AutoSamplingTheory.SALD

No declaration docstring.

def saldPaperRoot : String := "/home/nitanda_sub/mark/repos/sald/paper"

AutoSamplingTheory/SALD.lean:27published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:29published source at 77184245109a

No declaration docstring.

def saldAppendixSource : SourceAnchor :=
  localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "appendix.tex"
    "Appendix proofs for SALD, VA-SALD, and discrete-time bounds."

AutoSamplingTheory/SALD.lean:33published source at 77184245109a

No declaration docstring.

def saldIterationSource : SourceAnchor :=
  localTexAnchor "sald-original-vp" (saldPaperRoot ++ "/iteration_complexity.tex") "iteration_complexity.tex"
    "VP reverse marginal verification under dissipativity."

AutoSamplingTheory/SALD.lean:37published source at 77184245109a

No declaration docstring.

def saldGronwallSource : SourceAnchor :=
  localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "lem:gronwall"
    "appendix.tex lines 47-71: Gronwall inequality and integrating-factor proof."

AutoSamplingTheory/SALD.lean:41published source at 77184245109a

No declaration docstring.

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)."

AutoSamplingTheory/SALD.lean:45published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:49published source at 77184245109a

No declaration docstring.

def saldPiSource : SourceAnchor :=
  localTexAnchor "sald-original-appendix" (saldPaperRoot ++ "/appendix.tex") "def:PI"
    "appendix.tex lines 86-94: Poincare inequality vocabulary."

AutoSamplingTheory/SALD.lean:53published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:57published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:61published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:65published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:69published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:73published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:77published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:81published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:85published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:89published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:93published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:97published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:101published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:105published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:109published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:113published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:117published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:121published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:125published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:129published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:133published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:137published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:141published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:145published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:149published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:153published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:157published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:161published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:165published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:169published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:173published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:177published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:181published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:185published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:189published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:193published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:197published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:201published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:205published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:209published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:213published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:217published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:221published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:225published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:229published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:233published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:241published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:249published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:257published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:265published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:273published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:277published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:281published source at 77184245109a

No declaration docstring.

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."

AutoSamplingTheory/SALD.lean:285published source at 77184245109a

No declaration docstring.

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.
-/

AutoSamplingTheory/SALD.lean:289published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:309published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:330published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:351published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:369published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:380published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:393published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:406published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:425published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:452published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:469published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:488published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:510published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:532published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:553published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:574published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:591published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:612published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:645published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:725published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:807published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:844published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:866published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:904published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:941published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:975published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:1024published source at 77184245109a

- 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]`.
-/

AutoSamplingTheory/SALD.lean:1050published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1064published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1096published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:1122published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:1136published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1149published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1170published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1187published source at 77184245109a

- 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)`.
-/

AutoSamplingTheory/SALD.lean:1224published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1247published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:1272published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1296published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1322published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1359published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1392published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1410published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1428published source at 77184245109a

- 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

AutoSamplingTheory/SALD.lean:1444published source at 77184245109a

No declaration docstring.

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.
-/

AutoSamplingTheory/SALD.lean:1462published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1475published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1494published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1508published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:1524published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1543published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1567published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1611published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:1628published source at 77184245109a

- 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`.
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:1652published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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`.
-/

AutoSamplingTheory/SALD.lean:1697published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:1728published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1742published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1757published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:1769published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1784published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1804published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1837published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1854published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1876published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1913published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1952published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:1993published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:2035published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:2079published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:2103published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:2128published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:2168published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:2210published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:2234published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:2261published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:2296published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:2327published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:2375published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 =>
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:2464published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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))
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:2552published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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`.
-/

AutoSamplingTheory/SALD.lean:2622published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:2660published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:2742published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 :
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:2804published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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) +
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:2932published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:3008published source at 77184245109a

- 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,
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:3046published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:3150published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 ∧
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:3254published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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) =
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:3377published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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) =
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:3455published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 •
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:3546published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 •
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:3614published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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) +
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:3681published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:3758published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:3795published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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`.
-/

AutoSamplingTheory/SALD.lean:3864published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:3895published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 :
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:3943published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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))
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:4019published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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))
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:4098published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:4193published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:4288published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:4366published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:4465published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:4571published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:4695published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:4713published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:4740published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:4759published source at 77184245109a

