AutoSamplingTheory.TechnicalLemmas.Registry

import AutoSamplingTheory.TechnicalLemmas.Geometry.UnitSphereAngularCost
import AutoSamplingTheory.TechnicalLemmas.Geometry.QuadraticFeatureLawGradient
import AutoSamplingTheory.TechnicalLemmas.Probability.GaussianPolynomialDifferentiation
import AutoSamplingTheory.TechnicalLemmas.Geometry.QuadraticFeatureTangentGradient
import AutoSamplingTheory.TechnicalLemmas.Measure.BochnerMeanTransport
import AutoSamplingTheory.TechnicalLemmas.Analysis.JointEmpiricalSegmentRisk
import AutoSamplingTheory.TechnicalLemmas.Analysis.ProbabilitySegmentRisk
import AutoSamplingTheory.TechnicalLemmas.Probability.GaussianPolynomialAveraging
import AutoSamplingTheory.TechnicalLemmas.Probability.GaussianPolynomialRisk
import AutoSamplingTheory.TechnicalLemmas.Probability.GaussianPolynomialRidge
import AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticFeatureRisk
import AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.BoundedPerturbation
import AutoSamplingTheory.TechnicalLemmas.Geometry.SphereRayleigh
import AutoSamplingTheory.TechnicalLemmas.Geometry.SphereGradient
import AutoSamplingTheory.TechnicalLemmas.Geometry.CurveExtension
import AutoSamplingTheory.TechnicalLemmas.Geometry.AffineGradient
import AutoSamplingTheory.Core
import AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability
import AutoSamplingTheory.TechnicalLemmas.Analysis.SubgradientMethod
import AutoSamplingTheory.TechnicalLemmas.Analysis.GradientAECongruence
import AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexFirstOrder
import AutoSamplingTheory.TechnicalLemmas.Analysis.ChebyshevLobattoQuadrature
import AutoSamplingTheory.TechnicalLemmas.Analysis.ChebyshevLobattoPositiveWeights
import AutoSamplingTheory.TechnicalLemmas.Analysis.HessianSecantOperator
import AutoSamplingTheory.TechnicalLemmas.Analysis.MonotoneProximalMap
import AutoSamplingTheory.TechnicalLemmas.Analysis.ChebyshevLobattoMomentum
import AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff
import AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence
import AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient
import AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian
import AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Taylor
import AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator
import AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.CanonicalLogSobolev
import AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.LogSobolev
import AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare
import AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay
import AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.ForcedSemigroupDecay
import AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.BakryEmeryInterpolation
import AutoSamplingTheory.TechnicalLemmas.Geometry.EuclideanSpaceCoordinates
import AutoSamplingTheory.TechnicalLemmas.Geometry.GeodesicConvexity
import AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity
import AutoSamplingTheory.TechnicalLemmas.Geometry.MetricCurve
import AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity
import AutoSamplingTheory.TechnicalLemmas.InformationTheory.DonskerVaradhan
import AutoSamplingTheory.TechnicalLemmas.InformationTheory.KLDensity
import AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi
import AutoSamplingTheory.TechnicalLemmas.InformationTheory.TiltedLogRatio
import AutoSamplingTheory.TechnicalLemmas.InformationTheory.TiltedKL
import AutoSamplingTheory.TechnicalLemmas.InformationTheory.QuadraticTiltFisher
import AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinLipschitzMoment
import AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs
import AutoSamplingTheory.TechnicalLemmas.Measure.GibbsIntegral
import AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity
import AutoSamplingTheory.TechnicalLemmas.Measure.KantorovichDual
import AutoSamplingTheory.TechnicalLemmas.Measure.Product
import AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym
import AutoSamplingTheory.TechnicalLemmas.Probability.ConditionalKernel
import AutoSamplingTheory.TechnicalLemmas.Probability.KernelHybridTelescope
import AutoSamplingTheory.TechnicalLemmas.Probability.KernelInvariance
import AutoSamplingTheory.TechnicalLemmas.Probability.KernelL2
import AutoSamplingTheory.TechnicalLemmas.Probability.KernelRetainedReversibility
import AutoSamplingTheory.TechnicalLemmas.Measure.Transport
import AutoSamplingTheory.TechnicalLemmas.Probability.LawMap
import AutoSamplingTheory.TechnicalLemmas.Probability.UnitExponentialProduct
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualNonaccumulation
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhysicalTimeCover
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhysicalTimeMeasurability
import AutoSamplingTheory.ExampleCases.ProximalBPS.IdealHalfTurnKernel
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualSmallTimeContinuity
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualBoundedTestContinuity
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualOuterBoundedL2Continuity
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhaseTransitionKernel
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualFiniteFirstEventLaw
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualFiniteHistoryTailLaw
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualConditionalNextWaitSurvival
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualConditionalNextWaitDensity
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualFiniteHistoryExtension
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualAbsoluteNextEventTime
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualOrderedPrefix
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualFiniteCountPath
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedPathDensity
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualBoundedPathExpansion
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhaseReversalVolume
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualEndpointTimeReversal
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedGibbsSurvival
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedStateReversal
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedPathReversal
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedWeightedDensity
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualWeightedSector
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhaseNormalization
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualNormalizedEndpointSector
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualCountableEndpointLaw
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualBoundedGibbsDuality
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualL2Expectation
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualL2FlipDuality
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPositionL2Averaging
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualL2StrongContinuity
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualKernelL2Realization
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualHalfTurnReversibility
import AutoSamplingTheory.ExampleCases.ProximalBPS.ActualAugmentedKernelInvariance
import AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian
import AutoSamplingTheory.TechnicalLemmas.StochasticProcesses

/-!
# Technical lemma memory registry

The registry is the compiled Lean side of ASTIS technical lemma memory.  It
records which reusable lemmas exist locally, what upstream/source material
motivated them, and which proof backend they support.
-/

namespace AutoSamplingTheory
namespace TechnicalLemmas

inductive LemmaMemoryStatus where
  | formalizedLocal
  | portCandidate
  | sourceGap
  | referenceOnly
deriving Repr, DecidableEq

/-- Metadata for a lemma-memory entry.  The executable proof is the declaration
named in `localDecl`; this structure is only the retrieval record used by
agents and documentation exports. -/
structure LemmaMemoryEntry where
  key : String
  localDecl : String
  upstreamDecl : String
  upstreamFile : String
  status : LemmaMemoryStatus
  tags : List String
  saldUse : String
  note : String
deriving Repr, DecidableEq

def sltSourceAnchor (file decl note : String) : SourceAnchor :=
  sourceAnchor
    ("slt-" ++ file ++ "-" ++ decl)
    "externalLean"
    "https://github.com/YuanheZ/lean-stat-learning-theory"
    (file ++ ":" ++ decl)
    note

def analysisMemory : List LemmaMemoryEntry := [
  {
    key := "optlib-subgradient-norm",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.SubgradientMethod.norm_le_of_supportsAt",
    upstreamDecl := "Lipschitz_to_bounded_subgradient",
    upstreamFile := "Optlib/Algorithm/SubgradientMethod.lean@5da27c5f95aa6a8a45b8c14b968ade4c13ff18c3:80-118",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["convex", "subgradient", "Lipschitz", "optimisation"],
    saldUse := "Bounds supplied supporting vectors inside the actual subgradient energy estimate",
    note := "Native Lean 4.33.0 port over existing SupportsAt; no subgradient existence or random-oracle claim. Compilation, source review and integration are separately recorded."
  },
  {
    key := "optlib-subgradient-energy",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.SubgradientMethod.sum_weighted_gap_le",
    upstreamDecl := "subgradient_method_converge (extracted weighted intermediate)",
    upstreamFile := "Optlib/Algorithm/SubgradientMethod.lean@5da27c5f95aa6a8a45b8c14b968ade4c13ff18c3:171-232",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["convex", "subgradient", "energy", "optimisation"],
    saldUse := "Actual update telescopes weighted gaps; consumed by the finite best-iterate theorem",
    note := "Retains the terminal distance. Nonnegative steps and empty horizon explicitly generalize the source intermediate; not a claim of full Optlib or Chewi chapter completion."
  },
  {
    key := "optlib-subgradient-best",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.SubgradientMethod.exists_gap_le",
    upstreamDecl := "subgradient_method_converge",
    upstreamFile := "Optlib/Algorithm/SubgradientMethod.lean@5da27c5f95aa6a8a45b8c14b968ade4c13ff18c3:137-232",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["convex", "subgradient", "finite-horizon", "optimisation"],
    saldUse := "Finite best-iterate guarantee for a supplied nonsmooth subgradient run and arbitrary comparator",
    note := "Positive steps and nonempty horizon justify division. Native index witness replaces attained finite sInf; no minimizer, iterate convergence, stochastic oracle or sampling rate is claimed."
  },
  {
    key := "quadratic-probability-segment",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.ProbabilitySegmentRisk.quadratic_feature_probability_segment",
    upstreamDecl := "Original actual probability and Gaussian feature regularity supplement",
    upstreamFile := "Pinned Mathlib probability, Gaussian moments, L2 representatives and Bochner calculus",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["probability", "L2", "risk", "shared"],
    saldUse := "Generic probability-law calculus and feature averaging; no model completion",
    note := "Actual two-probability-law endpoint calculus and integrable potential pairings. Independent fresh compilation and whole-module current-context semantic review accepted. Final root/reader gates and postmerge purification separate. No spatial gradient regularity, SDE or stochastic convergence claim."
  },
  {
    key := "joint-empirical-segment-risk",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.JointEmpiricalSegmentRisk.joint_empirical_probability_segment_derivative",
    upstreamDecl := "Original actual probability and Gaussian feature regularity supplement",
    upstreamFile := "Pinned Mathlib probability, Gaussian moments, L2 representatives and Bochner calculus",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["probability", "L2", "risk", "shared"],
    saldUse := "Generic probability-law calculus and feature averaging; no model completion",
    note := "Actual joint empirical loss and endpoint derivative using only selected finite feature lists. Independent fresh compilation and whole-module current-context semantic review accepted. Final root/reader gates and postmerge purification separate. No spatial gradient regularity, SDE or stochastic convergence claim."
  },
  {
    key := "analysis.quadratic-feature-risk-1",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticFeatureRisk.empirical_squared_feature_risk_derivatives",
    upstreamDecl := "Samplinglib original quadratic feature-risk supplement v1, interface 1",
    upstreamFile := "AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticFeatureRisk.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["quadratic-risk", "empirical-measure", "L2", "optimisation"],
    saldUse := "Reusable squared-loss, genuine integral and indexed replacement calculations",
    note := "Compiled generic library interface. Exact hypotheses, actual integrals and remaining global-law boundaries are exposed in the authored lesson. Current public context-bound independent semantic reviews are accepted; serialized stabilization remains pending; no stochastic algorithm or model theorem is claimed."
  },
  {
    key := "analysis.quadratic-feature-risk-2",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticFeatureRisk.empirical_squared_risk_eq_L2_norm",
    upstreamDecl := "Samplinglib original quadratic feature-risk supplement v1, interface 2",
    upstreamFile := "AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticFeatureRisk.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["quadratic-risk", "empirical-measure", "L2", "optimisation"],
    saldUse := "Reusable squared-loss, genuine integral and indexed replacement calculations",
    note := "Compiled generic library interface. Exact hypotheses, actual integrals and remaining global-law boundaries are exposed in the authored lesson. Current public context-bound independent semantic reviews are accepted; serialized stabilization remains pending; no stochastic algorithm or model theorem is claimed."
  },
  {
    key := "analysis.quadratic-feature-risk-3",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticFeatureRisk.empirical_law_squared_risk_eq_L2_norm",
    upstreamDecl := "Samplinglib original quadratic feature-risk supplement v1, interface 3",
    upstreamFile := "AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticFeatureRisk.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["quadratic-risk", "empirical-measure", "L2", "optimisation"],
    saldUse := "Reusable squared-loss, genuine integral and indexed replacement calculations",
    note := "Compiled generic library interface. Exact hypotheses, actual integrals and remaining global-law boundaries are exposed in the authored lesson. Current public context-bound independent semantic reviews are accepted; serialized stabilization remains pending; no stochastic algorithm or model theorem is claimed."
  },
  {
    key := "analysis.quadratic-feature-risk-4",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticFeatureRisk.empirical_squared_loss_replacement",
    upstreamDecl := "Samplinglib original quadratic feature-risk supplement v1, interface 4",
    upstreamFile := "AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticFeatureRisk.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["quadratic-risk", "empirical-measure", "L2", "optimisation"],
    saldUse := "Reusable squared-loss, genuine integral and indexed replacement calculations",
    note := "Compiled generic library interface. Exact hypotheses, actual integrals and remaining global-law boundaries are exposed in the authored lesson. Current public context-bound independent semantic reviews are accepted; serialized stabilization remains pending; no stochastic algorithm or model theorem is claimed."
  },
  {
    key := "analysis.quadratic-feature-risk-5",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticFeatureRisk.quadratic_feature_mean_calculus",
    upstreamDecl := "Samplinglib original quadratic feature-risk supplement v1, interface 5",
    upstreamFile := "AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticFeatureRisk.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["quadratic-risk", "empirical-measure", "L2", "optimisation"],
    saldUse := "Reusable squared-loss, genuine integral and indexed replacement calculations",
    note := "Compiled generic library interface. Exact hypotheses, actual integrals and remaining global-law boundaries are exposed in the authored lesson. Current public context-bound independent semantic reviews are accepted; serialized stabilization remains pending; no stochastic algorithm or model theorem is claimed."
  },
  {
    key := "analysis.quadratic-feature-risk-6",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticFeatureRisk.joint_empirical_squared_risk_replacement",
    upstreamDecl := "Samplinglib original quadratic feature-risk supplement v1, interface 6",
    upstreamFile := "AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticFeatureRisk.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["quadratic-risk", "empirical-measure", "L2", "optimisation"],
    saldUse := "Reusable squared-loss, genuine integral and indexed replacement calculations",
    note := "Compiled generic library interface. Exact hypotheses, actual integrals and remaining global-law boundaries are exposed in the authored lesson. Current public context-bound independent semantic reviews are accepted; serialized stabilization remains pending; no stochastic algorithm or model theorem is claimed."
  },
  {
    key := "pbps.actualAugmentedKernelInvariance"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualAugmentedKernelInvariance.actual_augmented_kernel_invariant"
    upstreamDecl := "Actual augmented PBPS one-step invariant kernel"
    upstreamFile := "arXiv2609.06905v1 Proposition3.4, equation3.11 and Algorithm2"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "kernel", "invariance", "conditional", "Tonelli"]
    saldUse := "Actual normalized augmented kernel for the planned PBPS Theorem3.5 convergence route; no SALD claim"
    note := "Independent source/compiler verification at6fbb3f42689e294cbb3f994fee26455baf4afef9; accepted audit ASTIS-RT-20261017-PBPSActualAugmentedKernelInvariance. Original six analytic inputs and exactly source fixed rho in(0,1). Retain the SAME literal actual Z/R/H and every-y conditional reversibility. Derive normalized Gibbs/augmentation probability and the same explicit conditional R, lift fiber invariance through measurable-section Tonelli, construct the fixed probability mixture and compose after Gaussian reflection to obtain exact eq3.11 and K.Invariant J. Three private declarations are implementation, not additional leaves. Exceptional-clock representative is justified parameterwise AE, not by a uniform null event. No process restart/semigroup, augmented all-L2 realization, hypocoercivity or convergence rate, implementation error/caps/actual cost, initialization/composition, PURIFIED or full-paper claim."
  },
  {
    key := "pbps.actualHalfTurnReversibility"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualHalfTurnReversibility.actual_half_turn_reversible"
    upstreamDecl := "Actual PBPS conditional half-turn detailed balance"
    upstreamFile := "arXiv2609.06905v1 Proposition3.2, A.1 bounded duality and A.2 independent-reference averaging"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "kernel", "reversibility", "Fubini", "conditional"]
    saldUse := "Actual augmented PBPS one-step invariance and half-turn operator consumer; no SALD claim"
    note := "Independent source/compiler verification atcffd16e6a5613cd924bee9dc18ab0af6e5f67a3f; accepted audit ASTIS-RT-20261017-PBPSActualHalfTurnReversibility. Under exactly the original six analytic binders, retain the SAME globally measurable actual returned-position kernel, raw process and local initialization outputs, and prove every-y all-measurable-rectangle detailed balance. Identify raw process witnesses, absorb momentum flips on position tests, derive finite signed Fubini and convert indicator pairings to ENNReal fluxes internally. Planned paper consumer: actual augmented one-step kernel in Proposition3.4. No temporal Markov/restart/semigroup, augmented all-L2 realization, hypocoercivity, simulation errors/caps, actual expected costs, warm start, composition or complete paper. Private adapters are not additional leaves."
  },
  {
    key := "pbps.actualKernelL2Realization"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualKernelL2Realization.actual_kernel_l2_realization"
    upstreamDecl := "Actual PBPS invariant phase kernel and real-L2 realization"
    upstreamFile := "arXiv2609.06905v1 Appendix A.1 and Definition D.2, fixed-time observable realization only"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "kernel", "invariance", "representative"]
    saldUse := "Same-process PBPS transition/operator bridge; no SALD claim"
    note := "Independent source/compiler verification at52d523aca2aa825fa23d0fc997592b9128468bd2; accepted audit ASTIS-RT-20261016-PBPSActualKernelL2Realization. Original six analytic assumptions only. Identify the joint phase probability kernel with the SAME raw process Z using covered-arc/fallback equality, derive invariant fixed-parameter state fibers, and identify every canonical real-L2 representative with its kernel expectation AE, including fiber integrability. Derive AE order positivity and fixed real constant classes internally, retaining prior zero-time continuity, flip and Gaussian averaging. No process Markovness/restart, Chapman--Kolmogorov/semigroup, adjoint-density statement, independent-reference half-turn, hypocoercivity, simulation errors, actual expected cost, composition or complete paper."
  },
  {
    key := "pbps.actualL2StrongContinuity"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualL2StrongContinuity.actual_l2_strong_continuity"
    upstreamDecl := "Actual PBPS real-L2 strong continuity at zero"
    upstreamFile := "arXiv2609.06905v1 Appendix A.1 Proposition A.1(ii), zero-time density argument only"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "strong-continuity", "dense-core"]
    saldUse := "Actual same-process PBPS semigroup prerequisite; no SALD claim"
    note := "Independent exact-commit verification eb5f3e75d062428f8ddb219d9dcd3caf56c799ae and accepted source-blind audit ASTIS-RT-20261015-PBPSActualL2StrongContinuity. Under the original six analytic assumptions, preserve the same actual Z, phase family A, flip J and Gaussian position averaging. Identify bounded-continuous-core representatives, prove their L2 norm-square zero-time limit using the actual clock law, then extend by bounded-continuous density and the closed locus of an equicontinuous contraction family. Every real L2 observable and rank0 included. This adds only zero-time strong continuity: no restart, Chapman--Kolmogorov, semigroup, positive-time continuity, kernel/L2 realization, positivity/constants, independent-reference averaging, hypocoercivity, simulation error, actual expected cost, composition or full-paper completion."
  },
  {
    key := "pbps.actualPositionL2Averaging"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPositionL2Averaging.actual_position_l2_averaging"
    upstreamDecl := "Actual PBPS Gaussian momentum averaging and position L2 contraction"
    upstreamFile := "arXiv2609.06905v1 Appendix A.1(iii) final averaging proof"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "Gaussian-average", "self-adjoint"]
    saldUse := "Actual PBPS position averaging; no SALD claim"
    note := "Independent science 86e2da02b6b1a855916adcd46363e34d8a6ecb5b. Under the original six assumptions, retain the same108 global actual process Z, phase family A and canonical flip J. Construct the position lift B and Gaussian adjoint M=B*. For every arbitrary MemLp phase representative, derive Gaussian section integrability, output L2 membership and the literal Gaussian mean class. Prove JB=B and MJ=M, then K_T=M A_T B is selfadjoint, norm at most1 and K_0=id. Identify its literal Gaussian-outer clock-inner position expectation, with both nested AE integrability statements. Rank0 included. No new source premise. Markov positivity/constants, strong continuity, restart/semigroup, separate kernel witnesses, independent-reference averaging, hypocoercivity, errors, actual expected costs, composition and all four complete papers remain open."
  },
  {
    key := "pbps.actualL2FlipDuality"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualL2FlipDuality.actual_l2_flip_duality"
    upstreamDecl := "Actual PBPS full real-L2 momentum-flip adjoint identity"
    upstreamFile := "arXiv2609.06905v1 Appendix A.1 Eq A.3 and density extension"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "adjoint", "momentum-flip"]
    saldUse := "Actual same-process full real-L2 flip duality; no SALD claim"
    note := "Independent science 25a3e47e3e171f2d5584a6568ce685c6f546e529. Under the original six assumptions, the same global actual PBPS process and the same real L2 expectation family from107 retain all path, Borel, Gibbs, bounded-duality, integrability, contraction and initialization conclusions. The canonical momentum-flip pullback J is an involutive L2 isometry and A_T.adjoint = J A_T J for every nonnegative T, on all real L2 classes. Gaussian reflection and product maps prove phase preservation. Bounded simple representatives give the signed pairing; continuity and two-variable density extend it to all L2, then adjoint uniqueness closes Eq A.3. No new source premise. Restart/Markov/semigroup, strong continuity, separate kernel witnesses, actual position averaging, hypocoercivity, errors, actual expected costs, composition and four complete papers remain open."
  },
  {
    key := "pbps.actualL2Expectation"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualL2Expectation.actual_l2_expectation_contraction"
    upstreamDecl := "Actual PBPS L2 expectation contraction and zero-time identity"
    upstreamFile := "arXiv2609.06905v1 Appendix A.1 expectation operator and L2 domain toward Eq A.3"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "expectation", "contraction"]
    saldUse := "Actual same-process L2 expectation contraction; no SALD claim"
    note := "Independent science 2b657840a2f21c5d8001c83e4625a1ab13ef70ba. Under the original six assumptions, the same global jointly measurable actual PBPS Z retains live arcs, fallback, parameterwise initialization, Gibbs preservation and bounded flip duality. Every Borel test has a joint Borel total clock expectation; a uniform finite bound proves every section L1. For each y,r an internally constructed real L2 contraction family A_T realizes the literal clock expectation of every MemLp representative, with almost-everywhere section L1, output L2 membership, exact quotient identity, square-integral contraction and A_0=id. Actual106, normalization103 and UnitExponentialProduct are the three producers. Adjoint pullbacks, product L1, signed Fubini and indicator uniqueness are internal dependencies. Representative independence is a consequence of universal quotient identification; the planned direct-AE route remains an unselected alternative. Full L2 flip identity A.3, restart/Markov/semigroup, separate kernel witness identification, strong continuity, hypocoercivity, errors, actual expected costs, composition and all four papers remain open."
  },
  {
    key := "pbps.actualBoundedGibbsDuality"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualBoundedGibbsDuality.actual_bounded_gibbs_duality"
    upstreamDecl := "Actual PBPS bounded signed Gibbs duality and fixed-time invariance"
    upstreamFile := "arXiv2609.06905v1 Appendix A.1 final bounded signed bilinear identity and Gibbs stationarity"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "Gibbs", "endpoint", "reversal"]
    saldUse := "Actual bounded duality and fixed-time invariance; no SALD claim"
    note := "Independent science edc917ec4c792bfe1308eee35a5d183c3768703c. Under the original six assumptions, one global jointly measurable actual PBPS process retains its exact arc/fallback/zero-time contract. For every y,r,T its endpoint pushes the actual Gibbs phase law times countable exponential clocks to the same Gibbs phase law. For every pair of bounded Borel real F,G, both outer tests are integrable and integral F op(G) equals integral (G composed R) op(F composed R). Actual105, actual normalization103 and UnitExponentialProduct are the three direct producers. Momentum symmetry and signed Fubini are internally discharged. The selected direct-marginal proof and printed F=1 route have separate independently admitted source graph alternatives. L2 operators/domains, restart/Markovness, separate kernel witness identification, hypocoercivity, errors, actual expected costs, full composition and all four full papers remain open."
  },
  {
    key := "pbps.actualCountableEndpointLaw"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualCountableEndpointLaw.actual_countable_endpoint_law"
    upstreamDecl := "Actual PBPS countable endpoint probability and reversal"
    upstreamFile := "arXiv2609.06905v1 Appendix A.1 A7 and nonnegative Gibbs averaging before A8"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "Gibbs", "endpoint", "reversal"]
    saldUse := "Actual countable endpoint law; no SALD claim"
    note := "Independent science a6060928c393d3efe4c85d4cab2b9af166ab6b37. Under the original six assumptions, one global jointly measurable actual clock process Z retains the exact arc/fallback/zero-time contract. For every y,r,T, its Gibbs-initialized endpoint law Q is the countable sum of all actual normalized fixed-count endpoint sectors, is a probability, and is preserved by J(z0,zT)=(RzT,Rz0). Actual93, normalization103, UnitExponentialProduct and sector reversal104 are the four direct producers. Fixed-initial measurable count fibers, jointly measurable sector integrals and nonnegative Tonelli close assembly without a joint-count premise or signed uniform domination. Signed bilinear duality, marginal phase invariance, restart/Markovness, hypocoercivity, implementation errors, expected costs, full composition and all four full papers remain open."
  },
  {
    key := "pbps.actualNormalizedEndpointSector"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualNormalizedEndpointSector.actual_normalized_endpoint_sector_reversal"
    upstreamDecl := "Actual normalized PBPS endpoint-pair sector reversal"
    upstreamFile := "arXiv2609.06905v1 Appendix A.1 fixed-count reversal and Eq3.1 averaging"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "Gibbs", "endpoint", "reversal"]
    saldUse := "Actual normalized endpoint-pair sector; no SALD claim"
    note := "Independent science 1a0fb58a061845fc5ac7623e4d62b5e301d7eb78. Under the original six assumptions and every y,r,n,T, the actual density q_n times terminal survival weights the product of the normalized phase law and strict-simplex measure. Its pushforward through B(z,t)=(z,Gamma(z,t)(T)) is preserved by J(z0,zT)=(RzT,Rz0). Four actual public parents supply finite-count density measurability, endpoint reversal, unnormalized weighted substitution and phase normalization. Same reversal witness, simplex sigma-finiteness and normalized-sector preservation are derived internally. No additional premise or claim that a fixed-count sector is a probability measure. Countable assembly, signed bilinear identity, process invariance, Markovness, hypocoercivity, errors, expected costs and whole-paper/composition remain open."
  },
  {
    key := "pbps.actualPhaseNormalization"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhaseNormalization.actual_phase_gibbs_normalization"
    upstreamDecl := "Actual PBPS phase Gibbs normalization"
    upstreamFile := "arXiv2609.06905v1 conditional energy and Eq3.1"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "Gibbs", "normalization", "Gaussian"]
    saldUse := "Actual conditional position-momentum phase law; no SALD claim"
    note := "Independent science 030f4f70feb3641cafca981ee81c48a2687db1ee. Under the original six analytic assumptions, for every y the actual position Gibbs factor is integrable with positive Zx, and the phase partition is Zphase=ofReal(Zx*(sqrt(2*pi))^dim), positive and finite. The normalized phase withDensity is a probability measure exactly equal to the canonical tilted position law times stdGaussian momentum. Includes rank zero; no minimizer, critical point, normalizer or integrability premise. Existing shared Hessian, Gibbs tail, Gaussian integral/density and Gibbs normalization producers are consumed. Analytically independent of weighted sector102; both are parents only of later normalized averaging. No process invariance, countable assembly, Markovness, hypocoercivity, error, expected cost, main result or composition."
  },
  {
    key := "pbps.actualWeightedSector"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualWeightedSector.actual_weighted_sector_lintegral"
    upstreamDecl := "Actual PBPS weighted finite-sector substitution"
    upstreamFile := "arXiv2609.06905v1 Eq3.1 and Appendix A1 A7/A8 sector substitution"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "weighted-sector", "reversal", "lintegral"]
    saldUse := "Actual finite PBPS weighted sector substitution; no SALD claim"
    note := "Independent science 320ad09ccfd3bc05a9ec3d909c9c19d236c9a232. Original six analytic assumptions and actual density rho=q*terminal survival. On the phase-first Lebesgue times strict-simplex measure mu, actual endpoint/time reversal Theta satisfies integral W(q)F(Theta(q)) dmu=integral W(q)F(q) dmu for every measurable ENNReal F, where W=ofReal(exp(-U))*rho. Three actual public parents93/97/101 supply joint density measurability, measure preservation and weight matching; all ingredients are derived internally. Explicit nonnegative-test extension permits infinite integrals and includes n=0,T=0,rank0,zero normals/rates. No normalization, invariant probability, Markovness, hypocoercivity, error, cost, main result or composition."
  },
  {
    key := "pbps.actualPrescribedWeightedDensity"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedWeightedDensity.actual_prescribed_weighted_density_balance"
    upstreamDecl := "Actual PBPS weighted prescribed density matching"
    upstreamFile := "arXiv2609.06905v1 Eq3.1 and Appendix A1 A7/A8 weighted density"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "weighted-density", "reversal", "finite-path"]
    saldUse := "Actual finite PBPS weighted density matching; no SALD claim"
    note := "Independent science 4713623d3edbc5d258e563e8521271c316f0dc8c. Original six analytic assumptions and exact actual density rho=q*terminalweight. One reversed event-time map yields both w(z)dReal(z,t)=w(RA)dReal(RA,rho(t)) and ofReal(w(z))*rho(z,t)=ofReal(w(RA))*rho(RA,rho(t)). Four public actual parents94/98/99/100 discharge density bridge, Gibbs survival, incoming product and reflected survival. Includes n=0,T=0,rank0,zero normals/rates; no rate division. No weighted sector/integral invariance, normalized invariant probability, Markovness, hypocoercivity, errors, costs, main results or composition."
  },
  {
    key := "pbps.actualPrescribedPathReversal"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedPathReversal.actual_prescribed_path_reversal_and_survival"
    upstreamDecl := "Actual PBPS path reversal and survival-integral substitution"
    upstreamFile := "arXiv2609.06905v1 Appendix A1 A8 off-event reversal and survival substitution"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "reversal", "survival", "finite-path"]
    saldUse := "Actual finite PBPS path reversal and survival substitution; no SALD claim"
    note := "Independent science d9850090b15210e29ecd9b932f02c358d2c9e786. Original six analytic assumptions and literal actual path definitions. On 0<=s<=T away from finitely many reversed event times, barGamma(s)=R Gamma(T-s). Both actual scalar rates are interval integrable and their survival integrals agree. Actual state reversal, path interpolation, harmonic flow, phase reversal and prescribed-density integrability discharge all private premises. Endpoints, n=0,T=0,rank0,zero normals/rates included. No genuine-jump pointwise equality, weighted density/sector, invariant probability, Markovness, hypocoercivity, errors, costs, main results or composition."
  },
  {
    key := "pbps.actualPrescribedStateReversal"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedStateReversal.actual_prescribed_state_reversal_and_rate_product"
    upstreamDecl := "Actual PBPS finite-state reversal and incoming-rate product"
    upstreamFile := "arXiv2609.06905v1 Appendix A1 A5 and A8 finite incoming-state reversal"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "reversal", "incoming-rates", "finite-path"]
    saldUse := "Actual finite PBPS paired-state reversal and rate product; no SALD claim"
    note := "Independent science ce675a4cfad2d35cc8c4fec2c26979728742ac27. Original six analytic assumptions and actual path definitions. Reversing the strict event tuple gives barPost_j=R(arrive_(n-j)) and barArrive_j=R(post_(n-j)) on all n+1 arcs; products of the n incoming event rates agree. Actual phase reversal and endpoint laws discharge all private premises. Includes n=0,T=0,rank0,zero normals and rates. No all-time right-continuous path equality, survival time substitution, weighted density equality, invariant probability, Markovness, hypocoercivity, errors, expected costs, main results or composition."
  },
  {
    key := "pbps.actualPrescribedGibbsSurvival"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedGibbsSurvival.actual_prescribed_gibbs_survival_balance"
    upstreamDecl := "Actual PBPS finite-path energy and Gibbs/survival balance"
    upstreamFile := "arXiv2609.06905v1 Appendix A1 A4 and its integrated Gibbs/survival identity"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "energy", "Gibbs", "survival", "finite-path"]
    saldUse := "Actual finite PBPS target-energy and survival balance; no SALD claim"
    note := "Independent science 530a58f7c036a67787956b20d2d390890ec2fec7. Under the original six analytic assumptions, both actual prescribed-path rate integrals are finite and U_y(A)-U_y(z)=Iplus-Iminus, hence exp(-U_y(z))*exp(-Iplus)=exp(-U_y(A))*exp(-Iminus). The reverse rate uses momentum reversal along the same path. Actual flow, bounce, finite-count path and endpoint laws discharge all private premises. Includes n=0,T=0,rank0,zero normals and zero rates. No reverse-event density matching, weighted Theta preservation, invariant probability, Markovness, hypocoercivity, errors, actual expected costs, main results or composition."
  },
  {
    key := "pbps.actualEndpointTimeReversal"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualEndpointTimeReversal.actual_endpoint_time_reversal_laws"
    upstreamDecl := "Actual PBPS finite endpoint and event-time reversal"
    upstreamFile := "arXiv2609.06905v1 Appendix A1 endpoint/time substitution surrounding A8"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "endpoint", "simplex", "reversal", "volume"]
    saldUse := "Actual finite PBPS endpoint/time change of variables; no SALD claim"
    note := "Independent science 7089dfb0698c50d3ffd0180fb58d06bb32e5d7dd. Under the original six analytic assumptions, rho(t)_i=T-t_(n-1-i) is a measurable involution preserving exact strict-simplex comap volume m. A=Gamma(T) equals the final flow after the recursive post-bounce state, is jointly Borel and preserves phase volume ell for each event tuple. A(R A(z,t),rho t)=Rz. Theta(z,t)=(R A(z,t),rho t) is jointly Borel, involutive and preserves ell tensor m. Includes n=0, T=0, rank0 and zero normals. Actual OrderedPrefix, FiniteCountPath and PhaseReversalVolume supply dependencies internally. Gibbs-weight balance, stationary averaging/invariance, restart/Markovness, hypocoercivity, implementation errors, expected costs, main results and actual-input composition remain open."
  },
  {
    key := "pbps.actualPhaseReversalVolume"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhaseReversalVolume.actual_phase_reversal_volume_laws"
    upstreamDecl := "Actual PBPS phase product-volume and reversal laws"
    upstreamFile := "arXiv2609.06905v1 Appendix A1 flow A5 A6 and elementary A8 mechanics"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "volume", "reflection", "reversal", "integrability"]
    saldUse := "Actual PBPS phase-volume changes of variables; no SALD claim"
    note := "Independent science 2f2e2c01856c902b07406db87026c50639454d71. Under the original six analytic assumptions, the literal harmonic flow, state-dependent bounce and momentum reversal preserve canonical phase product volume. Joint Borel maps, both flow inverses, bounce/reversal involutions, R Phi_u R=Phi_-u, R S=S R, rate(R S z)=rate(z), and the real integrable-test A6 identity are proved. Covers rank0, zero normals and all real times without an unscaled isometry claim. Volume and regularity are derived, not provider assumptions. Finite prescribed endpoint/time reversal, stationary averaging, invariance, Markovness, hypocoercivity, implementation errors, expected costs, main results and actual-input composition remain open."
  },
  {
    key := "pbps.actualBoundedPathExpansion"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualBoundedPathExpansion.actual_bounded_path_functional_expansion"
    upstreamDecl := "Actual PBPS bounded real path functional expansion"
    upstreamFile := "arXiv2609.06905v1 Appendix A1 A7"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "path", "density", "integrability", "summation"]
    saldUse := "Actual PBPS bounded path expectations; no SALD claim"
    note := "Independent science 5df604f6c9ddab63e552272b2e9056752616621e. Under the six original analytic assumptions, the literal stopped-clock process satisfies the full A7 expectation expansion for every bounded real functional measurable for the coordinate-product path sigma algebra. Every weighted finite-simplex term is integrable and the signed component integrals are absolutely summable. Actual finite-count joint laws and the whole-path density identity are consumed internally. Covers n=0,T=0,rank0,zero rates and infinite waits. ASTIS uses countable probability-partition integration as an alternative to the printed rate-cap/factorial proof. No Skorokhod assertion, unbounded-test/cost transfer, restart/Markovness, invariance, hypocoercivity, implementation error, main result or actual-input composition completion."
  },
  {
    key := "pbps.actualPrescribedPathDensity"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedPathDensity.actual_prescribed_path_density_rewrite"
    upstreamDecl := "Actual PBPS prescribed-path hazard and density identity"
    upstreamFile := "arXiv2609.06905v1 Appendix A1 A7 prescribed trajectory ingredient"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "path", "hazard", "left-limit", "density"]
    saldUse := "Finite PBPS prescribed-path density consumer; no SALD claim"
    note := "Independent science ad3b80a339bf1cf0dd88685f202632a92eeb0797. Under exactly the six original analytic assumptions and the literal actual93 prescribed interpolation, every finite strict event tuple has integrable path rate, an exact n+1-arc hazard integral sum, incoming phase and rate left limits, and rho=ofReal(exp(-whole-path hazard integral) times product of incoming jump rates). Terminal survival, n=0, T=0, rank0 and zero rates are retained. No additional path/integrability/limit provider. This deterministic ingredient does not yet compose the actual finite-count measure with the printed density or sum full A7. Restart/Markovness, invariance, hypocoercivity, implementation errors, expected cost, paper mains, actual-input composition and four-paper completion remain OPEN."
  },
  {
    key := "pbps.actualFiniteCountPath"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualFiniteCountPath.actual_finite_count_path_law"
    upstreamDecl := "Actual PBPS finite-count complete-history and coordinate-path law"
    upstreamFile := "arXiv2609.06905v1 Appendix A1 A7 finite-count ingredient and Appendix A2"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "actual-input", "finite-count", "history", "path-law"]
    saldUse := "Actual PBPS finite path-law substrate; no SALD claim"
    note := "Independent science 8ec33d055fb8a9b3ba376d4113e388b00882026f. Six source analytic assumptions and actual stopped iid exponential-clock recursion unchanged. For every finite n and deterministic finite nonnegative horizon T, the unnormalized joint complete-history/coordinate-product-path law on N(T)=n is the pushforward of ordered-prefix density times terminal survival. One jointly measurable actual phase supplies pointwise interpolation on raw survived events, fixed-T endpoint nullity, measurable count, AE finite cardinalities, raw disjointness and full-probability union. All nonnegative tests measurable for the stated sigma algebras and count-sector mass identities are proved. No extra probability, density, measurability, nonexplosion or integrability provider. Hazard stitching, pre-jump left limits, infinite full A7, Skorokhod compatibility, restart/Markovness, invariance/hypocoercivity, errors/cost/main/composition/four-paper completion remain OPEN."
  },
  {
    key := "pbps.actualOrderedPrefix"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualOrderedPrefix.actual_ordered_prefix_law"
    upstreamDecl := "Actual PBPS strict ordered-prefix complete-history law"
    upstreamFile := "arXiv2609.06905v1 Appendix A1 A1 A2 A7 finite iteration ingredient and Appendix A2"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "actual-input", "ordered-simplex", "finite-history", "density"]
    saldUse := "Actual PBPS finite path-law substrate; no SALD claim"
    note := "Independent science a4599f28131fcf91d6b0754003facbc46e23ad2e. With the six original analytic hypotheses, canonical iid Exp(1) product P and literal stopped recursion, every finite n and T>=0 has a strict ordered-prefix complete-history law. Delta_n(T) is the open ordered time simplex with actual subtype comap of product Lebesgue measure. K gives all n+1 live time/state records; q is the product of n pre-bounce rates and elapsed segment survivals. Derive joint Borel K/q, actual history/observer/guard measurability, nonnegative real product, pointwise full-history reconstruction, H_*(P restricted to B)=K_*(q m), all nonnegative extended Borel test identities and integral q=P(B)<=1. Covers n0/T0/rank0/stopped/top/zero-input boundaries without new density or integrability providers. No terminal survival: not an exactly-n count law or full A7. Terminal survival/path/count assembly, restart/Markovness, invariance/hypocoercivity/errors/unbounded costs/main/composition/four-paper Goal remain OPEN."
  },
  {
    key := "pbps.actualAbsoluteNextEventTime"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualAbsoluteNextEventTime.actual_absolute_next_event_time_law"
    upstreamDecl := "Actual PBPS next absolute-event-time law"
    upstreamFile := "arXiv2609.06905v1 Appendix A1 A1 A2 A7 coordinate ingredient and Appendix A2"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "actual-input", "absolute-time", "finite-history", "density"]
    saldUse := "Actual PBPS ordered event-time iteration substrate; no SALD claim"
    note := "Independent science d62f7b2c89e749953a36d8922124865a4651e55e. Under the six original analytic assumptions, actual U reads the next stopped/live record. For each deterministic depth and every Borel finite-valued history horizon b, derive Borel observers/absolute append, all-prefix preservation, strict event equality Brel=B, actual guarded U=a+wait and complete-history append. The actual old-history/U law restricted to B has density D(h,u)=1_(a(h)<u<b(h))*rho(h_last,u-a(h)) against mu_n product real Lebesgue. Every-history density integral and total mass are at most1. Derive next-history joint/marginal transport and all joint ENNReal tests on all old-history events, including null events, without normalization or integrability premises. Direct parents90,89 and canonical exponential-product probability. Fiberwise translation plus Tonelli implements the needed weighted law; no separate broad unweighted shear theorem is claimed. Ordered iteration, final survival/path/count/fullA7, restart/Markov/invariance/hypocoercivity/errors/unbounded costs/main/composition/wholepaper/Goal remain OPEN."
