointWeakFpDependencyNames ++ cycle84ActiveEmBackendDependencyNames ++ cycle85EmConditionalKernelBoundaryDependencyNames ++ cycle86EmWeakFpGeneratorBoundaryDependencyNames ++ cycle87EmKlLogRatioBoundaryDependencyNames ++ cycle88EmKlLogRatioAdmissibilityDependencyNames ++ cycle91EmConditionalKernelDependencyNames ++ cycle92EmWeakFpGeneratorSplitDependencyNames ++ cycle93EmKlMassDerivativeDependencyNames ++ cycle94EmWeakFpDriftActionDependencyNames ++ cycle95DiscreteForwardKlPressureDependencyNames ++ cycle96EmCondexpGeneratorPairingDependencyNames ++ cycle97EmCanonicalCondDistribPairingDependencyNames ++ cycle98EmBarBDivergenceNoBoundaryDependencyNames ++ cycle99EmRawKlDerivativeDependencyNames ++ cycle100EmBarBWeakGradDefDependencyNames ++ cycle101DiscreteForwardKlPressureDependencyNames ++ cycle102DiscreteForwardKlZeroFluxTraceBoundaryDependencyNames ++ cycle103EmConditionalKernelComponentVersionDependencyNames ++ cycle104EmWeakFpNamedLawTransportDependencyNames ++ cycle105EmPureRawKlDerivativeDependencyNames ++ cycle106EmCanonicalCondDistribDriftDependencyNames ++ cycle107DiscreteForwardKlBoundaryFluxDependencyNames ++ cycle108DiscreteForwardKlHatRhoBarBFluxContinuityDependencyNames ++ cycle108DiscreteForwardKlHatRhoBarBFluxDerivativeDependencyNames ++ cycle109EmNamedBarBSourceDefDependencyNames ++ cycle110EmNamedBarBEqMeasDependencyNames ++ cycle110EmWeakFpDominatedGeneratorDependencyNames ++ cycle111EmKlTargetTimeDerivativeDependencyNames ++ cycle112EmNamedBarBCondExpRepresentativeDependencyNames ++ cycle113EmNamedBarBStateFieldRegularityDependencyNames ++ cycle114EmCanonicalBarBStateEventSetIntegralDependencyNames ++ cycle115EmNamedBarBSelectedVersionDependencyNames ++ cycle116EmCanonicalBarBCondExpDependencyNames ++ cycle117EmNamedBarBVersionSelectionDependencyNames ++ cycle118EmCanonicalBarBDownstreamDependencyNames ++ cycle119EmCanonicalBarBWeakFpConsumerDependencyNames ++ cycle120EmPathDerivativeDominationDependencyNames ++ cycle121EmSampleMeasDependencyNames ++ cycle122EmSampleIntDependencyNames ++ cycle123EmSampleDerivMeasDependencyNames ++ cycle124EmSampleDerivBoundDependencyNames ++ cycle124EmBoundIntDependencyNames ++ cycle125EmPathDerivDependencyNames ++ cycle126EmDerivValueDependencyNames ++ cycle127EmDriftActionDependencyNames ++ cycle128EmNoBoundaryTraceDependencyNames ++ cycle129EmDiffusionSourceDependencyNames ++ cycle130EmLaplacianIbPDependencyNames ++ cycle131EmSecondGreenNoBoundaryFluxDependencyNames ++ cycle132EmSecondGreenTestTraceZeroDependencyNames ++ cycle133EmSecondGreenPointwiseTestTraceZeroDependencyNames ++ cycle134EmTestLaplacianSourcePullbackDependencyNames ++ cycle135EmTestLaplacianStdBasisDependencyNames ++ cycle136EmWeakFpStdBasisSourceDensityDependencyNames ++ cycle137EmWeakFpDensityLaplacianActionDependencyNames ++ cycle138EmFirstGreenPointwiseBoundaryFluxDependencyNames ++ cycle139EmWeakFpSourceLaplacianDependencyNames ++ cycle140EmLaplacianSourceStateIntegralDependencyNames ++ cycle141EmGeneratorStdBasisSourceDependencyNames ++ cycle142EmGeneratorTraceStateIntegralDependencyNames ++ cycle143EmGeneratorLaplacianStateEventDependencyNames ++ cycle144EmGeneratorLaplacianStdBasisEventDependencyNames ++ cycle145EmGeneratorLaplacianStdBasisLawDependencyNames ++ cycle146EmGeneratorTraceEventTotalEventDependencyNames ++ cycle147EmGeneratorPointwiseActionDefTraceLaplacianDependencyNames ++ cycle148EmGeneratorPointwiseStateEventTraceLaplacianDependencyNames ++ cycle150EmGeneratorTotalEventTraceLawIntegralDependencyNames ++ cycle151EmGeneratorTotalEventTraceLaplacianStateIntegralPointwiseEventDependencyNames ++ cycle152EmGeneratorEventSourceFieldStdBasisDependencyNames ++ cycle153EmGeneratorLaplacianStateIntegralSourceFunctionalDependencyNames ++ cycle154EmGeneratorLaplacianStateIntegralTraceFieldSourceFunctionalDependencyNames ++ cycle155EmGeneratorTraceStateIntegralLaplacianFieldDependencyNames ++ cycle156EmGeneratorTraceFieldPointwiseDependencyNames ++ cycle157EmGeneratorLaplacianEventFieldPointwiseStdBasisDependencyNames ++ cycle158EmGeneratorLaplacianEventFieldPointwiseScalarAuditDependencyNames ++ cycle159EmGeneratorLaplacianEventFieldBrownianPointwiseDependencyNames ++ cycle160EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDependencyNames ++ cycle161EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDependencyNames ++ cycle162EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorRemainderDependencyNames ++ cycle163EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseTaylorDependencyNames ++ cycle164EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoMeasDependencyNames ++ cycle165EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHBoundDependencyNames ++ cycle166EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHFirstDependencyNames ++ cycle167EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorReflectDependencyNames ++ cycle168EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceContDiffDependencyNames ++ cycle169EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondDependencyNames ++ cycle170EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianOpNormDependencyNames ++ cycle171EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDependencyNames ++ cycle172EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDependencyNames ++ cycle173EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDependencyNames ++ cycle174EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticVariationDependencyNames ++ cycle175EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationDependencyNames ++ cycle176EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneDependencyNames ++ cycle177EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSecondTaylorCoeffDependencyNames ++ cycle178EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceDependencyNames ++ cycle179EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianAuditScalarLineCoeffDependencyNames ++ cycle180EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentDependencyNames ++ cycle183EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralDependencyNames ++ cycle184EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLawDependencyNames ++ cycle185EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderLimitDependencyNames ++ cycle186EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandDependencyNames ++ cycle187EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceQuadraticTermDependencyNames ++ cycle188EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandDefDependencyNames ++ cycle189EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralRawTermsDependencyNames ++ cycle190EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementCoordinateLineDependencyNames ++ cycle191EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralScalarPushforwardDependencyNames ++ cycle192EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralScalarPushforwardDependencyNames ++ cycle53UnifiedDiscreteGeneralDependencyNames ++ cycle58UnifiedDiscreteGeneralDependencyNames ++ cycle63UnifiedDiscreteGeneralDependencyNames ++ cycle68UnifiedDiscreteGeneralDependencyNames ++ ["SALD.cycle44MainSkeletonAnalyticInterfaceLedger", "SALD.cycle44MainSkeletonAnalyticInterfaceObligation", "SALD.cycle44MainSkeletonAnalyticInterfaceDag", "SALD.cycle48UnifiedDiscreteSkeletonUpperPacket", "SALD.cycle48UnifiedDiscreteSkeletonObligation", "SALD.cycle48UnifiedDiscreteSkeletonMiddleContract", "SALD.cycle48UnifiedDiscreteSkeletonMiddleObligation", "SALD.cycle48UnifiedDiscreteSkeletonDag", "ASTIS.SALD.general_moving_target_discrete.cycle48_theorem_skeleton_route", "ASTIS.SALD.unified_discrete_general.cycle48_middle_route_audit", "sald.unified_discrete_general.cycle48_theorem_skeleton_route", "sald.unified_discrete_general.cycle48_middle_route_audit", "SALD.cycle16GeneralVaSaldUpperPacket", "SALD.cycle20GeneralVaSaldUpperPacket", "SALD.cycle20GeneralVaSaldMiddleContract", "SALD.cycle28GeneralVaSaldUpperPacket", "SALD.cycle28GeneralVaSaldMiddleContract", "SALD.generalMovingTargetDiscreteGronwallSideConditionContract", "SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar", "SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar", "SALD.generalMovingTargetDiscreteFrozenResidualAlgebraVector", "SALD.generalMovingTargetDiscreteYoungFisherShareScalar", "SALD.generalMovingTargetDiscreteTwoYoungFisherBudgetScalar", "SALD.generalMovingTargetDiscreteResidualYoungCoefficientScalar", "thm:general-moving-target-SALD", "eq:SALD_general_EM", "eq:general_moving_target_SALD_frozen_interp", "eq:general_discrete_delta_def", "lem:frozen_delta_cross_lip", "lem:dv_variation", "lem:gronwall", "sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit", "SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff", "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation", "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation", "sald.general_moving_target_discrete.em_interpolation_fp", "sald.general_moving_target_discrete.frozen_delta_cross_lip", "sald.general_moving_target_discrete.cycle28_derivative_side_middle", "sald.general_moving_target_discrete.cycle28_derivative_side_lower", "sald.general_moving_target_discrete.derivative_side_conditions", "ASTIS.SALD.general_moving_target_discrete.cycle28_upper_derivative_side_conditions", "ASTIS.SALD.general_moving_target_discrete.cycle28_middle_derivative_side_conditions", "ASTIS.SALD.general_moving_target_discrete.cycle28_lower_derivative_side_conditions", "sald.general_moving_target_discrete.kl_derivative", "sald.general_moving_target_discrete.dv_finite_log_mgf_witness", "sald.general_moving_target_discrete.dv_m_energy", "sald.general_moving_target_discrete.gronwall_application", "sald.general_moving_target_discrete.cycle20_gronwall_middle", "sald.general_moving_target_discrete.gronwall_side_conditions", "sald.general_va_sald.guided_path_middle"] else [] def saldReusedByForLabel (label : String) : List String := if label = "lem:gronwall" || label = "lem:dv_variation" then ["thm:forward-KL", "thm:forward-KL-discrete", "thm:general-moving-target-SALD", "thm:general-moving-target-SALD-discrete"] else if label = "eq:LSI-KL-FI" then ["thm:forward-KL", "thm:forward-KL-discrete", "thm:general-moving-target-SALD", "thm:general-moving-target-SALD-discrete"] else if label = "def:PI" then ["lem:velocity-norm-bound"] else if label = "prop:guided_path_residual" then ["thm:unified-forward-KL"] else if label = "thm:general-moving-target-SALD" then ["thm:unified-forward-KL", "thm:general-moving-target-SALD-discrete"] else [] def saldStatusForLabel (label : String) : ProofStatus := if label = "lem:gronwall" then ProofStatus.obligation else if label = "lem:dv_variation" then ProofStatus.sourceCited else if label = "eq:LSI-KL-FI" then ProofStatus.obligation else if label = "def:PI" then ProofStatus.contractOnly else ProofStatus.contractOnly def saldFirstProofDag : List ProofDagBlock := firstFaithfulLabels.map fun label => { id := "ASTIS.SALD." ++ label interface := "Translate and formalize source label `" ++ label ++ "` without changing the paper statement." source := saldSourceForLabel label targetLean := saldLeanTargetForLabel label dependsOn := saldDependenciesForLabel label reusedBy := saldReusedByForLabel label status := saldStatusForLabel label } def saldExcludedFiles : List String := ["sald_version_2.tex"] end SALD end AutoSamplingTheory