- 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.
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:4783published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:4828published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:4889published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:4930published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:4971published source at 77184245109a

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- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:5024published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:5070published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:5133published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:5220published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 =
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:5319published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:5431published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:5524published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:5594published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:5672published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:5752published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:5882published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:5925published source at 77184245109a

- `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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:5969published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- `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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:6021published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:6085published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 :
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:6145published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- `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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:6214published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:6267published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:6330published source at 77184245109a

- `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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:6369published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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) :=
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:6426published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:6488published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:6534published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:6615published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:6795published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:6944published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:7092published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:7238published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:7387published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:7547published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:7715published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:7876published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:8047published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:8319published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:8534published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:8796published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 :
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:9028published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:9239published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:9280published source at 77184245109a

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:9315published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:9426published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:9584published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:9737published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:9900published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:10058published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:10217published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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`.
-/

AutoSamplingTheory/SALD.lean:10370published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:10390published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:10421published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:10453published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:10471published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:10520published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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`.
-/

AutoSamplingTheory/SALD.lean:10572published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:10612published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:10651published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:10669published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:10712published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:10741published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:10797published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:10940published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:11054published source at 77184245109a

- 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,
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:11092published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:11216published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:11265published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:11312published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 =
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:11364published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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`.
-/

AutoSamplingTheory/SALD.lean:11432published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:11454published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:11476published source at 77184245109a

- 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 excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:11513published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:11562published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:11618published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:11677published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 =
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:11740published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:11808published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:11858published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:11917published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:11982published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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,
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:12035published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:12097published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 :
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:12169published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:12241published source at 77184245109a

- 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]) :
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:12285published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:12357published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:12401published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:12456published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:12498published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:12575published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:12648published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:12690published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:12717published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:12744published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:12795published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:12853published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:12891published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:12949published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:13024published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:13100published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 =
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:13178published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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]
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:13258published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 :
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:13311published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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`,
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:13394published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 =>
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:13440published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:13521published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:13555published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:13595published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:13622published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:13663published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:13704published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:13739published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:13780published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:13807published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:13821published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:13851published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:13878published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:13916published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:13936published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:13971published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:13999published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:14042published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:14055published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:14075published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:14090published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:14138published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:14237published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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,
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:14301published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:14375published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:14429published source at 77184245109a

- 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`.
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:14464published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:14509published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:14554published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:14611published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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,
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:14657published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:14739published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:14785published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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]
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:14835published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:14887published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:15043published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:15073published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:15097published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:15138published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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) :=
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:15187published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:15262published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:15336published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 :
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:15399published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:15460published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 :
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:15526published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:15597published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:15678published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 :
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:15834published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:15904published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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))
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:15983published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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,
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:16105published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:16302published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:16335published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:16383published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:16436published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:16478published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:16525published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:16541published source at 77184245109a

- 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.
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:16567published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 :
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:16612published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 _ _
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:16676published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:16736published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:16782published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:16830published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:16867published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:16946published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:16983published source at 77184245109a

- 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.
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:17023published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:17074published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:17121published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:17143published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:17195published source at 77184245109a

- 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]
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:17231published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:17352published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:17391published source at 77184245109a

- 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.
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:17427published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:17472published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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`.
-/

AutoSamplingTheory/SALD.lean:17518published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:17556published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:17582published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:17605published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:17624published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:17636published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:17661published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:17695published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:17744published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:17782published source at 77184245109a

- 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))) +
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:17821published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:17888published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:17928published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:18005published source at 77184245109a

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:18042published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:18108published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:18170published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:18224published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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,
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:18286published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:18367published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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]) :
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:18443published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:18546published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:18616published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:18650published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:18693published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:18733published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:18759published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:18796published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:18860published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:18932published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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]
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:19007published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 =
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:19061published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 =
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:19138published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:19215published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:19295published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:19342published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:19391published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:19450published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 :
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:19630published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:19793published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:19812published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:19839published source at 77184245109a