  },
  {
    key := "pbps.actualFiniteHistoryExtension"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualFiniteHistoryExtension.actual_finite_history_extension"
    upstreamDecl := "Actual PBPS complete finite-history extension"
    upstreamFile := "arXiv2609.06905v1 AppendixA1 A1 A2 A7 update iteration ingredient and AppendixA2"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "actual-input", "finite-history", "measurability", "restricted-law"]
    saldUse := "Actual PBPS ordered finite-event iteration substrate; no SALD claim"
    note := "Independent science e8d38b3002f4562ef254205135c0d213f9efe410. Same six original analytic hypotheses and literal stopped iid exponential-clock recursion. Derive Borel actual full next history H_(n+1), total complete-history append A(h,s) and lift L(h,s)=(A(h,s),s); preserve every old coordinate on every abstract history/real s. Live append uses cumulative T+max(s,0), raw-real harmonic flow and actual reflection; stopped append absorbs. Negative time is an auxiliary totalization only. On B={0<W_n<d(H_n)} for every fixed Borel finite-valued history cap d (possibly globally unbounded), derive pointwise actual H_(n+1)=A(H_n,finiteWait), restricted joint/marginal next-history pushforward laws and mass<=1, plus all jointly Borel ENNReal next-history/wait test identities localized to every Borel old-history event. No bounded/integrable test, positive event mass, normalization, density/finite-wait/no-atom/constructor provider. Four actual compiled public parents: conditional density89, history/tail87, harmonic flow and bounce rate; survival88 is not an AND dependency. Only one actual complete-history extension edge within S87-I5/G86-11 closes. Unrestricted next-history laws, ordered absolute-time iteration/final survival/path/count assembly/fullA7, physical-time residual restart/Markovness, global nonexplosion/invariance/hypocoercivity/errors/unbounded costs/main/composition/wholepaper/Goal remain OPEN."
  },
  {
    key := "pbps.actualConditionalNextWaitDensity"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualConditionalNextWaitDensity.actual_conditional_next_wait_density"
    upstreamDecl := "Actual PBPS conditional finite next-wait density"
    upstreamFile := "arXiv2609.06905v1 AppendixA1 A1 A2 A7 strict finite density and AppendixA2"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "actual-input", "conditional-density", "finite-history", "subprobability-kernel"]
    saldUse := "Actual PBPS ordered finite-event iteration substrate; no SALD claim"
    note := "Independent science 58ddf518b7c75537680526fa809624ac7fe0b4c7. Same six original analytic hypotheses and literal stopped actual recursion. For every deterministic event depth n, every fixed Borel finite-valued cutoff d on full histories (no global bound), derive jointly Borel strict finite next-wait density D(h,s), the genuine Mathlib withDensity kernel K(h)=D(h,s)ds on every abstract history, uniform mass<=1/IsFiniteKernel, actual joint history/time measure identity, and all jointly Borel nonnegative extended test identities on every Borel history event A. No positive event probability, normalization, finite-wait/no-atom/density provider or bounded/integrable test premise. Restrict actual86 Ioc law further by Iio; never equate endpoint masses. Stopped histories have zero finite kernel; infinite waits remain outside finite component. Actual87 history/tail product, actual86 deterministic density, actual75 hazard, actual flow/rate and actual77 product are six genuine compiled parents; adjacent survival88 is not an AND dependency. Only density component of S87-I5/G86-11 closes. Ordered-time iteration, final survival/path factor, fullA7, residual restart/Markovness, global nonexplosion/invariance/hypocoercivity, implementation/errors/unbounded costs/main/composition/wholepaper/Goal OPEN."
  },
  {
    key := "pbps.actualConditionalNextWaitSurvival"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualConditionalNextWaitSurvival.actual_conditional_next_wait_survival"
    upstreamDecl := "Actual PBPS history-dependent next-wait conditional survival"
    upstreamFile := "arXiv2609.06905v1 AppendixA1 A1 A2 survival implicit A7 and AppendixA2"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "actual-input", "conditional-survival", "finite-history", "random-duration"]
    saldUse := "Actual PBPS finite-event iteration substrate; no SALD claim"
    note := "Independent science 456e13f9561abcf01a285133e91b3951bf2ed892. Same six original analytic hypotheses and literal actual stopped recursion. For every deterministic event depth n, Borel duration d of complete history H_n and Borel event A, derive joint next-wait event measurability, Borel weight q_d in [0,1], and P(H_n in A and live next wait>d(H_n))=integral_A q_d d(map H_n P). Stopped histories have zero weight; live infinite waits survive all finite durations. No normalization, boundedness of d or law/measurability provider premise. Actual87 joint history/tail law, actual75 hazard survival and actual77 coordinate law are compiled parents. Only S87-I4 closes. S87-I5 conditional finite-wait density, ordered-time disintegration/iteration, final factor/fullA7, random-depth and physical-time residual restart/Markov/invariance/cost/main/composition/wholepaper/Goal remain OPEN."
  },
  {
    key := "pbps.actualFiniteHistoryTailLaw"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualFiniteHistoryTailLaw.actual_finite_history_tail_law"
    upstreamDecl := "Actual PBPS deterministic finite history and unused entire clock tail"
    upstreamFile := "arXiv2609.06905v1 AppendixA1 A1 A2 deterministic iteration implicit A7 and AppendixA2"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "actual-input", "finite-history", "prefix-measurability", "infinite-tail", "independence"]
    saldUse := "Actual PBPS conditional-iteration substrate; no SALD claim"
    note := "Independent science ada2d073d04a78493555b2ac35344b9c5a264bfc. Same six source analytic hypotheses and literal actual76 stopped recursion driven by actual77 iidExp1. For every deterministic n, actual history R0..Rn consumes raw coordinates k<n: derive ambient and F_n measurability, pointwise prefix causality, unchanged full infinite raw tail starting n, history-tail independence/product law and unnormalized restricted-tail pushforward, including n0/stopped/null A. Truncation is only a measurable proof factorization. Closes only probability substrate of G86-11/S87I1-I3; conditional random-hazard next-wait survival/density/final survival/iteration S87I4-I5/fullA7, random stopping/physical-time residual memorylessness/restart/Markov/invariance/allL2/main/error/unboundedcost/composition/wholepaper/Goal remain OPEN."
  },
  {
    key := "pbps.actualFiniteFirstEventLaw"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualFiniteFirstEventLaw.actual_finite_first_event_law"
    upstreamDecl := "Actual PBPS first finite event density and post-bounce marked law"
    upstreamFile := "arXiv2609.06905v1 AppendixA1 A1 A2 first density factor implicit A7"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "actual-input", "first-event-density", "finite-subprobability", "post-bounce-law", "bounded-Borel-test"]
    saldUse := "Actual PBPS first density factor for source path law; no SALD claim"
    note := "Independent science 0bb3cd64ba1235114feaeed99ba22e675b23db76. Under the six original analytic hypotheses retain literal actual P/epsilon/c/Phi/S/rate/Lambda/tau and derive U from actual coordinate0. On A_T={0<U<=T}, prove exact finite first-event time density and SAME post-bounce marked law, mass1-exp(-LambdaT)<=1, three bounded Borel real-test L1 outputs and exact weighted Lebesgue integral. Finite mass is not renormalized; infinity/default0 excluded before projection. T0/M0/rank0/flat hazard/zero-normal identity remain included. Five key-lemma and six public BODY regions. Unused-tail conditioning, fullA7/path/reversal/invariance/restart/Markov/allL2/hypocoercivity/implementation/error/unboundedcost/main/composition remain OPEN. No TV-to-unbounded-cost inference; browser/main/PURIFIED/live/wholepaper/Goal separate."
  },
  {
    key := "pbps.actualPhaseTransitionKernel"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhaseTransitionKernel.actual_phase_transition_kernel"
    upstreamDecl := "Actual PBPS full-phase finite-time law and bounded measurable tests"
    upstreamFile := "arXiv2609.06905v1 AppendixA1 p3.1 p5.2 p6.1"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "actual-input", "physical-time", "probability-kernel", "bounded-Borel-test"]
    saldUse := "Actual PBPS source transition-law interface; no SALD claim"
    note := "Independent science 6d7d42f996a4fbb159c518623fd35c1ac7717fe9. Under the original six analytic hypotheses retain actual80 single jointly measurable physical phase with all eleven literal definitions, live-arc/fallback and fixed-parameter common-AE finite-cover/initialization contract. Construct its full-phase probability kernel jointly indexed by y,reference,initial phase and finite time using actual Exp1 clocks. Exact pushforward/event law, Dirac zero-time law and dual L1 plus integral transfer for every bounded Borel real test are derived. IsMarkovKernel means probability fibers only. Actual process Markov/restart/Chapman-Kolmogorov, path reversal/invariance, all-L2 operators/contraction/density, hypocoercivity, implemented errors/unbounded costs/main/composition remain OPEN; actual browser/main/PURIFIED/live separate."
  },
  {
    key := "pbps.actualOuterBoundedL2Continuity"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualOuterBoundedL2Continuity.actual_outer_bounded_l2_continuity"
    upstreamDecl := "Actual PBPS bounded-test outer square-integral prerequisite"
    upstreamFile := "arXiv2609.06905v1 AppendixA1 Ex22 p6.2"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "actual-input", "physical-time", "bounded-test", "dominated-convergence"]
    saldUse := "Source Ex22 bounded-test outer square-integral ingredient; no SALD claim"
    note := "Independently verified science dd3a23011b91569acbcd591ccd7b06301147d339. Under the original six analytic hypotheses, retain the entire actual83 phase and derive exact q_y=volume.tilted(-V-quadratic) and nu_y=q_y.prod(stdGaussian) probability. For every continuous real f with |f|<=M and M>=0, its actual independent-clock expectation A_t f is state measurable, its square discrepancy is nu_y-integrable and <=4M\u00b2, and the outer square integral tends0 at ordinary NNReal0. C_b/4M\u00b2 are ASTIS elaborations of source C_c bounded-test ingredient; no phase-invariance premise. Full all-L2 AE operator/invariance/Jensen/contraction/density/Markov/restart/semigroup/hypocoercivity/implementation/main/error/unbounded cost/composition and actual browser/main/PURIFIED/live remain OPEN."
  },
  {
    key := "pbps.actualBoundedTestContinuity"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualBoundedTestContinuity.actual_bounded_test_expectation_continuity"
    upstreamDecl := "Actual PBPS bounded-test clock expectation prerequisite"
    upstreamFile := "arXiv2609.06905v1 AppendixA1 Ex22 and p6.1-p6.2"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "actual-input", "physical-time", "bounded-test", "integrability"]
    saldUse := "Source Ex22 pointwise clock-expectation ingredient; no SALD claim"
    note := "Independently verified science 43b698b4d8dbdb7c881ff41b9c26ca8f0135c781. For the actual physical phase, every continuous real test f with global |f|<=M and M>=0 has measurable integrable clock pullbacks, |E f(Z_t)-f(Phi_t z0)|<=2M(1-exp(-Lambda_t)), and E f(Z_t)->f(z0) at ordinary NNReal0 for each fixed initial tuple. Original six analytic hypotheses, literal actual definitions and all prior physical/stochastic clauses retained. C_b and explicit2M are ASTIS elaboration of source C_c small-time step. Outer L2/invariant law/Jensen/contraction/density/Markov/restart/semigroup/hypocoercivity/implementation/main/error/cost/composition and actual browser/main/PURIFIED/live remain OPEN."
  },
  {
    key := "pbps.actualSmallTimeContinuity"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualSmallTimeContinuity.actual_small_time_stochastic_continuity"
    upstreamDecl := "Actual PBPS first-event defect and zero-time stochastic continuity prerequisite"
    upstreamFile := "arXiv2609.06905v1 AppendixA1 Ex22 and p6.1-p6.2"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "actual-input", "physical-time", "first-event", "stochastic-continuity"]
    saldUse := "Source Ex22 pointwise small-time ingredient; no SALD claim"
    note := "Independently verified science 8e182aeb384e5b9edb922c02fc14ceeca6f9b090. The actual jointly measurable phase satisfies measurable phase-flow defect <=1-exp(-Lambda_t), via the actual firstwait survival law and initial live arc. Each fixed y,xRef,z0 and positive real delta has measurable norm-tail probability tending to0 as NNReal t tends to0, including value0 by actual AE initialization. Original six conditions and full physical representative clauses retained. Full L2 strong continuity/Markov/restart/semigroup/invariance/hypocoercivity/implementation/main/cost/composition and visual/main/PURIFIED/live remain OPEN."
  },
  {
    key := "pbps.idealHalfTurnKernel"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.IdealHalfTurnKernel.ideal_half_turn_returned_position_kernel"
    upstreamDecl := "Ideal PBPS H_y half-turn returned-position kernel and actual product initialization"
    upstreamFile := "arXiv2609.06905v1 Algorithm1, Eq2.8, AppendixA1 SS2 p3.1-p4.1"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "ideal-reference", "kernel", "physical-time", "initialization"]
    saldUse := "Source ideal H_y invariance/mixing prerequisite; no SALD claim"
    note := "Independently verified science 7f815975f0ac55a211af4a4ba8c446b0d7f69a9f. Original six analytic conditions and literal actual recurrence give ideal exact-reference H_y at pi as a jointly measurable probability kernel. Exact q_y normalization is derived from Gibbs/conditional-kernel parents. Its independent reference x standard Gaussian x actual exponential product supports actual terminal arc and phase0 law dirac(x) x Gaussian. Full random all-time path law, version uniqueness, phase Markov/semigroup/invariance/mixing, reference implementation/errors/cost and actual PBPS-SPHMC composition remain OPEN. Local gates/visual/main/PURIFIED/live and whole-paper/Goal completion are distinct."
  },
  {
    key := "pbps.actualPhysicalTimeMeasurability"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhysicalTimeMeasurability.actual_physical_time_measurable_phase"
    upstreamDecl := "Actual PBPS joint physical-time phase measurability and AE initialization"
    upstreamFile := "arXiv2609.06905v1 AppendixA1 A2, E2 and between-jump formula p2.2"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "actual-input", "physical-time", "measurable", "initialization"]
    saldUse := "PBPS actual physical-time law/kernel construction prerequisite; no SALD claim"
    note := "Independently verified science ac7cabf30e17a322ec187b7eb13e1a9d4a57695d. Original six analytic source conditions and literal actual recursion yield a total phase representative jointly Borel in deterministic parameters, finite physical time and actual input stream. It agrees with every covering live harmonic arc; uncovered exceptional inputs use explicitly attributed initial-phase fallback. For each fixed deterministic parameter tuple, on one actual-product AE event it agrees with a live arc for every finite time and starts at z0. No uniform parameter AE event or correlated random-parameter substitution follows. Path regularity/adaptedness/Markov/invariance/kernel/hypocoercivity/main/error/expected-cost/composition remain open. Aggregate/visual/main/PURIFIED/live and whole-paper/Goal completion remain separate."
  },
  {
    key := "pbps.actualPhysicalTimeCover"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhysicalTimeCover.actual_fixed_reference_physical_time_cover"
    upstreamDecl := "Actual PBPS finite physical-time interval coverage"
    upstreamFile := "arXiv2609.06905v1 AppendixA1 A2, E2 and between-jump formula p2.2"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "actual-input", "physical-time", "interval", "stopped"]
    saldUse := "PBPS measurable physical-time interpolation prerequisite; no SALD claim"
    note := "Independently verified science 56e4b7e101a1016e03df2971018b1f018f03e6c1. With the original six analytic binders and canonical actual input, every finite physical time almost surely lies in a unique actual half-open event interval and selects a unique actual live record with stored time<=t and finite elapsed offset<actual next wait. No positive-wait/index/live-record/clock/nonaccumulation provider premise. Last live arc remains available when next wait is infinity. No global interpolation/measurability/origin/Markov/invariance/kernel/hypocoercivity/main/error/expected-cost/composition or whole-paper/Goal completion. Aggregate/visual/main/PURIFIED/live remain separate."
  },
  {
    key := "pbps.actualNonaccumulation"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualNonaccumulation.actual_fixed_reference_event_time_nonaccumulation"
    upstreamDecl := "Actual PBPS event-time nonaccumulation"
    upstreamFile := "arXiv2609.06905v1 AppendixA1 Ex8-Ex9 and SLLN/nonaccumulation"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "actual-input", "nonaccumulation", "exponential", "stopped"]
    saldUse := "PBPS global-path prerequisite; no SALD claim"
    note := "Independently verified science bbcad09376c51bbf27c0ed4c16be0dc053bb01c5. Actual fixed-reference PBPS A2 event times under the canonical Exp1 product escape every finite horizon almost surely and have finite bounded-horizon index sets, including index0. Original six source analytic binders; no supplied iid/cap/recurrence/divergence provider. Zero cap stops and absorbs; positive cap controls clocks by actual divergent partial sums. No global physical-time path/Markov/invariance/kernel/hypocoercivity/main/error/expected-cost/composition or whole-paper/Goal completion. Aggregate/visual/main/PURIFIED/live remain distinct."
  },
  {
    key := "probability.unitExponentialProduct"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.UnitExponentialProduct.unit_exponential_product_laws"
    upstreamDecl := "Countable unit-exponential input law and divergent threshold sums"
    upstreamFile := "arXiv2609.06905v1 AppendixA1 independent Exp1 inputs and Ex9/SLLN"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["probability", "exponential", "infinite-product", "independence", "SLLN"]
    saldUse := "PBPS clock consumer; no SALD claim"
    note := "Independently verified at 4f88383540a865aea304c63c40de5a699ea61611 by /root/exact_verify77. Actual countable Exp(1) product, measurable coordinates/clamps, exact coordinate marginals and mutual independence, simultaneous almost-sure positivity/clamp equality and divergent threshold sums. No public probability/iid/integrability/convergence provider premises. ASTIS bounded-indicator SLLN route is separate from the author's open direct Exp mean-one expansion. PBPS finite-recursion spacing is the next consumer; nonaccumulation/global path/Markov/invariance/main/error/cost/actual-input composition and whole-paper/Goal completion remain open. Aggregate/reader, purification/main/live states remain separately evidence-bound."
  },
  {
    key := "pbps.actualFiniteJumpRecursion"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualFiniteJumpRecursion.actual_fixed_reference_finite_jump_recursion"
    upstreamDecl := "Actual finite PBPS jump recursion and original-energy spacing"
    upstreamFile := "arXiv2609.06905v1 AppendixA1 equation(A2) and Ex8"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "recursion", "Borel", "energy", "waiting-time", "stopped"]
    saldUse := "PBPS consumer; no SALD admission"
    note := "Independent exactSCI76 e1f1d85d34426954829a97a46b563ea8e1dab8f1. Original six analytic callers, eleven literal definitions and ten conclusion groups: actual finite stopped postjump recursion(A.2) using actual73 flow, actual74 bounce/rate and actual75 first clock; joint Borel update/records/event times; monotone and absorbing stopping; original initial energy on records and pre-jump arcs; original-energy uniform waiting increment including stopped records; zero-cap/zero-threshold and strictly increasing finite-successor guards. No phase at infinity or global physical-time path for arbitrary zero thresholds. iid Exp1 realization, a.s. positivity/divergence and nonaccumulation remain next independent boundaries; full PDMP/Markov/invariance/kernel/hypocoercivity/main/errors/expected-cost/composition, Exposition/PURIFIED/main/live/Goal remain open."
  },
  {
    key := "pbps.actualHazardClock"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualHazardClock.actual_integrated_hazard_clock_laws"
    upstreamDecl := "Actual PBPS integrated hazard and first exponential clock"
    upstreamFile := "arXiv2609.06905v1 Algorithm1 and AppendixA1 equation(A1)"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "hazard", "first-clock", "exponential", "Borel", "energy"]
    saldUse := "PBPS consumer; no SALD admission"
    note := "Independent exactSCI75 51d3a65f65b189b0afaaf91a248f8c2f58162ef2. Original six analytic callers, eight literal actual definitions and ten conclusions: continuous/Borel integrated actual hazard, finite-interval integrability/nonnegative monotonicity, continuous-time closed first crossing with finite-value and infinity/e=0 guards, joint Borel clock, actual Exp(1) pushforward probability law and exact strict survival including infinity, original-state energy cap and extended waiting lower/zero-cap branches. No arbitrary cap/law provider or almost-sure finite wait premise. Recursive actual PDMP paths/iid nonaccumulation/Markov/invariance/terminal kernel/hypocoercivity/main/errors/expected costs/composition and full Exposition/PURIFIED/main/live/Goal remain open."
  },
  {
    key := "pbps.actualBounceRate"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualBounceRate.actual_bounce_rate_energy_laws"
    upstreamDecl := "Actual PBPS zero-safe bounce, rate and energy-layer bound"
    upstreamFile := "arXiv2609.06905v1 Section2 Algorithm1 and AppendixA1 Proposition3.1 construction"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "reflection", "bounce", "rate", "energy", "Borel"]
    saldUse := "PBPS consumer; no SALD admission"
    note := "Independent exactSCI74 d556a7550f0395d149720da6478bfdfff98368a7. All original six analytic callers, no new premise; actual zero-safe Borel bounce, continuous nonnegative rate, norm/involution/pairing and exact weighted SUM energy preservation, flipped-rate difference and pointwise same-energy-layer Lambda majorant. Internal beta-Lipschitz producer, rank0/alphaeta1/zero-energy/zero-normal legal. Actual clocks/path/nonexplosion/invariance/Markov/terminal kernel/main/errors/expected-query cost/composition and full Exposition/PURIFIED/main/live/Goal remain open."
  },
  {
    key := "pbps.actualHarmonicFlow"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualHarmonicFlow.actual_harmonic_flow_laws"
    upstreamDecl := "Actual PBPS harmonic flow, derivative and conserved energy"
    upstreamFile := "arXiv2609.06905v1 Section2 Algorithm1 and AppendixA1 Proposition3.1 construction"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "harmonic-flow", "energy", "Borel"]
    saldUse := "PBPS consumer; no SALD admission"
    note := "Independent exactSCI73 d7e00a7c0e8b0f37fcc2dbe99f6b646d3a7b1de6. All original six analytic callers, no new premise; joint continuity/Borel, zero/group/inverse, actual two-equation derivative, nonnegative conserved weighted SUM energy, and exact pi endpoint on finite-dimensional real E. Rank0/alphaeta1/zero-energy legal. No random path/clock/bounce/Markov law, invariance/nonexplosion/main/errors/cost/composition or full Exposition/PURIFIED/main/live/Goal credit."
  },
  {
    key := "pbps.hilbertCorrectorPerturbation"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.HilbertCorrectorPerturbation.quadratic_corrector_perturbation"
    upstreamDecl := "Hilbert corrector perturbation algebra with actual PBPS consumer"
    upstreamFile := "arXiv2609.06905v1 B20 and LemmaB4 Ex28-Ex34"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "Hilbert", "corrector", "perturbation"]
    saldUse := "PBPS consumer; no SALD admission"
    note := "Independent exactSCI72 18183c58eee62145b6059ded11c7be05a4cb82de. Explicit auxiliary structural assumptions on one complete real Hilbert space; exact C(u+Gr,v-Ar)-C(u,v)=inner(u,Inv r)+norm(r)^2/2. Actual72 supplies all conditions internally. No probabilistic or actual-r_rho producer. Rank0/alphaeta1 legal; no extra caller, onto or higher regularity. Full Exposition/PURIFIED/main/live/Goal not inferred."
  },
  {
    key := "analysis.hilbertSharpQuadraticCorrectorBound"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.HilbertCorrectorBound.quadratic_corrector_bound_of_square_identity"
    upstreamDecl := "Sharp quadratic corrector bound from a selfadjoint square identity"
    upstreamFile := "arXiv2609.06905v1 Appendix B3 (B23); ASTIS auxiliary Hilbert-space estimate"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["Hilbert", "selfadjoint", "quadratic-form", "sharp-corrector"]
    saldUse := "PBPS actual consumer; no SALD admission"
    note := "Independently exact-science68 VERIFIED 3ad3b127b5a645be9cf71b3d14520b2d8fea3122. Arbitrary complete real Hilbert space including zero: selfadjoint K,D, I+K\u00b2=D\u00b2 and norm(D)<=c with c>=0 yield the exact c/2 two-component bound. Uses the sum of squared component norms. No c>=1, finite dimension, nontriviality, operator positivity or K,D commutation premise. Actual PBPS supplies D=Inv and K=A0 Inv. Not a separately printed paper theorem."
  },
  {
    key := "gibbs.gradient.actual-zero-mean"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GibbsGradientMean.integrable_gradient_and_integral_eq_zero"
    upstreamDecl := "integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable / ASTIS actual Gibbs domain producers"
    upstreamFile := "Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts; ASTIS authored Gibbs cancellation"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["Gibbs", "gradient", "mean-zero", "IBP", "SPHMC"]
    saldUse := "Actual SPHMC true posterior smoothed-score mean identity; generic MCMC reuse planned only"
    note := "Exact true volume tilt, internally derived probability/L1/weighted fullspace IBP products, positive lower and finite upper Hessian bounds; alpha<=beta not assumed, dimension0 retained. Independent exactcommit f91b82a2920b40299e57ccb821583a91e0a3e188. No Gaussian LSI/T2/bias/main/cost credit."
  },
  {
    key := "analysis.hessian-secant.actual-integral",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.HessianSecantOperator.hessian_secant_operator",
    upstreamDecl := "Genuine Hessian segment integral represents full gradient difference",
    upstreamFile := "arXiv:2609.06906v1 Lemma4.6 proof HTML568-572",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["analysis", "Hessian", "gradient", "Bochner integral", "SPHMC"],
    saldUse := "SPHMC Lemma4.6 actual Picard-node H_j then averaged H; Chewi Proposition1.6/Exercise3.2 shared gradient-segment mechanism (planned source consumers)",
    note := "Actual C2 Riesz Hessian, ordinary unit-interval CLM integral, full secant equality, symmetry, signed quadratic bounds/norm on complete real Hilbert space including0. No supplied operator/secant/integrability. Smoothed potential producer, actual averaged H, discrete K expansion/contraction and both main/composition remain separate."
  },
  {
    key := "analysis.chebyshev-lobatto.momentum-positive",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.ChebyshevLobattoPositiveWeights.nonnegative_momentum_weights",
    upstreamDecl := "Actual ordinary integral momentum weights are nonnegative",
    upstreamFile := "arXiv:2609.06906v1 Appendix B.1 Proposition B.1 (B2)",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["analysis", "interpolation", "quadrature", "Chebyshev-Lobatto", "SPHMC"],
    saldUse := "SPHMC Lemma4.6 actual averaged Hessian bounds (planned), then Proposition4.7",
    note := "All J>=2,h>0; actual source cardinal polynomials/ordinary Lebesgue integrals. Derive finite cosine formula, moments, both parities, positive bracket and source affine scaling. No positivity/formula premise. Private helpers no separate credit. LogLambda/B2 and actual average-H/kernel/main remain independent."
  },
  {
    key := "analysis.monotone-proximal.nonexpansive",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.MonotoneProximalMap.nonexpansive_of_monotone_optimality",
    upstreamDecl := "Monotone exact optimality implies nonexpansiveness (existing proof extraction)",
    upstreamFile := "arXiv:2609.06906v1 Lemma 4.2 / Appendix D.1 proximal prerequisite",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["analysis", "proximal", "monotone", "nonexpansive", "SPHMC"],
    saldUse := "actual SPHMC exact-estimator input stability and full-range proximal phase comparison",
    note := "eta>=0, monotonicity and exact optimality imply LipschitzWith1. Extracted existing private proof once; no new mathematical completion credit. No existence, measurability or sampler claim."
  },
  {
    key := "analysis.chebyshev-lobatto.momentum-absolute-norm",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.ChebyshevLobattoMomentum.momentum_absolute_sum_le",
    upstreamDecl := "Absolute integral norm of actual SPHMC momentum coefficients",
    upstreamFile := "arXiv:2609.06906v1 Appendix B.1 coefficients / D.1 synchronous comparison",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["analysis", "interpolation", "quadrature", "Chebyshev-Lobatto", "SPHMC"],
    saldUse := "actual two-layer SPHMC proximal phase comparison; depth-K coefficient perturbation (planned)",
    note := "Actual J>=2,h>0 source basis and compact Lambda give sum|b_j|<=hLambda. Continuity/interval integrability derived. Does not prove individual b positivity, logarithmic Lambda or B2 remainder."
  },
  {
    key := "analysis.chebyshev-lobatto.coefficients",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.ChebyshevLobattoQuadrature.chebyshev_lobatto_coefficients",
    upstreamDecl := "SPHMC 3.6 / Proposition B.1 node, row and constant-integral components",
    upstreamFile := "arXiv:2609.06906v1 Appendix B.1; Mathlib.LinearAlgebra.Lagrange",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["analysis", "interpolation", "quadrature", "Chebyshev-Lobatto", "SPHMC"],
    saldUse := "actual SPHMC counted phase source coefficients; depth-K exact Picard integrator (planned consumer)",
    note := "J>=2 and h>0. Actual distinct nodes, cardinal/partition/endpoint laws, compact Lambda>=1, sharp row bound h^2*Lambda/2 and sum momentum weights=h. This coefficient theorem omits logarithmic Lambda, individual weight positivity and B2 remainder; the separate admitted positive-weights leaf closes positivity only. Not complete B1 or a main theorem."
  },
  {
    key := "analysis.gradient.ae-congruence-obstruction",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientAECongruence.not_gradient_ae_congr_for_arbitrary_measure",
    upstreamDecl := "representative-choice obstruction for classical gradients",
    upstreamFile := "Mathlib.Analysis.Calculus.Gradient.Basic; Mathlib.MeasureTheory.Measure.Dirac",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "FI", "gradient", "almost-everywhere", "representative", "typed-blocker", "SPHMC"],
    saldUse := "prevent unsound transfer of a classical score from an explicit smooth log-ratio representative to an arbitrary a.e.-equal canonical llr representative",
    note := "Dirac counterexample: the zero and identity functions agree delta_0-a.e. but their classical gradients disagree delta_0-a.e. A valid adapter needs local equality, a selected differentiable representative, or weak-gradient semantics."
  },
  {
    key := "analysis.optimisation.uniform-regularization",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.UniformRegularization.uniform_accuracy_and_query_bound",
    upstreamDecl := "QuadraticRegularizationTransfer.exists_minimizer_radius_and_accuracy; QuadraticRegularizationFirstOrder.curvature_gradient_and_smoothness; QuadraticRegularizationOracle.simulate_regularized",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1 Section4.1 Lemma4.2",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "optimisation", "regularization", "oracle", "uniform-complexity"],
    saldUse := "Uniform convex solver accuracy and actual normalized query bound from a uniform strongly convex program.",
    note := "Proper Hilbert; positive small-error regime; natural-valued budget and finite deterministic execution. No Phi monotonicity. Full source model/domain adapters remain."
  },
  {
    key := "analysis.optimisation.regularization-oracle",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticRegularizationOracle.simulate_regularized",
    upstreamDecl := "QuadraticRegularizationFirstOrder.curvature_gradient_and_smoothness; Nat.succ_le_succ",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1 Section1.1 and Section4.1 Lemma4.2",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "optimisation", "regularization", "oracle", "query-count"],
    saldUse := "Actual adaptive first-order oracle execution with one original reply per regularized reply.",
    note := "Finite deterministic query-fuel semantics; exact state/outcome/count preservation. Halt and exhaustion distinct. No class-uniform solver/budget or full Lemma4.2 closure."
  },
  {
    key := "analysis.optimisation.regularization-first-order",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticRegularizationFirstOrder.curvature_gradient_and_smoothness",
    upstreamDecl := "strongConvexOn_iff_convex; LinearMap.convexOn; HasFDerivAt.norm_sq; LipschitzWith.dist_le_mul",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1 Section4.1 Lemma4.2 proof",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "convexity", "optimisation", "regularization", "smoothness"],
    saldUse := "Actual regularized curvature and gradient Lipschitz shift at first-order regularity.",
    note := "Differentiable convex Hilbert objective with genuine beta-Lipschitz gradient. Nonnegative precision includes zero. No C2/Hessian, minimizer or oracle/class theorem."
  },
  {
    key := "analysis.optimisation.regularization-transfer",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticRegularizationTransfer.exists_minimizer_radius_and_accuracy",
    upstreamDecl := "IsCompact.exists_isMinOn; isCompact_closedBall; Metric.mem_closedBall; dist_eq_norm",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1 Section4.1 Lemma4.2 proof",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "optimisation", "regularization", "minimizer", "accuracy"],
    saldUse := "Construct actual regularized minimum and transfer radius and objective approximation.",
    note := "Continuous proper normed group, supplied original minimum, positive radius and precision. No convexity/C2 assumed; no uniqueness, curvature/smoothness, solver or oracle/class closure."
  },
  {
    key := "analysis.optimisation.restart-log-complexity",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.RestartLogComplexity.logarithmic_accuracy_and_cost",
    upstreamDecl := "RestartReduction.radius_accuracy_and_cost; Nat.le_ceil; Nat.ceil_eq_zero; Real.log_le_log_iff; Real.log_pow",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1 Section4.1 Lemma4.1 proof",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "convexity", "optimisation", "restart", "complexity"],
    saldUse := "Actual logarithmic restart horizon, final accuracy and explicit certified call-cost bounds.",
    note := "Positive radius, C1 Hilbert, supplied minimizer/base solver. N0 retains final call. Global two-term cost; pure-log3/log4 bound only under phi comparison and q>=4. No formal oracle/class realization."
  },
  {
    key := "analysis.optimisation.restart-reduction",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.RestartReduction.radius_accuracy_and_cost",
    upstreamDecl := "StrongConvexFirstOrder.firstOrder_lower_bound_of_strongConvexOn; IsLocalMin.fderiv_eq_zero; Nat.rec",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1 Section4.1 Lemma4.1 proof",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "convexity", "optimisation", "restart"],
    saldUse := "Actual radius-halving restart, final objective accuracy and cumulative certified solver cost.",
    note := "Positive upper radius, C1 Hilbert, supplied minimizer/base solver; exact finite N phi(8kappa)+phi(kappa). No phi monotonicity, oracle-machine or full logarithmic reduction claim; N0 retains polishing."
  },
  {
    key := "analysis.gradient-descent.distance-complexity",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentComplexity.distance_le_of_log_bound",
    upstreamDecl := "GradientDescentContraction.gradient_descent_distance_bound; Real.log_le_iff_le_exp",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1 Section3 Theorem3.3 following rate paragraph",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "convexity", "optimisation", "gradient-descent"],
    saldUse := "Actual reciprocal-step gradient iterates reach distance accuracy under the exact logarithmic budget.",
    note := "Positive alpha,beta,epsilon; threshold only R>0, zero radius and zero iterations retained. C1 Hilbert upper-model generalization; supplied minimizer. Sufficient count, not optimal complexity or companion-paper progress."
  },
  {
    key := "analysis.gradient-descent.convex-normalized-value",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentRates.convex_value_le",
    upstreamDecl := "Theorem3.4; GradientDescentValue.gradient_descent_weighted_value_bound",
    upstreamFile := "Chewi Lectures on Optimization arXiv2605.07006v1 Section3",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "convexity", "optimisation", "gradient-descent"],
    saldUse := "Normalized actual-iterate function-value rate from existing weighted parent.",
    note := "Positive h,N; arbitrary comparator; source beta reciprocal specialization tested; no analytic alpha limit."
  },
  {
    key := "analysis.gradient-descent.strong-normalized-value",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentRates.strongly_convex_value_le",
    upstreamDecl := "Theorem3.4; GradientDescentValue.gradient_descent_weighted_value_bound",
    upstreamFile := "Chewi Lectures on Optimization arXiv2605.07006v1 Section3",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "convexity", "optimisation", "gradient-descent"],
    saldUse := "Normalized actual-iterate function-value rate from existing weighted parent.",
    note := "Rational q0 extension and exact inverse power only q>0; source singular domain independently reviewed."
  },
  {
    key := "analysis.gradient-flow.last-time-lyapunov",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowLastIterate.lyapunov_and_rates",
    upstreamDecl := "Exercise2.1; ConvexityC2.gradient_mono_iff_fderiv2_lower; StrongConvexFirstOrder.firstOrder_lower_bound_of_strongConvexOn",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1 Section2; Mathlib.Analysis.ODE.Gronwall",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "convexity", "optimisation", "gradient-flow"],
    saldUse := "Actual convex-flow Lyapunov antitonicity and last-time gradient-square/value upper bounds.",
    note := "Hessian positivity derived; C2 Hilbert and right-time generalization explicit. Positive-time normalized coefficients 1/t squared and 1/(4t); nonsmooth sharpness and Exercise2.2 remain separate."
  },
  {
    key := "analysis.gradient-flow.attained-stationarity",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowStationarity.exists_min_norm_le",
    upstreamDecl := "Corollary2.8; IsCompact.exists_isMinOn; le_gronwallBound_of_liminf_deriv_right_le",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1 Section2; Mathlib.Analysis.ODE.Gronwall",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "optimisation", "gradient-flow", "stationarity"],
    saldUse := "Attained minimum of actual gradient norm over a positive time interval, bounded by the square root of initial objective gap divided by time.",
    note := "C1 Hilbert and right-time generalization explicit; global minimum and actual flow supplied. No convexity, PL, existence, final-time gradient or full trajectory convergence claim."
  },
  {
    key := "analysis.gradient-flow.convex-value-rate",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowValue.value_le",
    upstreamDecl := "Theorem2.4; firstOrder_lower_bound_of_strongConvexOn; le_gronwallBound_of_liminf_deriv_right_le",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1 Section2; Mathlib.Analysis.ODE.Gronwall",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "convexity", "optimisation", "gradient-flow"],
    saldUse := "Actual convex gradient trajectory objective rate with direct zero-curvature branch on finite forward intervals.",
    note := "Positive observation time: printed t0 quotient singularity has independent domain-clarification review. Hilbert/differentiability/right-time generalization explicit; minimum and flow supplied; no analytic parameter-limit or existence claim."
  },
  {
    key := "analysis.gradient-flow.pair-contraction",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowContraction.norm_sub_le",
    upstreamDecl := "Theorem2.2; gradient_inner_lower_bound_of_strongConvexOn; HasDerivWithinAt.norm_sq; le_gronwallBound_of_liminf_deriv_right_le",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1 Section2; Mathlib.Analysis.ODE.Gronwall",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "convexity", "optimisation", "gradient-flow"],
    saldUse := "Two actual gradient trajectories contract with exact curvature rate on finite forward intervals.",
    note := "Source alpha>=0 retained. C2 Euclidean setting generalized to differentiable Hilbert objective and right derivatives; no minimum or nonzero distance needed. No existence, extension or stochastic-flow claim."
  },
  {
    key := "analysis.gradient-flow.pl-dissipation-decay",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowPL.dissipation_and_decay",
    upstreamDecl := "Lemma2.1 and Corollary2.6; HasFDerivAt.comp_hasDerivWithinAt; le_gronwallBound_of_liminf_deriv_right_le",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1 Section2; Mathlib.Analysis.ODE.Gronwall",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "PL", "optimisation", "gradient-flow"],
    saldUse := "Actual Hilbert gradient dynamics yield objective dissipation and PL exponential value decay on a finite forward interval.",
    note := "Source C2 Euclidean setting generalized to differentiable Hilbert objective and right derivatives. Terminal continuity retained; no negative-time dynamics, existence/uniqueness, point-distance or stochastic-flow claim."
  },
  {
    key := "analysis.strong-convex.nonlinear-pl-pullback",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexPLPullback.exists_minimizer_and_pl",
    upstreamDecl := "Exercise2.3 (numerical PL inequality component)",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "PL", "optimisation"],
    saldUse := "Strong convexity and adjoint Jacobian coercivity yield a numerical PL bound for a differentiable nonlinear composite, with a lifted global minimizer.",
    note := "Differentiable Hilbert extension; sigma0 is degenerate. Does not assert composite C1 regularity, full positive-modulus PL definition, convexity, uniqueness or algorithm convergence."
  },
  {
    key := "analysis.gradient-descent.convex-gap-sharpness",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexGradientGapSharpness.quadratic_gap_lower_bound",
    upstreamDecl := "Exercise3.3 (convex function-value order in Theorem3.4)",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "quadratic", "optimisation"],
    saldUse := "Horizon-dependent admissible quadratic gives exact actual GD gap and beta/(16(N+1)) lower bound at step1/beta.",
    note := "Class beta need not be tight. Witness depends on N; no fixed-objective reciprocal tail, optimal constant, full exercise or adaptive/oracle lower bound."
  },
  {
    key := "analysis.gradient-descent.sharpness",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentSharpness.exists_quadratic_worst_case",
    upstreamDecl := "Exercise3.3 (fixed-step distance sharpness of Exercise3.2)",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "quadratic", "optimisation"],
    saldUse := "Actual scalar quadratic witnesses the endpoint envelope at every iteration count and its balanced minimax factor, with all function-class certificates proved.",
    note := "Positive alpha<=beta are valid class bounds, not both tight constants of the witness. Fixed-step distance sharpness only; no full-section or variable/adaptive/oracle lower bound."
  },
  {
    key := "analysis.quadratic-gd.iterate",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticGradientDescent.quadratic_gradient_iterate",
    upstreamDecl := "Exercise3.3 (exact trajectories component)",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "quadratic", "optimisation"],
    saldUse := "Actual centered-quadratic gradient derived by differentiation and symmetry; trajectory is the update operator power.",
    note := "Symmetric continuous Hilbert operator generalizes the positive-definite Euclidean source. Identities only: no stability, spectral endpoint existence, full-section sharpness or companion-paper completion."
  },
  {
    key := "analysis.quadratic-gd.eigenmode",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticGradientDescent.quadratic_eigenmode",
    upstreamDecl := "Exercise3.3 (exact trajectories component)",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "quadratic", "optimisation"],
    saldUse := "Given a mode relation, exact trajectory, absolute-value norm factor and squared objective factor; zero vector and unstable steps included.",
    note := "Symmetric continuous Hilbert operator generalizes the positive-definite Euclidean source. Identities only: no stability, spectral endpoint existence, full-section sharpness or companion-paper completion."
  },
  {
    key := "analysis.gradient-step.endpoint-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentOptimalStep.gradient_step_endpoint_bound",
    upstreamDecl := "Exercise3.2",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "smoothness", "optimisation"],
    saldUse := "Actual-gradient endpoint curvature bound from C2 Hessian bounds, symmetry, Rayleigh norm and explicit segment FTC.",
    note := "C2 actual-gradient Hilbert extension; explicit parameter domains. Uniform curvature envelope, not an objective-specific optimal step, iterate convergence or companion-paper completion."
  },
  {
    key := "analysis.gradient-step.optimal-step",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentOptimalStep.optimal_gradient_step",
    upstreamDecl := "Exercise3.2",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "smoothness", "optimisation"],
    saldUse := "Balanced step 2/(alpha+beta) gives factor (beta-alpha)/(alpha+beta), minimizing the endpoint envelope over all real steps.",
    note := "C2 actual-gradient Hilbert extension; explicit parameter domains. Uniform curvature envelope, not an objective-specific optimal step, iterate convergence or companion-paper completion."