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:19883published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:19986published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:20129published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:20214published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:20302published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:20383published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:20458published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:20516published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:20622published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:20701published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:20786published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:20846published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:20994published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:21021published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:21051published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:21086published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:21104published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:21142published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:21173published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:21214published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:21231published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:21255published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:21295published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:21324published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:21378published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:21421published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:21453published source at 77184245109a

- 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. -/

AutoSamplingTheory/SALD.lean:21495published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:21527published source at 77184245109a

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:21555published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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. -/

AutoSamplingTheory/SALD.lean:21615published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:21643published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:21669published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:21712published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:21778published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:21834published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:21905published source at 77184245109a

- 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 →
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:21944published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:22010published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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)
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:22167published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:22326published source at 77184245109a

- 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)`.
-/

AutoSamplingTheory/SALD.lean:22344published source at 77184245109a

- 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. -/

AutoSamplingTheory/SALD.lean:22375published source at 77184245109a

- 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. -/

AutoSamplingTheory/SALD.lean:22393published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:22401published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:22415published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:22436published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:22448published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:22467published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:22485published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:22511published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:22548published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:22587published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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) -
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:22635published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:22700published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:22738published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:22771published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:22795published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:22837published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:22855published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:22870published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:22889published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:22910published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:22930published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:22952published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:22971published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:22989published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23013published source at 77184245109a

- 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)`.
-/

AutoSamplingTheory/SALD.lean:23034published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23055published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:23080published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23113published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23150published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23177published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:23210published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23246published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23287published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23301published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23316published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23335published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23358published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:23374published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:23394published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23409published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:23425published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23457published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:23481published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23496published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23511published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23533published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:23567published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23601published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23640published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23674published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23709published source at 77184245109a

- 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.
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:23750published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:23795published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23817published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23841published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23863published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23900published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23940published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:23981published source at 77184245109a

- 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)`.
-/

AutoSamplingTheory/SALD.lean:24019published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24044published source at 77184245109a

- 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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:24072published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

- 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.
-/

AutoSamplingTheory/SALD.lean:24120published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24162published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24182published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24207published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24226published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24249published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24265published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24284published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24304published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24320published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24337published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24354published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24374published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24398published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24414published source at 77184245109a

- 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`. -/

AutoSamplingTheory/SALD.lean:24437published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24454published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:24474published source at 77184245109a

- 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`. -/

AutoSamplingTheory/SALD.lean:24499published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24516published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24535published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24559published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24588published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24608published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24630published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24652published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24678published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24698published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24721published source at 77184245109a

- 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. -/

AutoSamplingTheory/SALD.lean:24743published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24757published source at 77184245109a

- 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`. -/

AutoSamplingTheory/SALD.lean:24781published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24802published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24825published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24849published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24871published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:24898published source at 77184245109a

- 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. -/

AutoSamplingTheory/SALD.lean:24919published source at 77184245109a

- 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. -/

AutoSamplingTheory/SALD.lean:24938published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24957published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:24976published source at 77184245109a

- 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. -/

AutoSamplingTheory/SALD.lean:25003published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:25019published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:25041published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:25065published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:25091published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:25113published source at 77184245109a

- 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. -/

AutoSamplingTheory/SALD.lean:25136published source at 77184245109a

- 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. -/

AutoSamplingTheory/SALD.lean:25155published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:25170published source at 77184245109a

- 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})`.
-/

AutoSamplingTheory/SALD.lean:25195published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:25221published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:25247published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:25274published source at 77184245109a

- 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)`.
-/

AutoSamplingTheory/SALD.lean:25298published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:25324published source at 77184245109a

- 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. -/

AutoSamplingTheory/SALD.lean:25350published source at 77184245109a

- 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`.
-/

AutoSamplingTheory/SALD.lean:25370published source at 77184245109a

- 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.
-/

AutoSamplingTheory/SALD.lean:25392published source at 77184245109a

- 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

AutoSamplingTheory/SALD.lean:25416published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:25430published source at 77184245109a

No declaration docstring.

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"
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:25453published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:25500published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:25526published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:25533published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:25540published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:25547published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:25568published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:25608published source at 77184245109a

No declaration docstring.

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 <phi,psi>_{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

AutoSamplingTheory/SALD.lean:25628published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:25651published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:25683published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:25693published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:25725published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:25757published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:25778published source at 77184245109a

No declaration docstring.

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 0<alpha<=alpha0 from the alpha0-complexity assumption before invoking DV."
  commonSpaceAndAbsoluteContinuity := "The DV step requires rho_{s(t)} and pi_t on the same measurable space with rho_{s(t)} absolutely continuous with respect to pi_t wherever KL(rho_{s(t)}||pi_t) is used."
  measurabilityInterface := "v_t must be measurable and ||v_t||^2 must be a valid real-valued DV test function under both rho_{s(t)} and pi_t."
  scalingStep := "After DV gives E_rho[alpha*||v_t||^2] <= KL(rho||pi)+log E_pi[exp(alpha*||v_t||^2)], divide by alpha>0."
  outputBound := "||v_t||_{L2(rho_{s(t)})}^2 <= alpha^(-1)*K(t)+E_alpha(pi_t,v_t)."
  coefficientUse := "Multiplying by (1/2)*dot{s}(t)^(-1) yields the scalar terms used in appendix.tex:239-241 without changing the factor 1/2."
  dependencies := [
    "lem:dv_variation",
    "def:alpha-complexity",
    "SALD.cycle26ForwardKlMiddleContract",
    "SALD.saldDvFiniteLogMgfContract",
    "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"
  ]
  sourceGaps := [
    "the source theorem states finite alpha0-complexity pointwise in t, but does not state the monotonicity lemma from alpha0 to alpha",
    "the appendix does not isolate rho_{s(t)} << pi_t before using both KL and the DV formula",
    "measurability of v_t and the squared-norm test function is implicit in the paper's vector-field vocabulary"
  ]
  lowerPacket := [
    "Formalize only the alpha0-to-alpha finite-log-mgf monotonicity for Z=alpha*||v_t||^2, or keep it as a precise ProofObligation if no local exponential-moment API is ready.",
    "Do not change the theorem's alpha range or add a new finite-mgf assumption to thm:forward-KL.",
    "Use SALD.forwardKlDvPositiveAlphaScalingScalar and SALD.forwardKlDvPositiveAlphaCoefficientScalar for the pure real positive-alpha division, but keep the finite-log-mgf, common-space, and measurability backends as obligations."
  ]
  status := ProofStatus.obligation

AutoSamplingTheory/SALD.lean:25798published source at 77184245109a

No declaration docstring.

def forwardKlDvAlphaMonotonicityContract :
    ForwardKlDvAlphaMonotonicityContract where
  sourceBlock := saldForwardKlDvEnergySource
  theoremStatement := saldForwardKlSource
  sourceAssumption := "main_body.tex:240-241 assumes E_{alpha0}(pi_t,v_t)<+infty, i.e. alpha0^(-1)*log E_{pi_t}[exp(alpha0*||v_t||^2)] is finite for every t."
  alphaRange := "The theorem allows any alpha in (0,alpha0]; the proof must keep alpha>0 for division and alpha<=alpha0 for monotonicity."
  nonnegativeEnergy := "The scalar q_t(x)=||v_t(x)||^2 is nonnegative, so alpha*q_t(x) <= alpha0*q_t(x) pointwise when alpha<=alpha0."
  pointwiseDomination := "Use exp monotonicity to show exp(alpha*q_t(x)) <= exp(alpha0*q_t(x)) pointwise, with no change to the source test function Z_t."
  expectationBridge := "Transfer the pointwise domination through the pi_t integral/expectation to obtain finite E_{pi_t}[exp(alpha*q_t)] from the finite alpha0 expectation."
  logMgfBridge := "Show log E_{pi_t}[exp(alpha*q_t)] is finite and well-defined before applying lem:dv_variation with Z_t=alpha*q_t."
  alphaComplexityRewrite := "Rewrite alpha^(-1)*log E_{pi_t}[exp(alpha*q_t)] as E_alpha(pi_t,v_t), matching the definition in main_body.tex:218-223."
  downstreamCoefficientUse := "After multiplying the DV output by (1/2)*dot{s}(t)^(-1), preserve the source coefficient (1/2)*dot{s}(t)^(-1)*alpha^(-1) in appendix.tex:239-241."
  dependencies := [
    "def:alpha-complexity",
    "SALD.saldDvFiniteLogMgfContract",
    "sald.dv_variation.finite_log_mgf_interface"
  ]
  sourceGaps := [
    "the source invokes alpha0-to-alpha finite-log-mgf monotonicity implicitly",
    "the Lean backend still needs a measure-integral theorem turning pointwise exponential domination into finite expectation",
    "no local theorem currently proves the log finiteness and alpha-complexity rewrite"
  ]
  status := ProofStatus.obligation