  },
  {
    key := "analysis.gradient-descent.sum-sq",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentStationarity.gradient_descent_sum_sq_bound",
    upstreamDecl := "Theorem3.7 and proof",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "smoothness", "optimisation"],
    saldUse := "Actual-iterate accumulated squared-gradient bound via canonical descent and finite telescoping; no minimum needed.",
    note := "Actual gradient and global quadratic upper model; Hilbert algebraic generalization. Explicit domain overlay independently accepted. Best iterate, not last iterate, global optimality or companion-paper completion."
  },
  {
    key := "analysis.gradient-descent.stationarity",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentStationarity.exists_gradient_descent_norm_le",
    upstreamDecl := "Theorem3.7 and proof",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "smoothness", "optimisation"],
    saldUse := "Nonconvex best-iterate gradient-norm guarantee from actual updates, supplied global minimum, positive step and nonempty horizon.",
    note := "Actual gradient and global quadratic upper model; Hilbert algebraic generalization. Explicit domain overlay independently accepted. Best iterate, not last iterate, global optimality or companion-paper completion."
  },
  {
    key := "analysis.gradient-descent.step-descent",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentBasic.gradient_step_descent_of_quadratic_upper_bound",
    upstreamDecl := "Lemma3.1",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "smoothness", "optimisation"],
    saldUse := "Canonical public extraction of existing actual-gradient descent; reused by energy/value and PL consumers.",
    note := "Actual totalized gradient in global upper/PL models, h>=0,beta*h<=1; real Hilbert and signed-moduli algebraic extension. Source step repair independently accepted. No convexity, optimizer existence or companion-paper claim."
  },
  {
    key := "analysis.gradient-descent.pl-value",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentPL.gradient_descent_pl_value_bound",
    upstreamDecl := "Theorem3.6",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "smoothness", "optimisation"],
    saldUse := "Actual iterate PL final-value rate; supplied global minimum handles both signs of geometric coefficient.",
    note := "Actual totalized gradient in global upper/PL models, h>=0,beta*h<=1; real Hilbert and signed-moduli algebraic extension. Source step repair independently accepted. No convexity, optimizer existence or companion-paper claim."
  },
  {
    key := "analysis.gradient-descent.comparator-energy",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentValue.gradient_step_energy_bound",
    upstreamDecl := "Theorem3.4 (3.1),(3.3)",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "convexity", "smoothness", "optimisation"],
    saldUse := "Arbitrary-comparator energy inequality and actual parent of weighted final value.",
    note := "C1 complete real Hilbert space and global curvature/upper models; h>=0,beta*h<=1, weighted alpha*h<=1. Whole-module review includes private descent; source step gap separately reviewed. No full inverse-rate or companion-paper claim."
  },
  {
    key := "analysis.gradient-descent.weighted-value",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentValue.gradient_descent_weighted_value_bound",
    upstreamDecl := "Theorem3.4 weighted proof after Lemma3.5",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "convexity", "smoothness", "optimisation"],
    saldUse := "Actual Nth gradient iterate, signed comparator gap and finite geometric weights; normalized rates exercised in Tests.",
    note := "C1 complete real Hilbert space and global curvature/upper models; h>=0,beta*h<=1, weighted alpha*h<=1. Whole-module review includes private descent; source step gap separately reviewed. No full inverse-rate or companion-paper claim."
  },
  {
    key := "analysis.gradient-descent.step-contraction",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentContraction.gradient_step_contraction",
    upstreamDecl := "Theorem3.3 single-step inequality",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "convexity", "smoothness", "optimisation"],
    saldUse := "Actual parent of the iterated distance theorem.",
    note := "C1 complete real inner-product space; global strong convexity and quadratic upper model. Nonnegative step and division-free beta*h<=1 explicit. Source gap independently reviewed; no logarithmic complexity or companion-paper theorem."
  },
  {
    key := "analysis.gradient-descent.distance-rate",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentContraction.gradient_descent_distance_bound",
    upstreamDecl := "Theorem3.3 iterated distance paragraph",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "convexity", "smoothness", "optimisation"],
    saldUse := "Actual Nth gradient iterate around a global minimizer, with geometric and exponential distance control.",
    note := "C1 complete real inner-product space; global strong convexity and quadratic upper model. Nonnegative step and division-free beta*h<=1 explicit. Source gap independently reviewed; no logarithmic complexity or companion-paper theorem."
  },
  {
    key := "analysis.convex-smooth.bregman-gradient",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexSmoothGradient.gradient_gap_sq_le_bregman",
    upstreamDecl := "Exercise3.1 (3.4), positive reciprocal domain",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "convexity", "smoothness", "optimisation"],
    saldUse := "Actual parent of cocoercivity; positive-domain reciprocal consumer.",
    note := "C1 convex global quadratic upper model on a complete real inner-product space. Positive reciprocal scope and Hilbert generalization explicit; scaled cocoercivity and Lipschitz include beta=0. Not a SALD or companion-paper theorem."
  },
  {
    key := "analysis.convex-smooth.cocoercivity",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexSmoothGradient.gradient_cocoercive",
    upstreamDecl := "Exercise3.1 (3.5), including zero modulus",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "convexity", "smoothness", "optimisation"],
    saldUse := "Actual parent of Lipschitz; nonexpansive gradient-step consumer.",
    note := "C1 convex global quadratic upper model on a complete real inner-product space. Positive reciprocal scope and Hilbert generalization explicit; scaled cocoercivity and Lipschitz include beta=0. Not a SALD or companion-paper theorem."
  },
  {
    key := "analysis.convex-smooth.gradient-lipschitz",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexSmoothGradient.gradient_lipschitz",
    upstreamDecl := "Exercise3.1 Cauchy-Schwarz conclusion",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "convexity", "smoothness", "optimisation"],
    saldUse := "Actual consumer of prior C1 single-sided smoothness equivalence with convexity; zero-modulus gradient constancy.",
    note := "C1 convex global quadratic upper model on a complete real inner-product space. Positive reciprocal scope and Hilbert generalization explicit; scaled cocoercivity and Lipschitz include beta=0. Not a SALD or companion-paper theorem."
  },
  {
    key := "analysis.smoothness.c1-upper-equivalence",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.SmoothnessEquivalences.upper_model_iff_gradient_upper",
    upstreamDecl := "Definition1.12 (1.7), Proposition1.13 C1 clause",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "smoothness", "Hessian", "optimisation"],
    saldUse := "Actual parent of C2 equivalence; beta0 non-Lipschitz concave boundary test",
    note := "Complete real inner-product space and signed beta explicitly generalize the source. Uses actual derivatives and signed convexity parents on -f. No convexity assumption or Lipschitz-gradient conclusion; independent source review accepted the boundary."
  },
  {
    key := "analysis.smoothness.c2-hessian-equivalence",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.SmoothnessEquivalences.upper_model_iff_fderiv2_upper",
    upstreamDecl := "Proposition1.13 C2 Hessian clause",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "smoothness", "Hessian", "optimisation"],
    saldUse := "Actual gradient-step descent consumer from Hessian upper bound; signed sharp quadratic test",
    note := "Complete real inner-product space and signed beta explicitly generalize the source. Uses actual derivatives and signed convexity parents on -f. No convexity assumption or Lipschitz-gradient conclusion; independent source review accepted the boundary."
  },
  {
    key := "analysis.calculus.segment-hessian-integral",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC2.gradient_sub_inner_eq_integral_fderiv2",
    upstreamDecl := "Proposition1.6 final Hessian FTC identity",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "CONV", "Hessian", "optimisation"],
    saldUse := "Actual parent of gradient-Hessian equivalence; reversed quadratic segment test",
    note := "Complete real inner-product-space generalization; C2 derives genuine continuous, interval-integrable Hessian pairings."
  },
  {
    key := "analysis.strong-convexity.gradient-hessian-equivalence",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC2.gradient_mono_iff_fderiv2_lower",
    upstreamDecl := "Proposition1.6 (1.5)-(1.6), derivative limit and FTC",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "CONV", "Hessian", "optimisation"],
    saldUse := "Actual parent of source C2 specialization; signed quadratic curvature test",
    note := "Signed modulus permitted. Cancels only positive directional parameter; independent derivative and source audits completed."
  },
  {
    key := "analysis.strong-convexity.c2-hessian-equivalence",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC2.strongConvexOn_iff_fderiv2_lower",
    upstreamDecl := "Proposition1.6 part2, (1.3)-(1.6)",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "CONV", "Hessian", "optimisation"],
    saldUse := "Source C2 theorem; joins unchanged C1 equivalences and stationary-point growth consumer",
    note := "Whole finite-dimensional Euclidean space, nonnegative modulus, genuine C2. Completes the selected part2 boundary, not a chapter or Riemannian analogue."
  },
  {
    key := "analysis.calculus.segment-gradient-integral",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC1.sub_eq_integral_gradient",
    upstreamDecl := "Proposition 1.6 proof: segment FTC identity",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "gradient", "FTC", "shared"],
    saldUse := "Actual parent of the two-segment integral converse; reversed-segment quadratic test",
    note := "C1 derives continuous, interval-integrable gradient pairing. Complete real inner-product-space generalization; source boundary separately reviewed."
  },
  {
    key := "analysis.strong-convexity.gradient-integral-converse",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC1.strongConvexOn_univ_of_gradient_mono_integral",
    upstreamDecl := "Proposition 1.6 (1.5) to (1.3)",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "CONV", "gradient", "FTC", "shared"],
    saldUse := "Actual parent of the exact C1 equivalence adapter; signed quadratic normalization tested",
    note := "Source integral proof with explicit Hilbert-space and signed-modulus generalizations. Cancels only s(1-t)>0; endpoints and integrability handled. No C2 claim."
  },
  {
    key := "analysis.strong-convexity.c1-equivalences",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC1.convexity_equivalences",
    upstreamDecl := "Proposition 1.6 part 1, (1.3)-(1.5)",
    upstreamFile := "Chewi Lectures on Optimization arXiv:2605.07006v1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "CONV", "gradient", "equivalence", "optimisation"],
    saldUse := "Whole-space Euclidean C1 source theorem; stationary-point quadratic-growth and zero-dimension consumers",
    note := "Nonnegative modulus retained; exact source C1 equivalence independently reviewed. Reuses the shared first-order bound and new integral converse. The separate ConvexityC2 module now supplies the C2/Hessian equivalence; this entry itself covers C1 only."
  },
  {
    key := "analysis.strong-convexity.first-order-lower-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexFirstOrder.firstOrder_lower_bound_of_strongConvexOn",
    upstreamDecl := "Strong_Convex_second_lower",
    upstreamFile := "Optlib/Convex/StronglyConvex.lean@5da27c5f95aa6a8a45b8c14b968ade4c13ff18c3",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "CONV", "strong-convexity", "gradient", "first-order", "shared"],
    saldUse := "Parent of gradient_inner_lower_bound_of_strongConvexOn; intended Optimization Proposition 1.6 (1.4) and sampling convex-potential adapters",
    note := "Existing ASTIS theorem admitted to Registry during PR #248 integration, not a new proof. Domain-local Hilbert-space supplied-gradient interface with arbitrary real modulus; not the whole source equivalence. Optlib mathematical provenance: Chenyi Li and Ziyu Wang, Apache-2.0; ASTIS proof uses the existing geodesic first-order lemma."
  },
  {
    key := "analysis.strong-convexity.gradient-inner-lower-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexFirstOrder.gradient_inner_lower_bound_of_strongConvexOn",
    upstreamDecl := "Strong_Convex_lower",
    upstreamFile := "Optlib/Convex/StronglyConvex.lean@5da27c5f95aa6a8a45b8c14b968ade4c13ff18c3",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "CONV", "strong-convexity", "gradient", "monotonicity", "shared"],
    saldUse := "Positive-modulus gradient injectivity in Tests.Shared.StrongConvexFirstOrder; intended Optimization Proposition 1.6 (1.5) and gradient-flow contraction adapters",
    note := "PR #248 by andyjm3: sum the ASTIS first-order bounds in both directions. Arbitrary real modulus; only the injectivity consumer needs positivity. No reverse implication, Hessian equivalence, flow theorem, Gibbs invariance or new conceptual transport certificate is claimed."
  },
  {
    key := "analysis.strong-convexity.minimizer-existence",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexMinimizer.exists_isMinOn_and_gradient_eq_zero",
    upstreamDecl := "StrongConvexFirstOrder.firstOrder_lower_bound_of_strongConvexOn / IsCompact.exists_isMinOn / IsLocalMin.fderiv_eq_zero",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexMinimizer; Mathlib",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "CONV", "SPHMC", "strong-convexity", "minimizer", "gradient", "finite-dimensional"],
    saldUse := "SPHMC Lemma 4.16 hidden minimizer-selection prerequisite: construct the minimizer used by the initialization argument",
    note := "Independently verified at exact commit 8be837c. Requires positive modulus, genuine differentiability and finite-dimensional compactness. No uniqueness, Gibbs normalization, concentration or all-q statement is claimed."
  },
  {
    key := "analysis.strong-convexity.of-gradient-inner-lower-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexGradientConverse.strongConvexOn_of_gradient_inner_lower_bound",
    upstreamDecl := "Lower_Strong_Convex",
    upstreamFile := "Optlib/Convex/StronglyConvex.lean@5da27c5f95aa6a8a45b8c14b968ade4c13ff18c3",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["CALC", "CONV", "strong-convexity", "gradient", "monotonicity", "shared"],
    saldUse := "Shared converse for optimization gradient criteria; signed quadratic and closed-domain midpoint consumers in Tests.Shared.StrongConvexGradientConverse",
    note := "Domain-local ambient-gradient criterion with arbitrary real modulus. Optlib statement provenance: Chenyi Li and Ziyu Wang, Apache-2.0. ASTIS proof uses Mathlib scalar derivative monotonicity along a segment. No complete Chewi equivalence or textbook integral proof is claimed."
  },
  {
    key := "analysis.integrability.of-real-lintegral-finite",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_ofReal_ne_top_of_integrable_nonneg",
    upstreamDecl := "lintegral_ofReal_ne_top_iff_integrable",
    upstreamFile := "Mathlib.MeasureTheory.Function.L1Space.Integrable",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "integrability", "lintegral", "ENNReal", "ofReal", "nonnegative"],
    saldUse := "Chewi DENS/ANALYSIS root: turn real-valued tail integrability estimates into finite ENNReal density integrals",
    note := "Generic bridge used by Gibbs and future Renyi/Hellinger finite-integral leaves."
  },
  {
    key := "analysis.integrability.gaussian-quadratic-tail",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.integrable_exp_neg_mul_norm_sq",
    upstreamDecl := "GaussianFourier.integrable_cexp_neg_mul_sq_norm_add",
    upstreamFile := "Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "integrability", "Lebesgue", "Gaussian-tail", "quadratic", "finite-dimensional"],
    saldUse := "Chewi DENS/CONV root: finite-dimensional Lebesgue integrability of `exp (-a * ‖x‖^2)` for coercive Gibbs envelopes",
    note := "Mathlib-backed high-dimensional Gaussian tail integrability leaf; extracted from the complex Fourier Gaussian API."
  },
  {
    key := "analysis.integrability.shifted-gaussian-quadratic-tail",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.integrable_exp_neg_add_mul_norm_sq",
    upstreamDecl := "integrable_exp_neg_mul_norm_sq / Real.exp_add",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "integrability", "Lebesgue", "Gaussian-tail", "quadratic", "shifted"],
    saldUse := "Chewi DENS/CONV root: integrability of shifted quadratic envelopes `exp (-(a‖x‖^2+b))`",
    note := "Keeps additive constants in coercive lower potentials separate from the Gaussian tail theorem."
  },
  {
    key := "analysis.integrability.centered-gaussian-quadratic-tail",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.integrable_exp_neg_add_mul_norm_sub_sq",
    upstreamDecl := "integrable_exp_neg_add_mul_norm_sq / Integrable.comp_sub_right",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability; Mathlib.MeasureTheory.Group.Integral",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "integrability", "Lebesgue", "Gaussian-tail", "quadratic", "centered"],
    saldUse := "Chewi DENS/CONV root: integrability of translated quadratic envelopes `exp (-(a‖x-m‖^2+b))`",
    note := "Fills the center-translation gap for mode-centered strongly convex Gibbs envelopes; does not claim a general coercive-tail theorem."
  },
  {
    key := "analysis.integrability.laplace-absolute-linear-tail",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.integrable_exp_neg_add_mul_abs",
    upstreamDecl := "integrableOn_exp_mul_Ioi / integrableOn_exp_mul_Iic / integrableOn_union",
    upstreamFile := "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals; Mathlib.MeasureTheory.Integral.IntegrableOn",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "integrability", "Lebesgue", "Laplace-tail", "absolute-value", "one-dimensional"],
    saldUse := "Chewi DENS/CONV root: integrability of one-dimensional Laplace tails `exp (-(a|x|+b))` for log-concave non-strongly-convex examples",
    note := "Splits the real line into `Iic 0` and `Ioi 0`; this is a one-dimensional nonquadratic envelope, not a general coercive theorem."
  },
  {
    key := "analysis.integrability.gaussian-quadratic-tail-normalizer",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_exp_neg_mul_norm_sq_eq",
    upstreamDecl := "GaussianFourier.integral_rexp_neg_mul_sq_norm / ofReal_integral_eq_lintegral_ofReal",
    upstreamFile := "Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform; Mathlib.MeasureTheory.Integral.Bochner.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "integrability", "Lebesgue", "Gaussian-tail", "quadratic", "normalizer"],
    saldUse := "Chewi DENS/CONV root: exact ENNReal normalizer for finite-dimensional quadratic Gaussian tails",
    note := "Stronger than finiteness; exposes the normalizing constant needed by explicit Gibbs densities."
  },
  {
    key := "analysis.integrability.shifted-gaussian-quadratic-tail-normalizer",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_exp_neg_add_mul_norm_sq_eq",
    upstreamDecl := "lintegral_exp_neg_mul_norm_sq_eq / integral_const_mul / Real.exp_add",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "integrability", "Lebesgue", "Gaussian-tail", "quadratic", "shifted", "normalizer"],
    saldUse := "Chewi DENS/CONV root: exact ENNReal normalizer for shifted quadratic Gibbs envelopes",
    note := "Tracks additive constants explicitly instead of hiding them in finite-envelope hypotheses."
  },
  {
    key := "analysis.integrability.centered-gaussian-quadratic-tail-normalizer",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_exp_neg_add_mul_norm_sub_sq_eq",
    upstreamDecl := "lintegral_exp_neg_add_mul_norm_sq_eq / lintegral_sub_right_eq_self",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability; Mathlib.MeasureTheory.Group.Integral",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "integrability", "Lebesgue", "Gaussian-tail", "quadratic", "centered", "normalizer"],
    saldUse := "Chewi DENS/CONV root: exact ENNReal normalizer for translated finite-dimensional quadratic Gibbs envelopes",
    note := "Uses Lebesgue translation invariance to reuse the explicit quadratic normalizer; still only covers quadratic tails."
  },
  {
    key := "analysis.integrability.laplace-absolute-linear-tail-finite-lintegral",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_exp_neg_add_mul_abs_ne_top",
    upstreamDecl := "integrable_exp_neg_add_mul_abs / lintegral_ofReal_ne_top_of_integrable_nonneg",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "integrability", "lintegral", "ENNReal", "Laplace-tail", "absolute-value"],
    saldUse := "Chewi DENS/CONV root: finite ENNReal integral for one-dimensional absolute-linear lower-potential envelopes",
    note := "Finite-tail theorem for Laplace-type examples; the exact one-dimensional normalizer is recorded as a separate stronger leaf."
  },
  {
    key := "analysis.integrability.laplace-absolute-linear-tail-real-normalizer",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.integral_exp_neg_add_mul_abs_eq",
    upstreamDecl := "integral_exp_mul_Ioi / integral_exp_mul_Iic / setIntegral_union",
    upstreamFile := "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals; Mathlib.MeasureTheory.Integral.Bochner.Set",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "integrability", "Lebesgue", "Laplace-tail", "absolute-value", "normalizer", "one-dimensional"],
    saldUse := "Chewi DENS/CONV root: exact real integral `∫ exp (-(a|x|+b)) = 2 exp(-b)/a` for Laplace examples",
    note := "Splits the real line into the two exponential half-lines; this is exact only in one dimension."
  },
  {
    key := "analysis.integrability.laplace-absolute-linear-tail-ennreal-normalizer",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_exp_neg_add_mul_abs_eq",
    upstreamDecl := "integral_exp_neg_add_mul_abs_eq / ofReal_integral_eq_lintegral_ofReal",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability; Mathlib.MeasureTheory.Integral.Bochner.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "integrability", "lintegral", "ENNReal", "Laplace-tail", "absolute-value", "normalizer"],
    saldUse := "Chewi DENS root: exact ENNReal normalizer for normalized one-dimensional Laplace Gibbs densities",
    note := "Consumer-facing normalizer for `withDensity`; does not imply any multidimensional coercive theorem."
  },
  {
    key := "measure.gibbs-density.explicit-laplace-normalized-probability",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.isProbabilityMeasure_withDensity_exp_neg_add_mul_abs",
    upstreamDecl := "lintegral_exp_neg_add_mul_abs_eq / isProbabilityMeasure_withDensity_normalized_gibbs",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability; AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "withDensity", "Laplace-tail", "absolute-value", "normalizer", "probability-measure"],
    saldUse := "Chewi DENS/CONV root: construct explicitly normalized one-dimensional Laplace-type Gibbs laws",
    note := "Exact-probability version of the Laplace example; base measure is Lebesgue on `ℝ`."
  },
  {
    key := "measure.gibbs-density.explicit-quadratic-normalized-probability",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.isProbabilityMeasure_withDensity_exp_neg_add_mul_norm_sq",
    upstreamDecl := "lintegral_exp_neg_add_mul_norm_sq_eq / isProbabilityMeasure_withDensity_normalized_gibbs",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability; AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "withDensity", "Lebesgue", "quadratic", "normalizer", "probability-measure"],
    saldUse := "Chewi DENS/CONV root: construct the explicitly normalized finite-dimensional quadratic Gibbs law",
    note := "Useful consumer-facing version for Gaussian-reference Gibbs targets and Langevin invariant-law statements."
  },
  {
    key := "measure.gibbs-density.explicit-centered-quadratic-normalized-probability",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.isProbabilityMeasure_withDensity_exp_neg_add_mul_norm_sub_sq",
    upstreamDecl := "lintegral_exp_neg_add_mul_norm_sub_sq_eq / isProbabilityMeasure_withDensity_normalized_gibbs",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability; AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "withDensity", "Lebesgue", "quadratic", "centered", "normalizer", "probability-measure"],
    saldUse := "Chewi DENS/CONV root: construct explicitly normalized mode-centered quadratic Gibbs laws",
    note := "Consumer-facing centered Gaussian law bridge; log-concavity geometry and later stationarity are separate leaves."
  },
  {
    key := "measure.gibbs-density.integral-finite-quadratic-lower-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_gibbsDensityENNReal_ne_top_of_ae_quadratic_lower_bound",
    upstreamDecl := "lintegral_exp_neg_add_mul_norm_sq_ne_top / lintegral_gibbsDensityENNReal_ne_top_of_ae_potential_ge",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability; AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "density", "Lebesgue", "coercivity", "quadratic-lower-bound", "finite"],
    saldUse := "Chewi DENS/CONV root: prove finite Gibbs normalizer on finite-dimensional Lebesgue space from a quadratic lower bound on the potential",
    note := "First concrete Lebesgue coercivity envelope leaf; stronger nonquadratic/coercive tails remain future analysis leaves."
  },
  {
    key := "measure.gibbs-density.integral-finite-centered-quadratic-lower-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_gibbsDensityENNReal_ne_top_of_ae_centered_quadratic_lower_bound",
    upstreamDecl := "lintegral_exp_neg_add_mul_norm_sub_sq_ne_top / lintegral_gibbsDensityENNReal_ne_top_of_ae_potential_ge",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability; AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "density", "Lebesgue", "coercivity", "quadratic-lower-bound", "centered", "finite"],
    saldUse := "Chewi DENS/CONV root: prove finite Gibbs normalizer from a mode-centered quadratic lower bound on the potential",
    note := "Adds the translated quadratic lower-bound case needed by strongly convex targets with nonzero minimizer; general coercive envelopes remain future work."
  },
  {
    key := "measure.gibbs-density.normalized-probability-quadratic-lower-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.isProbabilityMeasure_withDensity_normalized_gibbs_of_ae_quadratic_lower_bound",
    upstreamDecl := "lintegral_gibbsDensityENNReal_ne_zero / lintegral_gibbsDensityENNReal_ne_top_of_ae_quadratic_lower_bound",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability; AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "withDensity", "Lebesgue", "coercivity", "quadratic-lower-bound", "probability-measure"],
    saldUse := "Chewi DENS/CONV root: construct normalized finite-dimensional Lebesgue Gibbs targets from measurable potentials with quadratic lower bounds",
    note := "Turns the first concrete coercivity normalizer into the normalized target law consumed by Langevin and KL/FI branches."
  },
  {
    key := "measure.gibbs-density.normalized-probability-centered-quadratic-lower-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.isProbabilityMeasure_withDensity_normalized_gibbs_of_ae_centered_quadratic_lower_bound",
    upstreamDecl := "lintegral_gibbsDensityENNReal_ne_zero / lintegral_gibbsDensityENNReal_ne_top_of_ae_centered_quadratic_lower_bound",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability; AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "withDensity", "Lebesgue", "coercivity", "quadratic-lower-bound", "centered", "probability-measure"],
    saldUse := "Chewi DENS/CONV root: construct normalized finite-dimensional Lebesgue Gibbs targets from measurable potentials with centered quadratic lower bounds",
    note := "This is the normalized target-law handoff for strongly convex potentials after a minimizer/mode has been exposed."
  },
  {
    key := "measure.gibbs-density.integral-finite-strong-convex-minimizer",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_gibbsDensityENNReal_ne_top_of_strongConvexOn_minimizer",
    upstreamDecl := "centered_quadratic_lower_bound_of_strongConvexOn_minimizer / lintegral_gibbsDensityENNReal_ne_top_of_ae_centered_quadratic_lower_bound",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity; AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "density", "strong-convexity", "minimizer", "quadratic-lower-bound", "finite"],
    saldUse := "Chewi DENS/CONV root: prove finite Gibbs normalizer from strong convexity once a global minimizer is exposed",
    note := "Uses the compiled midpoint `k/4` lower envelope; the sharper first-order `k/2` envelope remains a separate future leaf."
  },
  {
    key := "measure.gibbs-density.normalized-probability-strong-convex-minimizer",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.isProbabilityMeasure_withDensity_normalized_gibbs_of_strongConvexOn_minimizer",
    upstreamDecl := "lintegral_gibbsDensityENNReal_ne_zero / lintegral_gibbsDensityENNReal_ne_top_of_strongConvexOn_minimizer",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability; AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "withDensity", "strong-convexity", "minimizer", "probability-measure"],
    saldUse := "Chewi DENS/CONV/SDE root: construct normalized finite-dimensional Gibbs targets for strongly convex potentials with a known minimizer",
    note := "Direct consumer-facing target-law constructor for strongly log-concave finite-dimensional Langevin branches."
  },
  {
    key := "measure.gibbs-density.integral-finite-absolute-linear-lower-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_gibbsDensityENNReal_ne_top_of_ae_abs_linear_lower_bound",
    upstreamDecl := "lintegral_exp_neg_add_mul_abs_ne_top / lintegral_gibbsDensityENNReal_ne_top_of_ae_potential_ge",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability; AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "density", "Laplace-tail", "absolute-linear-lower-bound", "finite"],
    saldUse := "Chewi DENS/CONV root: prove finite one-dimensional Gibbs normalizer from an absolute-linear lower bound",
    note := "First compiled nonquadratic Lebesgue Gibbs envelope; full multidimensional/general coercive envelopes remain todo-red."
  },
  {
    key := "measure.gibbs-density.normalized-probability-absolute-linear-lower-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.isProbabilityMeasure_withDensity_normalized_gibbs_of_ae_abs_linear_lower_bound",
    upstreamDecl := "lintegral_gibbsDensityENNReal_ne_zero / lintegral_gibbsDensityENNReal_ne_top_of_ae_abs_linear_lower_bound",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability; AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "withDensity", "Laplace-tail", "absolute-linear-lower-bound", "probability-measure"],
    saldUse := "Chewi DENS/CONV/SDE root: construct normalized one-dimensional Gibbs targets with Laplace-type tails",
    note := "Source-facing probability bridge for Chewi's log-concave but non-strongly-log-concave Laplace examples."
  },
  {
    key := "analysis.finite-power-series.iterated-derivative-budget",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.FinitePowerSeries.abs_iteratedDeriv_finitePowerSeries_le_mass",
    upstreamDecl := "iteratedDeriv_pow / iteratedDeriv_fun_sum",
    upstreamFile := "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["analysis", "polynomial", "iterated-derivative", "coefficient-budget", "C4"],
    saldUse := "Reusable signed finite-polynomial derivative budget for smooth kernels, moment features, and local truncation estimates.",
    note := "All derivative orders are covered through Nat.descFactorial; coefficients may have arbitrary signs and the bound holds on the closed unit interval."
  },
  {
    key := "analysis.fourth-order-composition-expression",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.FinitePowerSeries.fourthOrderCompositionExpression_abs_le",
    upstreamDecl := "one-dimensional fourth-order chain rule coefficients",
    upstreamFile := "Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["analysis", "C4", "chain-rule", "Faa-di-Bruno", "remainder"],
    saldUse := "Termwise fourth-order composition control after derivative bounds for the outer and inner functions are available.",
    note := "This is the scalar algebraic estimate with coefficients 1,6,3,4,1; identifying actual iterated derivatives of a composition is a separate Mathlib theorem."
  }
]

def gaussianMemory : List LemmaMemoryEntry := [
  {
    key := "gaussian.lipschitz.sharp-standard-mgf"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.GaussianLipschitzExponential.integral_exp_centered_le_stdGaussian_of_lipschitz"
    upstreamDecl := "IsGaussian.map_rotation_eq_self / actual weighted Stein / Herbst / same-L bump approximation"
    upstreamFile := "Mathlib.Probability.Distributions.Gaussian.Fernique; ASTIS authored concentration route"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["gaussian", "Lipschitz", "sharp-concentration", "signed", "SPHMC"]
    saldUse := "Actual SPHMC Gaussian gradient-output own-mean Laplace bound4.3 with coefficient eta*norm(a)^2/2"
    note := "Finite real Hilbert stdGaussian, nonsmooth observables, all signed t, L0 and dimension0. Independent exact commit2f4e0ba9. No Gaussian LSI/T2, posterior score mean identity/bias, full Lemma4.2, main or cost/composition credit."
  },
  {
    key := "gaussian.lipschitz.signed-exponential-domain"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.GaussianLipschitzExponential.integrable_and_integrable_exp_centered_of_lipschitz"
    upstreamDecl := "IsGaussian.integrable_id / exists_integrable_exp_sq / Young domination"
    upstreamFile := "Mathlib.Probability.Distributions.Gaussian.Fernique"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["gaussian", "Lipschitz", "first-moment", "exponential-domain", "signed", "SPHMC"]
    saldUse := "SPHMC actual Gaussian gradient-output centered Laplace domain; general Gaussian-observable background"
    note := "Real Banach/Borel Gaussian including degenerate laws and zero Lipschitz constants. Actual same-producer consumer independently verified1f746dc6. Domain only: no sharp coefficient, bias, posterior, main or cost credit."
  },
  {
    key := "gaussian.std.norm-square-second-moment"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.StdGaussianMoment.integrable_norm_sq_and_integral_stdGaussian"
    upstreamDecl := "IsGaussian.memLp_two_id / covarianceBilin_stdGaussian / finite-dimensional Parseval"
    upstreamFile := "Mathlib.Probability.Distributions.Gaussian.Multivariate"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["gaussian", "stdGaussian", "integrability", "second-moment", "finite-dimensional", "sampling"]
    saldUse := "shared Gaussian second-moment leaf for OU momentum refreshes and stochastic-gradient moment estimates"
    note := "Canonical public owner replacing future repetition of two older private SPHMC helper proofs; no independence or higher-moment claim."
  },
  {
    key := "gaussian.product.coordinate-law",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian.map_eval_stdGaussianPi",
    upstreamDecl := "map_eval_stdGaussianPi",
    upstreamFile := "SLT/GaussianMeasure.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["gaussian", "coordinate-law", "brownian-increment"],
    saldUse := "normalized scalar coordinate law in the Brownian/Ito EM backend",
    note := "ASTIS-native product-Gaussian coordinate projection theorem."
  },
  {
    key := "gaussian.product.coordinate-integrable",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian.integrable_eval_stdGaussianPi",
    upstreamDecl := "integrable_eval_stdGaussianPi",
    upstreamFile := "SLT/GaussianMeasure.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["gaussian", "integrability", "coordinate", "brownian-increment"],
    saldUse := "Brownian/Ito coordinate integrability for scalar Taylor moment and generator leaves",
    note := "Cycle 203 lower_3 ASTIS-owned port; uses Mathlib Gaussian exponential integrability and Measure.map transport."
  },
  {
    key := "gaussian.product.coordinate-square-integrable",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian.integrable_sq_eval_stdGaussianPi",
    upstreamDecl := "integrable_sq_eval_stdGaussianPi",
    upstreamFile := "SLT/GaussianMeasure.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["gaussian", "integrability", "quadratic-moment", "brownian-increment"],
    saldUse := "Brownian/Ito coordinate square integrability for polynomial moment leaves",
    note := "Cycle 203 lower_3 ASTIS-owned port; reuses the local quadratic Gaussian integrability lemma."
  },
  {
    key := "gaussian.product.linear-form-integrable",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian.integrable_linearForm_stdGaussianPi",
    upstreamDecl := "finite-sum closure of integrable_eval_stdGaussianPi",
    upstreamFile := "Mathlib.MeasureTheory.Function.L1Space.Integrable; ASTIS Gaussian",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["gaussian", "product-gaussian", "linear-form", "integrability", "brownian-increment", "Esscher"],
    saldUse := "finite-dimensional Gaussian linear-function integrability before MGF, Esscher tilt, and Brownian increment packaging",
    note := "Small ASTIS-owned closure lemma; keeps product-Gaussian linear forms separate from exponential tilt/MGF work."
  },
  {
    key := "gaussian.product.coordinate-mean-zero",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian.integral_eval_stdGaussianPi",
    upstreamDecl := "integral_eval_stdGaussianPi",
    upstreamFile := "SLT/GaussianMeasure.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["gaussian", "mean", "coordinate", "brownian-increment"],
    saldUse := "coordinate mean-zero rewrite for Brownian/Ito scalar Taylor moment leaves",
    note := "Cycle 203 lower_3 ASTIS-owned port; combines product-coordinate law with the local centered Gaussian mean."
  },
  {
    key := "gaussian.product.linear-form-mean-zero",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian.integral_linearForm_stdGaussianPi",
    upstreamDecl := "finite-sum closure of integral_eval_stdGaussianPi",
    upstreamFile := "Mathlib.MeasureTheory.Integral.Bochner.Basic; ASTIS Gaussian",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["gaussian", "product-gaussian", "linear-form", "mean-zero", "brownian-increment", "Esscher"],
    saldUse := "zero-mean finite-dimensional Gaussian linear forms before centered tilt/MGF and generator algebra",
    note := "Compiled finite-sum closure of centered coordinate means; full Gaussian Esscher density identity remains a separate target."
  },
  {
    key := "gaussian.product.linear-form-mgf",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian.integral_exp_linearForm_stdGaussianPi",
    upstreamDecl := "mgf_gaussianReal / integral_fintype_prod_eq_prod",
    upstreamFile := "Mathlib.Probability.Distributions.Gaussian.Real; Mathlib.MeasureTheory.Integral.Pi; external AST GaussianMGF.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["gaussian", "product-gaussian", "linear-form", "MGF", "Esscher", "Chewi"],
    saldUse := "Chewi GAUSS root: compute finite product-Gaussian exponential moments for tilt normalizers and Brownian increment laws",
    note := "ASTIS-owned finite-product MGF leaf distilled from the external GaussianMGF proof route; full density shift remains separate."
  },
  {
    key := "gaussian.product.centered-esscher-normalizer",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian.integral_exp_centered_linearForm_stdGaussianPi",
    upstreamDecl := "integral_exp_linearForm_stdGaussianPi plus elementary exponential algebra",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["gaussian", "product-gaussian", "Esscher", "tilt", "normalization", "Chewi"],
    saldUse := "Chewi GAUSS root: certify the centered exponential tilt has mass one before proving the shifted density/change-of-measure identity",
    note := "This is the compiled normalizer half of finite-dimensional Esscher; it does not yet prove the withDensity shifted-Gaussian identity."
  },
  {
    key := "gaussian.scalar.esscher-shift-density",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian.gaussianReal_withDensity_exp_shift",
    upstreamDecl := "gaussianReal_of_var_ne_zero / gaussianPDFReal algebra",
    upstreamFile := "Mathlib.Probability.Distributions.Gaussian.Real; external AST GaussianShift.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["gaussian", "Esscher", "withDensity", "density-shift", "Chewi"],
    saldUse := "Chewi GAUSS root: one-dimensional Gaussian exponential tilt shifts the mean",
    note := "ASTIS-owned scalar density identity; product versions consume the finite-pi withDensity decomposition leaf."
  },
  {
    key := "gaussian.product.esscher-shift-density",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian.stdGaussianPi_withDensity_exp_shift",
    upstreamDecl := "gaussianReal_withDensity_exp_shift / pi_withDensity_prod",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian; external AST GaussianShift.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["gaussian", "product-gaussian", "Esscher", "withDensity", "density-shift", "Chewi"],
    saldUse := "Chewi GAUSS root: finite product-Gaussian centered exponential tilt equals the shifted product Gaussian law",
    note := "Completes the shifted-density half of finite-dimensional Esscher; integral change-of-measure and Euclidean/path-space packaging remain separate."
  },
  {
    key := "gaussian.product.esscher-change-of-measure",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian.stdGaussianPi_shift_integral",
    upstreamDecl := "stdGaussianPi_withDensity_exp_shift / integral_withDensity_eq_integral_toReal_smul",
    upstreamFile := "Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap; external AST GaussianShift.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["gaussian", "product-gaussian", "Esscher", "change-of-measure", "Girsanov", "Chewi"],
    saldUse := "Chewi GAUSS/PATH root: rewrite integrals under finite shifted product Gaussians as centered product-Gaussian weighted integrals",
    note := "Finite-dimensional product-Gaussian change-of-measure is compiled; EuclideanSpace pushforward, stdGaussian, and path-space/Girsanov packaging are separate entries."
  },
  {
    key := "gaussian.euclidean.pushforward-esscher-change-of-measure",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian.stdGaussianPi_shift_integral_map_toLp",
    upstreamDecl := "pi_gaussianReal_shift_integral / integral_map / WithLp.toLp",
    upstreamFile := "Mathlib.Probability.Distributions.Gaussian.Multivariate; AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["gaussian", "EuclideanSpace", "product-gaussian", "Esscher", "change-of-measure", "Girsanov", "Chewi"],
    saldUse := "Chewi GAUSS/PATH root: transport finite shifted product-Gaussian change-of-measure through the EuclideanSpace `WithLp.toLp 2` interface",
    note := "Compiled coordinate-exponent pushforward bridge; canonical `stdGaussian` inner-product form and path-space/Girsanov packaging are separate entries."
  },
  {
    key := "gaussian.euclidean.stdGaussian-esscher-change-of-measure",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian.stdGaussian_shift_integral_map_toLp",
    upstreamDecl := "stdGaussianPi_shift_integral_map_toLp / map_pi_eq_stdGaussian / EuclideanSpace inner and norm coordinates",
    upstreamFile := "Mathlib.Probability.Distributions.Gaussian.Multivariate; Mathlib.Analysis.InnerProductSpace.PiL2",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["gaussian", "stdGaussian", "EuclideanSpace", "Esscher", "change-of-measure", "Girsanov", "Chewi"],
    saldUse := "Chewi GAUSS/PATH root: express finite-dimensional Gaussian Esscher change-of-measure against Mathlib `stdGaussian` with inner-product/norm exponent",
    note := "Compiled canonical finite-dimensional Hilbert-space form; Brownian/path-space Girsanov packaging remains separate."
  },
  {
    key := "gaussian.unit-variance.nnreal",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian.nnrealVarianceOneOfGaussianRealUnitLaw",
    upstreamDecl := "variance_id_gaussianReal",
    upstreamFile := "Mathlib.Probability.Distributions.Gaussian.Real",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["gaussian", "variance", "NNReal"],
    saldUse := "turn scalar Gaussian law and variance-field definition into normalized variance one",
    note := "Uses Mathlib Gaussian variance locally; no SLT import."
  },
  {
    key := "gaussian.quadratic-bound-integrable",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian.integrable_const_mul_sq_gaussianReal_zero",
    upstreamDecl := "integrable_exp_mul_gaussianReal / integrable_pow_of_integrable_exp_mul",
    upstreamFile := "Mathlib.Probability.Distributions.Gaussian.Real",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["gaussian", "integrability", "quadratic-bound", "brownian-increment"],
    saldUse := "supply normalized-remainder bound integrability once the source identifies remainderBound as C * z^2",
    note := "Cycle 196 lower_3 ASTIS-owned Gaussian quadratic integrability support; no SLT import."
  }
]

def taylorMemory : List LemmaMemoryEntry := [
  {
    key := "taylor.hessian.source-field-to-opnorm",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Taylor.hessianOpNormOfSourceHessianField",
    upstreamDecl := "deriv2_bounded_of_compactlySupported",
    upstreamFile := "SLT/GaussianPoincare/TaylorBound.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["taylor", "hessian", "source-contract"],
    saldUse := "convert source-supplied selected-test Hessian representative into downstream Hessian operator-norm bound",
    note := "Does not prove that the SALD source supplies the Hessian field; it only packages the local bridge once supplied."