AutoSamplingTheory/SALD.lean:25834published source at 77184245109a

No declaration docstring.

def forwardKlGronwallInstantiationContract : ForwardKlGronwallInstantiationContract where
  sourceBlock := saldForwardKlGronwallSource
  quantityK := "K(t)=KL(rho_{s(t)} || pi_t)."
  gronwallA := "a(t)=dot{s}(t)*C_LSI(t) - (1/2)*dot{s}(t)^(-1)*alpha^(-1)."
  gronwallB := "b(t)=(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_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."
  splitBound := "Separate the exponent into the negative LSI term and the positive alpha term, then drop the nonpositive LSI part in the residual exponential to match main_body.tex lines 243-246."
  requiredRegularity := [
    "continuity or interval-integrability of a(t) and b(t) on [0,T]",
    "differentiability of K(t) on the interval in the sense required by the Gronwall backend",
    "positivity of dot{s}(t) so dot{s}(t)^(-1) is defined"
  ]
  sourceGaps := [
    "the source theorem does not separately state the Gronwall regularity hypotheses for K, a, and b; the proof uses them implicitly",
    "the residual exponential simplification uses nonnegativity of dot{s}(u)*C_LSI(u) to discard its negative contribution"
  ]
  dependencies := [
    "lem:gronwall",
    "sald.forward_kl.kl_derivative",
    "sald.forward_kl.dv_energy_bound",
    "sald.forward_kl.gronwall_side_conditions"
  ]
  status := ProofStatus.obligation

AutoSamplingTheory/SALD.lean:25858published source at 77184245109a

No declaration docstring.

def forwardKlMovingTargetDependencyContract :
    ForwardKlMovingTargetDependencyContract where
  sourceStatement := saldForwardKlSource
  sourceProof := saldForwardKlProofSource
  slowedTargetPath := "main_body.tex:238 and appendix.tex:187-197 use tilde_pi_s=pi_{t(s)} with inverse slowdown s=s(t)."
  transportVelocityInterface := "The theorem assumes v is the transport velocity of pi; appendix.tex:187-197 requires partial_t pi_t+div(v_t*pi_t)=0 and tilde_v_s=dot{t}(s)*v_{t(s)}."
  saldLawInterface := "rho_s is the law of SALD eq:SALD, and appendix.tex:176-185 uses the corresponding Fokker--Planck equation before any LSI, DV, or Gronwall step."
  lsiBridge := "main_body.tex:240-241 assumes C_LSI(t)>=0; appendix.tex:210-217 uses eq:LSI-KL-FI to convert the remaining -(1/2)*FI term into -C_LSI(t(s))*KL."
  dvBridge := "appendix.tex:230-241 applies lem:dv_variation with Z=alpha*||v_t||^2 and the finite alpha0-complexity assumption from main_body.tex:241."
  gronwallBridge := "appendix.tex:244-252 applies lem:gronwall after DV 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)."
  terminalIdentification := "K(T)=KL(rho_S||pi_T) and K(0)=KL(rho_0||pi_0) rely on S=s(T), s(0)=0, and the slowed target identity."
  lowerFormalizationTarget := "Refine the named moving-target dependency-chain obligation before attempting the KL derivative, DV-energy, or Gronwall theorem proof."
  dependencies := [
    "eq:SALD",
    "eq:FP-eq",
    "TransportVelocityContract",
    "eq:LSI-KL-FI",
    "def:alpha-complexity",
    "lem:dv_variation",
    "lem:gronwall",
    "sald.forward_kl.endpoint_schedule_identities",
    "sald.forward_kl.density_boundary_regular",
    "sald.forward_kl.schedule_time_change",
    "sald.forward_kl.kl_derivative",
    "sald.forward_kl.dv_energy_bound",
    "sald.forward_kl.gronwall_application"
  ]
  sourceGaps := [
    "the source states smooth monotone slowdown but not endpoint identities s(0)=0 and S=s(T) as a separate lemma",
    "the transport-velocity and Fokker--Planck interfaces are used in the appendix proof without a local analytic backend",
    "the theorem statement does not isolate continuity or integrability assumptions for C_LSI, dot{s}, and E_alpha needed by Gronwall"
  ]
  status := ProofStatus.obligation

AutoSamplingTheory/SALD.lean:25882published source at 77184245109a

No declaration docstring.

def forwardKlDependencyChainAuditContract :
    ForwardKlDependencyChainAuditContract where
  sourceBlock := saldForwardKlDependencyChainSource
  postYoungInequality := "appendix.tex:210-217 combines the KL derivative terms after Young: d/ds KL(rho_s||tilde_pi_s) <= -(1/2)*FI(rho_s||tilde_pi_s)+(1/2)*||tilde_v_s||_{L2(rho_s)}^2."
  lsiCoefficientStep := "Using eq:LSI-KL-FI, (1/2)*FI(rho_s||tilde_pi_s) >= C_LSI(t(s))*KL(rho_s||tilde_pi_s), giving the source coefficient -C_LSI(t(s))*K_s."
  timeChangeStep := "appendix.tex:218-228 multiplies by dot{s}(t) and uses dot{s}(t)*dot{t}(s(t))^2=dot{s}(t)^(-1), producing the velocity term (1/2)*dot{s}(t)^(-1)*||v_t||_{L2(rho_{s(t)})}^2."
  dvInstantiation := "appendix.tex:230-241 applies lem:dv_variation with nu=rho_{s(t)}, mu=pi_t, and Z=alpha*||v_t||^2, rewriting alpha^(-1)*log E_{pi_t}[exp(alpha||v_t||^2)] as E_alpha(pi_t,v_t)."
  scalarDifferentialInequality := "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)."
  gronwallA := "a(t)=dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1)."
  gronwallB := "b(t)=(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t)."
  exponentSplit := "appendix.tex:244-250 splits exp(-int a) into exp(-int dot{s}*C_LSI)*exp(int (1/2)*dot{s}^(-1)*alpha^(-1))."
  residualExponentSimplification := "appendix.tex:248-251 drops the nonpositive contribution -int_t^T dot{s}(u)*C_LSI(u) du in the residual exponential, using C_LSI(u)>=0 and dot{s}(u)>0."
  terminalEndpointBridge := "The theorem bound identifies K(T)=KL(rho_S||pi_T) and K(0)=KL(rho_0||pi_0), requiring S=s(T), s(0)=0, and tilde_pi_{s(t)}=pi_t."
  sourceLineLedger := [
    "appendix.tex:210-217: the post-Young inequality leaves exactly -(1/2)*FI plus (1/2)*||tilde v_s||^2, then eq:LSI-KL-FI turns the FI term into -C_LSI(t(s))*K_s.",
    "appendix.tex:218-228: the time change multiplies by dot{s}(t) and rewrites dot{s}(t)*dot t(s(t))^2 as dot{s}(t)^(-1), preserving the factor 1/2.",
    "appendix.tex:230-241: DV with Z=alpha*||v_t||^2 contributes (1/2)*dot{s}(t)^(-1)*alpha^(-1)*K(t) and (1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t).",
    "appendix.tex:244-247: Gronwall is applied 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:248-252 and main_body.tex:243-246: the initial exponent is split into the LSI contraction factor and the positive alpha factor, while the residual exponent drops only the nonpositive LSI contribution."
  ]
  scalarSideConditions := [
    "alpha>0 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"
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:25916published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:25973published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:26010published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:26047published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:26072published source at 77184245109a

No declaration docstring.

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"
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:26109published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:26169published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:26206published source at 77184245109a

No declaration docstring.

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",
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:26243published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:26308published source at 77184245109a

No declaration docstring.

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

AutoSamplingTheory/SALD.lean:26347published source at 77184245109a

No declaration docstring.

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
-- Source excerpt truncated; follow the exact source link.

AutoSamplingTheory/SALD.lean:26375published source at 77184245109a

Excerpt truncated; the exact source link is authoritative.