  },
  {
    key := "taylor.fderiv-hessian-to-iterated",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Taylor.iteratedFDerivTwoOpNormOfFDerivFDerivOpNorm",
    upstreamDecl := "taylor_order_one / TaylorBound proof idiom",
    upstreamFile := "SLT/GaussianPoincare/TaylorBound.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["taylor", "iteratedFDeriv", "hessian"],
    saldUse := "feed selected-line Taylor bounds from a Hessian operator-norm field",
    note := "ASTIS-owned Mathlib bridge."
  },
  {
    key := "brownian.quadratic-variation-normalization",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Taylor.quadraticVariationNormalizationOfCoeffDefAndVarianceOne",
    upstreamDecl := "not upstream; extracted from SALD local proof needs",
    upstreamFile := "ASTIS/SALD cycles 174-176",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["brownian", "ito", "quadratic-variation", "normalization"],
    saldUse := "assemble quadratic coefficient and variance-one fields without re-assuming the downstream normalization",
    note := "Pure algebraic bridge made reusable for later SDE papers."
  }
]

def calculusMemory : List LemmaMemoryEntry := [
  {
    key := "analysis.calculus.smooth-unit-cutoff-eq-smoothTransition",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_eq_smoothTransition",
    upstreamDecl := "GaussianSobolev.smoothCutoff_eq_smoothTransition",
    upstreamFile := "SLT/GaussianSobolevDense/Defs.lean@d0f506f; Mathlib.Analysis.Calculus.BumpFunction.InnerProduct",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "smoothTransition", "unit-scale"],
    saldUse := "log-concave sampling Ch.1 cutoff root: expose the unit cutoff through Mathlib's smooth-transition formula",
    note := "Formula leaf only. It does not assert a scaled family, derivative bounds, tail passage, weighted IBP, or invariant law."
  },
  {
    key := "analysis.calculus.smooth-unit-cutoff-contDiff",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_contDiff",
    upstreamDecl := "GaussianSobolev.smoothCutoff_contDiff",
    upstreamFile := "SLT/GaussianSobolevDense/Defs.lean@d0f506f; Mathlib.Analysis.Calculus.BumpFunction.InnerProduct",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "ContDiff", "unit-scale"],
    saldUse := "log-concave sampling Ch.1 cutoff root: global smoothness of the reusable one-dimensional unit cutoff",
    note := "Smoothness leaf only. It gives no derivative-size estimate or integration theorem."
  },
  {
    key := "analysis.calculus.smooth-unit-cutoff-one-of-abs-le-one",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_eq_one_of_abs_le_one",
    upstreamDecl := "GaussianSobolev.smoothCutoff_eq_one_of_le",
    upstreamFile := "SLT/GaussianSobolevDense/Defs.lean@d0f506f; Mathlib.Analysis.SpecialFunctions.SmoothTransition",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "plateau", "unit-scale"],
    saldUse := "log-concave sampling Ch.1 cutoff root: identify the unit cutoff plateau on the closed unit interval",
    note := "Unit-scale value leaf only; no scaled derivative or tail claim."
  },
  {
    key := "analysis.calculus.smooth-unit-cutoff-zero-of-two-le-abs",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_eq_zero_of_two_le_abs",
    upstreamDecl := "GaussianSobolev.smoothCutoff_eq_zero_of_ge",
    upstreamFile := "SLT/GaussianSobolevDense/Defs.lean@d0f506f; Mathlib.Analysis.SpecialFunctions.SmoothTransition",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "support", "unit-scale"],
    saldUse := "log-concave sampling Ch.1 cutoff root: the unit cutoff vanishes beyond radius two",
    note := "Pointwise vanishing leaf only; compactness and topological support are separate leaves."
  },
  {
    key := "analysis.calculus.smooth-unit-cutoff-mem-Icc",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_mem_Icc",
    upstreamDecl := "ContDiffBumpBase.mem_Icc",
    upstreamFile := "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "range", "unit-scale"],
    saldUse := "log-concave sampling Ch.1 cutoff root: pointwise [0,1] range of the unit cutoff",
    note := "Range leaf only; it does not provide monotonicity or derivative bounds."
  },
  {
    key := "analysis.calculus.smooth-unit-cutoff-deriv-bounded",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_deriv_bounded",
    upstreamDecl := "GaussianSobolev.smoothCutoff_deriv_bounded",
    upstreamFile := "SLT/GaussianSobolevDense/Cutoff.lean@d0f506f; Mathlib.Analysis.Calculus.ContDiff.Deriv",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "deriv", "uniform-bound", "unit-scale"],
    saldUse := "log-concave sampling Ch.1 cutoff root: choose one positive derivative bound before introducing the exhaustion scale",
    note := "One-dimensional compact-support bound. It supplies the scale-independent constant used by the radial first-derivative leaf."
  },
  {
    key := "analysis.calculus.smooth-unit-cutoff-second-deriv-continuous",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_secondDeriv_continuous",
    upstreamDecl := "ContDiff.iterate_deriv'",
    upstreamFile := "SLT/GaussianPoincare/TaylorBound.lean@d0f506f; Mathlib.Analysis.Calculus.ContDiff",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "second-derivative", "continuity", "SLT-port"],
    saldUse := "log-concave sampling Ch.1 second-order cutoff root: continuity input for a unit-scale Hessian bound",
    note := "ASTIS-owned adaptation of the audited SLT C_c^2 second-derivative continuity leaf."
  },
  {
    key := "analysis.calculus.smooth-unit-cutoff-second-deriv-compact-support",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_secondDeriv_hasCompactSupport",
    upstreamDecl := "HasCompactSupport.deriv",
    upstreamFile := "SLT/GaussianPoincare/TaylorBound.lean@d0f506f; Mathlib.Topology.Algebra.Support",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "second-derivative", "compact-support", "SLT-port"],
    saldUse := "log-concave sampling Ch.1 second-order cutoff root: compact-support input for a global derivative bound",
    note := "The fixed unit cutoff is differentiated twice; no radial scaling or Hessian estimate is claimed here."
  },
  {
    key := "analysis.calculus.smooth-unit-cutoff-second-deriv-bounded",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_secondDeriv_bounded",
    upstreamDecl := "HasCompactSupport.exists_bound_of_continuous",
    upstreamFile := "SLT/GaussianPoincare/TaylorBound.lean@d0f506f; Mathlib.Topology.Algebra.Support",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "second-derivative", "uniform-bound", "SLT-port"],
    saldUse := "log-concave sampling Ch.1: choose the unit-scale constant before proving O(R^-2) radial Hessian/Laplacian bounds",
    note := "Compiled unit-scale bound only. The chain rule and radial Hessian/Laplacian scaling remain a separate red leaf."
  },
  {
    key := "analysis.calculus.radial-smooth-cutoff-one-of-norm-le",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_eq_one_of_norm_le",
    upstreamDecl := "GaussianSobolev.smoothCutoffR_eq_one_of_norm_le",
    upstreamFile := "SLT/GaussianSobolevDense/Defs.lean@d0f506f",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "radial", "plateau", "closed-ball"],
    saldUse := "log-concave sampling Ch.1 exhaustion base: the scale-R radial cutoff equals one on the radius-R closed ball",
    note := "Plateau-value leaf only. Uniform derivative and tail estimates remain red."
  },
  {
    key := "analysis.calculus.radial-smooth-cutoff-zero-of-two-mul-le-norm",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_eq_zero_of_two_mul_le_norm",
    upstreamDecl := "GaussianSobolev.smoothCutoffR_eq_zero_of_norm_ge",
    upstreamFile := "SLT/GaussianSobolevDense/Defs.lean@d0f506f",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "radial", "support", "closed-ball"],
    saldUse := "log-concave sampling Ch.1 exhaustion base: the scale-R radial cutoff vanishes at norm at least 2R",
    note := "Pointwise vanishing leaf only. It does not identify support equality."
  },
  {
    key := "analysis.calculus.radial-smooth-cutoff-mem-Icc",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_mem_Icc",
    upstreamDecl := "GaussianSobolev.smoothCutoffR_nonneg plus smoothCutoffR_le_one",
    upstreamFile := "SLT/GaussianSobolevDense/Defs.lean@d0f506f",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "radial", "range"],
    saldUse := "log-concave sampling Ch.1 exhaustion base: every radial cutoff value lies in [0,1]",
    note := "Range leaf only; scale positivity is not needed for this statement."
  },
  {
    key := "analysis.calculus.fderiv-norm-div-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.fderiv_norm_div_bound",
    upstreamDecl := "GaussianSobolev.fderiv_norm_div_bound",
    upstreamFile := "SLT/GaussianSobolevDense/Cutoff.lean@d0f506f; Mathlib.Analysis.Calculus.Deriv.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "fderiv", "Lipschitz", "scale"],
    saldUse := "log-concave sampling Ch.1 exhaustion base: the totalized derivative of x maps to norm x divided by R has operator norm at most 1/R",
    note := "Valid on real normed spaces, including the origin through Mathlib's totalized fderiv. Genuine norm differentiability at the origin is not asserted."
  },
  {
    key := "analysis.calculus.radial-smooth-cutoff-contDiff",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_contDiff",
    upstreamDecl := "GaussianSobolev.smoothCutoffR_contDiff",
    upstreamFile := "SLT/GaussianSobolevDense/Defs.lean@d0f506f; Mathlib.Analysis.Calculus.BumpFunction.InnerProduct",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "radial", "ContDiff", "inner-product-space"],
    saldUse := "log-concave sampling Ch.1 exhaustion base: smoothness of the positive-scale radial cutoff on real inner-product spaces",
    note := "Requires an inner-product norm; no claim is made for arbitrary nonsmooth norms or for derivative magnitude."
  },
  {
    key := "analysis.calculus.radial-smooth-cutoff-fderiv-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_fderiv_bound",
    upstreamDecl := "GaussianSobolev.smoothCutoffR_fderiv_bound, with scale-uniform quantifier order",
    upstreamFile := "SLT/GaussianSobolevDense/Cutoff.lean@d0f506f; Mathlib.Analysis.Calculus.FDeriv.Comp",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "radial", "fderiv", "O-R-inverse", "inner-product-space"],
    saldUse := "log-concave sampling Ch.1 exhaustion base: one constant controls the first derivative of every positive-scale radial cutoff by C/R",
    note := "The existential constant precedes the radius quantifier. Hessian, Laplacian, dominated-tail, weighted-IBP, and invariance leaves remain separate."
  },
  {
    key := "analysis.calculus.radial-smooth-cutoff-iterated-fderiv-two-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_iteratedFDeriv_two_bound",
    upstreamDecl := "ContinuousLinearMap.iteratedFDeriv_comp_right; HasCompactSupport.iteratedFDeriv; ContinuousMultilinearMap.norm_compContinuousLinearMap_le",
    upstreamFile := "Mathlib.Analysis.Calculus.ContDiff.{Bounds,FTaylorSeries}; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "radial", "iteratedFDeriv", "second-order", "O-R-inverse-squared"],
    saldUse := "log-concave sampling Ch.1 second-order exhaustion branch: one constant controls the second iterated Frechet derivative at every positive scale by C/R^2",
    note := "The unit-scale derivative field is bounded by continuity and compact support, then transported by exact linear dilation. Laplacian trace and any integral consumer remain separate."
  },
  {
    key := "analysis.calculus.radial-smooth-cutoff-fderiv-zero-of-two-mul-le-norm",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_fderiv_eq_zero_of_two_mul_le_norm",
    upstreamDecl := "IsLocalMin.fderiv_eq_zero; radialSmoothCutoff_eq_zero_of_two_mul_le_norm; radialSmoothCutoff_mem_Icc",
    upstreamFile := "Mathlib.Analysis.Calculus.LocalExtr.Basic; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "radial", "fderiv", "derivative-support", "outer-radius", "boundary"],
    saldUse := "log-concave sampling Ch.1 cutoff exhaustion: the totalized first derivative vanishes on the closed outer region with norm at least 2R",
    note := "The boundary case uses the global-minimum characterization and Mathlib's totalized fderiv. It does not assert genuine differentiability of an arbitrary norm there, an integral tail limit, weighted IBP, or invariance."
  },
  {
    key := "analysis.calculus.radial-smooth-cutoff-support-subset-closedBall",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_support_subset_closedBall",
    upstreamDecl := "GaussianSobolev.smoothCutoffR_support_subset",
    upstreamFile := "SLT/GaussianSobolevDense/Defs.lean@d0f506f",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "radial", "Function.support", "closed-ball"],
    saldUse := "log-concave sampling Ch.1 exhaustion base: plain support of the scale-R cutoff is contained in the radius-2R closed ball",
    note := "Support containment, not equality. Box support and boundary-face handoffs remain separate."
  },
  {
    key := "analysis.calculus.radial-smooth-cutoff-tsupport-subset-closedBall",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_tsupport_subset_closedBall",
    upstreamDecl := "radialSmoothCutoff_support_subset_closedBall plus closure_minimal",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "radial", "tsupport", "closed-ball"],
    saldUse := "log-concave sampling Ch.1 exhaustion base: topological support remains inside the radius-2R closed ball",
    note := "Topological-support containment only; it does not claim a box-shaped tsupport or support equality."
  },
  {
    key := "analysis.calculus.radial-smooth-cutoff-hasCompactSupport",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_hasCompactSupport",
    upstreamDecl := "GaussianSobolev.smoothCutoffR_hasCompactSupport",
    upstreamFile := "SLT/GaussianSobolevDense/Defs.lean@d0f506f",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "radial", "HasCompactSupport", "finite-dimensional"],
    saldUse := "log-concave sampling Ch.1 exhaustion base: positive-scale radial cutoffs have compact support in finite dimension",
    note := "Compact-support leaf only. No derivative, domination, integration, or invariant-law conclusion."
  },
  {
    key := "analysis.calculus.radial-smooth-cutoff-tendsto-one",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_tendsto_one",
    upstreamDecl := "GaussianSobolev.smoothCutoffR_tendsto_one",
    upstreamFile := "SLT/GaussianSobolevDense/Defs.lean@d0f506f",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "radial", "exhaustion", "Tendsto"],
    saldUse := "log-concave sampling Ch.1 exhaustion base: at each fixed point the radial cutoff is eventually one as R tends to infinity",
    note := "Pointwise eventual constancy only. It does not prove dominated convergence for cutoff errors or derivative terms."
  },
  {
    key := "analysis.calculus.exists-contDiff-eq-one-tsupport-subset",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.exists_contDiff_eq_one_tsupport_subset",
    upstreamDecl := "exists_compact_between; IsOpen.exists_contDiff_support_eq; IsCompact.exists_forall_le'; Real.smoothTransition",
    upstreamFile := "Mathlib.Topology.Compactness.LocallyCompact; Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension; Mathlib.Topology.Order.Compact; Mathlib.Analysis.SpecialFunctions.SmoothTransition",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "plateau", "compact", "open-set", "tsupport", "ContDiff"],
    saldUse := "log-concave sampling Ch.1 cutoff root: a compact set inside an open finite-dimensional neighborhood admits a smooth [0,1] plateau with compact support inside that neighborhood",
    note := "Generic compact-in-open plateau leaf. It supplies no exhausting sequence, derivative bounds, tail passage, weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.exists-contDiff-cutoff-eq-one-on-Icc-tsupport-subset-outer-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.exists_contDiff_cutoff_eq_one_on_Icc_tsupport_subset_outer_univ_pi_Ioo",
    upstreamDecl := "Cutoff.exists_contDiff_eq_one_tsupport_subset plus Icc_subset_univ_pi_Ioo_of_strict_bounds",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "cutoff", "plateau", "closed-box", "open-box", "tsupport", "ContDiff"],
    saldUse := "log-concave sampling Ch.1 finite-box route: one smooth compactly supported cutoff equals one on the entire inner closed Pi-box and is topologically supported in the outer open Pi-box",
    note := "One-cutoff box specialization only. Exhausting families with derivative bookkeeping, tail limits, whole-space weighted IBP, generator domains, invariant Gibbs law, reversibility, and KL/FI remain red."
  },
  {
    key := "analysis.calculus.gradient-exp-neg-potential-chain-rule",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.hasGradientAt_expNegPotential_of_hasGradientAt",
    upstreamDecl := "HasGradientAt.hasFDerivAt / HasFDerivAt.neg / HasFDerivAt.exp",
    upstreamFile := "Mathlib.Analysis.Calculus.Gradient.Basic; Mathlib.Analysis.SpecialFunctions.ExpDeriv",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Gibbs-weight", "gradient", "chain-rule", "exp-neg-potential"],
    saldUse := "Chewi SDE/DENS root: turn a supplied potential gradient `∇V` into the Gibbs-weight gradient `∇ exp(-V) = -exp(-V) • ∇V` before finite-coordinate Langevin divergence algebra",
    note := "Pointwise gradient chain-rule leaf only. It does not prove weighted divergence, product rules for `rho ∇f`, integration by parts, boundary decay, stationarity, reversibility, or invariant Gibbs law."
  },
  {
    key := "analysis.calculus.gradient-exp-neg-potential-mathlib-gradient-from-hasGradientAt",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.gradient_expNegPotential_eq_of_hasGradientAt",
    upstreamDecl := "HasGradientAt.gradient plus hasGradientAt_expNegPotential_of_hasGradientAt",
    upstreamFile := "Mathlib.Analysis.Calculus.Gradient.Basic; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Gibbs-weight", "gradient", "HasGradientAt", "Mathlib-gradient", "exp-neg-potential"],
    saldUse := "Chewi SDE/DENS root: rewrite Mathlib's total `gradient (exp(-V)) x` from a supplied `HasGradientAt V gradV x`, keeping the potential-gradient representative explicit",
    note := "Pointwise Mathlib-gradient selection leaf only. It uses uniqueness of gradients after the local chain rule; divergence, product rule, IBP, generator domains, invariant law, reversibility, and KL/FI remain red."
  },
  {
    key := "analysis.calculus.gradient-exp-neg-potential-coordinate-from-hasGradientAt",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.gradient_expNegPotential_coordinate_eq_of_hasGradientAt",
    upstreamDecl := "gradient_expNegPotential_eq_of_hasGradientAt plus EuclideanSpace coordinate evaluation of scalar multiplication",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Gibbs-weight", "gradient", "HasGradientAt", "EuclideanSpace", "coordinate"],
    saldUse := "Chewi SDE/DENS root: coordinate Gibbs-weight chain-rule equality from a supplied potential gradient representative before finite Euclidean weighted-divergence algebra",
    note := "Coordinate Mathlib-gradient display only. It does not prove coordinate product rules, the a.e. bridge or box-integrability assumptions needed by the compiled box divergence wrapper, Laplacian identities, IBP, stationarity, reversibility, invariant Gibbs law, or KL/FI dissipation."
  },
  {
    key := "analysis.calculus.gradient-exp-neg-potential-mathlib-gradient",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.gradient_expNegPotential_eq_of_differentiableAt",
    upstreamDecl := "HasGradientAt.gradient plus hasGradientAt_expNegPotential_of_hasGradientAt",
    upstreamFile := "Mathlib.Analysis.Calculus.Gradient.Basic; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Gibbs-weight", "gradient", "DifferentiableAt", "Mathlib-gradient", "exp-neg-potential"],
    saldUse := "Chewi SDE/DENS root: turn `DifferentiableAt ℝ V x` into the Mathlib total-gradient identity `gradient (exp(-V)) x = -exp(-V x) • gradient V x` before weighted-divergence handoffs",
    note := "Pointwise Mathlib-gradient display only. It removes the separate Gibbs-weight chain-rule hypothesis when `V` is differentiable at the point; divergence, product rule, IBP, generator domains, invariant law, reversibility, and KL/FI remain red."
  },
  {
    key := "analysis.calculus.gradient-exp-neg-potential-coordinate",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.gradient_expNegPotential_coordinate_eq_of_differentiableAt",
    upstreamDecl := "gradient_expNegPotential_eq_of_differentiableAt plus EuclideanSpace coordinate evaluation of scalar multiplication",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Gibbs-weight", "gradient", "EuclideanSpace", "coordinate", "DifferentiableAt"],
    saldUse := "Chewi SDE/DENS root: supply the coordinate Gibbs-weight chain-rule equality used by finite Euclidean Langevin weighted-divergence handoffs",
    note := "Coordinate chain-rule display only. It does not prove coordinate product rules, the a.e. bridge or box-integrability assumptions needed by the compiled box divergence wrapper, Laplacian identities, IBP, stationarity, reversibility, invariant Gibbs law, or KL/FI dissipation."
  },
  {
    key := "analysis.calculus.continuous-gradient-of-contDiff-one",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.continuous_gradient_of_contDiff_one",
    upstreamDecl := "ContDiff.continuous_fderiv plus the continuous inverse Riesz equivalence for `gradient`",
    upstreamFile := "Mathlib.Analysis.Calculus.ContDiff.Basic; Mathlib.Analysis.Calculus.Gradient.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "gradient", "ContDiff", "Continuous", "regularity", "test-function"],
    saldUse := "Chewi Ch.1 Langevin root: derive continuity of Mathlib's total gradient from global `C¹` regularity before the scalar generator display on boxes",
    note := "Global `C¹` to continuous Mathlib gradient only. It does not assert a closed-box `ContDiffOn` variant, field differentiability for `rho * fderiv f eᵢ`, weighted IBP, boundary cancellation, domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.gradient-coordinate-unit-line-derivative",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.hasGradientAt_coordinateUnit_hasLineDerivAt",
    upstreamDecl := "HasGradientAt.hasFDerivAt / HasFDerivAt.hasLineDerivAt / InnerProductSpace.toDual_apply_apply",
    upstreamFile := "Mathlib.Analysis.Calculus.Gradient.Basic; Mathlib.Analysis.Calculus.LineDeriv.Basic; AutoSamplingTheory.TechnicalLemmas.Geometry.EuclideanSpaceCoordinates",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "gradient", "line-derivative", "EuclideanSpace", "coordinate", "finite-dimensional"],
    saldUse := "Chewi SDE/ANALYSIS root: identify a coordinate-unit line derivative with the corresponding coordinate of a supplied Mathlib gradient before finite-coordinate Langevin generator algebra",
    note := "Pointwise gradient-coordinate bridge only. It does not prove divergence, weighted product rules, Laplacian identities, IBP, boundary decay, stationarity, reversibility, or invariant Gibbs law."
  },
  {
    key := "analysis.calculus.fderiv-apply-eq-inner-of-hasGradientAt",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.fderiv_apply_eq_inner_of_hasGradientAt",
    upstreamDecl := "HasGradientAt.hasFDerivAt plus HasFDerivAt.unique and InnerProductSpace.toDual_apply_apply",
    upstreamFile := "Mathlib.Analysis.Calculus.Gradient.Basic; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "gradient", "fderiv", "inner-product", "HasGradientAt", "pointwise"],
    saldUse := "Chewi Ch.1 calculus root: rewrite a pointwise Frechet derivative application through a supplied Mathlib gradient representative before coordinate Langevin displays",
    note := "Pointwise fderiv/gradient bridge only. It does not choose coordinate bases, define divergence, prove product rules, IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.fderiv-apply-eq-inner-gradient",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.fderiv_apply_eq_inner_gradient_of_differentiableAt",
    upstreamDecl := "DifferentiableAt.hasGradientAt plus fderiv_apply_eq_inner_of_hasGradientAt",
    upstreamFile := "Mathlib.Analysis.Calculus.Gradient.Basic; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "gradient", "fderiv", "inner-product", "DifferentiableAt", "pointwise"],
    saldUse := "Chewi Ch.1 calculus root: under `DifferentiableAt ℝ f x`, rewrite `fderiv ℝ f x v` as `inner ℝ (gradient f x) v`",
    note := "Pointwise total-gradient display only. It does not prove global smoothness, divergence, coordinate sums, IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.fderiv-coordinate-eq-gradient-coordinate",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.fderiv_apply_coordinate_eq_gradient_coordinate_of_differentiableAt",
    upstreamDecl := "fderiv_apply_eq_inner_gradient_of_differentiableAt plus EuclideanSpace.inner_basisFun_real",
    upstreamFile := "Mathlib.Analysis.Calculus.Gradient.Basic; Mathlib.Analysis.InnerProductSpace.PiL2; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "gradient", "fderiv", "EuclideanSpace", "coordinate", "DifferentiableAt"],
    saldUse := "Chewi Ch.1 Langevin root: discharge the local bridge `fderiv ℝ f x eᵢ = (gradient f x) i` used by finite-coordinate generator displays",
    note := "Pointwise coordinate bridge only. It requires `DifferentiableAt ℝ f x`; differentiability of `fun y => fderiv ℝ f y` alone is not a substitute. It does not define divergence, prove that a coordinate sum is divergence, prove IBP/no-boundary terms, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.coordinate-divergence-sum-lineDeriv",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence_eq_sum_lineDeriv",
    upstreamDecl := "definition of coordinateDivergence",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence; Mathlib.Analysis.BoxIntegral.DivergenceTheorem pointwise summand shape",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "coordinate-divergence", "lineDeriv", "EuclideanSpace", "pointwise"],
    saldUse := "Chewi Ch.1 calculus root: name the finite-dimensional coordinate divergence convention `sum_i lineDeriv F_i x e_i` before Langevin divergence-form displays",
    note := "Definition/unfolding leaf only. It is pointwise and coordinate-dependent; it does not prove a divergence theorem, integration by parts, boundary cancellation, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.coordinate-divergence-fderiv-trace-sum",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence_eq_sum_fderiv_apply_of_hasFDerivAt",
    upstreamDecl := "HasFDerivAt.hasLineDerivAt plus PiLp.proj component projection",
    upstreamFile := "Mathlib.Analysis.Calculus.LineDeriv.Basic; Mathlib.Analysis.Normed.Lp.PiLp; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "coordinate-divergence", "fderiv", "trace-style", "EuclideanSpace", "pointwise"],
    saldUse := "Chewi Ch.1 calculus root: align ASTIS coordinate divergence with the Mathlib divergence-theorem summand shape `sum_i F' e_i i` under `HasFDerivAt F F' x`",
    note := "Pointwise Frechet-derivative expansion only. It does not assert coordinate independence, prove the Mathlib divergence theorem, instantiate box/all-space integrability, prove IBP/no-boundary terms, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.coordinate-divergence-fderiv-default-summand",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence_eq_sum_fderiv_apply_of_differentiableAt",
    upstreamDecl := "coordinateDivergence_eq_sum_fderiv_apply_of_hasFDerivAt plus DifferentiableAt.hasFDerivAt",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence; Mathlib.MeasureTheory.Integral.DivergenceTheorem summand shape",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "coordinate-divergence", "fderiv", "DifferentiableAt", "EuclideanSpace", "pointwise"],
    saldUse := "Chewi Ch.1 calculus root: rewrite ASTIS `coordinateDivergence F x` to the exact default Mathlib `fderiv ℝ F x` summand shape before any integral divergence-theorem instantiation",
    note := "Pointwise bridge to the Bochner divergence theorem integrand only. It does not prove integrability on boxes, face terms, no-boundary cancellation, weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.withlp-continuousLinearEquiv-euclidean-single",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.continuousLinearEquiv_apply_euclideanSpace_single",
    upstreamDecl := "PiLp.continuousLinearEquiv_apply",
    upstreamFile := "Mathlib.Analysis.Calculus.FDeriv.WithLp; Mathlib.Analysis.Normed.Lp.PiLp",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "WithLp", "PiLp", "EuclideanSpace", "coordinate-unit"],
    saldUse := "Chewi Ch.1 calculus root: identify the Euclidean coordinate unit with the Pi-space `Pi.single` unit before transporting derivative traces through `WithLp`",
    note := "Representation bridge only. It does not prove differentiability, divergence, a.e. equality, integrability, IBP, boundary cancellation, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.radial-smooth-cutoff-comp-toLp-hasFDerivAt",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.hasFDerivAt_radialSmoothCutoff_comp_toLp",
    upstreamDecl := "PiLp.hasFDerivAt_toLp; HasFDerivAt.comp; radialSmoothCutoff_contDiff",
    upstreamFile := "Mathlib.Analysis.Calculus.FDeriv.WithLp; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "cutoff", "WithLp", "PiLp", "EuclideanSpace", "HasFDerivAt", "chain-rule"],
    saldUse := "Chewi Ch.1 cutoff-smul route: expose the Euclidean radial-cutoff derivative as a raw finite-Pi-space HasFDerivAt producer",
    note := "Pointwise derivative transport only. It proves no support, measurability, integrability, tail convergence, IBP, generator-domain, or invariant-law statement."
  },
  {
    key := "analysis.calculus.pilp-radial-cutoff-gradient-L1-tendsto-zero",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.tendsto_integral_norm_fderiv_radialSmoothCutoff_comp_toLp_apply",
    upstreamDecl := "radialSmoothCutoff_fderiv_bound; hasFDerivAt_radialSmoothCutoff_comp_toLp; MeasureTheory.tendsto_integral_filter_of_dominated_convergence",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.{Cutoff,Divergence}; Mathlib.MeasureTheory.Integral.DominatedConvergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "cutoff", "WithLp", "PiLp", "L1", "Integrable", "dominated-convergence"],
    saldUse := "Chewi Ch.1 cutoff-smul route: make the cutoff-gradient cross term vanish in L1 for every integrable finite-Pi vector field",
    note := "Generic cutoff-gradient L1 limit only. The PiLp inverse-equivalence operator norm is retained explicitly; the theorem proves neither Gibbs/source-field integrability, main-term convergence, Gibbs tails, whole-space IBP, generator domains, nor invariant law."
  },
  {
    key := "analysis.calculus.pilp-radial-cutoff-main-integral-tendsto",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.tendsto_integral_radialSmoothCutoff_comp_toLp_smul",
    upstreamDecl := "radialSmoothCutoff_contDiff; radialSmoothCutoff_mem_Icc; radialSmoothCutoff_tendsto_one; MeasureTheory.tendsto_integral_filter_of_dominated_convergence",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.{Cutoff,Divergence}; Mathlib.MeasureTheory.Integral.DominatedConvergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "cutoff", "WithLp", "PiLp", "Integrable", "Bochner-integral", "dominated-convergence"],
    saldUse := "Chewi Ch.1 cutoff-smul route: pass the main integrable field through the PiLp-wrapped radial cutoff by dominated convergence",
    note := "Generic cutoff main-term limit only. It assumes the source field is Integrable and proves neither integrability of the concrete Langevin generator display, Gibbs tails, the cutoff-gradient cross term, whole-space IBP, generator domains, nor invariant law."
  },
  {
    key := "analysis.calculus.pilp-integrable-field-norm-tail-tendsto-zero",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.tendsto_setIntegral_norm_norm_ge_comp_toLp",
    upstreamDecl := "MeasureTheory.tendsto_setIntegral_of_antitone",
    upstreamFile := "Mathlib.MeasureTheory.Integral.Bochner.Set; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "tail", "WithLp", "PiLp", "L1", "Integrable", "set-integral"],
    saldUse := "Chewi Ch.1 cutoff route: make the L1 norm of any integrable raw-Pi field vanish outside expanding Euclidean balls",
    note := "Generic antitone-set tail theorem only. It has no Gibbs, generator, weighted-IBP, generator-domain, stationarity, invariance, reversibility, or KL/FI semantics."
  },
  {
    key := "analysis.calculus.compact-support-whole-space-coordinate-divergence-zero",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_wrapped_eq_zero_of_contDiff_of_hasCompactSupport",
    upstreamDecl := "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable; HasCompactSupport; Bornology.IsBounded.subset_ball_lt; setIntegral_eq_integral_of_forall_compl_eq_zero",
    upstreamFile := "Mathlib.MeasureTheory.Integral.DivergenceTheorem; Mathlib.Topology.MetricSpace.Bounded; Mathlib.MeasureTheory.Integral.Bochner.Set; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "coordinate-divergence", "whole-space", "compact-support", "ContDiff", "zero-boundary", "PiLp"],
    saldUse := "Chewi Ch.1 weighted-IBP root: integrate the divergence of any compactly supported C1 finite-dimensional vector field over the whole space and obtain zero",
    note := "Reusable analytic no-boundary theorem. It encloses the compact support in a strict Pi-box and invokes the finite-box divergence theorem; it contains no Gibbs, generator-domain, semigroup, invariant-law, reversibility, or KL/FI semantics."
  },
  {
    key := "analysis.calculus.sum-smulRight-apply-pi-single-eq-apply",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.sum_smulRight_apply_pi_single_eq_apply",
    upstreamDecl := "pi_eq_sum_univ'; ContinuousLinearMap.map_sum; ContinuousLinearMap.map_smul",
    upstreamFile := "Mathlib.Algebra.BigOperators.Pi; Mathlib.Topology.Algebra.Module.FiniteDimension",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "trace", "smulRight", "Pi.single", "finite-dimensional", "cross-term"],
    saldUse := "Chewi Ch.1 divergence product route: collapse the standard-basis trace of the cutoff cross derivative to the scalar derivative applied to the vector field",
    note := "Finite-dimensional linear algebra only. It proves no product-rule hypothesis, measurability, integrability, convergence, boundary cancellation, IBP, or invariant-law result."
  },
  {
    key := "analysis.calculus.coordinate-divergence-wrapped-toPi-trace",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence_wrapped_toPi_trace_of_hasFDerivAt",
    upstreamDecl := "PiLp.hasFDerivAt_ofLp / PiLp.hasFDerivAt_toLp / HasFDerivAt.comp",
    upstreamFile := "Mathlib.Analysis.Calculus.FDeriv.WithLp; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "coordinate-divergence", "WithLp", "Pi-space", "fderiv", "trace-style", "pointwise"],
    saldUse := "Chewi Ch.1 calculus root: transport a Pi-space `HasFDerivAt F F' x` into the ASTIS wrapped `EuclideanSpace` coordinate divergence trace formula",
    note := "Pointwise interface bridge only. It does not prove the derivative exists a.e., box integrability, face terms, no-boundary cancellation, weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.eventuallyEq-restrict-Icc-open-box-diff-countable",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.eventuallyEq_restrict_Icc_of_eqOn_univ_pi_Ioo_diff_countable",
    upstreamDecl := "Measure.univ_pi_Ioo_ae_eq_Icc plus Set.Countable.ae_notMem",
    upstreamFile := "Mathlib.MeasureTheory.Integral.DivergenceTheorem; Mathlib.MeasureTheory.Constructions.Polish.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "measure", "eventuallyEq", "closed-box", "open-box", "countable-exception"],
    saldUse := "Chewi Ch.1 calculus root: turn equality on the open Pi-box away from a countable exceptional set into equality a.e. on the restricted closed box",
    note := "A.e. transport leaf only. It does not prove differentiability, integrability, a divergence theorem, IBP, no-boundary limits, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.coordinate-divergence-wrapped-toPi-trace-ae",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence_wrapped_toPi_trace_ae_of_ae_hasFDerivAt",
    upstreamDecl := "coordinateDivergence_wrapped_toPi_trace_of_hasFDerivAt",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "coordinate-divergence", "WithLp", "Pi-space", "a.e.", "HasFDerivAt"],
    saldUse := "Chewi Ch.1 calculus root: discharge the finite-box face-term wrapper's `hdiv_ae` equality from an explicit a.e. Pi-space differentiability hypothesis",
    note := "A.e. equality bridge only. It assumes the derivative exists a.e.; it does not prove box integrability, face terms, no-boundary cancellation, weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.coordinate-divergence-wrapped-toPi-trace-ae-off-countable",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence_wrapped_toPi_trace_ae_of_hasFDerivAt_off_countable",
    upstreamDecl := "coordinateDivergence_wrapped_toPi_trace_of_hasFDerivAt plus eventuallyEq_restrict_Icc_of_eqOn_univ_pi_Ioo_diff_countable",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence; Mathlib.MeasureTheory.Integral.DivergenceTheorem",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "coordinate-divergence", "WithLp", "Pi-space", "a.e.", "open-box", "countable-exception"],
    saldUse := "Chewi Ch.1 calculus root: discharge the finite-box face-term wrapper's `hdiv_ae` equality from Mathlib-style open-box/off-countable differentiability data",
    note := "This closes only the a.e. bridge assumption for the box wrapper. Box integrability, whole-space/no-boundary cancellation, weighted IBP, generator domains, invariant Gibbs law, reversibility, stationarity, and KL/FI dissipation remain red."
  },
  {
    key := "analysis.calculus.integrableOn-coordinate-divergence-wrapped-of-trace",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integrableOn_coordinateDivergence_wrapped_of_integrableOn_trace_of_hasFDerivAt_off_countable",
    upstreamDecl := "IntegrableOn.congr_fun_ae plus coordinateDivergence_wrapped_toPi_trace_ae_of_hasFDerivAt_off_countable",
    upstreamFile := "Mathlib.MeasureTheory.Integral.IntegrableOn; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "coordinate-divergence", "WithLp", "Pi-space", "IntegrableOn", "trace-style", "a.e.", "open-box"],
    saldUse := "Chewi Ch.1 calculus root: transfer Mathlib trace-summand `IntegrableOn` across the compiled a.e. bridge to the ASTIS wrapped coordinate-divergence integrand",
    note := "Integrability transfer only. It assumes trace integrability and differentiability off a countable set; it does not prove trace integrability for a concrete Langevin field, weighted IBP, no-boundary terms, generator domains, invariant law, reversibility, stationarity, or KL/FI."
  },
  {
    key := "analysis.calculus.integral-coordinate-divergence-toPi-box-trace-integrable",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_of_integrableOn_trace_of_hasFDerivAt_off_countable",
    upstreamDecl := "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable plus integrableOn_coordinateDivergence_wrapped_of_integrableOn_trace_of_hasFDerivAt_off_countable",
    upstreamFile := "Mathlib.MeasureTheory.Integral.DivergenceTheorem; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "coordinate-divergence", "box-divergence-theorem", "face-terms", "IntegrableOn", "trace-style", "finite-dimensional"],
    saldUse := "Chewi Ch.1 calculus root: finite-box signed face-term formula for ASTIS coordinateDivergence with only Mathlib trace-integrability as the integrability input",
    note := "Box-level signed face-term wrapper only. It derives the a.e. bridge and ASTIS-side integrability internally, but still assumes trace integrability. It does not prove trace integrability for the explicit Langevin field, whole-space/no-boundary cancellation, weighted IBP, generator domains, invariant Gibbs law, reversibility, stationarity, or KL/FI."
  },
  {
    key := "analysis.calculus.integral-coordinate-divergence-toPi-box-zero-face",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_integrableOn_trace_of_hasFDerivAt_off_countable",
    upstreamDecl := "integral_coordinateDivergence_toPi_box_of_integrableOn_trace_of_hasFDerivAt_off_countable plus supplied zero signed-face term",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "coordinate-divergence", "box-divergence-theorem", "face-terms", "zero-face", "finite-dimensional"],
    saldUse := "log-concave sampling Ch.1 finite-box weighted-IBP route: once the signed face term is explicitly zero, the ASTIS coordinate-divergence box integral is zero",
    note := "Conditional zero-face handoff only. It assumes the face term vanishes; it does not prove compact support, boundary decay, whole-space limits, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.signed-face-term-sum-zero-of-boundary-component-zero",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.signedFaceTermSum_eq_zero_of_boundary_component_eq_zero",
    upstreamDecl := "integral_zero / Finset.sum_eq_zero via componentwise zero values on lower and upper faces",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "face-terms", "zero-face", "finite-box", "coordinate-divergence"],
    saldUse := "log-concave sampling Ch.1 boundary route: explicit zero normal components on all finite-box faces imply the signed face-term sum vanishes",
    note := "Boundary-value producer only. It assumes componentwise zero values on the faces; it does not prove compact support, tail decay, whole-space limits, weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.integral-coordinate-divergence-toPi-box-zero-boundary-component",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_boundary_component_eq_zero",
    upstreamDecl := "integral_coordinateDivergence_toPi_box_eq_zero_of_integrableOn_trace_of_hasFDerivAt_off_countable plus signedFaceTermSum_eq_zero_of_boundary_component_eq_zero",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "coordinate-divergence", "box-divergence-theorem", "zero-face", "finite-dimensional"],
    saldUse := "log-concave sampling Ch.1 finite-box weighted-IBP route: if the vector field's normal component is zero on each lower and upper face, the ASTIS coordinate-divergence box integral is zero",
    note := "Finite-box conditional boundary handoff only. It still assumes trace integrability and open-box/off-countable differentiability; it does not derive compact support, tail decay, whole-space weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.signed-face-term-sum-zero-of-update-boundary-component-zero",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.signedFaceTermSum_eq_zero_of_update_boundary_component_eq_zero",
    upstreamDecl := "signedFaceTermSum_eq_zero_of_boundary_component_eq_zero plus Function.update_eq_self on Fin.insertNth faces",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "Function.update", "face-terms", "zero-face", "finite-box"],
    saldUse := "log-concave sampling Ch.1 boundary route: update-to-boundary zero component hypotheses imply the finite-box signed face-term sum vanishes",
    note := "Update-shaped boundary-value producer only. It assumes zero components after replacing a coordinate by the face endpoint; it does not prove compact support, tail decay, whole-space limits, weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.integral-coordinate-divergence-toPi-box-zero-update-boundary-component",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_update_boundary_component_eq_zero",
    upstreamDecl := "integral_coordinateDivergence_toPi_box_eq_zero_of_integrableOn_trace_of_hasFDerivAt_off_countable plus signedFaceTermSum_eq_zero_of_update_boundary_component_eq_zero",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "Function.update", "coordinate-divergence", "box-divergence-theorem", "zero-face"],
    saldUse := "log-concave sampling Ch.1 finite-box weighted-IBP route: update-to-boundary zero component hypotheses imply the ASTIS coordinate-divergence box integral is zero",
    note := "Finite-box conditional update-boundary handoff only. It still assumes trace integrability and open-box/off-countable differentiability; it does not derive compact support, tail decay, whole-space weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.update-boundary-component-zero-of-eq-zero-off-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.update_boundary_component_eq_zero_of_eq_zero_off_univ_pi_Ioo",
    upstreamDecl := "Set.mem_univ_pi plus endpoint self-inequality contradiction for Function.update boundary points",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "open-box", "Function.update", "zero-boundary", "finite-box"],
    saldUse := "log-concave sampling Ch.1 boundary route: off-open-box vanishing implies update-to-boundary normal components are zero",
    note := "Off-open-box boundary producer only. It assumes the vector field is already zero outside the open Pi-box; it does not prove compact support, cutoff support, tail decay, whole-space weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.signed-face-term-sum-zero-of-eq-zero-off-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.signedFaceTermSum_eq_zero_of_eq_zero_off_univ_pi_Ioo",
    upstreamDecl := "update_boundary_component_eq_zero_of_eq_zero_off_univ_pi_Ioo plus signedFaceTermSum_eq_zero_of_update_boundary_component_eq_zero",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "open-box", "face-terms", "zero-face", "finite-box"],
    saldUse := "log-concave sampling Ch.1 boundary route: off-open-box vanishing implies Mathlib's finite-box signed face-term sum is zero",
    note := "Signed face-term producer only. It assumes off-open-box vanishing; it does not prove compact support, tail decay, whole-space limits, weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.integral-coordinate-divergence-toPi-box-zero-off-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_eq_zero_off_univ_pi_Ioo",
    upstreamDecl := "integral_coordinateDivergence_toPi_box_eq_zero_of_integrableOn_trace_of_hasFDerivAt_off_countable plus signedFaceTermSum_eq_zero_of_eq_zero_off_univ_pi_Ioo",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "open-box", "coordinate-divergence", "box-divergence-theorem", "zero-face"],
    saldUse := "log-concave sampling Ch.1 finite-box weighted-IBP route: off-open-box vanishing implies the ASTIS coordinate-divergence box integral is zero",
    note := "Finite-box off-open-box handoff only. It still assumes trace integrability and open-box/off-countable differentiability; it does not derive compact support, tail decay, whole-space weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.eq-zero-off-univ-pi-Ioo-of-support-subset-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.eq_zero_off_univ_pi_Ioo_of_support_subset_univ_pi_Ioo",
    upstreamDecl := "Function.support subset plus contradiction outside target set",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "support", "open-box", "finite-box"],
    saldUse := "log-concave sampling Ch.1 boundary route: a support subset of the open Pi-box implies the vector field is zero outside that open Pi-box",
    note := "Plain support-to-off-open-box producer only. It assumes the support subset; it does not prove compact support construction, cutoff support, tail decay, whole-space weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.exists-contDiff-cutoff-tsupport-subset-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.exists_contDiff_cutoff_tsupport_subset_univ_pi_Ioo",
    upstreamDecl := "exists_contDiff_tsupport_subset specialized to finite Pi-open boxes",
    upstreamFile := "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "smooth-cutoff", "ContDiff", "HasCompactSupport", "tsupport", "open-box", "finite-dimensional"],
    saldUse := "log-concave sampling Ch.1 cutoff route: construct a local smooth real-valued cutoff whose topological support is contained in a finite Pi-open box and which equals one at a chosen interior point",
    note := "Local smooth-cutoff existence leaf only. It does not choose an exhausting cutoff family, prove concrete derivative formulas for the cutoff, pass to whole-space tail limits, prove weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.support-subset-univ-pi-Ioo-of-tsupport-subset-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.support_subset_univ_pi_Ioo_of_tsupport_subset_univ_pi_Ioo",
    upstreamDecl := "subset_tsupport composed with finite Pi-open-box containment",
    upstreamFile := "Mathlib.Topology.Support; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "support", "tsupport", "open-box", "finite-dimensional"],
    saldUse := "log-concave sampling Ch.1 cutoff route: convert Mathlib topological-support containment for a scalar cutoff into the plain Function.support containment consumed by finite-box zero-face handoffs",
    note := "Support-API bridge only. It does not construct a cutoff, prove compactness, choose an exhausting cutoff family, prove derivative formulas, tail limits, weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.exists-contDiff-cutoff-support-subset-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.exists_contDiff_cutoff_support_subset_univ_pi_Ioo",
    upstreamDecl := "exists_contDiff_cutoff_tsupport_subset_univ_pi_Ioo plus support_subset_univ_pi_Ioo_of_tsupport_subset_univ_pi_Ioo",
    upstreamFile := "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "smooth-cutoff", "ContDiff", "HasCompactSupport", "support", "tsupport", "open-box", "finite-dimensional"],
    saldUse := "log-concave sampling Ch.1 cutoff route: construct a local smooth cutoff with both topological support and plain Function.support contained in the finite Pi-open box",
    note := "Local smooth-cutoff packaging leaf only. It does not choose an exhausting cutoff family, prove concrete derivative formulas, pass to whole-space tail limits, prove weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.exists-contDiff-support-eq-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.exists_contDiff_support_eq_univ_pi_Ioo",
    upstreamDecl := "IsOpen.exists_contDiff_support_eq specialized to finite Pi-open boxes",
    upstreamFile := "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "smooth-cutoff", "ContDiff", "support", "support-eq", "open-box", "finite-dimensional"],
    saldUse := "log-concave sampling Ch.1 cutoff/exhaustion route: construct a smooth [0,1]-valued function whose plain support is exactly a finite Pi-open box",
    note := "Exact plain-support open-box smooth function only. It does not prove compact support, topological-support containment, plateau equal to one on an inner closed box, an exhausting family, derivative bounds, tail limits, weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.positive-on-univ-pi-Ioo-of-support-eq-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.positive_on_univ_pi_Ioo_of_support_eq_univ_pi_Ioo",
    upstreamDecl := "Function.support equality plus range subset Set.Icc 0 1",
    upstreamFile := "Mathlib.Topology.Support; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "support", "support-eq", "positivity", "open-box", "finite-dimensional"],
    saldUse := "log-concave sampling Ch.1 cutoff/exhaustion route: turn exact open-box plain support and [0,1] range into strict positivity on the finite Pi-open box",
    note := "Support/range consequence only. It does not construct a cutoff, prove compact support, prove plateau behavior, choose an exhausting family, prove derivative bounds, tail limits, weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.Icc-subset-univ-pi-Ioo-of-forall-lt",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.Icc_subset_univ_pi_Ioo_of_strict_bounds",
    upstreamDecl := "coordinatewise order on Pi spaces plus strict outer bounds",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "closed-box", "open-box", "exhaustion", "finite-dimensional"],
    saldUse := "log-concave sampling Ch.1 cutoff/exhaustion route: an inner closed Pi-box is contained in any coordinatewise strictly larger open Pi-box",
    note := "Closed-box/open-box containment bookkeeping only. It does not construct a cutoff, choose an exhausting family, prove derivative bounds, tail limits, weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.exists-contDiff-cutoff-support-subset-outer-univ-pi-Ioo-of-mem-Icc",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.exists_contDiff_cutoff_support_subset_outer_univ_pi_Ioo_of_mem_Icc",
    upstreamDecl := "Icc_subset_univ_pi_Ioo_of_strict_bounds plus exists_contDiff_cutoff_support_subset_univ_pi_Ioo",
    upstreamFile := "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "smooth-cutoff", "closed-box", "open-box", "support", "tsupport", "exhaustion", "finite-dimensional"],
    saldUse := "log-concave sampling Ch.1 cutoff/exhaustion route: every point of an inner closed Pi-box has a local smooth cutoff whose support and topological support lie in a strictly larger open Pi-box",
    note := "Local pointwise cutoff in an outer open box only. It does not construct one cutoff equal to one on the whole inner closed box, choose an exhausting cutoff family, prove derivative bounds, pass to whole-space tail limits, prove weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.signed-face-term-sum-zero-of-support-subset-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.signedFaceTermSum_eq_zero_of_support_subset_univ_pi_Ioo",
    upstreamDecl := "eq_zero_off_univ_pi_Ioo_of_support_subset_univ_pi_Ioo plus signedFaceTermSum_eq_zero_of_eq_zero_off_univ_pi_Ioo",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "support", "open-box", "face-terms", "zero-face", "finite-box"],
    saldUse := "log-concave sampling Ch.1 boundary route: a support subset of the open Pi-box implies Mathlib's finite-box signed face-term sum is zero",
    note := "Support-subset face-term producer only. It assumes the support subset; it does not prove compact support construction, cutoff support, tail decay, whole-space limits, weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.integral-coordinate-divergence-toPi-box-zero-support-subset-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_support_subset_univ_pi_Ioo",
    upstreamDecl := "integral_coordinateDivergence_toPi_box_eq_zero_of_integrableOn_trace_of_hasFDerivAt_off_countable plus signedFaceTermSum_eq_zero_of_support_subset_univ_pi_Ioo",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "support", "open-box", "coordinate-divergence", "box-divergence-theorem", "zero-face"],
    saldUse := "log-concave sampling Ch.1 finite-box weighted-IBP route: a support subset of the open Pi-box implies the ASTIS coordinate-divergence box integral is zero",
    note := "Finite-box support-subset handoff only. It still assumes trace integrability and open-box/off-countable differentiability, and it assumes the support subset; it does not derive compact support, tail decay, whole-space weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.support-smul-subset-univ-pi-Ioo-of-cutoff-eq-zero-off-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.support_smul_subset_univ_pi_Ioo_of_eq_zero_off_univ_pi_Ioo",
    upstreamDecl := "scalar zero outside open Pi-box implies zero smul vector field outside open Pi-box",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "support", "smul", "open-box", "finite-box"],
    saldUse := "log-concave sampling Ch.1 cutoff route: a scalar cutoff vanishing outside the open Pi-box forces the cutoff-smul vector field to be supported in the open Pi-box",
    note := "Plain support-containment bridge only. It does not construct a smooth cutoff, prove HasCompactSupport, prove cutoff-smul regularity, tail decay, whole-space weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.support-smul-subset-univ-pi-Ioo-of-scalar-support-subset-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.support_smul_subset_univ_pi_Ioo_of_scalar_support_subset_univ_pi_Ioo",
    upstreamDecl := "scalar Function.support subset plus support_smul_subset_univ_pi_Ioo_of_eq_zero_off_univ_pi_Ioo",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "support", "smul", "open-box", "finite-box"],
    saldUse := "log-concave sampling Ch.1 cutoff route: a scalar cutoff supported in the open Pi-box forces the cutoff-smul vector field to be supported in the open Pi-box",
    note := "Scalar-support-to-vector-support bridge only. It uses Function.support, not topological compact support, and does not prove cutoff construction, cutoff-smul regularity, tail decay, whole-space weighted IBP, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.support-smul-subset-univ-pi-Ioo-of-scalar-tsupport-subset-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.support_smul_subset_univ_pi_Ioo_of_scalar_tsupport_subset_univ_pi_Ioo",
    upstreamDecl := "support_subset_univ_pi_Ioo_of_tsupport_subset_univ_pi_Ioo plus support_smul_subset_univ_pi_Ioo_of_scalar_support_subset_univ_pi_Ioo",
    upstreamFile := "Mathlib.Topology.Support; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "support", "tsupport", "smul", "open-box", "finite-box"],
    saldUse := "log-concave sampling Ch.1 cutoff route: a scalar cutoff with topological support inside the open Pi-box forces the cutoff-smul vector field to be plain-supported in that open box",
    note := "Direct tsupport-to-cutoff-smul support bridge only. It does not construct a cutoff, prove cutoff-smul regularity, tail decay, whole-space weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.continuousOn-smul-vectorField-of-continuousOn",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.continuousOn_smul_vectorField_of_continuousOn",
    upstreamDecl := "ContinuousOn.smul",
    upstreamFile := "Mathlib.Topology.Algebra.Module.Basic; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "smul", "ContinuousOn", "closed-box", "regularity"],
    saldUse := "log-concave sampling Ch.1 cutoff route: derive closed-box continuity of a cutoff-smul vector field from separate cutoff and vector-field continuity",
    note := "Closed-box continuity bridge only. It does not prove smooth cutoff construction, differentiability, trace integrability, boundary cancellation, weighted IBP, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.hasFDerivAt-smul-vectorField-of-hasFDerivAt",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.hasFDerivAt_smul_vectorField_of_hasFDerivAt",
    upstreamDecl := "HasFDerivAt.smul",
    upstreamFile := "Mathlib.Analysis.Calculus.FDeriv.Basic; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "smul", "HasFDerivAt", "product-rule", "regularity"],
    saldUse := "log-concave sampling Ch.1 cutoff route: pointwise product-rule Frechet derivative for the cutoff-smul vector field",
    note := "Pointwise derivative bridge only. It does not prove continuity, open-box/off-countable coverage, trace integrability, boundary cancellation, weighted IBP, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.hasFDerivAt-smul-vectorField-off-countable",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.hasFDerivAt_smul_vectorField_off_countable",
    upstreamDecl := "hasFDerivAt_smul_vectorField_of_hasFDerivAt applied pointwise on open box minus a countable set",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "smul", "HasFDerivAt", "open-box", "countable-exception", "regularity"],
    saldUse := "log-concave sampling Ch.1 cutoff route: derive the divergence-theorem off-countable derivative hypothesis for a cutoff-smul field from separate cutoff and vector-field derivative hypotheses",
    note := "Off-countable derivative bridge only. It assumes a shared exceptional set and does not prove trace integrability, closed-box continuity, boundary cancellation, weighted IBP, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.signed-face-term-sum-smul-zero-of-cutoff-eq-zero-off-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.signedFaceTermSum_smul_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo",
    upstreamDecl := "support_smul_subset_univ_pi_Ioo_of_eq_zero_off_univ_pi_Ioo plus signedFaceTermSum_eq_zero_of_support_subset_univ_pi_Ioo",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "smul", "face-terms", "zero-face", "finite-box"],
    saldUse := "log-concave sampling Ch.1 cutoff route: a scalar cutoff vanishing outside the open Pi-box implies the cutoff-smul finite-box signed face-term sum is zero",
    note := "Finite-box cutoff-smul face-term producer only. It does not prove smooth cutoff existence, regularity, trace integrability, tail limits, whole-space weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.signed-face-term-sum-smul-zero-of-scalar-support-subset-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.signedFaceTermSum_smul_eq_zero_of_scalar_support_subset_univ_pi_Ioo",
    upstreamDecl := "support_smul_subset_univ_pi_Ioo_of_scalar_support_subset_univ_pi_Ioo plus signedFaceTermSum_eq_zero_of_support_subset_univ_pi_Ioo",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "support", "smul", "face-terms", "zero-face", "finite-box"],
    saldUse := "log-concave sampling Ch.1 cutoff route: scalar cutoff support inside the open Pi-box implies the cutoff-smul finite-box signed face-term sum is zero",
    note := "Finite-box scalar-support face-term producer only. It does not prove smooth cutoff existence, regularity, trace integrability, tail limits, whole-space weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.signed-face-term-sum-smul-zero-of-scalar-tsupport-subset-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.signedFaceTermSum_smul_eq_zero_of_scalar_tsupport_subset_univ_pi_Ioo",
    upstreamDecl := "support_subset_univ_pi_Ioo_of_tsupport_subset_univ_pi_Ioo plus signedFaceTermSum_smul_eq_zero_of_scalar_support_subset_univ_pi_Ioo",
    upstreamFile := "Mathlib.Topology.Support; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "support", "tsupport", "smul", "face-terms", "zero-face", "finite-box"],
    saldUse := "log-concave sampling Ch.1 cutoff route: scalar topological support inside the open Pi-box implies the cutoff-smul finite-box signed face-term sum is zero",
    note := "Finite-box scalar-tsupport face-term producer only. It does not prove smooth cutoff construction, cutoff-smul regularity, trace integrability, tail limits, whole-space weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.integral-coordinate-divergence-toPi-box-zero-cutoff-eq-zero-off-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo",
    upstreamDecl := "integral_coordinateDivergence_toPi_box_eq_zero_of_support_subset_univ_pi_Ioo plus support_smul_subset_univ_pi_Ioo_of_eq_zero_off_univ_pi_Ioo",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "smul", "coordinate-divergence", "box-divergence-theorem", "zero-face"],
    saldUse := "log-concave sampling Ch.1 finite-box cutoff route: scalar cutoff vanishing outside the open Pi-box implies the cutoff-smul coordinate-divergence box integral is zero under the existing trace/differentiability assumptions",
    note := "Finite-box cutoff-smul handoff only. It still assumes continuity, open-box/off-countable differentiability, and trace integrability for the cutoff-smul field; it does not prove those regularity facts, smooth cutoff existence, tail decay, whole-space weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.integral-coordinate-divergence-toPi-box-zero-scalar-support-subset-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_support_subset_univ_pi_Ioo",
    upstreamDecl := "integral_coordinateDivergence_toPi_box_eq_zero_of_support_subset_univ_pi_Ioo plus support_smul_subset_univ_pi_Ioo_of_scalar_support_subset_univ_pi_Ioo",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "support", "smul", "coordinate-divergence", "box-divergence-theorem", "zero-face"],
    saldUse := "log-concave sampling Ch.1 finite-box cutoff route: scalar cutoff support inside the open Pi-box implies the cutoff-smul coordinate-divergence box integral is zero under the existing trace/differentiability assumptions",
    note := "Finite-box scalar-support cutoff handoff only. It still assumes continuity, open-box/off-countable differentiability, and trace integrability for the cutoff-smul field; it does not prove those regularity facts, smooth cutoff existence, tail decay, whole-space weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.integral-coordinate-divergence-toPi-box-zero-scalar-tsupport-subset-univ-pi-Ioo",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_tsupport_subset_univ_pi_Ioo",
    upstreamDecl := "support_subset_univ_pi_Ioo_of_tsupport_subset_univ_pi_Ioo plus integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_support_subset_univ_pi_Ioo",
    upstreamFile := "Mathlib.Topology.Support; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "support", "tsupport", "smul", "coordinate-divergence", "box-divergence-theorem", "zero-face"],
    saldUse := "log-concave sampling Ch.1 finite-box cutoff route: scalar cutoff topological support inside the open Pi-box implies the cutoff-smul coordinate-divergence box integral is zero under the existing trace/differentiability assumptions",
    note := "Finite-box scalar-tsupport cutoff handoff only. It still assumes continuity, open-box/off-countable differentiability, and trace integrability for the cutoff-smul field; it does not prove smooth cutoff construction, cutoff-smul regularity, tail decay, whole-space weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.integral-coordinate-divergence-toPi-box-zero-cutoff-eq-zero-off-univ-pi-Ioo-regularity",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo_of_regularity",
    upstreamDecl := "continuousOn_smul_vectorField_of_continuousOn plus hasFDerivAt_smul_vectorField_off_countable plus integral_coordinateDivergence_toPi_box_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "smul", "regularity", "coordinate-divergence", "box-divergence-theorem", "zero-face"],
    saldUse := "log-concave sampling Ch.1 finite-box cutoff route: derive the cutoff-smul continuity and off-countable derivative hypotheses before applying the zero-face coordinate-divergence handoff",
    note := "Finite-box regularity handoff only. Trace integrability for the product-rule derivative remains explicit; it does not prove smooth cutoff construction, tail decay, whole-space weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.integral-coordinate-divergence-toPi-box-zero-scalar-support-subset-univ-pi-Ioo-regularity",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_support_subset_univ_pi_Ioo_of_regularity",
    upstreamDecl := "continuousOn_smul_vectorField_of_continuousOn plus hasFDerivAt_smul_vectorField_off_countable plus integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_support_subset_univ_pi_Ioo",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "support", "smul", "regularity", "coordinate-divergence", "box-divergence-theorem", "zero-face"],
    saldUse := "log-concave sampling Ch.1 finite-box cutoff route: derive cutoff-smul regularity hypotheses before applying the scalar-support zero-face coordinate-divergence handoff",
    note := "Finite-box scalar-support regularity handoff only. Trace integrability for the product-rule derivative remains explicit; it does not prove smooth cutoff construction, tail decay, whole-space weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.integral-coordinate-divergence-toPi-box-zero-scalar-support-subset-univ-pi-Ioo-fderiv",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_support_subset_univ_pi_Ioo_of_fderiv",
    upstreamDecl := "scalar-support regularity handoff specialized to canonical fderiv field",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "support", "smul", "fderiv", "regularity", "coordinate-divergence", "box-divergence-theorem", "zero-face"],
    saldUse := "log-concave sampling Ch.1 finite-box cutoff route: replace the supplied vector-field derivative parameter by the canonical `fderiv ℝ G` in the scalar-support zero integral handoff",
    note := "Finite-box canonical-fderiv handoff only. It still assumes cutoff derivative data, continuity, trace integrability, and scalar support containment; it does not construct an exhausting cutoff family, prove tail decay, whole-space weighted IBP, generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.integrableOn-smul-vectorField-trace-of-continuousOn",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integrableOn_smul_vectorField_trace_of_continuousOn",
    upstreamDecl := "ContinuousOn.integrableOn_compact on isCompact_Icc",
    upstreamFile := "Mathlib.MeasureTheory.Integral.Bochner.Basic; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "smul", "trace-style", "IntegrableOn", "ContinuousOn", "closed-box"],
    saldUse := "log-concave sampling Ch.1 finite-box cutoff route: discharge compact-box integrability of the cutoff-smul product-rule trace from closed-box trace continuity",
    note := "Compact-box integrability handoff only. It assumes trace continuity and does not prove trace continuity from component assumptions, smooth cutoff construction, tail decay, whole-space weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.integral-coordinate-divergence-toPi-box-zero-cutoff-eq-zero-off-univ-pi-Ioo-trace-continuous",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo_of_trace_continuous",
    upstreamDecl := "integrableOn_smul_vectorField_trace_of_continuousOn plus integral_coordinateDivergence_toPi_box_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo_of_regularity",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "smul", "trace-style", "ContinuousOn", "coordinate-divergence", "box-divergence-theorem", "zero-face"],
    saldUse := "log-concave sampling Ch.1 finite-box cutoff route: derive cutoff-smul regularity and compact-box trace integrability before applying the zero-face coordinate-divergence handoff",
    note := "Finite-box trace-continuity handoff only. It still assumes closed-box trace continuity and cutoff vanishing outside the open box; it does not prove smooth cutoff construction, tail decay, whole-space weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.integral-coordinate-divergence-toPi-box-zero-scalar-support-subset-univ-pi-Ioo-trace-continuous",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_support_subset_univ_pi_Ioo_of_trace_continuous",
    upstreamDecl := "integrableOn_smul_vectorField_trace_of_continuousOn plus integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_support_subset_univ_pi_Ioo_of_regularity",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "support", "smul", "trace-style", "ContinuousOn", "coordinate-divergence", "box-divergence-theorem", "zero-face"],
    saldUse := "log-concave sampling Ch.1 finite-box cutoff route: derive cutoff-smul regularity and compact-box trace integrability before applying the scalar-support zero-face coordinate-divergence handoff",
    note := "Finite-box scalar-support trace-continuity handoff only. It still assumes closed-box trace continuity and scalar Function.support containment; it does not prove smooth cutoff construction, tail decay, whole-space weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.continuousOn-smul-vectorField-trace-of-component-continuousOn",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.continuousOn_smul_vectorField_trace_of_component_continuousOn",
    upstreamDecl := "continuousOn_finset_sum plus scalar ContinuousOn.mul/add after expanding smulRight trace",
    upstreamFile := "Mathlib.Topology.Algebra.Group.Basic; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "smul", "trace-style", "ContinuousOn", "component", "closed-box"],
    saldUse := "log-concave sampling Ch.1 finite-box cutoff route: derive closed-box continuity of the cutoff-smul product-rule trace from exactly the diagonal component continuity hypotheses used by the trace summand",
    note := "Trace-continuity assembly only. It does not prove that the component fields are derivatives, does not identify the product-rule operator with canonical fderiv, and does not prove smooth cutoff construction, tail decay, whole-space weighted IBP, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.continuousOn-smul-vectorField-trace-of-components",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.continuousOn_smul_vectorField_trace_of_components",
    upstreamDecl := "continuousOn_smul_vectorField_trace_of_component_continuousOn plus ContinuousOn.clm_apply/component projection",
    upstreamFile := "Mathlib.Analysis.Normed.Operator.BoundedLinearMaps; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "smul", "trace-style", "ContinuousOn", "closed-box"],
    saldUse := "log-concave sampling Ch.1 finite-box cutoff route: convenience wrapper deriving trace continuity from CLM-valued continuity of χ' and G'",
    note := "Convenience trace-continuity wrapper only. It is stronger than the component-continuity leaf and still does not prove derivative existence, smooth cutoff construction, tail decay, weighted IBP, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.integral-coordinate-divergence-toPi-box-zero-cutoff-eq-zero-off-univ-pi-Ioo-component-continuous",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo_of_component_continuous",
    upstreamDecl := "continuousOn_smul_vectorField_trace_of_component_continuousOn plus trace-continuous zero-face handoff",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "smul", "component", "trace-style", "ContinuousOn", "coordinate-divergence", "box-divergence-theorem", "zero-face"],
    saldUse := "log-concave sampling Ch.1 finite-box cutoff route: component trace continuity, regularity hypotheses, and cutoff vanishing imply the finite-box cutoff-smul coordinate-divergence integral is zero",
    note := "Finite-box component-continuity handoff only. It assumes derivative hypotheses and cutoff vanishing; it does not prove smooth cutoff construction, tail decay, whole-space weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.integral-coordinate-divergence-toPi-box-zero-scalar-support-subset-univ-pi-Ioo-component-continuous",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_support_subset_univ_pi_Ioo_of_component_continuous",
    upstreamDecl := "continuousOn_smul_vectorField_trace_of_component_continuousOn plus scalar-support trace-continuous zero-face handoff",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Langevin", "boundary", "cutoff", "support", "smul", "component", "trace-style", "ContinuousOn", "coordinate-divergence", "box-divergence-theorem", "zero-face"],
    saldUse := "log-concave sampling Ch.1 finite-box cutoff route: component trace continuity, regularity hypotheses, and scalar support containment imply the finite-box cutoff-smul coordinate-divergence integral is zero",
    note := "Finite-box scalar-support component-continuity handoff only. It assumes derivative hypotheses and scalar Function.support containment; it does not prove smooth cutoff construction, tail decay, whole-space weighted IBP, generator domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.integral-coordinate-divergence-toPi-box",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_of_hasFDerivAt_off_countable",
    upstreamDecl := "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable plus coordinateDivergence_eq_sum_fderiv_apply_of_differentiableAt",
    upstreamFile := "Mathlib.MeasureTheory.Integral.DivergenceTheorem; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "coordinate-divergence", "box-divergence-theorem", "face-terms", "IntegrableOn", "ae-equality", "finite-dimensional"],
    saldUse := "Chewi Ch.1 calculus root: integrate ASTIS coordinateDivergence over a finite Mathlib box and obtain only the signed face-term formula, assuming the a.e. bridge to Mathlib's trace integrand and box integrability explicitly",
    note := "Box-level signed face-term wrapper only. It assumes the finite-box continuity/off-countable differentiability hypotheses, an a.e. bridge from ASTIS coordinateDivergence to Mathlib's trace integrand, and IntegrableOn for the coordinate-divergence integrand. It does not derive that a.e. bridge, prove box integrability, prove whole-space/no-boundary cancellation, weighted IBP, generator domains, invariant Gibbs law, reversibility, stationarity, or KL/FI dissipation."
  },
  {
    key := "analysis.calculus.line-derivative-product-rule",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.hasLineDerivAt_mul",
    upstreamDecl := "HasDerivAt.mul applied to the line curve t ↦ x + t • v",
    upstreamFile := "Mathlib.Analysis.Calculus.LineDeriv.Basic; Mathlib.Analysis.Calculus.Deriv.Mul",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "line-derivative", "product-rule", "coordinate", "weighted-divergence", "NormedAlgebra"],
    saldUse := "Chewi SDE/ANALYSIS root: expose the generic algebra-valued line-derivative product rule before finite Euclidean weighted-divergence algebra",
    note := "Generic product-rule leaf only. It does not by itself discharge the current Langevin `hdiv` hypothesis; it does not identify `g` with a coordinate derivative of a test function, prove Hessian/iterated-derivative identities, define divergence, prove divergence-sum identities, IBP, boundary decay, stationarity, reversibility, invariant Gibbs law, or KL/FI dissipation."
  },
  {
    key := "analysis.calculus.line-derivative-rho-product-rule",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.hasLineDerivAt_rho_mul",
    upstreamDecl := "hasLineDerivAt_mul specialized to real-valued `rho * g`, reordered for weighted-divergence algebra",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "line-derivative", "product-rule", "rho", "coordinate", "weighted-divergence"],
    saldUse := "Chewi SDE/ANALYSIS root: isolate the product-rule component of a coordinate calculation `∂ᵢ (rho * g) = rho * ∂ᵢ g + (∂ᵢ rho) * g` before finite Euclidean weighted-divergence algebra",
    note := "Real-valued product-rule specialization only. It does not by itself discharge the current Langevin `hdiv` hypothesis; `g = ∂ᵢ f`, the diagonal Hessian/iterated derivative, divergence-sum identity, IBP, stationarity, reversibility, invariant Gibbs law, and KL/FI remain red."
  },
  {
    key := "analysis.calculus.line-derivative-rho-product-rule-lineDeriv",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.lineDeriv_rho_mul_eq_of_hasLineDerivAt",
    upstreamDecl := "HasLineDerivAt.lineDeriv applied to hasLineDerivAt_rho_mul",
    upstreamFile := "Mathlib.Analysis.Calculus.LineDeriv.Basic; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "line-derivative", "product-rule", "rho", "coordinate", "weighted-divergence", "lineDeriv"],
    saldUse := "Chewi SDE/ANALYSIS root: convert supplied coordinate derivative facts for `rho` and `g` into the exact `lineDeriv` equality shape used by weighted-divergence displays",
    note := "Equality-form product-rule leaf only. It does not identify `g` with a coordinate derivative of a test function, prove Hessian/iterated-derivative identities, define divergence, prove divergence-sum identities, IBP, boundary decay, stationarity, reversibility, invariant Gibbs law, or KL/FI dissipation."
  },
  {
    key := "analysis.calculus.line-derivative-exp-neg-potential-product-coordinate",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.lineDeriv_expNegPotential_mul_eq_of_differentiableAt",
    upstreamDecl := "DifferentiableAt.hasGradientAt; hasGradientAt_expNegPotential_of_hasGradientAt; hasGradientAt_coordinateUnit_hasLineDerivAt; lineDeriv_rho_mul_eq_of_hasLineDerivAt",
    upstreamFile := "Mathlib.Analysis.Calculus.LineDeriv.Basic; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Gibbs-weight", "line-derivative", "product-rule", "exp-neg-potential", "EuclideanSpace", "coordinate"],
    saldUse := "Chewi SDE/DENS root: compute the coordinate-unit line derivative of `fun y => exp (-V y) * g y` from `DifferentiableAt ℝ V x` and a supplied coordinate derivative of `g`, narrowing the remaining Langevin `hdiv` branch to `g = ∂ᵢ f` and Hessian-coordinate wiring",
    note := "Pointwise coordinate product-rule leaf for the explicit Gibbs weight. It does not identify `g` with a coordinate derivative of a test function, prove the diagonal Hessian/iterated derivative, define divergence, prove divergence-sum identities, IBP, no-boundary terms, stationarity, reversibility, invariant Gibbs law, or KL/FI dissipation."
  },
  {
    key := "analysis.calculus.line-derivative-fderiv-apply-const-from-hasFDerivAt",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.hasLineDerivAt_fderiv_apply_const_of_hasFDerivAt_fderiv",
    upstreamDecl := "HasFDerivAt.clm_apply plus HasFDerivAt.hasLineDerivAt",
    upstreamFile := "Mathlib.Analysis.Calculus.FDeriv.CompCLM; Mathlib.Analysis.Calculus.LineDeriv.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "line-derivative", "fderiv", "second-derivative", "Hessian", "coordinate"],
    saldUse := "Chewi SDE/ANALYSIS root: a supplied derivative of `fun y => fderiv ℝ f y` gives the line derivative of the first-derivative slice `fun y => fderiv ℝ f y v`",
    note := "Second-derivative wiring leaf only. It does not identify the supplied derivative with `iteratedFDeriv`, replace `fderiv ℝ f y eᵢ` by `(gradient f y) i`, define divergence, prove IBP, stationarity, reversibility, invariant Gibbs law, or KL/FI dissipation."
  },
  {
    key := "analysis.calculus.line-derivative-fderiv-apply-const-lineDeriv",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.lineDeriv_fderiv_apply_const_eq_of_hasFDerivAt_fderiv",
    upstreamDecl := "HasLineDerivAt.lineDeriv applied to hasLineDerivAt_fderiv_apply_const_of_hasFDerivAt_fderiv",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "lineDeriv", "fderiv", "second-derivative", "Hessian", "coordinate"],
    saldUse := "Chewi SDE/ANALYSIS root: equality-form line derivative of a first-derivative slice, usable as an input to later supplied weighted-product/divergence displays",
    note := "Equality-form second-derivative wiring leaf only. It does not identify `fderiv ℝ f` with a gradient coordinate, prove divergence, prove IBP, stationarity, reversibility, invariant Gibbs law, or KL/FI dissipation."
  },
  {
    key := "analysis.calculus.line-derivative-fderiv-apply-iteratedFDeriv-two",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.lineDeriv_fderiv_apply_const_eq_iteratedFDeriv_two",
    upstreamDecl := "iteratedFDeriv_two_apply plus lineDeriv_fderiv_apply_const_eq_of_hasFDerivAt_fderiv",
    upstreamFile := "Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "lineDeriv", "fderiv", "iteratedFDeriv", "Hessian", "second-derivative"],
    saldUse := "Chewi SDE/ANALYSIS root: rewrite the line derivative of `fun y => fderiv ℝ f y v` as Mathlib's `iteratedFDeriv ℝ 2 f x ![w, v]` under differentiability of the total `fderiv` map at `x`",
    note := "Pointwise total-fderiv/lineDeriv leaf only. It rewrites a fixed `fderiv ℝ f · v` slice into Mathlib's total `iteratedFDeriv ℝ 2` representative at `x`. It does not assert `f` is globally C², does not prove classical Hessian symmetry, does not replace `fderiv ℝ f x eᵢ` by `(gradient f x) i`, does not define or sum divergence, and does not prove IBP, boundary decay, invariant Gibbs law, reversibility, or KL/FI dissipation."
  },
  {
    key := "analysis.calculus.line-derivative-fderiv-coordinate-iteratedFDeriv-two",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.lineDeriv_fderiv_apply_coordinate_eq_iteratedFDeriv_two",
    upstreamDecl := "lineDeriv_fderiv_apply_const_eq_iteratedFDeriv_two specialized to EuclideanSpace coordinate units",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "EuclideanSpace", "coordinate", "lineDeriv", "fderiv", "iteratedFDeriv", "Hessian"],
    saldUse := "Chewi Ch.1 Langevin root: coordinate-unit line derivative of a first-derivative slice equals the diagonal `iteratedFDeriv` term used by finite-coordinate generator displays",
    note := "Pointwise total-fderiv/lineDeriv coordinate leaf only. It rewrites the coordinate-unit line derivative of the fixed `fderiv ℝ f · eᵢ` slice into Mathlib's `iteratedFDeriv ℝ 2` representative at `x`. It does not assert global C² regularity, classical Hessian symmetry, gradient-coordinate replacement, divergence, IBP, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.line-derivative-exp-neg-potential-fderiv-coordinate",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.lineDeriv_expNegPotential_mul_fderiv_coordinate_eq",
    upstreamDecl := "lineDeriv_expNegPotential_mul_eq_of_differentiableAt plus lineDeriv_fderiv_apply_coordinate_eq_iteratedFDeriv_two",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Gibbs-weight", "lineDeriv", "fderiv", "iteratedFDeriv", "Hessian", "EuclideanSpace", "coordinate"],
    saldUse := "Chewi Ch.1 Langevin root: compute the coordinate line derivative of `exp(-V) * fderiv f eᵢ`, producing the Gibbs-weighted diagonal iterated-derivative term and the potential-gradient product term",
    note := "Pointwise Gibbs-weight product-rule leaf for `lineDeriv_i (exp(-V) * fderiv f eᵢ)`. It discharges only the local `exp(-V)` derivative and the local total-fderiv slice derivative under the stated pointwise assumptions. Gradient-coordinate replacement, divergence operator/sum, Laplacian identification, IBP/no-boundary terms, stationarity, reversibility, invariant Gibbs law, and KL/FI remain separate obligations."
  },
  {
    key := "analysis.calculus.laplacian-std-orthonormal-basis",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian.laplacian_eq_sum_stdOrthonormalBasis",
    upstreamDecl := "InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis",
    upstreamFile := "Mathlib.Analysis.InnerProductSpace.Laplacian",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Laplacian", "finite-dimensional", "stdOrthonormalBasis", "iteratedFDeriv"],
    saldUse := "Chewi SDE/ANALYSIS root: identify Mathlib's finite-dimensional Laplacian with the standard orthonormal-basis second-derivative sum used in Langevin generator displays",
    note := "Coordinate Laplacian bridge only. It does not prove gradients, divergence, integration by parts, boundary decay, stationarity, reversibility, or invariant Gibbs law."
  },
  {
    key := "analysis.calculus.laplacian-functional-std-orthonormal-basis",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian.laplacianFunctional_eq_of_stdOrthonormalBasis_sum",
    upstreamDecl := "laplacian_eq_sum_stdOrthonormalBasis",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Laplacian", "weak-generator", "finite-dimensional", "test-function"],
    saldUse := "Chewi SDE/ANALYSIS root: rewrite weak-generator or source-defined test-function Laplacian actions from coordinate second-derivative sums to Mathlib `Laplacian.laplacian`",
    note := "Functional handoff for definitions only; analytic weak-FP, IBP, and invariant-law statements remain separate red branches."
  },
  {
    key := "analysis.calculus.continuous-laplacian-of-contDiff-two",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian.continuous_laplacian_of_contDiff_two",
    upstreamDecl := "ContDiff.iteratedFDeriv_right plus Mathlib finite-dimensional Laplacian basis display",
    upstreamFile := "Mathlib.Analysis.InnerProductSpace.Laplacian; Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Laplacian", "ContDiff", "Continuous", "iteratedFDeriv", "regularity"],
    saldUse := "Chewi Ch.1 Langevin root: derive continuity of Mathlib's finite-dimensional total Laplacian from global `C²` regularity before closed-box integrability handoffs",
    note := "Global `C²` to continuous Mathlib Laplacian only. It does not assert a closed-box `ContDiffOn` variant, field differentiability, weighted IBP, boundary cancellation, domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "analysis.calculus.laplacian-finrank-iterated-fderiv-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian.norm_laplacian_le_finrank_mul_norm_iteratedFDeriv_two",
    upstreamDecl := "InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis; ContinuousMultilinearMap.le_opNorm",
    upstreamFile := "Mathlib.Analysis.InnerProductSpace.Laplacian",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Laplacian", "iteratedFDeriv", "finrank", "trace-bound"],
    saldUse := "log-concave sampling Ch.1: turn an operator-norm second-derivative estimate into a Laplacian estimate with an explicit dimension factor",
    note := "Pointwise finite-dimensional trace bound only; no compact support, integrability, cutoff, or invariant-law statement is included."
  },
  {
    key := "analysis.calculus.radial-smooth-cutoff-laplacian-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian.radialSmoothCutoff_laplacian_bound",
    upstreamDecl := "radialSmoothCutoff_iteratedFDeriv_two_bound; norm_laplacian_le_finrank_mul_norm_iteratedFDeriv_two",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.{Cutoff,Laplacian}",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "cutoff", "radial", "Laplacian", "O-R-inverse-squared", "finrank"],
    saldUse := "log-concave sampling Ch.1 second-order exhaustion branch: radial cutoff Laplacian bound with explicit finrank times C/R^2",
    note := "Compiled second-order cutoff estimate. It remains independent of the external-blocked concrete Langevin semigroup/domain construction."
  }
]

def measureMemory : List LemmaMemoryEntry := [
  {
    key := "bochner-mean-quadratic-transport",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.BochnerMeanTransport.bounded_bochner_mean_quadratic_transport_control",
    upstreamDecl := "Original generic feature differentiation and transport supplement",
    upstreamFile := "Pinned Mathlib Gaussian moments, Frechet calculus, Hilbert adjoints, Bochner integrals and transport infimum",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["feature", "derivative", "transport", "shared"],
    saldUse := "Generic probability, optimisation and sphere-feature prerequisites; concrete model compositions remain separate",
    note := "Actual Bochner means controlled by the infimum of quadratic costs over all actual couplings, with cost finiteness and residual norm bounds produced. No optimizer or cost-minimizer assumption. Fresh independent source and compiler audits accepted. Final root/reader gates and postmerge purification remain separate; no whole-model or convergence credit."
  },
  {
    key := "pbps.actualCorrectorPerturbation"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualCorrectorPerturbation.actual_corrector_perturbation"
    upstreamDecl := "Actual PBPS corrector perturbation on the same centered space"
    upstreamFile := "arXiv2609.06905v1 B20 and LemmaB4 Ex28-Ex34"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "conditional-expectation", "corrector", "perturbation"]
    saldUse := "PBPS consumer; no SALD admission"
    note := "Independent exactSCI72 18183c58eee62145b6059ded11c7be05a4cb82de. Same original six analytic callers/twelve witnesses/all actual71 clauses; same C/A0/Gamma0/Inv exact perturbation for every u,v,r in HP0. Generic conditions produced internally. Arbitrary r is not actual r_rho; actual H/K/B27/B28, dynamics/nonexplosion/main/errors/cost/composition remain open. Rank0/alphaeta1 legal; no extra caller, onto or higher regularity. Full Exposition/PURIFIED/main/live/Goal not inferred."
  },
  {
    key := "pbps.actualCorrectorChange"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualCorrectorChange.actual_corrector_change"
    upstreamDecl := "Actual PBPS exact corrector change under the reflected step"
    upstreamFile := "arXiv2609.06905v1 Appendix B3 B20/B21; Lemma B4/B28 ingredient"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "reflection", "conditional-expectation", "corrector"]
    saldUse := "PBPS consumer; no SALD admission"
    note := "Independently exact-science71 VERIFIED 4e7ce5d2996ffe1d1b0ab570778e02425c6be34e. Same six original analytic callers and twelve witnesses, actual g=U(P-Pperp)f and its actual conditional gP/polar gV satisfy C(gP,gV)-C(fP,fV)=-norm(fP)^2+norm(fV)^2, with C(u,v)=(norm(u)^2-norm(v)^2)/2-inner(A0(Inv u),v). Exact half, signs and same inverse. No extra smoothness/domain, onto-V0, mean premise or sharp-energy68 dependency. Rank0/alphaeta1 remain legal. B4/B27/B28 actual-update perturbation, H1/B2 dynamics, invariance/nonexplosion, mains, implementation errors/caps, actual-input expected-query costs/composition and full Exposition/PURIFIED remain open."
  },
  {
    key := "pbps.actualProjectedRotation"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualProjectedRotation.actual_projected_rotation"
    upstreamDecl := "Actual PBPS projected reflection rotation and two-component energy"
    upstreamFile := "arXiv2609.06905v1 Appendix B1/B3; B21 and Lemma B4 prerequisites"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "reflection", "conditional-expectation", "projected-rotation"]
    saldUse := "PBPS consumer; no SALD admission"
    note := "Independently exact-science70 VERIFIED c46af8a55e89419109f654c4553cf527993cbeed. Same six original analytic callers and twelve witnesses yield actual g=U(P-Pperp)f, internally mean(g)=0, actual conditional gP and polar gV, gP=A0fP-Gamma0fV and gV=Gamma0fP+A0fV, with exact pair-energy conservation. No right-hand-side-defined output, mean premise, extra regularity or sharp-energy68 premise. Rank0/alphaeta1 remain legal. B21 corrector change, B4 dynamics, main, invariance/nonexplosion, errors/caps, expected-query costs/composition and full Exposition/PURIFIED remain open."
  },
  {
    key := "pbps.actualReflectionIntertwining"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ReflectionIntertwining.actual_reflection_intertwining"
    upstreamDecl := "Actual PBPS reflection intertwining on the full conditional complement"
    upstreamFile := "arXiv2609.06905v1 Appendix B1/B3; B21 and Lemma B4 prerequisites"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "reflection", "adjoint", "full-micro-space"]
    saldUse := "PBPS consumer; no SALD admission"
    note := "Independently exact-science69 VERIFIED 2d286c283a6fb5dfc13180204bb0da54531a5c67. Same six original analytic callers and twelve witnesses yield D=R U inclusion and V0*D=-A0V0* on ALL ker P. Equality of actual AE reflection actions identifies U2=U before consuming the ambient block theorem. No range-V0 restriction, surjectivity, extra regularity or sharp-energy68 premise. Rank0/alphaeta1 remain legal. Next actual projected rotation/mean preservation, B21 corrector change and B4 dynamics remain unproved; main, invariance/nonexplosion, errors/caps, expected-query costs/composition and full Exposition/PURIFIED remain open."
  },
  {
    key := "pbps.actualSharpCorrectorEnergy"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.SharpCorrectorEnergy.actual_sharp_corrector_bound"
    upstreamDecl := "Sharp corrector bound and actual PBPS modified energy equivalence"
    upstreamFile := "arXiv2609.06905v1 Appendix B3 Lemma B3; B19,B20,B22,B23,B24"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "same-root", "corrector", "modified-energy"]
    saldUse := "PBPS actual consumer; no SALD admission"
    note := "Independently exact-science68 VERIFIED 3ad3b127b5a645be9cf71b3d14520b2d8fea3122. Original finite real Hilbert/Borel/C2/two Hessians/positive capped eta inputs produce SAME actual root/inverse/coefficient and sharp |C(u,v)|<=(norm(u)\u00b2+norm(v)\u00b2)/(2gamma). Every globally centered actual f has SAME fP/fV, the actual norm budget and |C(fP,fV)|<=norm(f)\u00b2/(2gamma). Genuine original-input Test proves half/three-halves energy equivalence and exact perturbation bounds for every0<omegaWeight<=gamma. Rank0/alphaeta1 retained; literal private Props are representations only. B21 reflection rotation, H1/B2, B4 dynamics, main results, implementation errors, costs and composition remain open."
  },
  {
    key := "measure.realL2PositiveSquareCommutation"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.L2RealSquareCommute.positive_square_commutation"
    upstreamDecl := "Positive-square commutation on arbitrary real L2"
    upstreamFile := "arXiv2609.06905v1 Appendix D1 functional calculus background; ASTIS auxiliary corollary for B23/B21"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["L2", "positive", "CFC", "square-root", "commutation"]
    saldUse := "PBPS real consumer; no SALD admission"
    note := "Independently exact-science67 VERIFIED 3da29415011a971a65f749502a625e416213f487. Arbitrary measure and positive real-L2 G,D: Commute(G\u00b2,D) implies Commute(G,D), by canonical complexification and the same unique positive square root. No probability/finiteness/nontriviality/caller CFC premise. Actual PBPS internally produces positive D=I+T. Not a separately printed paper theorem."
  },
  {
    key := "pbps.actualRootInverseCommutation"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ActualRootCommutation.actual_same_root_inverse_commutation"
    upstreamDecl := "Same actual PBPS root and centered inverse commutation"
    upstreamFile := "arXiv2609.06905v1 Appendix B3 proof ingredient for Lemma B3/B23 and B21"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "same-root", "centered-inverse", "corrector"]
    saldUse := "PBPS real consumer; no SALD admission"
    note := "Independently exact-science67 VERIFIED 3da29415011a971a65f749502a625e416213f487. Original C2/two Hessians/positive capped eta produce SAME actual Gamma/T/A/HP0/Inv. Internal I+T positivity yields root commutation; exact A0 restriction is selfadjoint, commutes with Gamma0 and Inv, and A0\u00b2+Gamma0\u00b2=I. The SAME inverse is selfadjoint. Genuine original-input Test gives K=A0Inv selfadjoint and I+K\u00b2=Inv\u00b2. Rank0/alphaeta1 retained. Two literal private Props are representations, never providers/premises. Sharp B23/LemmaB3 energy/B21/H1/dynamics/main/errors/cost/composition remain open."
  },
  {
    key := "pbps.ambientAdjointCorrector"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.AmbientAdjointCorrector.actual_ambient_adjoint_centered_decomposition"
    upstreamDecl := "Actual PBPS ambient adjoint and globally centered corrector"
    upstreamFile := "arXiv2609.06905v1 Appendix B3 first corrector after B16 before B20"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "ambient-adjoint", "centering", "corrector"]
    saldUse := "Actual PBPS consumer; no SALD admission"
    note := "Independently exact-science66 VERIFIED a115115d42b3fa2b67885d87fe4d5300af36fcd1. SAME original C2/two Hessians/positive capped eta inputs produce canonical R:L2(J)->kerP with inclusion Rg=g-Pg and all-vector Bambient*=i0 B0*R. Every actual globally centered input produces fP in HP0,fperp=Rf,fV=V0*fperp with B* fperp=i0 Gamma0 fV=GammaP i0 fV, exact Pythagoras and contraction. Genuine Test gives original-input reconstruction,B*f=GammaP i0 fV and squared norm budget. Rank0/alphaeta1 retained. Two literal private Prop expansions are representations only; no extra premise/provider. Full B20/B21,halfturn/H1/dynamics/main/errors/expected costs/composition remain open."
  },
  {
    key := "pbps.actualPolarIsometry"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.PolarIsometry.actual_centered_polar_isometry"
    upstreamDecl := "Actual PBPS typed polar isometry and adjoint corrector"
    upstreamFile := "arXiv2609.06905v1 B16 and B3 first corrector"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "polar", "isometry", "adjoint-corrector"]
    saldUse := "Actual PBPS consumer; no SALD admission"
    note := "Independently exact-science65 VERIFIED ecd9d1f10ad0241312492cefacd5fe48c317e9c2. Original C2/two Hessians/positive capped eta produce SAME actual laws/e/U/T/root/centered inverse. Exact B0:HP0->kerP and V0=B0 Inv satisfy B0=V0 Gamma0,V0.adjoint composed V0=I and all-vector norm equality. Genuine original-input Test gives B0.adjoint=Gamma0 V0.adjoint, adjoint contraction and residual annihilation. No onto/reverse product. Rank0/alphaeta1 retained. B21 root commutation/rotation, B17/H1/operator inequalities,dynamics/main/errors/expected costs/composition remain open."
  },
  {
    key := "measure.realL2PositiveSquareOrder"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.L2RealSquareOrder.positive_square_order"
    upstreamDecl := "Arbitrary real L2 positive square order"
    upstreamFile := "PBPS arXiv2609.06905v1 D1 order/root background; ASTIS internal complexification completion"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["L2", "positive", "operator", "square-order"]
    saldUse := "Actual PBPS consumer; no SALD admission"
    note := "Independently exact-science64 verified 59fff63d320aa5e3dc4b45e81e40e2029ce42734. Arbitrary measure and real scalar-valued L2: positive A,B with positive B squared-A squared internally yield positive B-A, using three canonical complex lifts and internal complex CFC. No caller CFC/nontriviality/finite-L2 premise; zero/trivial case included. Genuine consumer is actual PBPS B15 centered root order."
  },
  {
    key := "pbps.actualCenteredRootOrderInverse"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.CenteredRootOrderInverse.actual_centered_root_order_inverse"
    upstreamDecl := "Actual PBPS centered root order and bounded inverse"
    upstreamFile := "arXiv2609.06905v1 B15; C2/C3/C4; B16 actual input"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "centered", "root-order", "bounded-inverse"]
    saldUse := "Actual PBPS consumer; no SALD admission"
    note := "Independently exact-science64 verified 59fff63d320aa5e3dc4b45e81e40e2029ce42734. Original C2/two global Hessian bounds,0<alpha<=beta,eta>0,betaeta<=1. SAME actual laws/e/U/T/GammaP, compact-gradient closure and literal conditional action internally give C4 sharp contraction, GammaP0 >= gamma I with gamma=2sqrt(alphaeta)/(1+alphaeta), exact HP0=ker inner(e1,.), unit, both inverse cancellations and norm inverse<=1/gamma. Rank0/alphaeta=1 retained. Full-space inverse excluded; completed polar, H1/B13/B14, dynamics/main/error/cost/composition remain open."
  },
  {
    key := "pbps.actualUniquePositiveMacroscopicDefectRoot"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.MacroscopicDefectRoot.actual_unique_positive_macroscopic_defect_root"
    upstreamDecl := "PBPS actual unique positive macroscopic defect root and typed Gram"
    upstreamFile := "arXiv2609.06905v1 AppendixB1-B5 first Gram/B10-B11; attributed ASTIS completion"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "macroscopic", "square-root", "Gram", "uniqueness"]
    saldUse := "Actual PBPS consumer; no SALD admission"
    note := "Independently exact-science63 verified 4d02622332d02d0bd6c977d3cee48fd535ebf203. Original C2/two global Hessian bounds,0<alpha<=beta,eta>0,betaeta<=1 internally produce SAME stationary reflected S/T/U, canonical M onto complete HP=lpMeas=ran(P), and scalar positive root transported by SAME e. A=e T e^-1, typed leakage B:HP-to-joint, GammaP^2=I_HP-A^2=B.adjoint composed B, norm/energy for EVERY macro class and ALL positive same-square macro uniqueness without alternative energy. Rank0 and alphaeta=1 retained. Fulljoint Gram remains P-AJ^2. Other B5, centered root order/inverse/polar,H1/dynamics/main/error/cost/composition remain open."
  },
  {
    key := "measure.positiveRealL2SquareRootUnique"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.L2RealSquareRootUnique.positive_square_roots_unique"
    upstreamDecl := "ASTIS positive real L2 square-root uniqueness"
    upstreamFile := "arXiv2609.06905v1 AppendixD1/B10-B11; attributed ASTIS background"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["L2", "positive", "square-root", "uniqueness"]
    saldUse := "Actual PBPS consumer; no SALD admission"
    note := "Independently exact-science62 verified 9d7f7b640c7cb18fea133ccbd300de129af40b83. On arbitrary-measure real L2, bounded positive A/B with equal squares are equal. Canonical complex lifts and actual complex positive-root uniqueness descend through the isometric real embedding. No finite/probability measure, finite L2, nontriviality or caller CFC certificate. Printed joint GammaP/ontoM/typed B*B, centered order/inverse/polar,H1/dynamics/main/error/cost/composition remain open."
  },
  {
    key := "pbps.actualUniquePositiveRealDefectRoot"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.RealDefectRootUnique.actual_unique_positive_real_defect_root"
    upstreamDecl := "PBPS actual unique positive real scalar defect root"
    upstreamFile := "arXiv2609.06905v1 AppendixD1/B10-B11; attributed ASTIS background"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "positive", "square-root", "uniqueness"]
    saldUse := "Actual PBPS consumer; no SALD admission"
    note := "Independently exact-science62 verified 9d7f7b640c7cb18fea133ccbd300de129af40b83. Original C2/two global Hessian bounds,0<alpha<=beta,eta>0,betaeta<=1 internally produce SAME stationary reflected S/T/D and positive REAL root Gamma. ALL positive alternative roots with same square equal Gamma; their energy is not a premise. Quantifier lies outside every-u energy. Genuine actual-input Test checks alternative-root contractivity. Rank zero and alphaeta=1 retained. Printed joint GammaP/ontoM/typed B*B, centered order/inverse/polar,H1/dynamics/main/error/cost/composition remain open."
  },
  {
    key := "measure.positiveRealL2SquareRoot"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.L2RealSquareRoot.exists_positive_real_square_root"
    upstreamDecl := "ASTIS genuine positive real L2 square root"
    upstreamFile := "arXiv2609.06905v1 AppendixD1/B10-B11; attributed ASTIS background"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["L2", "positive", "square-root", "real-descent"]
    saldUse := "Actual PBPS consumer; no SALD admission"
    note := "Independently exact-science61 verified bcd245d90b21b899acb9937fc54dffcea20e86ee. Arbitrary measure and bounded positive real D; actual complex CFC root, conjugated-root positivity and uniqueness establish pointwise conjugation preservation; full fixed-space descent produces bounded positive REAL Gamma, Gamma squared equals D and full-domain quadratic energy. No finite/probability measure, finite L2, nontriviality or caller CFC/root certificate. Exported real uniqueness, joint GammaP/ontoM/typed B*B, centered root order/inverse/polar, weakH1/dynamics/main/error/cost/composition remain open."
  },
  {
    key := "pbps.actualPositiveRealDefectRoot"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.RealDefectRoot.actual_positive_real_defect_root"
    upstreamDecl := "PBPS actual positive real scalar defect root"
    upstreamFile := "arXiv2609.06905v1 AppendixD1/B10-B11; attributed ASTIS background"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "positive", "square-root"]
    saldUse := "Actual PBPS consumer; no SALD admission"
    note := "Independently exact-science61 verified bcd245d90b21b899acb9937fc54dffcea20e86ee. Original C2, both global Hessian bounds, 0<alpha<=beta, eta>0, beta*eta<=1; genuine every-state normalized reflected Gaussian S, conditional law, stationary marginals and SAME scalar selfadjoint contraction T internally produced. Positive REAL Gamma squared equals I-T*T and every-u normGamma squared equals normu squared minus normTu squared; actual-input Test checks contractivity. Rank zero and alpha*eta=1 retained. Exported real uniqueness, joint GammaP/ontoM/typed B*B, centered root order/inverse/polar, weakH1/dynamics/main/error/cost/composition remain open."
  },
  {
    key := "measure.positiveRealL2ComplexLift"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.L2RealComplexOperator.exists_positive_complex_lift"
    upstreamDecl := "ASTIS positive real L2 operator complexification"
    upstreamFile := "arXiv2609.06905v1 AppendixD1/B10-B11; attributed ASTIS background"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["L2", "positive", "complexification", "fixed-range"]
    saldUse := "Actual PBPS consumer; no SALD admission"
    note := "Independently exact-science60 verified 0a77416f5ec38702c46ec1358966b9dd4846c8d3. Arbitrary measure real positive bounded D; actual compLpL scalar maps, norm-preserving embedding, full pointwise-conjugation fixed range, actual positive bounded complex lift with literal formula/intertwining/conjugation. Reused by actual PBPS consumer. Complex-root API test only; real-root preservation/descent/Gamma/B11/polar/fullH1/dynamics/main/error/cost/composition remain open."
  },
  {
    key := "pbps.actualPositiveDefectComplexLift"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.DefectComplexLift.actual_positive_defect_complex_lift"
    upstreamDecl := "PBPS actual positive squared-defect complex lift"
    upstreamFile := "arXiv2609.06905v1 AppendixD1/B10-B11; attributed ASTIS background"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "positive", "complexification"]
    saldUse := "Actual PBPS consumer; no SALD admission"
    note := "Independently exact-science60 verified 0a77416f5ec38702c46ec1358966b9dd4846c8d3. Original C2, both global Hessian bounds, 0<alpha<=beta, eta>0, beta*eta<=1; actual normalized Gaussian reflected kernel produces same verified59 T and D=1-T*T internally. No caller D/positivity/embedding/CFC certificate. Rank zero extension and alpha*eta=1 retained. Complex-root API test only; real-root preservation/descent/Gamma/B11/polar/fullH1/dynamics/main/error/cost/composition remain open."
  },
  {
    key := "pbps.actualCenteredSelfadjointDefect"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.CenteredDefectOperator.actual_centered_selfadjoint_defect"
    upstreamDecl := "PBPS actual centered selfadjoint squared defect"
    upstreamFile := "arXiv2609.06905v1 AppendixB B1-B5/D1-D2"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "centered", "selfadjoint", "squared-defect"]
    saldUse := "Actual PBPS consumer; no SALD admission"
    note := "Original C2/global two Hessian/positive capped eta. Actual stationary reflected Gaussian kernel produces real L2 selfadjoint mean operator and canonical AE constant; its whole closed centered restriction and actual full/centered positive squared defects are proved. Real Test consumes exact58 macro contraction for rho/delta and IsUnit of the squared defect only on the centered domain. Rank-zero extension and alpha*eta=1 retained; no finite-dimensional L2 or caller regularity/root certificate. Gamma/root/polar/fullH1/dynamics/main/error/cost/composition remain open. Independently verified 2d6cd0167adc4fae1d8d166068a2eda51cd3c3ad."
  },
  {
    key := "l2Pullback.rangeEqLpMeas"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.L2PullbackRange.l2_pullback_range_eq_lpMeas"
    upstreamDecl := "Canonical arbitrary-measure real L2 pullback range"
    upstreamFile := "arXiv2609.06905v1 AppendixB B1-B5/C3-C4"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["measure-preserving", "L2", "comap", "factorization"]
    saldUse := "Actual PBPS consumer; no SALD admission"
    note := "Arbitrary measurable spaces/measures; genuine measure-preserving map. Exact range equals comap AE strongly measurable L2 submodule, with internally constructed factor and MemLp transfer; no probability/finite-measure/StandardBorel premise. Actual PBPS consumer. Independently verified 8c8847715c1d4c3033224b069d8dd694f2a4bd30."
  },
  {
    key := "pbps.actualMacroscopicCenteredRange"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.MacroscopicRange.actual_macroscopic_centered_range"
    upstreamDecl := "PBPS B1-B5 actual macro range and centered image"
    upstreamFile := "arXiv2609.06905v1 AppendixB B1-B5/C3-C4"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "L2", "conditional-expectation", "centered-range"]
    saldUse := "Actual PBPS consumer; no SALD admission"
    note := "Original C2, both Hessian bounds, positive capped eta. Actual Gibbs/Gaussian J and nu; canonical snd pullback range equals true conditional projection range; mean transport and full centered image. Finite real Hilbert/Borel/rank0 extension explicit. Real Test joins same55 reflection/mean with57 contraction to give allmacro sharp contraction and squared defect gap. No Gamma/root/inverse/weakH1/dynamics/cost/composition closure. Independently verified 8c8847715c1d4c3033224b069d8dd694f2a4bd30."
  },
  {
    key := "gaussianMarginal.centeredDomainPoincare"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianMarginalPoincare.actual_gaussian_marginal_centered_poincare"
    upstreamDecl := "Authored actual Gaussian marginal centered closure Poincare and PBPS C.3 consumer"
    upstreamFile := "arXiv2609.06905v1; arXiv2609.06906v1 smoothing background"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["Gaussian-marginal", "Poincare", "centered-domain", "PBPS"]
    saldUse := "Reusable Gaussian smoothing PI with PBPS consumer; no SALD admission"
    note := "Actual normalized Gibbs/Gaussian marginal nu, one genuine dense closable compact-gradient G and closed closure BEFORE every centered domain z; exact alpha/(1+alpha eta) PI. Original C2/two Hessian/positive capped eta retained; finite Hilbert/Borel/rank0 extension explicit. Internal normalization, posteriorL2 and exact alpha/(1+alphaeta),beta/(1+betaeta) Hessian bounds use six real admitted parents; no caller law/moment/curvature/PI/domain certificate or private provider. SAME56 closed pair and actual55 stationary mean consumer derive centering, C3 and sharp rho=(1-alphaeta)/(1+alphaeta) contraction including endpoint. WeakH1/macro-range/fullB13/Gamma/dynamics/main/errors/cost/composition OPEN. Independently verified e8a9044ba5a945eaa4b4aecd110b63494fe6c68e."
  },
  {
    key := "pbps.actualL2.roughMeanGradient"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.RoughMeanGradient.actual_rough_mean_gradient"
    upstreamDecl := "Authored actual all-L2 PBPS closed mean gradient and C.2 sharp defect"
    upstreamFile := "arXiv2609.06905v1"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "all-L2", "closed-gradient", "sharp-defect"]
    saldUse := "Source-specific PBPS integration; no SALD admission"
    note := "SAME Gibbs/Gaussian mu/J/nu and literal S give one genuine dense closable compact-gradient G and bounded T/K BEFOREallrough L2 inputs. Actual closed graph membership, sharp c=(1-alpha*eta)²/[4(1+alpha*eta)] and source4eta norm defects hold; AE source mean/fiber integrability only. Original C2/two Hessian/capped eta retained; finite Hilbert/Borel/rank0 extension disclosed. Three actual admitted parents, no private/background copy or caller law/domain/convergence premise. Actual SAME-T joint BMu rough-difference and alphaeta1 full K=0 consumers tested. Separate weak-H1/full B13/Gamma/dynamics/main/errors/cost/composition open. Independently verified db2c1237cd56698ae560a8121abfcbf374ff8838."
  },
  {
    key := "pbps.actualL2.macroscopicMean"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.L2MacroscopicMean.actual_macroscopic_l2_mean"
    upstreamDecl := "Authored actual all-L2 PBPS conditional mean and B.9 variance defect"
    upstreamFile := "arXiv2609.06905v1"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "all-L2", "conditional-mean", "variance-defect"]
    saldUse := "Source-specific PBPS integration; no SALD admission"
    note := "Actual SAME Gibbs/Gaussian mu/J/nu and every-y literal reflected source S internally produce canonical snd isometry M and bounded T BEFORE all rough L2 inputs. PM=M, MT=AM, source conditional mean and fiber L1/squareL1 only AE; actual integrated variance is the joint reflection block/norm defect. Original C2/two Hessian/capped eta retained; finite Hilbert/rank0 extension disclosed. Actual rough same-T difference and noncentered rank0 class1 fixed/variance0 tested. Five real existing parents, no new private/background copy/caller law/domain/cost premise. Rough gradient/weakH1/B13/Gamma/dynamics/main/errors/cost/composition remain open. Independently verified 4f71a36d56500fda7f86e8080f695a514913950a."
  },
  {
    key := "pbps.literalSourceMean.closedGradientDomain"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.SourceMeanGradientDomain.literal_source_mean_in_closed_gradient"
    upstreamDecl := "Authored actual PBPS compact source mean canonical gradient-domain integration for C.1/C.2"
    upstreamFile := "arXiv2609.06905v1"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "actual-source-mean", "L2", "closed-gradient"]
    saldUse := "Source-specific PBPS integration; no SALD admission"
    note := "Actual literal source S on SAME mu/J/nu produces compact-observer whole mean Tf and gradientTf L2 witnesses; true whole C1 and the uniform genuine compact-gradient core yield canonical closed-gradient graph membership via the existing C1 cutoff/mollifier adapter. Source C2/two-sided Hessian/capped eta retained; finite Hilbert/rank0 and explicit core closure interpretation disclosed. PUP source mean and noncentered rank0 canonical1/0 tested; no Tf compactness or caller law/analytic certificate. Full rough B13, separately defined weak-H1 equivalence, Gamma/dynamics/main/errors/cost/composition remain separate. Four real existing parents reused; no new generic-background copy. Independently verified 16797326f3e06a853e2d047f67fde924ac8c1647."
  },
  {
    key := "pbps.literalReflectedSource.compactMeanC1"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.LiteralReflectedMean.reflected_gibbs_mean_c1"
    upstreamDecl := "Authored actual PBPS reflected source law and compact mean integration for C.1"
    upstreamFile := "arXiv2609.06905v1"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "reflected-source-law", "conditional-mean", "C1"]
    saldUse := "Source-specific PBPS integration; no SALD admission"
    note := "Actual source mu=volume.tilted(-V), R_y=mu quadratic tilt and S_y=volume.tilted(-V((y+u)/2)-norm(y-u)^2/(8eta)). Genuine Gibbs partition/L1/probability, posterior tilt composition, inverse half and normalized affine map produce EVERY-y S_y=(2x-y)#R_y; the whole compact signed-observer mean is C1. C2 lower curvature/allpositive eta/compact C1/finite Hilbert/rank0 are explicit sufficient-background extensions; original-source Test retains upper Hessian/capped eta/smoothcompact f and SAME mu/J Markov conditional, source-S probability, rank0 mean1/derivative0. Source-specific integration consumes existing reusable parents; no new generic-background copy or caller analytic/law certificates. Actual Tf closed-gradient membership, full rough B13/Gamma/dynamics/main/errors/cost/composition remain separate. Independently verified 9f0305db380966529eb3b0fc17a62aa9bcd47a85."
  },
  {
    key := "gaussian.reflectedPosterior.compactMeanC1"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.GaussianReflectedMean.gaussian_reflected_mean_c1"
    upstreamDecl := "Authored literal Gaussian posterior-reflection compact-observer regularity for PBPS C.1"
    upstreamFile := "arXiv2609.06905v1"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "Gaussian", "conditional-mean", "C1"]
    saldUse := "Shared Gaussian posterior compact-observer regularity; no SALD admission"
    note := "Any probability mu on finite real Hilbert/Borel E, eta>0 and signed compact C1 observer f, including rank0. Literal R_y=mu.tilted(-norm(x-y)^2/(2eta)), S_y=R_y.map(2x-y): its exact every-y mean is C1. Compact observer/fderiv bounds, genuine dominated Frechet numerator derivative and its continuity, likelihood/posterior/map L1, positive C2 normalizer and actual N/Z identity are produced internally. No moment/density/domination/normalizer/derivative-continuity/closed-gradient certificate. Actual PBPS same mu/J conditional-kernel consumer retains source curvature/step assumptions; noncentered rank0 constant has real mean1 and actual derivative0. Authored analytic background generalization for C.1. Source-volume SAME-S identity, Tf closure/full rough B13/Gamma/main/errors/expected costs/composition remain separate. Independently verified a18cd1cf8e8310f228391d4c53a2ac8f1a8900ec."
  },
  {
    key := "entropy.two-point.signed-squared"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.TwoPointEntropy.two_point_squared_entropy_le_half_sq_sub"
    upstreamDecl := "Pinned SLT TwoPoint.rothaus_lemma / BernoulliLSI.bernoulli_logSobolev; ASTIS authored signed extension / RMS induction"
    upstreamFile := "SLT d0f506f0a695018265dccb33bcb05e2f5ca1c876; no Lake upstream dependency"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["entropy", "Bernoulli", "log-Sobolev", "Gaussian-background", "SPHMC"]
    saldUse := "Actual SPHMC omitted Gaussian LSI background; no SALD or second-route admission"
    note := "Arbitrary signed a,b and zero cases; upper bound Ent(a squared,b squared) <= (a-b) squared/2. Minimal 26-helper positive-Rothaus port; upstream opposite-signed two_point_inequality unused. Independently verified 6d5df34cdb124e28022f16f566ff549145eb7e65. No GaussianLSI/T2/FIRST4.6/main/work/composition completion."
  },
  {
    key := "entropy.bernoulli.function-log-sobolev"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.BernoulliLogSobolev.bernoulli_function_logSobolev"
    upstreamDecl := "Pinned SLT TwoPoint.rothaus_lemma / BernoulliLSI.bernoulli_logSobolev; ASTIS authored signed extension / RMS induction"
    upstreamFile := "SLT d0f506f0a695018265dccb33bcb05e2f5ca1c876; no Lake upstream dependency"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["entropy", "Bernoulli", "log-Sobolev", "Gaussian-background", "SPHMC"]
    saldUse := "Actual SPHMC omitted Gaussian LSI background; no SALD or second-route admission"
    note := "Actual normalized count law on Fin n -> Bool, all signed functions and n0; actual full coordinate-flip squared energy, exact half coefficient and internally derived probability/three L1 domains. Authored RMS entropy induction; no Gaussian inequality or limit credit. Independently verified 6d5df34cdb124e28022f16f566ff549145eb7e65. No GaussianLSI/T2/FIRST4.6/main/work/composition completion."
  },
  {
    key := "gaussian.relative.sqrt-density-domains"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.GaussianSqrtDensityDomain.gaussian_sqrt_density_domain"
    upstreamDecl := "ProbabilityTheory.stdGaussian / MeasureTheory.integrable_tilted_iff / ASTIS Gaussian moment and curvature producers"
    upstreamFile := "Mathlib.Probability.Distributions.Gaussian.Multivariate; Mathlib.MeasureTheory.Measure.Tilted"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["Gaussian", "relative-density", "square-root", "entropy", "Fisher", "SPHMC"]
    saldUse := "Actual SPHMC standardized true RGO analytic domains before Gaussian LSI/T2; no SALD or second-consumer admission"
    note := "Actual covarianceI Gaussian, true volume/relative tilt equality, internally positive partition and L2/entropy domains, explicit gradients and exact energy1/4. Includes L0/dimension0; source consumes all eta>0. Independent exactcommit 1de412105042ebcfe147d50a3d1022fb0514faad. No weakSobolev/KL/LSI/T2/W2/main/cost credit."
  },
  {
    key := "measure.wasserstein.chewi-definition-1-3-4",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.wassersteinDistance",
    upstreamDecl := "Chewi Definition 1.3.4",
    upstreamFile := "Log-Concave Sampling, book page 20 / PDF page 32",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Wasserstein", "W2", "quadratic-cost", "coupling", "ENNReal"],
    saldUse := "Chewi Definition 1.3.4 root: define W2 as the positive square root of quadratic Kantorovich cost",
    note := "Exact extended-real value definition; metric properties and finite-second-moment finiteness are separate theorems."
  },
  {
    key := "measure.wasserstein.chewi-display-1-3-5",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.wassersteinDistance_sq",
    upstreamDecl := "Chewi display (1.3.5)",
    upstreamFile := "Log-Concave Sampling, book page 20 / PDF page 32",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Wasserstein", "W2", "quadratic-cost", "source-display"],
    saldUse := "Chewi display (1.3.5): rewrite W2 squared as the quadratic coupling infimum",
    note := "Actual ENNReal rpow calculation; no optimal coupling is assumed."
  },
  {
    key := "measure.wasserstein.chewi-definition-1-3-12",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.IsAbsolutelyContinuousFiniteSecondMoment",
    upstreamDecl := "Chewi Definition 1.3.12",
    upstreamFile := "Log-Concave Sampling, book page 25 / PDF page 37",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Wasserstein", "P2ac", "absolute-continuity", "second-moment", "Lebesgue"],
    saldUse := "Chewi Definition 1.3.12 root: package probability normalization, finite second moment, and Lebesgue absolute continuity",
    note := "Exact measure-class definition; no Wasserstein metric, optimal map, or gradient-flow theorem is asserted."
  },
  {
    key := "measure.wasserstein.lipschitz-observable-second-moment",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinLipschitzMoment.of_wassersteinDistance_sq_le",
    upstreamDecl := "quadratic coupling inequality for Lipschitz observables",
    upstreamFile := "standard Wasserstein consequence; SPHMC arXiv:2609.06906v1 Lemma D.4 prerequisite",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Wasserstein", "Lipschitz", "second-moment", "integrability", "SPHMC"],
    saldUse := "transfer square-integrability and a quadratic observable moment bound along a W2 estimate",
    note := "Uses approximate couplings rather than optimal-plan existence. A twisted paper metric such as SPHMC M_kappa still needs an explicit metric comparison."
  },
  {
    key := "measure.wasserstein.quadratic-optimal-coupling",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation.IsQuadraticOptimalCoupling",
    upstreamDecl := "Chewi Definition 1.3.25 / quadratic Kantorovich attainment",
    upstreamFile := "Log-Concave Sampling, book page 30 / PDF page 42",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Wasserstein", "optimal-coupling", "quadratic-cost", "attainment"],
    saldUse := "state that a prescribed coupling actually attains the quadratic Kantorovich infimum before constructing displacement interpolation",
    note := "This predicate does not prove that an optimizer exists; it records marginal feasibility and exact attainment."
  },
  {
    key := "measure.wasserstein.displacement-interpolation",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation.displacementInterpolation",
    upstreamDecl := "Chewi Definition 1.3.25 / law of (1-t)X0+tX1",
    upstreamFile := "Log-Concave Sampling, book page 30 / PDF page 42",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Wasserstein", "measure-map", "affine-interpolation", "McCann"],
    saldUse := "build the time-t law as the pushforward of a coupling by the affine endpoint map",
    note := "The endpoint identities are proved separately; constant speed and uniqueness remain Theorem 1.3.23."
  },
  {
    key := "measure.wasserstein.chewi-definition-1-3-25",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation.IsWassersteinGeodesic",
    upstreamDecl := "Chewi Definition 1.3.25",
    upstreamFile := "Log-Concave Sampling, book page 30 / PDF page 42",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Wasserstein", "geodesic", "displacement-interpolation", "P2ac"],
    saldUse := "Chewi Definition 1.3.25 root: package P2ac endpoints, an optimal quadratic coupling, and its affine-law curve on [0,1]",
    note := "Source-faithful definition only; optimal-plan existence, metric constant speed, and uniqueness are not inferred."
  },
  {
    key := "measure.transport.chewi-definition-1-3-1",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Transport.transportCost",
    upstreamDecl := "Chewi Definition 1.3.1",
    upstreamFile := "Log-Concave Sampling, book page 20 / PDF page 32",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Kantorovich", "optimal-transport", "coupling", "sInf", "ENNReal"],
    saldUse := "Chewi Definition 1.3.1 root: define the extended nonnegative Kantorovich transport cost over all couplings",
    note := "Exact optimization value; lower semicontinuity and existence of an optimal plan are separate theorem routes."
  },
  {
    key := "measure.transport.chewi-display-1-3-2",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Transport.transportCost_eq_sInf",
    upstreamDecl := "Chewi display (1.3.2)",
    upstreamFile := "Log-Concave Sampling, book page 20 / PDF page 32",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Kantorovich", "optimal-transport", "coupling", "source-display"],
    saldUse := "Chewi display (1.3.2): expose the exact infimum over coupling costs",
    note := "Definitional source equality; no minimizer is claimed."
  },
  {
    key := "measure.transport.chewi-definition-1-3-6",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.KantorovichDual.dualTransportValue",
    upstreamDecl := "Chewi Definition 1.3.6",
    upstreamFile := "Log-Concave Sampling, book page 21 / PDF page 33",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "optimal-transport", "Kantorovich-dual", "potentials", "definition"],
    saldUse := "Chewi Definition 1.3.6 root: define the supremum of integrable dual-potential objectives under the product-a.e. cost constraint",
    note := "Strong duality, boundedness, and attainment are separate theorems; the source P2 quadratic setting supplies their analytic hypotheses."
  },
  {
    key := "measure.transport.chewi-display-1-3-7",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.KantorovichDual.dualTransportValue_eq_sSup",
    upstreamDecl := "Chewi display (1.3.7)",
    upstreamFile := "Log-Concave Sampling, book page 21 / PDF page 33",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "optimal-transport", "Kantorovich-dual", "sSup", "display"],
    saldUse := "Chewi display (1.3.7): expand the dual transport value into its exact feasible-potential supremum",
    note := "Definitional equality; no primal-dual equality or optimizer is asserted."
  },
  {
    key := "measure.transport.coupling-probability",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Transport.isProbabilityMeasure_of_isCoupling_left",
    upstreamDecl := "Measure.fst_univ",
    upstreamFile := "Mathlib.MeasureTheory.Measure.Prod",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "coupling", "probability-measure", "marginal", "optimal-transport"],
    saldUse := "Chewi Definition 1.3.1: recover the joint probability-measure interface from a prescribed probability marginal",
    note := "The first marginal suffices because its mass on the whole space equals the joint measure's total mass; the symmetric result follows from Measure.snd_univ if needed."
  },
  {
    key := "measure.transport.product-coupling",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Transport.isCoupling_prod",
    upstreamDecl := "Measure.fst_prod / Measure.snd_prod",
    upstreamFile := "Mathlib.MeasureTheory.Measure.Prod",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "coupling", "optimal-transport", "product-measure", "Wasserstein"],
    saldUse := "Chewi Definition 1.3.1 and Chapters 1/4: named coupling contract with a canonical independent-product witness",
    note := "Defines the reusable marginal contract and proves nonemptiness for probability marginals; it does not define transport cost, prove existence of an optimal plan, or establish Wasserstein metric properties."
  },
  {
    key := "measure.law-map.integral",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.LawMap.lawMapIntegral",
    upstreamDecl := "Mathlib.MeasureTheory.Measure.map / integral_map",
    upstreamFile := "Mathlib measure/integration APIs",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["measure-map", "law", "integral"],
    saldUse := "rewrite weak-test integrals under endpoint laws and EM interpolation laws",
    note := "Alias to a compiled ASTIS theorem in Probability.lean."
  },
  {
    key := "measure.law-map.dominated-derivative",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.LawMap.lawMapIntegralHasDerivAtOfDominated",
    upstreamDecl := "hasDerivAt_integral_of_dominated_loc_of_deriv_le",
    upstreamFile := "Mathlib.Analysis.Calculus.ParametricIntegral",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["parametric-integral", "dominated-convergence", "weak-test"],
    saldUse := "transport dominated sample-space derivatives to law-level weak-test derivatives",
    note := "Key backend for EM weak Fokker--Planck and KL-derivative handoffs."
  },
  {
    key := "measure.conditional-distribution.integral",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.ConditionalKernel.condDistribIntegralNamedLawIntegral",
    upstreamDecl := "condDistrib / condExpKernel map orientation",
    upstreamFile := "Mathlib.Probability.Kernel.CondDistrib and Condexp",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["conditional-law", "kernel", "Bochner-integral"],
    saldUse := "conditional frozen drift and named-law conditional integral interface",
    note := "Compiled ASTIS conditional-law bridge from Probability.lean."
  },
  {
    key := "probability.kernel.invariant-powers",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.KernelInvariance.invariant_pow",
    upstreamDecl := "Kernel.Invariant.comp / Measure.id_comp / pow_succ",
    upstreamFile := "Mathlib.Probability.Kernel.Invariance; Mathlib.Probability.Kernel.Composition.Comp",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["KERN", "MCMC", "DiscreteSampling", "invariance", "iteration"],
    saldUse := "Shared MCMC/discrete kernel floor: propagate one-step target invariance through any finite number of transitions",
    note := "Source-neutral reusable edge; no ergodicity, mixing or complexity conclusion. bind_pow_eq is its notation adapter and is not counted separately."
  },
  {
    key := "conditional-kernel.named-field-integral",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.ConditionalKernel.condDistribIntegralNamedFieldIntegral",
    upstreamDecl := "condDistrib / compProd_map_condDistrib / integral_congr_ae",
    upstreamFile := "Mathlib.Probability.Kernel.CondDistrib and MeasureTheory integral APIs",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["conditional-law", "kernel", "Bochner-integral", "versioning"],
    saldUse := "turn a named conditional drift component version into the joint-law weak-generator integral",
    note := "Pro-assimilated leaf: conditional-pairing/versioning backend separated from weak FP and Ito."
  },
  {
    key := "measure.with-density.probability-normalization",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.isProbabilityMeasure_withDensity_of_lintegral_eq_one",
    upstreamDecl := "withDensity_apply / IsProbabilityMeasure",
    upstreamFile := "Mathlib.MeasureTheory.Measure.WithDensity; Mathlib.MeasureTheory.Measure.Typeclasses.Probability",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "withDensity", "normalization", "probability-measure", "density"],
    saldUse := "Chewi DENS/MEAS root: turn normalized Gibbs or tilted density contracts into a probability measure",
    note := "Small measure-normalization wrapper; concrete Gibbs integrability remains a separate analytic leaf."
  },
  {
    key := "measure.with-density.ofReal-exp-probability-normalization",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.isProbabilityMeasure_withDensity_ofReal_exp_of_integral_eq_one",
    upstreamDecl := "ofReal_integral_eq_lintegral_ofReal / isProbabilityMeasure_withDensity_of_lintegral_eq_one",
    upstreamFile := "Mathlib.MeasureTheory.Integral.Bochner.Basic; AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym; SLT/GaussianLSI/DualityEntropy.lean as proof-pattern provenance",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "SLT", "withDensity", "exponential-tilt", "probability-measure", "entropy-duality", "Girsanov"],
    saldUse := "log-concave sampling DENS/MEAS/PATH root: normalize real exponential tilts for entropy duality, Gibbs variational formulas, and finite-dimensional Girsanov/RN routes",
    note := "ASTIS-owned exponential-tilt normalization leaf. It assumes integrability and unit mass of `exp U`; it does not prove DV duality, Girsanov, or any entropy inequality."
  },
  {
    key := "measure.density.normalized-lintegral-one",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.lintegral_inv_lintegral_mul_eq_one",
    upstreamDecl := "lintegral_const_mul' / ENNReal.inv_mul_cancel",
    upstreamFile := "Mathlib.MeasureTheory.Integral.Lebesgue.Add; Mathlib.Data.ENNReal.Inv",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "density", "lintegral", "normalization", "ENNReal"],
    saldUse := "Chewi DENS/MEAS root: normalize any finite nonzero density before specializing to Gibbs densities",
    note := "Generic reciprocal-lintegral normalization; no concrete measurability or integrability proof is hidden."
  },
  {
    key := "measure.with-density.normalized-probability",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.isProbabilityMeasure_withDensity_normalized_lintegral",
    upstreamDecl := "lintegral_inv_lintegral_mul_eq_one / withDensity_apply / IsProbabilityMeasure",
    upstreamFile := "Mathlib.MeasureTheory.Measure.WithDensity; Mathlib.MeasureTheory.Integral.Lebesgue.Add",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "withDensity", "density", "normalization", "probability-measure"],
    saldUse := "Chewi DENS/MEAS root: construct probability targets from finite nonzero unnormalized densities",
    note := "Bridges generic density normalization to the `withDensity` probability-measure contract."
  },
  {
    key := "measure.pi.lintegral-product-factorization",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.lintegral_fintype_prod_eq_prod",
    upstreamDecl := "ENNReal analogue of integral_fintype_prod_eq_prod",
    upstreamFile := "Mathlib.MeasureTheory.Integral.Pi; external AST PiWithDensity.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Measure.pi", "lintegral", "product", "ENNReal", "Fubini"],
    saldUse := "Chewi MEAS/DENS root: factor finite-product ENNReal densities before tensorized withDensity or Gaussian shift leaves",
    note := "Compiled ASTIS-owned finite-product ENNReal Fubini leaf; useful beyond Gaussian tilts."
  },
  {
    key := "measure.pi.with-density-product",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.pi_withDensity_prod",
    upstreamDecl := "lintegral_fintype_prod_eq_prod / withDensity_apply",
    upstreamFile := "Mathlib.MeasureTheory.Measure.WithDensity; external AST PiWithDensity.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Measure.pi", "withDensity", "product-density", "tensorization"],
    saldUse := "Chewi MEAS/DENS root: decompose finite product density tilts into coordinatewise withDensity measures",
    note := "Shared root for product Gaussian Esscher, product Gibbs, and tensorization-style density constructions."
  },
  {
    key := "measure.pi.update-coordinate-map",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Product.map_update_prod_pi",
    upstreamDecl := "SLT.EfronStein.map_update_prod_pi",
    upstreamFile := "SLT/EfronStein.lean; Mathlib.MeasureTheory.Constructions.Pi",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "SLT", "Measure.pi", "product-measure", "Function.update", "finite-coordinate"],
    saldUse := "Chewi MEAS/FI/SDE root: replace one coordinate of a finite product sample by an independent coordinate draw while preserving the product law",
    note := "Finite product probability map leaf only; no conditional expectation theorem, entropy, LSI, kernel selection, weak-FP, or stationarity statement."
  },
  {
    key := "measure.pi.update-coordinate-map-preserving",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Product.measurePreserving_update_prod_pi",
    upstreamDecl := "SLT.EfronStein.map_update_prod_pi",
    upstreamFile := "SLT/EfronStein.lean; Mathlib.MeasureTheory.Constructions.Pi",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "SLT", "Measure.pi", "product-measure", "Function.update", "Fubini", "conditional-slice"],
    saldUse := "Chewi tensorization and coordinate-slice roots: treat coordinate replacement as a `MeasurePreserving` map before moving between product integrals and slices",
    note := "Measure-preserving wrapper for finite product probability laws only; coordinate slice integrability, entropy subadditivity, LSI, and conditional kernels remain separate leaves."
  },
  {
    key := "measure.pi.update-coordinate-integral",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Product.integral_update_prod_pi_eq_integral",
    upstreamDecl := "SLT.EfronStein.integral_update_eq_integral",
    upstreamFile := "SLT/EfronStein.lean; Mathlib.MeasureTheory.Integral.Prod",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "SLT", "Measure.pi", "Bochner-integral", "product-measure", "Function.update", "Fubini"],
    saldUse := "Chewi MEAS/FI/SDE root: rewrite the two-step integral over a fresh coordinate and a product sample back to the original product-law integral",
    note := "Bochner integral transport leaf derived from the coordinate replacement map; does not prove conditional expectation identities, entropy subadditivity, LSI, or sampler weak-FP statements."
  },
  {
    key := "measure.pi.update-coordinate-slice-integrable-ae",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Product.integrable_update_slice_ae",
    upstreamDecl := "SLT.GaussianLSI.SubAddEnt.Basic.integrable_update_slice / Integrable.prod_left_ae",
    upstreamFile := "SLT/GaussianLSI/SubAddEnt/Basic.lean; Mathlib.MeasureTheory.Integral.Prod",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "SLT", "Measure.pi", "Bochner-integral", "product-measure", "Function.update", "Fubini", "slice-integrability"],
    saldUse := "Chewi FI/MEAS root: expose a.e. integrability of coordinate-replacement slices before tensorization, conditional entropy, and product functional-inequality arguments",
    note := "A.e. slice integrability only; no conditional expectation identity, entropy subadditivity, LSI, Markov kernel construction, or Gibbs invariant-law statement."
  },
  {
    key := "measure.with-density.absolute-continuity",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.withDensity_absolutelyContinuous_base",
    upstreamDecl := "withDensity_absolutelyContinuous",
    upstreamFile := "Mathlib.MeasureTheory.Measure.WithDensity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "withDensity", "absolute-continuity", "RN"],
    saldUse := "Chewi DENS/MEAS root before RN derivative and density-comparison leaves",
    note := "Callable ASTIS wrapper for the base absolute-continuity fact of withDensity."
  },
  {
    key := "measure.with-density.map-measurable-equiv",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.measurableEquiv_map_withDensity",
    upstreamDecl := "Measure.map_apply / withDensity_apply / setLIntegral_map",
    upstreamFile := "Mathlib.MeasureTheory.Integral.Lebesgue.Map; Mathlib.MeasureTheory.Measure.WithDensity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "withDensity", "measure-map", "measurable-equivalence", "density-transport"],
    saldUse := "Chewi MEAS/PATH root: transport explicit densities through coordinate changes such as product coordinates to EuclideanSpace cylinders",
    note := "Generic Mathlib-ready transport leaf; it keeps density pushforward separate from any Gaussian or path-space theorem."
  },
  {
    key := "measure.rn-deriv.reconstruction",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.withDensity_rnDeriv_eq_of_absolutelyContinuous",
    upstreamDecl := "Measure.withDensity_rnDeriv_eq",
    upstreamFile := "Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Radon-Nikodym", "withDensity", "absolute-continuity"],
    saldUse := "Chewi DENS/MEAS root for recognizing supplied density/RN derivative representations",
    note := "Thin Mathlib-facing RN reconstruction leaf."
  },
  {
    key := "measure.gibbs-density.pointwise-positive-finite",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.gibbsDensityENNReal_pos",
    upstreamDecl := "ENNReal.ofReal_pos / ENNReal.ofReal_lt_top / Real.exp_pos",
    upstreamFile := "Mathlib.Data.ENNReal.Real; Mathlib.Analysis.SpecialFunctions.Exp",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "density", "positivity", "ENNReal"],
    saldUse := "Chewi DENS/MEAS root: expose positivity and finiteness of the Gibbs shape before integral contracts",
    note := "Companion finite-value theorem is `gibbsDensityENNReal_lt_top`; integral nonzero remains a measure-level hypothesis."
  },
  {
    key := "measure.gibbs-density.aemeasurable",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.aemeasurable_gibbsDensityENNReal",
    upstreamDecl := "Real.measurable_exp / AEMeasurable.ennreal_ofReal",
    upstreamFile := "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic; Mathlib.MeasureTheory.Constructions.BorelSpace.Real",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "density", "measurability", "ENNReal"],
    saldUse := "Chewi DENS/MEAS root: turn a measurable/a.e.-measurable potential into a measurable Gibbs density",
    note := "Keeps the measurable potential contract separate from convexity and integrability."
  },
  {
    key := "measure.gibbs-density.normalized-toReal-logconcave-convex-potential",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity.logConcaveOn_normalized_gibbsDensityENNReal_toReal_of_convexOn",
    upstreamDecl := "logConcaveOn_const_mul_exp_neg_of_convexOn / ENNReal.toReal_mul / ENNReal.toReal_inv",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity; AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "density", "toReal", "log-concavity", "convex-potential", "normalization"],
    saldUse := "Chewi DENS/CONV root: view an ENNReal normalized Gibbs density with finite nonzero normalizer as a positive real log-concave density shape",
    note := "Requires supplied `Z ≠ 0` and `Z ≠ ∞`; it does not prove the normalizer exists or that a probability law has been constructed."
  },
  {
    key := "measure.gibbs-density.normalized-toReal-logconcave-strong-convex-potential",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity.logConcaveOn_normalized_gibbsDensityENNReal_toReal_of_strongConvexOn",
    upstreamDecl := "convexOn_of_strongConvexOn_nonneg / logConcaveOn_normalized_gibbsDensityENNReal_toReal_of_convexOn",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity; AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "density", "toReal", "log-concavity", "strong-convexity", "normalization"],
    saldUse := "Chewi DENS/CONV root: expose normalized ENNReal Gibbs densities of strongly convex potentials as real log-concave shapes once `Z` is finite and nonzero",
    note := "Geometry/typing bridge only; finite normalizer, minimizer existence, invariant law, and PI/LSI remain separate obligations."
  },
  {
    key := "measure.gibbs-density.lintegral-normalized-toReal-logconcave-convex-potential",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity.logConcaveOn_lintegral_normalized_gibbsDensityENNReal_toReal_of_convexOn",
    upstreamDecl := "logConcaveOn_normalized_gibbsDensityENNReal_toReal_of_convexOn with `Z = ∫⁻ gibbsDensity`",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "density", "toReal", "lintegral", "log-concavity", "convex-potential"],
    saldUse := "Chewi DENS/CONV root: match the source notation `Z^{-1} exp(-V)` when a finite nonzero Gibbs integral has already been proved",
    note := "The finite and nonzero lintegral hypotheses are inputs, not consequences of convexity."
  },
  {
    key := "measure.gibbs-density.lintegral-normalized-toReal-logconcave-strong-convex-potential",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity.logConcaveOn_lintegral_normalized_gibbsDensityENNReal_toReal_of_strongConvexOn",
    upstreamDecl := "logConcaveOn_normalized_gibbsDensityENNReal_toReal_of_strongConvexOn with `Z = ∫⁻ gibbsDensity`",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "density", "toReal", "lintegral", "log-concavity", "strong-convexity"],
    saldUse := "Chewi DENS/CONV root: expose strong-convex Gibbs density shapes under a supplied finite nonzero Gibbs integral",
    note := "Requires `0 ≤ k`; sharp growth, minimizer existence, and finite normalization remain separate leaves."
  },
  {
    key := "measure.gibbs-density.lintegral-normalized-toReal-logconcave-strong-convex-minimizer",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity.logConcaveOn_lintegral_normalized_gibbsDensityENNReal_toReal_of_strongConvexOn_minimizer",
    upstreamDecl := "lintegral_gibbsDensityENNReal_ne_zero / lintegral_gibbsDensityENNReal_ne_top_of_strongConvexOn_minimizer / logConcaveOn_lintegral_normalized_gibbsDensityENNReal_toReal_of_strongConvexOn",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity; AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["log-concave-sampling", "Gibbs", "density", "toReal", "lintegral", "log-concavity", "strong-convexity", "minimizer"],
    saldUse := "log-concave sampling DENS/CONV root: combine the strong-convex minimizer finite-normalizer leaf with the normalized real log-concavity shape",
    note := "Convenience composition for finite-dimensional strongly convex targets with an exposed minimizer. It does not prove minimizer existence, invariant law, PI/LSI, or sampler convergence."
  },
  {
    key := "measure.gibbs-density.explicit-laplace-toReal-logconcave",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity.logConcaveOn_normalized_laplace_gibbsDensityENNReal_toReal",
    upstreamDecl := "convexOn_univ_const_mul_abs_add / logConcaveOn_normalized_gibbsDensityENNReal_toReal_of_convexOn",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity; AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "Laplace-tail", "toReal", "log-concavity", "normalizer", "one-dimensional"],
    saldUse := "Chewi DENS/CONV root: source-facing one-dimensional Laplace `ENNReal` Gibbs density as a real log-concave normalized shape",
    note := "Uses the explicit positive Laplace constant; the exact integral equality is provided by the separate Integrability normalizer leaf."
  },
  {
    key := "measure.gibbs-density.withDensity-integral-rewrite",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsIntegral.integral_withDensity_inv_mul_gibbsDensityENNReal_eq_integral_inv_mul_exp_smul",
    upstreamDecl := "integral_withDensity_eq_integral_toReal_smul₀ / gibbsDensityENNReal_lt_top",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsIntegral; Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "withDensity", "Bochner-integral", "test-function-integral", "toReal"],
    saldUse := "Chewi MEAS/DENS/SDE root: rewrite Bochner integrals under a Gibbs withDensity measure as density-weighted base-measure integrals",
    note := "Algebraic Bochner-integral bridge only; stationarity, reversibility, KL/FI decay, and finite normalizer proofs remain separate leaves."
  },
  {
    key := "measure.gibbs-density.lintegral-withDensity-integral-rewrite",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsIntegral.integral_withDensity_lintegral_inv_mul_gibbsDensityENNReal_eq_integral_lintegral_inv_mul_exp_smul",
    upstreamDecl := "integral_withDensity_inv_mul_gibbsDensityENNReal_eq_integral_inv_mul_exp_smul with `Z = ∫⁻ gibbsDensity`",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsIntegral",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "withDensity", "Bochner-integral", "lintegral", "test-function-integral"],
    saldUse := "Chewi MEAS/DENS/SDE root: match the source density notation `Z^{-1} exp(-V(x)) dx` inside Bochner test-function integrals",
    note := "Consumes the nonzero normalizer hypothesis; if the goal needs a probability law, combine with the finite-normalizer probability leaf."
  },
  {
    key := "measure.gibbs-density.lintegral-withDensity-integral-rewrite-nonzero-base",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsIntegral.integral_withDensity_lintegral_inv_mul_gibbsDensityENNReal_eq_integral_lintegral_inv_mul_exp_smul_of_neZero",
    upstreamDecl := "integral_withDensity_lintegral_inv_mul_gibbsDensityENNReal_eq_integral_lintegral_inv_mul_exp_smul / lintegral_gibbsDensityENNReal_ne_zero",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Measure.GibbsIntegral; AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "withDensity", "Bochner-integral", "lintegral", "nonzero-base-measure"],
    saldUse := "Chewi MEAS/DENS/SDE root: use textbook-shaped `Z = ∫ exp(-V)` Gibbs density inside Bochner test-function integrals without separately passing the nonzero-normalizer proof",
    note := "Derives only nonzero normalizer from `[NeZero μ]` and a.e.-measurability; finite normalizer and probability-measure status remain separate."
  },
  {
    key := "measure.gibbs-density.normalized-probability",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.isProbabilityMeasure_withDensity_normalized_gibbs",
    upstreamDecl := "isProbabilityMeasure_withDensity_normalized_lintegral",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym; Mathlib.MeasureTheory.Measure.WithDensity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "withDensity", "normalization", "probability-measure"],
    saldUse := "Chewi DENS/MEAS root: construct normalized Gibbs target measures once finite nonzero integral contracts are supplied",
    note := "First concrete Gibbs-density bridge; still does not prove coercivity or integrability of a specific potential."
  },
  {
    key := "measure.gibbs-density.integral-nonzero",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.lintegral_gibbsDensityENNReal_ne_zero",
    upstreamDecl := "lintegral_eq_zero_iff' / Filter.eventually_false_iff_eq_bot / Real.exp_pos",
    upstreamFile := "Mathlib.MeasureTheory.Integral.Lebesgue.Basic; Mathlib.Order.Filter.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "density", "lintegral", "nonzero"],
    saldUse := "Chewi DENS/MEAS root: discharge the nonzero half of a Gibbs normalization constant over a nonzero base measure",
    note := "Requires a.e.-measurable potential and nonzero base measure; no growth or finiteness assumption is used."
  },
  {
    key := "measure.gibbs-density.integral-finite-envelope",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.lintegral_gibbsDensityENNReal_ne_top_of_ae_le",
    upstreamDecl := "lintegral_mono_ae / ne_top_of_le_ne_top",
    upstreamFile := "Mathlib.MeasureTheory.Integral.Lebesgue.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "density", "lintegral", "envelope", "finite"],
    saldUse := "Chewi DENS/MEAS root: reduce Gibbs integrability to a finite ENNReal envelope",
    note := "This is the target interface for later coercivity, tail, or quadratic-growth leaves."
  },
  {
    key := "measure.gibbs-density.potential-envelope-pointwise",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.gibbsDensityENNReal_le_of_potential_ge",
    upstreamDecl := "Real.exp_le_exp / ENNReal.ofReal_le_ofReal",
    upstreamFile := "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic; Mathlib.Data.ENNReal.Real",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "density", "potential", "envelope", "monotonicity"],
    saldUse := "Chewi DENS/CONV root: turn a potential lower bound into a pointwise density envelope",
    note := "If `W ≤ V`, then `exp (-V) ≤ exp (-W)` as an ENNReal density."
  },
  {
    key := "measure.gibbs-density.integral-finite-potential-envelope",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.lintegral_gibbsDensityENNReal_ne_top_of_ae_potential_ge",
    upstreamDecl := "gibbsDensityENNReal_ae_le_of_ae_potential_ge / lintegral_gibbsDensityENNReal_ne_top_of_ae_le",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs; Mathlib.MeasureTheory.Integral.Lebesgue.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "density", "potential", "lintegral", "envelope", "finite"],
    saldUse := "Chewi DENS/CONV root: prove finite normalization for `V` from an integrable lower-potential envelope `W`",
    note := "This is the compiled intermediate target before concrete coercivity or tail-growth leaves."
  },
  {
    key := "measure.gibbs-density.finite-measure-lower-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.isProbabilityMeasure_withDensity_normalized_gibbs_of_ae_ge_const",
    upstreamDecl := "lintegral_const_lt_top / lintegral_gibbsDensityENNReal_ne_top_of_ae_le / lintegral_gibbsDensityENNReal_ne_zero",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs; Mathlib.MeasureTheory.Integral.Lebesgue.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "density", "finite-measure", "lower-bound", "normalization", "compact-domain"],
    saldUse := "Chewi DENS/CONV root: normalize Gibbs laws on finite or truncated base measures from a measurable potential with an a.e. constant lower bound",
    note := "Concrete compact-domain envelope leaf; full Lebesgue coercivity/growth tail integrability remains a separate analytic theorem."
  },
  {
    key := "measure.gibbs-density.normalized-probability-envelope",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.isProbabilityMeasure_withDensity_normalized_gibbs_of_ae_le",
    upstreamDecl := "lintegral_gibbsDensityENNReal_ne_zero / lintegral_gibbsDensityENNReal_ne_top_of_ae_le",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs; Mathlib.MeasureTheory.Measure.WithDensity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "withDensity", "normalization", "envelope", "probability-measure"],
    saldUse := "Chewi DENS/MEAS root: turn a measurable potential plus finite envelope into a normalized Gibbs probability measure",
    note := "Keeps coercivity/growth work isolated as the proof of the envelope bound."
  },
  {
    key := "measure.gibbs-density.normalized-probability-potential-envelope",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.isProbabilityMeasure_withDensity_normalized_gibbs_of_ae_potential_ge",
    upstreamDecl := "lintegral_gibbsDensityENNReal_ne_zero / lintegral_gibbsDensityENNReal_ne_top_of_ae_potential_ge",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs; Mathlib.MeasureTheory.Measure.WithDensity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "withDensity", "potential", "normalization", "envelope", "probability-measure"],
    saldUse := "Chewi DENS/CONV root: construct the normalized target law once a lower-potential integrable envelope is supplied",
    note := "Useful for Chewi coercivity/growth proofs because the analytic tail estimate only has to provide `W ≤ V` and finite integral for `exp(-W)`."
  },
  {
    key := "samplewiki.sphmc.initial-gibbs-q2",
    localDecl := "AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.InitialGibbsPhaseTransport.initial_gibbs_phase_transport_q2",
    upstreamDecl := "GibbsPositionMoment.gibbs_position_moment / PhaseReferenceMoment.gaussianSmoothing_position_second_moment / InitialPhaseTransport.phaseWassersteinSq_initial_le_position_moment",
    upstreamFile := "AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.InitialGibbsPhaseTransport",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["SampleWiki", "SPHMC", "Gibbs", "Gaussian-smoothing", "phase-transport", "q2", "initialization"],
    saldUse := "SPHMC Lemma 4.16 q=2 initialization component: assemble the actual smoothed Gibbs position moment with the identical-momentum phase coupling",
    note := "Independently verified at commit c470a1b. Conditional on a supplied critical point; all-q concentration and critical-point existence remain separate open edges. The explicit factor 5 is ASTIS-derived, not source-literal."
  },
  {
    key := "samplewiki.sphmc.initial-gibbs-q2-unconditional",
    localDecl := "AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.InitialGibbsPhaseTransportUnconditional.initial_gibbs_phase_transport_q2_of_hessian_bounds",
    upstreamDecl := "HessianStrongConvexity.strongConvexOn_univ_of_fderiv2_lower / StrongConvexMinimizer.exists_isMinOn_and_gradient_eq_zero / InitialGibbsPhaseTransport.initial_gibbs_phase_transport_q2",
    upstreamFile := "AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.InitialGibbsPhaseTransportUnconditional",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["SampleWiki", "SPHMC", "Gibbs", "Gaussian-smoothing", "phase-transport", "q2", "initialization", "unconditional"],
    saldUse := "SPHMC Lemma 4.16 unconditional q=2 initialization edge: discharge the source's implicit minimizer witness from its Hessian bounds",
    note := "Independently verified at exact commit 8be837c. The source all-q concentration, Picard dynamics, final accuracy and query cost remain open. The explicit factor 5 is ASTIS-derived, not source-literal."
  }
]



def functionalInequalityMemory : List LemmaMemoryEntry := [
  {
    key := "entropy.measureDomination"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.BoundedPerturbation.entropy_le_mul_of_measure_le"
    upstreamDecl := "Canonical finite-integral supporting core for bounded perturbation"
    upstreamFile := "Chewi, Log-Concave Sampling, August20 2026, section2.3.1 pp60-61"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["entropy", "log-Sobolev", "change-of-measure", "shared"]
    saldUse := "For probability measures and nonnegative g with finite entropy integrals, nu <= b mu gives Ent_nu(g) <= b Ent_mu(g)."
    note := "Independently compiled at proof commit 9ba98ec42d9fa6abd78c045315cdd8d795e37c3c; current five packet-bound audits ASTIS-RT-20261020-BoundedEntropy-1 through -5 accept only this finite-integral scope. The zero-mean case is proved separately; no extended-real entropy claim."
  },
  {
    key := "logSobolev.energyMeasureComparison"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.BoundedPerturbation.energyLogSobolev_of_measure_comparison"
    upstreamDecl := "Canonical finite-integral supporting core for bounded perturbation"
    upstreamFile := "Chewi, Log-Concave Sampling, August20 2026, section2.3.1 pp60-61"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["entropy", "log-Sobolev", "change-of-measure", "shared"]
    saldUse := "For one fixed energy and test domain, opposite probability-measure comparisons transfer a finite-integral LSI from C to C*a*b."
    note := "Independently compiled at proof commit 9ba98ec42d9fa6abd78c045315cdd8d795e37c3c; current five packet-bound audits ASTIS-RT-20261020-BoundedEntropy-1 through -5 accept only this finite-integral scope. The test domain and energy are explicit; reference integrability is derived from target integrability."
  },
  {
    key := "logSobolev.boundedExponentialPerturbation"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.BoundedPerturbation.energyLogSobolev_tilted"
    upstreamDecl := "Canonical finite-integral supporting core for bounded perturbation"
    upstreamFile := "Chewi, Log-Concave Sampling, August20 2026, section2.3.1 pp60-61"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["entropy", "log-Sobolev", "change-of-measure", "shared"]
    saldUse := "A bounded measurable exponential tilt transfers the fixed-energy finite-integral LSI from C to C*exp(u-l)."
    note := "Independently compiled at proof commit 9ba98ec42d9fa6abd78c045315cdd8d795e37c3c; current five packet-bound audits ASTIS-RT-20261020-BoundedEntropy-1 through -5 accept only this finite-integral scope. Normalization, positive partition function and both density comparisons are derived. Geometric reference LSI and Sobolev extension remain separate."
  },
  {
    key := "gaussian.outerMarginal.closableGradient"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianMarginalGradient.gaussian_marginal_gradient_closable"
    upstreamDecl := "Authored actual Gaussian outer-gradient background for PBPS Appendix C.1 toward B.13"
    upstreamFile := "arXiv2609.06905v1"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "Gaussian", "gradient-domain", "closed-gradient"]
    saldUse := "Shared Gaussian outer-marginal gradient; no SALD admission"
    note := "Any probability mu on a finite real Hilbert/Borel space, eta>0, including rank0. For literal J=law(X,X+sqrt(eta)G), nu=J.snd, produce one dense genuine smooth-compact scalar/vector L2 gradient graph, closability and closed closure. Positive C2 Gaussian density, density L1, real normalizer1 and exact normalized tilted-law equality are internally derived. No caller moment/density/potential/integrability/normalization/closed-gradient certificate. Actual PBPS consumer retains original curvature and step assumptions; noncentered rank0 constant has mean1 and zero gradient. Authored background elaboration of C.1 density/closedness prerequisite. Literal Tf closure membership, full rough B13, Gamma/half-turn/main/errors/expected costs/composition remain separate. Independently verified 19f7e6bed7975fac9b7e1ea0b0b95d0c084083f6."
  },
  {
    key := "pbps.reflectedConditional.macroscopicBlockEnergy"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.MacroscopicEnergy.actual_macroscopic_gradient_energy_blocks"
    upstreamDecl := "PBPS Appendix B.1-B.9 genuine joint blocks and C.1 C.2 smooth energy"
    upstreamFile := "arXiv2609.06905v1"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "macroscopic-energy", "joint-blocks", "variance-defect"]
    saldUse := "PBPS actual macroscopic block energy; no SALD admission"
    note := "Actual same J/nu/Lambda/R/S and joint L2 conditional projection P and selfadjoint involution U. Produce actual A=PUP, B=(I-P)UP, D=(I-P)U(I-P), block adjoint identities and norm defect. For every signed smooth compact f, produce true differentiability/all L2/L1 domains, g=[f o snd], Pg=g, Ag=[Tf o snd], genuine second moments and integrated conditional variance=normBg squared. Sharp eta integral gradientTf squared <= (1-alpha eta)^2/[4(1+alpha eta)] normBg squared. AE S=S0 only transports means/classes, no derivative version transport. Positive alpha<=beta, global C2 Hessian bounds, eta>0 and beta eta<=1; finite Hilbert/rank0 extension explicit. No supplied law/operator/domain/bound or centering certificate. Full rough H1/closed-gradient/Gamma/half-turn/mains/errors/cost/composition remain separate. Independently verified 864ff305b78030d9e1c840938e82fefcf4185515."
  },
  {
    key := "pbps.reflectedConditional.outerGradientEnergy"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalGradientEnergy.reflected_conditional_gradient_energy"
    upstreamDecl := "PBPS Appendix C.1 formula C.2 smooth compact outer energy; B.9 variance defect"
    upstreamFile := "arXiv2609.06905v1"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "conditional-gradient", "outer-energy", "variance-defect"]
    saldUse := "PBPS actual outer gradient energy; no SALD admission"
    note := "Actual same reflected R/S with genuine probability laws and every-y density. For every signed smooth compact f, produce true differentiability, f/Tf/gradient L2, conditional variance L1 and exact variance defect. Genuine disintegration, reflection invariance and finite-domain Fubini give integrated variance=integral f squared minus integral Tf squared. Integrate true pointwise gradient domination to exact eta coefficient (1-alpha eta)^2/[4(1+alpha eta)]. Positive alpha<=beta, global C2 Hessian bounds, eta>0 and beta eta<=1; finite Hilbert/rank0 extension explicit. No supplied law, domain or desired-bound certificate. Full rough L2-to-H1/closed-gradient and literal block/Gamma adapters, mains/errors/cost/composition remain separate. Independently verified 5ba91a1c9a553b13a38d48f02f0725a69af146a4."
  },
  {
    key := "pbps.reflectedConditional.gradientVariance"
    localDecl := "AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalGradientVariance.reflected_conditional_gradient_variance"
    upstreamDecl := "PBPS Appendix C.1 A3.Ex7 pointwise gradient variance ingredient"
    upstreamFile := "arXiv2609.06905v1"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["PBPS", "conditional-gradient", "variance", "Poincare"]
    saldUse := "PBPS actual reflected conditional mean; no SALD admission"
    note := "Actual reflected conditional kernel and every-y density; true differentiability and exact pointwise gradient variance bound for every smooth compact observable. Identify both parent S internally from density, produce centered L2 domains, evaluate derivative at actual gradient and cancel zero-aware. Positive alpha<=beta, eta>0, beta*eta<=1; finite Hilbert/rank0 extension explicit. Actual Gaussian precision/rank0 Tests. No supplied law/coherence/domain/desired inequality certificate. Outer C.2/B.13/full L2-H1 and mains/error/cost separate. Independently verified 8d950c37e41bd5d816c4132c6a5cdde0e30dcbee."
  },
  {
    key := "sphmc.standardizedRGO.canonicalKLDimension"
    localDecl := "AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.StandardizedRGOKLDimension.standardized_rgo_unique_prox_and_kl_le_dimension"
    upstreamDecl := "SPHMC FIRST4.6 omitted Gaussian LSI and moment component"
    upstreamFile := "arXiv2609.06906v1 normalized C2/Hessian source; actual31/43"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["Gaussian", "relative-entropy", "dimension", "RGO", "SPHMC"]
    saldUse := "SPHMC actual standardized posterior; no SALD admission"
    note := "Actual same-prox canonical finite KL<=eta^2 real finrank/2. Actual43 unique proximal witness internally identifies actual31 stationary witness, transporting entire rho/Q/r/Fisher output. Retain true posterior probability/AC/finiteKL/gradientL2; compose actual halfFisher and actual numerical Fisher. Every positive eta, measurable family and rank0; no supplied law/coherence/domain/bound certificate. Actual eta2/unboundedfamily/rank0 Tests. GaussianT2/W2/FIRST/main/work/cost separate. Independently verified 76373366787499ebbc9e778fe568d0332233aef0."
  },
  {
    key := "sphmc.standardizedRGO.canonicalKLFisher"
    localDecl := "AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.StandardizedRGOKLFisher.standardized_rgo_unique_prox_and_kl_le_fisher"
    upstreamDecl := "SPHMC FIRST4.6 omitted Gaussian LSI component and actual S2 RGO"
    upstreamFile := "arXiv2609.06906v1 normalized C2/Hessian source; actual32/33/42"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["Gaussian", "relative-entropy", "Fisher", "RGO", "SPHMC"]
    saldUse := "SPHMC actual standardized posterior; no SALD admission"
    note := "Actual same-prox canonical finite KL<=half smooth rho Fisher. Internally reconcile actual32/33 proximal witnesses by uniqueness, transport all rho/Z/q/f/r, use actual42 noncompact GaussianLSI2 and actual32 quarter energy/mass1. True posterior probability/AC/finiteKL/scoreL2 outputs; no supplied law/coherence/domain/inequality certificate or pointwise canonicalRN derivative. Actual eta2/unboundedfamily/rank0 Tests. GaussianT2/FIRST/main/work/cost separate. Independently verified 19b569ae0fe37c97da0f98b4d1f4933ebabc7052."
  },
  {
    key := "gaussian.noncompact-logSobolev"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianLogSobolev.gaussian_logSobolev_of_contDiff"
    upstreamDecl := "SPHMC FIRST4.6 omitted Gaussian LSI; actual compact41 plus cutoff/DCT"
    upstreamFile := "Mathlib db584cd6 real DCT, Riesz gradient, mul-log and cutoff"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["Gaussian", "log-Sobolev", "Hilbert", "gradient", "SPHMC"]
    saldUse := "SPHMC Gaussian background/raw posterior; no SALD admission"
    note := "Actual noncompact C2 Gaussian function LSI, true stdGaussian E and Riesz gradient. Genuine fL2/gradientL2/PhiL1 domains. Actual radial cutoff/product-gradient/zero-aware entropy domination, three internal real DCT limits and closed-order passage retain2, signed/zero/rank0. Actual32 raw posterior consumer. Same33 canonicalKL/Fisher, T2/FIRST/fullW12/main/cost remain open. Independently verified 13556ea0eccd0069b83e7f7cb452a4ac7d7105ea."
  },
  {
    key := "gaussian.compact-hilbert-logSobolev"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianCompactHilbertLogSobolev.compact_stdGaussian_logSobolev"
    upstreamDecl := "SPHMC FIRST4.6 omitted Gaussian LSI; actual ASTIS40 with canonical Hilbert transport"
    upstreamFile := "Mathlib db584cd6 Gaussian.Multivariate/Basis/Gradient/PiL2/map integrals"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["Gaussian", "log-Sobolev", "Hilbert", "gradient", "SPHMC"]
    saldUse := "Actual SPHMC Gaussian background and standardized posterior cutoff consumer; no SALD admission"
    note := "Actual signed C2compact finiteHilbert Gaussian LSI. Literal stdGaussian law identified with canonical basis sum-map; continuous inverse basis pulls C2/support to actual40. True chain rule/Riesz/Parseval gives gradient norm squared, no Pi operator-norm shortcut or dimension factor. All3L1 internal, homogeneous entropy coefficient2 including rank0/zero mass. Actual32 posterior square-root-density radial cutoff Test uses same actual proximal witness; noncompact cutoff limits/coherent33KL/Fisher/T2/FIRST/main/cost open. Independently verified 8c44529058e9a7e3ad58951550f45785639d5036."
  },
  {
    key := "gaussian.compact-product-logSobolev"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianCompactProductLogSobolev.compact_gaussian_pi_logSobolev"
    upstreamDecl := "Pinned SLT TensorizedGLSI background; authored compact original-slice induction route"
    upstreamFile := "SLT d0f506f0 external reference; Mathlib db584cd6 Pi/FDeriv/Prod"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["Gaussian", "log-Sobolev", "coordinate-energy", "SPHMC"]
    saldUse := "Actual SPHMC finite GaussianLSI background consumer; no SALD/second-route admission"
    note := "Actual signed compact C2 all-dimensional product Gaussian function LSI. True square/Phi/all-coordinate/fullsum L1, Fin.cons Gaussian law and original-slice C2compact domains internally produced. Actual compactscalar38 and boundedbinary39 with strict dimensionIH preserve literal coordinate-square energy2, including0 dimension/mass. No public domain/slice/law/desiredinequality/limit certificates. FiniteHilbert/Parseval/noncompact32/33/T2/FIRST/main/cost open. Independently verified eaf29841eab8b3d4df5b9419f658fbcc1dda39f3."
  },
  {
    key := "entropy.product.bounded-subadditivity"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.InformationTheory.ProductEntropy.bounded_product_entropy_subadditivity"
    upstreamDecl := "Pinned SLT SubAddEnt source; authored bounded heterogeneous binary regularization route"
    upstreamFile := "SLT d0f506f0 external reference; Mathlib db584cd6 Prod/NegMulLog/DominatedConvergence"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["entropy", "probability-product", "zero-fibers", "SPHMC"]
    saldUse := "Actual SPHMC finite GaussianLSI background consumer; no SALD/second-route admission"
    note := "Actual bounded measurable nonnegative binary probability-product entropy with joint/all-slice/marginal L1 and zero mass/fibers. Positive AB/m comparison, logsum, genuine Fubini and xlogx DCT internally produced. No public positivity/domain/normalization/desired-inequality certificate. Finite tensorization/GaussianLSI/Hilbert/cutoff/T2/FIRST/main/cost open. Independently verified 913438726472dcac2483cefbe6b4b2f4746cbe74."
  },
  {
    key := "entropy.gaussian.compact-logSobolev"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianCompactLogSobolev.compact_gaussian_logSobolev"
    upstreamDecl := "Pinned compact GaussianLSI source route; actual ASTIS34/36/37 and Mathlib closed-order limit"
    upstreamFile := "SLT d0f506f0 external reference; Mathlib db584cd6 OrderClosed/RingCommute/MonoidDefs"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["entropy", "Gaussian-LSI-background", "compact-support", "SPHMC"]
    saldUse := "Actual SPHMC omitted Gaussian LSI background; no SALD/second-route admission"
    note := "Actual compact C2 scalar Gaussian homogeneous entropy<=2 derivative-square integral, signed/zero mass included. Internally consumes actual half Bernoulli LSI, true entropy convergence and fullflip limit4; only f/C2/compact caller inputs. Exact variance1; no law/domain/inequality/limit certificate. Continuous product/Hilbert/noncompact/T2/FIRST/main/cost open. Independently verified d8cbf270a3627d99971023d783c785d0585be18a."
  },
  {
    key := "energy.gaussian.compact-count-full-flip-limit"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianFlipEnergy.compact_count_gaussian_flip_energy_limit"
    upstreamDecl := "Pinned Mathlib compact derivative/MVT/count integral APIs; SLT full shift-energy source reference"
    upstreamFile := "Mathlib db584cd6; ASTIS authored Boolean secant alternative; SLT d0f506f0 reference only"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["energy", "Gaussian-background", "compact-support", "SPHMC"]
    saldUse := "Actual SPHMC omitted Gaussian LSI background; no SALD/second-route admission"
    note := "Actual compact C2 full Boolean coordinate-flip energy Gaussian limit4; true Gaussian derivative-square and allN finite-count full energy L1, N0 included. Internally produced derivative bounds/Lipschitz, actual signed displacement and secants, uniform16BK/sqrtN error and true integral passage. Sole direct ASTIS36 parent, no caller law/domain/bound/limit certificate. Compact GaussianLSI/Hilbert/noncompact/T2/main/cost remain open. Independently verified c3bbf938a6d46de7d43036d9f066ba68a3e6be6f."
  },
  {
    key := "entropy.gaussian.compact-count-observer-limits"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianCompactEntropy.compact_count_gaussian_entropy_limits"
    upstreamDecl := "Pinned Mathlib weak-integral/compact support APIs; SLT compact entropy background reference"
    upstreamFile := "Mathlib db584cd6; ASTIS authored actual count adapter; SLT d0f506f0 external reference only"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["entropy", "Gaussian-background", "compact-support", "SPHMC"]
    saldUse := "Actual SPHMC omitted Gaussian LSI background; no SALD/second-route admission"
    note := "Actual compact C2 Gaussian and allN normalized Bool-count L1 domains for f squared, f squared log f squared and derivative squared; successor three integral limits and homogeneous entropy limit. Actual35 law internally called, no caller domain/law/limit/positive-mass certificate. Fullflip-energy4, GaussianLSI/noncompact/T2/main/cost remain open. Independently verified 421496a5de7355830c0b4904ea10f33722ed3602."
  },
  {
    key := "functional-inequality.gibbs-local-l2",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedLocalL2.lp_locallyMemLp_volume",
    upstreamDecl := "memLp_two_iff_integrable_sq_norm; integrable_tilted_iff; absolutelyContinuous_tilted",
    upstreamFile := "Mathlib MeasureTheory; authored PBPS arXiv2609.06905v1 Appendix C.1 prerequisite",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["MEAS", "FI", "Gibbs", "local-integrability"],
    saldUse := "Actual weak-resolvent solution, gradient and forcing acquire compact unweighted L2 inputs.",
    note := "Continuous potential and finite Gibbs normalization; arbitrary normed additive target. No global L2 bound, H2, Poincare or paper completion."
  },
  {
    key := "functional-inequality.chewi-definition-1-2-19",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator.SatisfiesPoincare",
    upstreamDecl := "Chewi Definition 1.2.19",
    upstreamFile := "Log-Concave Sampling, book page 16 / PDF page 28",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Poincare", "generator", "Dirichlet-form", "variance", "definition"],
    saldUse := "Chewi Definition 1.2.19 root: state the general generator Poincare inequality on every finite-integral observable",
    note := "Exact reversible-generator formulation. The gradient-energy specialization and PI-to-decay equivalence are separate theorem routes."
  },
  {
    key := "functional-inequality.chewi-definition-1-2-25",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator.SatisfiesLogSobolev",
    upstreamDecl := "Chewi Definition 1.2.25",
    upstreamFile := "Log-Concave Sampling, book page 18 / PDF page 30",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "log-Sobolev", "generator", "Dirichlet-form", "density", "definition"],
    saldUse := "Chewi Definition 1.2.25 root: state KL(rho mu || mu) <= (C/2) E(rho,log rho) for every admissible normalized density",
    note := "Exact density-generator formulation with positivity, normalization, and finite entropy/energy conditions explicit. KL decay is a separate theorem route."
  },
  {
    key := "functional-inequality.canonical-lsi-kl-fisher",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.CanonicalLogSobolev.finite_klDiv_and_toReal_le_half_mul_information",
    upstreamDecl := "Chewi Definition 1.2.25 applied to the canonical RN density; canonical Dirichlet--Fisher bridge",
    upstreamFile := "Log-Concave Sampling, Definition 1.2.25; AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalDirichletFisher",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "SALD", "SPHMC", "log-Sobolev", "KL", "Fisher", "Radon-Nikodym", "finite-entropy"],
    saldUse := "turn an established generator LSI plus explicit canonical density/IBP/score domains into finite canonical KL and KL <= (C/2) FI",
    note := "Conditional interface bridge only. Finiteness comes from density admissibility. Strong-convex Gibbs LSI, the reciprocal-rate adapter, Talagrand T2 and Wasserstein contraction remain separate red nodes."
  },
  {
    key := "gronwall.chewi-lemma-1-2-20",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_lemma_1_2_20",
    upstreamDecl := "le_gronwallBound_of_liminf_deriv_right_le / gronwallBound_ε0",
    upstreamFile := "Mathlib.Analysis.ODE.Gronwall; Log-Concave Sampling, book page 16 / PDF page 28",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gronwall", "differential-inequality", "exponential-bound", "Chapter-1"],
    saldUse := "Chewi Lemma 1.2.20: turn g'(t) ≤ c g(t) on [0,T] into g(t) ≤ g(0) exp(ct)",
    note := "Source-faithful differentiable scalar specialization of Mathlib's more general one-sided-slope Gronwall theorem."
  },
  {
    key := "semigroup-decay.from-arbitrary-time",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.exponential_decay_of_scaled_dissipation_from",
    upstreamDecl := "le_gronwallBound_of_liminf_deriv_right_le / time translation",
    upstreamFile := "Mathlib.Analysis.ODE.Gronwall",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "functional-inequality", "semigroup", "dissipation", "shifted-time", "exponential-decay"],
    saldUse := "propagate a coercive energy-dissipation inequality exponentially between arbitrary starting and terminal times",
    note := "Shifted-time strengthening of the zero-time Gronwall leaf; the concrete energy and dissipation remain explicit inputs."
  },
  {
    key := "semigroup-decay.converse-from-shifted-exponential",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.scaled_dissipation_of_exponential_decay",
    upstreamDecl := "IsLocalMaxOn.hasFDerivWithinAt_nonpos / one_mem_posTangentConeAt_iff_frequently",
    upstreamFile := "Mathlib.Analysis.Calculus.LocalExtr.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "functional-inequality", "semigroup", "converse", "right-derivative", "Fermat"],
    saldUse := "recover instantaneous coercivity from exponential decay valid after every starting time",
    note := "Uses a one-sided local-maximum comparison with the exponential envelope; this is the scalar converse mechanism, not a concrete semigroup construction."
  },
  {
    key := "semigroup-decay.chewi-theorem-1-2-21-scalar-equivalence",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_21_scalar_equivalence",
    upstreamDecl := "Chewi Theorem 1.2.21, scalar energy-dissipation equivalence",
    upstreamFile := "Log-Concave Sampling, book page 16 / PDF page 28",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Poincare", "variance", "semigroup", "equivalence", "exponential-decay"],
    saldUse := "equate Poincare-style coercivity with shifted variance-style exponential decay under the exact -2 dissipation identity",
    note := "Full scalar equivalence. Instantiating variance for a reversible Markov semigroup and extending from a smooth core remain downstream."
  },
  {
    key := "semigroup-decay.chewi-theorem-1-2-22-scalar-equivalence",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_22_scalar_equivalence",
    upstreamDecl := "Chewi Theorem 1.2.22, scalar energy-dissipation equivalence",
    upstreamFile := "Log-Concave Sampling, book page 17 / PDF page 29",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Poincare", "chi-square", "semigroup", "equivalence", "exponential-decay"],
    saldUse := "equate Poincare-style coercivity with shifted chi-square-style exponential decay under the exact -2 dissipation identity",
    note := "The density/Radon-Nikodym realization of chi-square is not hidden in this scalar theorem."
  },
  {
    key := "semigroup-decay.chewi-theorem-1-2-26-scalar-equivalence",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_26_scalar_equivalence",
    upstreamDecl := "Chewi Theorem 1.2.26, scalar entropy-dissipation equivalence",
    upstreamFile := "Log-Concave Sampling, book page 18 / PDF page 30",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "log-Sobolev", "KL", "Fisher-information", "equivalence", "exponential-decay"],
    saldUse := "equate LSI-style coercivity with shifted KL exponential decay under the exact entropy-dissipation identity",
    note := "Full scalar equivalence. Concrete density regularity, KL/FI differentiation, and domain closure remain downstream."
  },
  {
    key := "semigroup-decay.scaled-dissipation",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.exponential_decay_of_scaled_dissipation",
    upstreamDecl := "le_gronwallBound_of_liminf_deriv_right_le / gronwallBound_ε0",
    upstreamFile := "Mathlib.Analysis.ODE.Gronwall",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "functional-inequality", "semigroup", "dissipation", "Gronwall", "exponential-decay"],
    saldUse := "generic scalar bridge from a coercive energy-dissipation inequality to exponential semigroup decay",
    note := "Uses an exact one-sided HasDerivWithinAt identity and Mathlib Gronwall; it does not construct the concrete variance, KL, or Fisher-information curve."
  },
  {
    key := "semigroup-decay.chewi-theorem-1-2-21-forward",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_21_forward",
    upstreamDecl := "Chewi Theorem 1.2.21, forward implication",
    upstreamFile := "Log-Concave Sampling, book page 16 / PDF page 28",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Poincare", "variance", "semigroup", "exponential-decay", "forward"],
    saldUse := "derive variance-style decay at rate 2/C from Poincare coercivity and the -2 Dirichlet-energy dissipation identity",
    note := "Source-faithful forward direction only; the converse and the concrete reversible-semigroup identification remain obligations."
  },
  {
    key := "semigroup-decay.chewi-theorem-1-2-22-forward",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_22_forward",
    upstreamDecl := "Chewi Theorem 1.2.22, forward implication",
    upstreamFile := "Log-Concave Sampling, book page 17 / PDF page 29",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Poincare", "chi-square", "semigroup", "exponential-decay", "forward"],
    saldUse := "derive chi-square-style decay at rate 2/C from Poincare coercivity and the -2 energy dissipation identity",
    note := "The theorem deliberately reuses the scalar PI decay mechanism and does not assert a concrete density evolution."
  },
  {
    key := "semigroup-decay.chewi-theorem-1-2-26-forward",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_26_forward",
    upstreamDecl := "Chewi Theorem 1.2.26, forward implication",
    upstreamFile := "Log-Concave Sampling, book page 18 / PDF page 30",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "log-Sobolev", "KL", "Fisher-information", "semigroup", "exponential-decay", "forward"],
    saldUse := "derive KL-style decay at rate 2/C from LSI coercivity and the entropy-dissipation identity KL'=-FI",
    note := "Source-faithful forward direction only; density regularity, concrete entropy dissipation, and the converse remain explicit."
  },
  {
    key := "semigroup-decay.forced-dissipation-from-arbitrary-time",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.forced_exponential_decay_of_scaled_dissipation_from",
    upstreamDecl := "le_gronwallBound_of_liminf_deriv_right_le / gronwallBound with nonzero forcing",
    upstreamFile := "Mathlib.Analysis.ODE.Gronwall",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["functional-inequality", "semigroup", "dissipation", "Gronwall", "forcing", "steady-state-floor"],
    saldUse := "turn a forced coercive energy-dissipation inequality into the exact exponential interpolation between the initial energy and forcing floor",
    note := "Scalar reusable leaf. Concrete carré-du-champ differentiation, Bakry–Émery curvature, and generator-domain closure remain separate prerequisites."
  },
  {
    key := "semigroup-decay.backward-interpolation-contraction",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.backward_interpolation_contraction_of_growth",
    upstreamDecl := "le_gronwallBound_of_liminf_deriv_right_le / Bakry--Émery backward interpolation",
    upstreamFile := "Mathlib.Analysis.ODE.Gronwall",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Bakry-Emery", "semigroup", "gradient-contraction", "interpolation", "Gronwall"],
    saldUse := "convert a lower differential bound for a backward interpolation quantity into the endpoint exponential contraction estimate",
    note := "Scalar interpolation leaf only. Concrete carré-du-champ derivative identities, semigroup order preservation, and generator-domain closure remain explicit inputs."
  },
  {
    key := "semigroup-decay.backward-interval-interpolation-contraction",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.backward_endpoint_contraction_of_growth_on_Icc",
    upstreamDecl := "derivative monotonicity / Bakry--Émery backward interpolation on a compact interval",
    upstreamFile := "Mathlib.Analysis.Calculus.Deriv.MeanValue",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Bakry-Emery", "semigroup", "gradient-contraction", "interpolation", "closed-interval"],
    saldUse := "derive exact endpoint contraction from a derivative-growth bound only on the interpolation interval",
    note := "Interval-scoped scalar leaf with genuine HasDerivAt hypotheses. The semigroup derivative identity, curvature bound, and terminal-energy identification remain explicit inputs."
  },
  {
    key := "semigroup-interpolation.exponential-kernel-endpoint",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.endpoint_le_of_deriv_le_exponential_kernel",
    upstreamDecl := "fundamental theorem / monotonicity from a nonnegative derivative",
    upstreamFile := "Mathlib.Analysis.Calculus.Deriv.MeanValue",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["functional-inequality", "semigroup", "entropy", "interpolation", "exponential-kernel"],
    saldUse := "integrate a backward exponential derivative bound into its exact endpoint convolution factor",
    note := "Reusable scalar analytic leaf. A concrete semigroup must still provide the entropy derivative identity and gradient-contraction estimate."
  }
]

def stochasticProcessMemory : List LemmaMemoryEntry := [
  {
    key := "bakry-emery.abstract-gradient-contraction",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.backwardInterpolation_contraction_of_expandedBakryEmery",
    upstreamDecl := "Bakry-Emery backward interpolation argument",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BakryEmeryGradientContraction",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Bakry-Emery", "semigroup", "positive-operator", "gradient-contraction"],
    saldUse := "combine curvature, positivity, the interpolation derivative identity, and scalar Gronwall into endpoint contraction",
    note := "Abstract adapter only. Concrete semigroups must discharge positivity, the exact interpolation derivative identity, continuity, and domain closure."
  },
  {
    key := "bakry-emery.interpolation-derivative-lower-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.interpolationDerivative_lower_bound_of_expandedBakryEmery",
    upstreamDecl := "CD(alpha, infinity) and the defining Gamma-two identity",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Bakry-Emery", "carre-du-champ", "Gamma2", "semigroup", "gradient-contraction"],
    saldUse := "derive the exact lower bound for the backward interpolation derivative expression from CD(alpha, infinity)",
    note := "Algebraic generator leaf only. A concrete semigroup must still prove the interpolation derivative identity, positivity/order preservation, and domain regularity."
  },
  {
    key := "localization.chewi-proposition-1-1-13",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalLocalizationTheorem.chewi_proposition_1_1_13",
    upstreamDecl := "Chewi Proposition 1.1.13",
    upstreamFile := "Log-Concave Sampling, book page 7 / PDF page 19",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "localization", "stopping-time", "progressive-L2"],
    saldUse := "package the canonical energy first-hitting times, terminal convergence, and stopped global-L2 bound into the exact Chapter 1 source result",
    note := "Display (1.1.14) and Proposition 1.1.16 remain separate theorem routes; both are now compiled, so this entry records only the canonical localization component."
  },
  {
  key := "localization.chewi-display-1-1-14",
  localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalStoppedItoIntegral.chewi_display_1_1_14",
  upstreamDecl := "Chewi display (1.1.14)",
  upstreamFile := "Log-Concave Sampling, book page 7 / PDF page 19",
  status := LemmaMemoryStatus.formalizedLocal,
  tags := ["Chewi", "Ito", "localization", "stopped-integral", "continuous-martingale"],
  saldUse := "package each canonical energy truncation as a globally L2 Ito integrand whose Ito process is adapted, continuous, martingale, and compatible with deterministic-time restriction",
  note := "This compiles the intermediate stopped-global-Ito display. Arbitrary random-stopping consistency and cross-horizon gluing are discharged separately by Proposition 1.1.16."
},
{
  key := "localization.chewi-proposition-1-1-16",
  localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiProposition1_1_16.chewi_proposition_1_1_16",
  upstreamDecl := "Chewi Proposition 1.1.16",
  upstreamFile := "Log-Concave Sampling, book page 7 / PDF page 19",
  status := LemmaMemoryStatus.formalizedLocal,
  tags := ["Chewi", "Ito", "local-martingale", "localization", "random-stopping", "continuous-paths"],
  saldUse := "package random-stopping consistency, cross-horizon overlap, localized martingale coherence, and pathwise gluing into the source local-Ito continuous-local-martingale result",
  note := "Uses the canonical localizers from Proposition 1.1.13; stopped-integral compatibility, stopping-graph nullity, and cross-horizon gluing are all compiled."
},
  {
    key := "localization.completed-integrand-progressive",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedIntegrand.completedIntegrand_stronglyProgressive",
    upstreamDecl := "Progressiveness after completing the null bad-path set",
    upstreamFile := "Log-Concave Sampling, Proposition 1.1.13, book page 7 / PDF page 19",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "localization", "completion", "progressive"],
    saldUse := "replace nonintegrable sample paths by zero while preserving the progressive sigma-algebra",
    note := "The source local-L2 domain assumes only almost-sure finite path energy. Usual-condition completeness makes the null replacement measurable at every filtration time."
  },
  {
    key := "localization.energy-stopped-path-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedIntegrand.integral_energyStoppedIntegrand_sq_le",
    upstreamDecl := "Pathwise square-energy bound after canonical stopping",
    upstreamFile := "Log-Concave Sampling, Proposition 1.1.13, book page 7 / PDF page 19",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "localization", "stopped-integrand", "energy-bound"],
    saldUse := "bound each stopped path by its canonical energy level before taking expectation",
    note := "The threshold representation is proved equivalent almost everywhere in time to stopping before the equality-level localizer. No expected-energy hypothesis is used."
  },
  {
    key := "localization.stopped-progressive-l2",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2.stoppedProgressiveL2",
    upstreamDecl := "Global progressive product-L2 membership after energy stopping",
    upstreamFile := "Log-Concave Sampling, Proposition 1.1.13, book page 7 / PDF page 19",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "localization", "progressive-L2", "Tonelli"],
    saldUse := "feed each canonical stopped integrand into the global Ito integral constructed in Theorem 1.1.8",
    note := "Tonelli integrates the pathwise level bound over the probability measure. The stopped-integral identity and local-martingale consistency remain downstream."
  },
  {
    key := "localization.completed-energy-process",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy.continuous_completedEnergy",
    upstreamDecl := "Everywhere-continuous completed accumulated energy",
    upstreamFile := "Log-Concave Sampling, Proposition 1.1.13, book page 7 / PDF page 19",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "localization", "completion", "continuous-path"],
    saldUse := "replace the null set of nonintegrable paths without losing filtration measurability",
    note := "Usual-condition completeness makes the null bad set measurable at every time; all completed paths are continuous, monotone, and nonnegative."
  },
  {
    key := "localization.canonical-energy-stopping-time",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyStoppingTime.canonicalLocalizingTime_isChewiStoppingTime",
    upstreamDecl := "Stopping-time property of the canonical integer energy localizer",
    upstreamFile := "Log-Concave Sampling, Proposition 1.1.13, book page 7 / PDF page 19",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "localization", "stopping-time", "first-hit"],
    saldUse := "supply the stopping-time component of Chewi's canonical localizing sequence",
    note := "The first equality-level time is characterized by a fixed-time completed-energy threshold event; the proof uses continuity and the intermediate value theorem."
  },
  {
    key := "localization.canonical-localizer-monotone-terminal",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.tendsto_canonicalLocalizingTime",
    upstreamDecl := "Monotone canonical localizers eventually equal the terminal horizon",
    upstreamFile := "Log-Concave Sampling, Proposition 1.1.13, book page 7 / PDF page 19",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "localization", "monotone", "Tendsto"],
    saldUse := "establish pathwise monotonicity and terminal convergence of the canonical localizing sequence",
    note := "Once the integer level exceeds finite terminal completed energy, its equality-level set is empty and the localizer equals T. Stopped-integrand global L2 and the stopped-integral identity remain downstream."
  },
  {
    key := "localization.canonical-stopped-energy-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.completedEnergy_at_canonicalLocalizingTime_le",
    upstreamDecl := "Canonical localizer bounds stopped completed energy by n+1",
    upstreamFile := "Log-Concave Sampling, Proposition 1.1.13, book page 7 / PDF page 19",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "localization", "energy-bound", "stopped-integrand"],
    saldUse := "provide the deterministic energy bound needed to place every stopped integrand in global product L2",
    note := "The bound is exact on reached levels and uses terminal fallback when the level is not reached. Progressive stopped-integrand measurability remains downstream."
  },
  {
    key := "analysis.prefix-integral-continuity",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral.continuous_prefixIntegral",
    upstreamDecl := "Continuity of finite-horizon moving prefix integrals",
    upstreamFile := "Log-Concave Sampling, Proposition 1.1.13, book page 7 / PDF page 19",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "analysis", "Bochner-integral", "continuity", "localization"],
    saldUse := "supply the analytic continuity theorem for accumulated square energy",
    note := "The proof uses dominated convergence on the repository's finite NNReal time measure and treats the moving endpoint singleton as a null set."
  },
  {
    key := "localization.energy-path-continuity",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity.continuous_accumulatedEnergyReal_ae",
    upstreamDecl := "Almost-sure continuity of accumulated progressive square energy",
    upstreamFile := "Log-Concave Sampling, Proposition 1.1.13, book page 7 / PDF page 19",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "localization", "energy", "continuity"],
    saldUse := "justify the first-hitting canonical localizer by a continuous monotone energy path",
    note := "The theorem starts from progressive measurability and almost-sure finite path energy; the stopping-time and stopped-integrand theorems remain downstream."
  },
  {
    key := "localization.fixed-time-energy-measurability",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2.accumulatedEnergyReal_stronglyMeasurable",
    upstreamDecl := "Fixed-time filtration measurability of accumulated square energy",
    upstreamFile := "Log-Concave Sampling, Proposition 1.1.13, book page 7 / PDF page 19",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "localization", "progressive", "filtration", "energy"],
    saldUse := "make canonical energy-hitting events measurable at each observation time",
    note := "The source domain assumes only almost-sure finite path energy, not finite expected energy. Continuity and the first-hitting stopping-time theorem remain downstream."
  },
  {
    key := "localization.accumulated-energy-monotonicity",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.AccumulatedEnergy.accumulatedEnergy_mono",
    upstreamDecl := "Monotonicity of accumulated pathwise square energy",
    upstreamFile := "Log-Concave Sampling, Proposition 1.1.13, book page 7 / PDF page 19",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "localization", "energy", "stopping-time"],
    saldUse := "supply the monotone nonnegative path-energy process used by the canonical localizing sequence",
    note := "This is an exact ENNReal integral leaf. Fixed-time filtration measurability, path continuity on the finite-energy set, and the first-hitting stopping time remain downstream."
  },
  {
    key := "brownian-motion.chewi-definition-1-1-1",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.IsStandardBrownianMotion",
    upstreamDecl := "ProbabilityTheory.HasLaw / ProbabilityTheory.iIndepFun / ProbabilityTheory.gaussianReal / Chewi Definition 1.1.1",
    upstreamFile := "Mathlib.Probability.Distributions.Gaussian.Basic; Mathlib.Probability.Independence.Basic; Log-Concave Sampling, book page 4 / PDF page 16",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Brownian-motion", "Gaussian", "independent-increments", "continuous-paths"],
    saldUse := "Chewi Definition 1.1.1 root: expose zero start, independent increments, centered isotropic Gaussian increment laws, and almost-sure path continuity",
    note := "The vector Gaussian law is stated through every continuous-linear projection, matching Mathlib's coordinate-free Gaussian interface. Existence and construction remain separate theorems."
  },
  {
    key := "ito-integral.usual-filtration-conditions",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.SatisfiesUsualConditions",
    upstreamDecl := "Complete right-continuous filtered probability space",
    upstreamFile := "Log-Concave Sampling, book page 4 / PDF page 16; Mathlib.Probability.Process.Filtration",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "filtration", "complete", "right-continuous"],
    saldUse := "package the usual filtration conditions required by continuous-time stochastic calculus",
    note := "Completeness is stated as inclusion of every ambient measure-null set at every time; right continuity reuses Mathlib's right-continuation interface."
  },
  {
    key := "ito-integral.progressive-l2-domain",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand",
    upstreamDecl := "Progressive globally square-integrable stochastic integrand",
    upstreamFile := "Log-Concave Sampling, Theorem 1.1.8, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "progressive", "L2", "product-measure"],
    saldUse := "retain filtration information together with the probability-time L2 representative used by Ito completion",
    note := "This is the source domain, not an Ito-integral existence theorem; adapted elementary density and continuous martingale construction remain downstream."
  },
  {
    key := "ito-integral.progressive-l2-algebra",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_smul",
    upstreamDecl := "Real vector-space closure of progressive L2 integrands",
    upstreamFile := "Log-Concave Sampling, construction in Theorem 1.1.8, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "progressive", "L2", "linearity"],
    saldUse := "assemble linear combinations of progressive square-integrable integrands while preserving their canonical product-space Lp representatives",
    note := "Zero, addition, negation, subtraction, and real scalar multiplication are constructed with exact toLp compatibility. This does not provide elementary density or the Ito extension."
  },
  {
    key := "ito-integral.progressive-l2-restriction",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.norm_restrictAt_le",
    upstreamDecl := "L2 contraction under terminal-time restriction",
    upstreamFile := "Log-Concave Sampling, Theorem 1.1.8, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "progressive", "restriction", "L2-contraction"],
    saldUse := "define time-indexed terminal Ito maps without increasing integrand energy",
    note := "Restriction uses a strict endpoint representative, equivalent to the closed interval convention under Lebesgue time measure."
  },
  {
    key := "ito-elementary.same-grid-algebra",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.elementaryItoIntegral_sub",
    upstreamDecl := "Linearity of elementary Ito integration on a fixed adapted time grid",
    upstreamFile := "Log-Concave Sampling, proof of Theorem 1.1.8, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "elementary-process", "linearity", "common-grid"],
    saldUse := "keep differences of same-grid elementary approximants inside the elementary Ito domain",
    note := "Zero, negation, scalar multiplication, addition, and subtraction are constructed with pointwise and finite-integral linearity. Common-grid refinement and density remain downstream obligations."
  },
  {
    key := "ito-elementary.progressive-l2-embedding",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding.toProgressiveL2",
    upstreamDecl := "Elementary adapted processes lie in the progressive L2 integrand domain",
    upstreamFile := "Log-Concave Sampling, proof of Theorem 1.1.8, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "elementary-process", "progressive", "L2"],
    saldUse := "compare elementary approximants and general progressive integrands in one product-space L2 domain",
    note := "Joint strong measurability, strong progressiveness, a finite coefficient bound, and product-space MemLp are proved before constructing the canonical embedding. Density remains open."
  },
  {
    key := "ito-elementary.terminal-l2-isometry",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2.norm_elementaryItoTerminalToLp",
    upstreamDecl := "Elementary terminal Ito map is an L2 isometry",
    upstreamFile := "Log-Concave Sampling, displays (1.1.5)--(1.1.6), book page 5 / PDF page 17",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "elementary-process", "L2", "linear-isometry"],
    saldUse := "transfer Cauchy control from same-grid elementary integrands to terminal stochastic sums",
    note := "Both sides are actual Lp elements and their norms are proved equal from the expectation-level isometry. Adapted density, common refinement, and completion remain downstream."
  },
  {
    key := "ito-elementary.terminal-l2-inner-isometry",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2.inner_elementaryItoTerminalToLp",
    upstreamDecl := "Elementary Ito inner-product isometry on a common grid",
    upstreamFile := "Log-Concave Sampling, displays (1.1.5)--(1.1.9), book pages 5--6 / PDF pages 17--18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "elementary-process", "L2", "inner-product", "isometry"],
    saldUse := "transport Hilbert-space Cauchy and orthogonality arguments from elementary integrands to terminal stochastic integrals",
    note := "Derived by polarization from the actual norm isometry after proving same-grid addition on both Lp sides. Different-grid refinement and general completion remain open."
  },
  {
    key := "ito-integral.coefficient-l2-truncation",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.tendsto_clipNat_toLp",
    upstreamDecl := "Bounded truncation approximation in L2",
    upstreamFile := "Log-Concave Sampling, proof of Theorem 1.1.8, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "L2", "truncation", "dominated-convergence"],
    saldUse := "replace square-integrable real coefficients by bounded measurable coefficients without changing their L2 limit",
    note := "The clipping map is proved measurable, norm dominated, eventually pointwise equal, and convergent in the actual Lp space. Adapted elementary density still requires time discretization and filtration-aware approximation."
  },
  {
    key := "ito-integral.progressive-l2-truncation",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation.tendsto_clipped_toLp",
    upstreamDecl := "Bounded progressive truncation in product-space L2",
    upstreamFile := "Log-Concave Sampling, proof of Theorem 1.1.8, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "progressive", "L2", "truncation", "product-measure"],
    saldUse := "reduce a progressive square-integrable process to uniformly bounded progressive processes in product-space L2",
    note := "Clipping is lifted to the full ProgressiveL2Integrand structure and convergence is proved in its actual Lp representative. Causal time regularization and elementary density remain open."
  },
  {
    key := "ito-integral.dyadic-clipped-elementary-construction",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.sampledClippedDyadic",
    upstreamDecl := "Dyadic bounded adapted left-step approximation object",
    upstreamFile := "Log-Concave Sampling, proof of Theorem 1.1.8, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "elementary-process", "dyadic-grid", "adapted", "truncation"],
    saldUse := "construct genuine bounded elementary adapted processes from deterministic left-endpoint samples of a progressive L2 process",
    note := "The strict grid, left-endpoint filtration measurability, coefficient bound, and terminal endpoint are proved. L2 time-regularization, convergence, and the diagonal density theorem remain open."
  },
  {
    key := "ito-integral.lagged-dyadic-adapted-average",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedDyadicApprox_isElementaryAdapted",
    upstreamDecl := "Lagged dyadic cell-average approximation by bounded adapted elementary processes",
    upstreamFile := "Log-Concave Sampling, proof of Theorem 1.1.8, book page 6 / PDF page 18; Mathlib.MeasureTheory.Integral.Prod",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "elementary-process", "dyadic-grid", "adapted", "Bochner-integral"],
    saldUse := "regularize progressive L2 integrands causally: average a clipped process over the preceding cell and use that coefficient on the next cell",
    note := "Parameterized Bochner measurability is proved from progressive measurability after an explicit zero extension, and every coefficient is bounded by the clipping level. L2 convergence of the lagged averages remains open."
  },
  {
    key := "ito-integral.lagged-dyadic-l2-convergence",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence.tendsto_laggedDyadicApprox_toLp_clipped",
    upstreamDecl := "Lagged dyadic adapted averages converge in product-space L2",
    upstreamFile := "Log-Concave Sampling, proof of Theorem 1.1.8, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "dyadic", "adapted", "L2-convergence"],
    saldUse := "replace a bounded progressive integrand by causal elementary adapted processes in the actual product L2 space",
    note := "Lebesgue differentiation, active-cell geometry, product-a.e. transfer, and dominated convergence are all explicit."
  },
  {
    key := "ito-integral.progressive-elementary-density",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.progressiveL2_elementary_dense",
    upstreamDecl := "Bounded dyadic elementary adapted processes are dense in progressive L2",
    upstreamFile := "Log-Concave Sampling, proof of Theorem 1.1.8, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "density", "progressive", "fast-diagonal"],
    saldUse := "supply fast nested-grid elementary approximants for terminal completion and pathwise Borel-Cantelli control",
    note := "The noncomputable diagonal chooses increasing truncation and dyadic levels with summable quantitative errors."
  },
  {
    key := "ito-integral.dyadic-refinement",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.norm_terminal_sub_eq_process_sub",
    upstreamDecl := "Common dyadic refinement preserves elementary processes and Ito terminal sums",
    upstreamFile := "Log-Concave Sampling, proof of Theorem 1.1.8, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "dyadic", "refinement", "isometry"],
    saldUse := "compare heterogeneous elementary approximants on a shared grid without changing either L2 representative",
    note := "Coefficient replication, filtration monotonicity, finite Brownian-increment telescoping, and the distance isometry are proved."
  },
  {
    key := "ito-integral.terminal-isometry",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_norm",
    upstreamDecl := "General terminal Ito integral by L2 completion",
    upstreamFile := "Log-Concave Sampling, Theorem 1.1.8 and display (1.1.9), book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "L2-completion", "terminal-map", "isometry"],
    saldUse := "extend elementary stochastic integration uniquely and linearly to every progressive globally square-integrable integrand",
    note := "The definition uses completeness of Lp; the universal approximation theorem removes dependence on the canonical choice sequence."
  },
  {
    key := "ito-integral.elementary-martingale",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.elementaryItoProcess_martingale",
    upstreamDecl := "Elementary Ito integral process is a continuous martingale",
    upstreamFile := "Log-Concave Sampling, proof of Theorem 1.1.8, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "elementary-process", "martingale", "continuous-paths"],
    saldUse := "lift terminal elementary sums to adapted time-indexed martingales before taking the uniform path limit",
    note := "Conditional mean-zero future increments prove the martingale identity; Brownian path continuity proves a.e. continuity."
  },
  {
    key := "ito-integral.doob-l2",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.doobL2_continuous",
    upstreamDecl := "Doob L2 maximal control for continuous martingales",
    upstreamFile := "Log-Concave Sampling, proof of Theorem 1.1.8, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "Doob", "maximal-inequality", "continuous-martingale"],
    saldUse := "turn summable terminal L2 approximation errors into summable uniform path-deviation events",
    note := "The finite discrete inequality is proved first and then extended to continuous paths through nested dyadic observation grids."
  },
  {
    key := "ito-integral.continuous-martingale",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_martingale",
    upstreamDecl := "Continuous adapted general Ito integral process",
    upstreamFile := "Log-Concave Sampling, Theorem 1.1.8, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "Borel-Cantelli", "continuous-martingale", "restriction"],
    saldUse := "construct the actual continuous martingale version and identify every fixed time with the restricted terminal completion",
    note := "A summable Doob-Borel-Cantelli argument gives a uniform path limit; usual-condition completeness makes the null-set patch adapted."
  },
  {
    key := "ito-integral.bounded-stopping-mean-zero",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BoundedStoppedItoMean.integral_itoIntegralProcess_at_boundedStopping_eq_zero",
    upstreamDecl := "RandomStoppingGeneralIto.itoIntegralTerminal_stop_ae; ItoIntegralProcess.itoIntegralProcess_martingale",
    upstreamFile := "AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingGeneralIto.lean; AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Ito", "random-stopping", "martingale", "integrability", "mean-zero"],
    saldUse := "replace an abstract optional-stopping or centered-martingale premise by the constructed completed Ito process at a bounded stopping time",
    note := "The companion theorem proves integrability of the randomly evaluated process. The stopping time must be pointwise bounded by the positive construction horizon; no claim is made for unbounded stopping times or local-L2 integrands before localization."
  },
  {
    key := "ito-integral.chewi-theorem-1-1-8",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.chewi_theorem_1_1_8",
    upstreamDecl := "Chewi Theorem 1.1.8",
    upstreamFile := "Log-Concave Sampling, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "progressive", "martingale", "uniqueness"],
    saldUse := "provide the source-facing existence, adaptedness, martingale, continuity, fixed-time compatibility, isometry, and indistinguishability interface",
    note := "This theorem packages constructed objects and proved properties; it assumes no abstract stochastic-integral contract."
  },
  {
    key := "ito-integral.chewi-display-1-1-9",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.chewi_display_1_1_9",
    upstreamDecl := "Chewi display (1.1.9)",
    upstreamFile := "Log-Concave Sampling, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "isometry", "fixed-time", "L2"],
    saldUse := "rewrite every fixed-time stochastic-integral second moment as the product-space energy of the time-restricted integrand",
    note := "The strict endpoint convention differs from the closed interval only on a null singleton and is the exact Lean representative used by restrictAt."
  },
  {
    key := "brownian-motion.filtration-contract",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.IsBrownianMotionWithFiltration",
    upstreamDecl := "Brownian motion relative to a filtration",
    upstreamFile := "Log-Concave Sampling, book page 5 / PDF page 17; Mathlib.Probability.BrownianMotion.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Brownian-motion", "filtration", "adapted", "independent-increments"],
    saldUse := "ensure future Brownian increments are independent of the whole past sigma-algebra used by adapted coefficients",
    note := "This strengthens bare independent increments by making the filtration explicit; it does not claim existence for an arbitrary enlarged filtration."
  },
  {
    key := "brownian-motion.conditional-increment-mean-zero",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.IsBrownianMotionWithFiltration.condExp_increment_eq_zero",
    upstreamDecl := "Conditional mean-zero Brownian increment",
    upstreamFile := "Log-Concave Sampling, book page 5 / PDF page 17; Mathlib.Probability.ConditionalExpectation",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Brownian-motion", "conditional-expectation", "filtration", "martingale-increment"],
    saldUse := "remove future increments after conditioning on the left-endpoint filtration",
    note := "Derived from filtration-level independence and the centered Brownian law under an explicit probability-measure instance."
  },
  {
    key := "brownian-motion.increment-second-moment",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.IsBrownianMotionWithFiltration.integral_increment_sq",
    upstreamDecl := "Brownian increment second moment",
    upstreamFile := "Log-Concave Sampling, book page 5 / PDF page 17; Mathlib.Probability.BrownianMotion.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Brownian-motion", "second-moment", "variance", "Ito"],
    saldUse := "replace an increment-square expectation by its elapsed time",
    note := "Uses Mathlib's Brownian covariance formula and keeps the ordered nonnegative-time subtraction explicit."
  },
  {
    key := "brownian-motion.conditional-increment-second-moment",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.IsBrownianMotionWithFiltration.condExp_increment_sq",
    upstreamDecl := "Conditional Brownian increment second moment",
    upstreamFile := "Log-Concave Sampling, book page 5 / PDF page 17; Mathlib.Probability.ConditionalExpectation",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Brownian-motion", "conditional-expectation", "second-moment", "filtration", "Ito"],
    saldUse := "evaluate the conditional variance of a future Brownian increment at the left endpoint",
    note := "The square-increment sigma-algebra is explicitly shown to be below the increment sigma-algebra before applying conditional-expectation independence."
  },
  {
    key := "ito-elementary.chewi-display-1-1-2",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral.chewi_display_1_1_2",
    upstreamDecl := "Chewi display (1.1.2)",
    upstreamFile := "Log-Concave Sampling, book page 4 / PDF page 16",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "elementary-process", "adapted", "finite-sum"],
    saldUse := "source-faithful elementary adapted-process root for the Ito construction",
    note := "The structure records a strict grid, left-endpoint filtration measurability, and bounded coefficients; no L2 completion is claimed."
  },
  {
    key := "ito-elementary.chewi-display-1-1-3",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral.chewi_display_1_1_3",
    upstreamDecl := "Chewi display (1.1.3)",
    upstreamFile := "Log-Concave Sampling, book page 5 / PDF page 17",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "Brownian-increment", "elementary-process", "finite-sum"],
    saldUse := "finite-sum definition used before proving orthogonality and the elementary Ito isometry",
    note := "This is the exact stopped Brownian-increment sum for an elementary process; the isometry and L2 extension remain downstream."
  },
  {
    key := "ito-elementary.weighted-increment-second-moment",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.integral_weightedIncrement_sq",
    upstreamDecl := "Adapted weighted Brownian-increment diagonal term",
    upstreamFile := "Log-Concave Sampling, book page 5 / PDF page 17",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "adapted", "Brownian-increment", "second-moment"],
    saldUse := "factor each diagonal Ito-isometry term into coefficient energy and clipped interval length",
    note := "The coefficient is left-endpoint measurable and independent of the future increment; both factors are proved integrable."
  },
  {
    key := "ito-elementary.weighted-increment-orthogonality",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.integral_weightedIncrement_mul_eq_zero",
    upstreamDecl := "Orthogonality of distinct adapted weighted Brownian increments",
    upstreamFile := "Log-Concave Sampling, book page 5 / PDF page 17",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "adapted", "orthogonality", "filtration"],
    saldUse := "eliminate cross terms in finite elementary stochastic-integral squares",
    note := "The proof orders the intervals, proves the earlier weighted term is measurable at the later left endpoint, and factors against the centered future increment."
  },
  {
    key := "ito-elementary.chewi-display-1-1-5",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.chewi_display_1_1_5",
    upstreamDecl := "Chewi display (1.1.5)",
    upstreamFile := "Log-Concave Sampling, book page 5 / PDF page 17",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "elementary-process", "orthogonality", "finite-sum"],
    saldUse := "expand the square of an elementary stochastic integral and retain only diagonal terms",
    note := "This is an expectation identity, not merely a finite-sum algebra lemma; integrability and cross-term orthogonality are proved."
  },
  {
    key := "ito-elementary.process-l2-energy-value",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.processL2Energy_value",
    upstreamDecl := "Elementary-process product-space L2 energy",
    upstreamFile := "Log-Concave Sampling, book page 5 / PDF page 17",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "L2", "product-measure", "time-measure"],
    saldUse := "evaluate elementary-process energy as the sum of coefficient energies times clipped cell lengths",
    note := "Disjoint time cells are integrated against the stopped NNReal Lebesgue measure and then Tonelli identifies the product-space energy."
  },
  {
    key := "ito-elementary.chewi-display-1-1-6",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.chewi_display_1_1_6",
    upstreamDecl := "Chewi display (1.1.6)",
    upstreamFile := "Log-Concave Sampling, book page 5 / PDF page 17",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "isometry", "elementary-process", "L2"],
    saldUse := "identify the second moment of the elementary Ito integral with its product-space integrand energy",
    note := "The equality is stated in ENNReal to match the repository's nonnegative process-energy interface; general L2 completion remains downstream."
  },
  {
    key := "ito-elementary.chewi-display-1-1-7",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral.chewi_display_1_1_7",
    upstreamDecl := "Chewi display (1.1.7)",
    upstreamFile := "Log-Concave Sampling, book page 5 / PDF page 17",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "L2", "product-measure", "Tonelli"],
    saldUse := "canonical product-space energy for square-integrable stochastic integrands",
    note := "Uses ENNReal and Tonelli under explicit joint a.e. measurability; finiteness remains a visible inequality rather than a totalized integral."
  },
  {
    key := "ito-local.chewi-display-1-1-10",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral.chewi_display_1_1_10",
    upstreamDecl := "Chewi display (1.1.10)",
    upstreamFile := "Log-Concave Sampling, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Ito", "localization", "L2", "almost-sure"],
    saldUse := "local square-integrability premise for canonical stopping and localized stochastic integration",
    note := "Exact ENNReal almost-sure finiteness condition; construction of the canonical localizing sequence and localized Ito integral remain downstream."
  },
  {
    key := "markov-semigroup.chewi-definition-1-2-1",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.markovOperator",
    upstreamDecl := "Chewi Definition 1.2.1",
    upstreamFile := "Log-Concave Sampling, book page 10 / PDF page 22",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Markov-semigroup", "transition-kernel", "conditional-expectation", "definition"],
    saldUse := "Chewi Definition 1.2.1 root: define P_t f(x) by integration against the conditional transition kernel",
    note := "Exact kernel conditional-law operator; constructing kernels from a concrete SDE remains separate."
  },
  {
    key := "martingale.chewi-definition-1-1-4",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Martingale.IsChewiMartingale",
    upstreamDecl := "MeasureTheory.Martingale / Chewi Definition 1.1.4",
    upstreamFile := "Mathlib.Probability.Martingale.Basic; Log-Concave Sampling, book page 5 / PDF page 17",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "martingale", "filtration", "conditional-expectation", "adapted"],
    saldUse := "Chewi Definition 1.1.4 root: connect the source adapted conditional-expectation law to Mathlib Martingale",
    note := "Exact real continuous-time predicate; continuity of sample paths and local martingale localization are separate properties."
  },
  {
    key := "stopping-time.chewi-definition-1-1-11",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingTime.IsChewiStoppingTime",
    upstreamDecl := "MeasureTheory.IsStoppingTime / Chewi Definition 1.1.11",
    upstreamFile := "Mathlib.Probability.Process.Stopping; Log-Concave Sampling, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "stopping-time", "filtration", "measurability", "continuous-time"],
    saldUse := "Chewi Definition 1.1.11 root: connect the source event measurability condition to Mathlib's stopping-time API",
    note := "Exact continuous nonnegative-time predicate with possible infinity; localization and stopped Ito integrals remain downstream."
  },
  {
    key := "localization.chewi-definition-1-1-12",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Localization.IsLocalizingSequence",
    upstreamDecl := "MeasureTheory.ProgMeasurable / MeasureTheory.IsStoppingTime / Chewi Definition 1.1.12",
    upstreamFile := "Mathlib.Probability.Process.Adapted; Mathlib.Probability.Process.Stopping; Log-Concave Sampling, book page 6 / PDF page 18",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "localizing-sequence", "stopping-time", "progressive", "L2", "almost-sure"],
    saldUse := "Chewi Definition 1.1.12 root: expose progressive measurability, stopped L2 finiteness, monotone stopping times, and the almost-sure terminal-time limit",
    note := "Exact definition over nonnegative time. Construction of the canonical hitting-time sequence is Proposition 1.1.13 and remains separate."
  },
  {
    key := "localization.chewi-definition-1-1-15",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Localization.IsLocalMartingale",
    upstreamDecl := "MeasureTheory.Adapted / MeasureTheory.stoppedProcess / MeasureTheory.Martingale / Chewi Definition 1.1.15",
    upstreamFile := "Mathlib.Probability.Martingale.Basic; Mathlib.Probability.Process.Stopping; Log-Concave Sampling, book page 7 / PDF page 19",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "local-martingale", "stopping-time", "stopped-process", "adapted"],
    saldUse := "Chewi Definition 1.1.15 root: define local martingales through an increasing a.s.-divergent stopping sequence and centered stopped martingales",
    note := "Exact definition; continuity and the theorem that localized Ito integrals satisfy it remain separate properties."
  },
  {
    key := "reversibility.chewi-definition-1-2-10",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Reversibility.IsReversible",
    upstreamDecl := "Chewi Definition 1.2.10",
    upstreamFile := "Log-Concave Sampling, book page 13 / PDF page 25",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "reversibility", "Hilbert-space", "self-adjoint", "semigroup"],
    saldUse := "Chewi Definition 1.2.10 root: state semigroup reversibility as symmetry on the L2(pi) Hilbert space",
    note := "Exact Hilbert-space predicate; constructing the concrete L2(pi) semigroup and invariant law remains downstream."
  },
  {
    key := "carre-du-champ.chewi-definition-1-2-12",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp.carreDuChamp",
    upstreamDecl := "Chewi Definition 1.2.12",
    upstreamFile := "Log-Concave Sampling, book page 14 / PDF page 26",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "carre-du-champ", "generator", "Dirichlet-form"],
    saldUse := "Chewi Definition 1.2.12 root: expose the bilinear generator expression used by reversibility and functional-inequality arguments",
    note := "Exact algebraic source definition for a real linear generator; positivity and diffusion identities are separate theorems."
  },
  {
    key := "carre-du-champ.chewi-lemma-1-2-13",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp.carreDuChamp_nonneg_of_markov_jensen_rightGenerator",
    upstreamDecl := "Chewi Lemma 1.2.13",
    upstreamFile := "Log-Concave Sampling, book page 14 / PDF page 26",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "carre-du-champ", "Markov-Jensen", "right-generator", "nonnegative"],
    saldUse := "Chewi Lemma 1.2.13: derive pointwise Gamma nonnegativity from the Markov Jensen inequality and the actual right-generator limits",
    note := "The proof takes the nonnegative Jensen-gap quotient to its right limit; it requires separate generator limits for f and f squared and right continuity of the orbit."
  },
  {
    key := "carre-du-champ.chewi-theorem-1-2-14",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp.fundamental_integration_by_parts",
    upstreamDecl := "Chewi Theorem 1.2.14",
    upstreamFile := "Log-Concave Sampling, book page 14 / PDF page 26",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "carre-du-champ", "integration-by-parts", "stationarity", "reversibility"],
    saldUse := "Chewi Theorem 1.2.14: derive symmetry of the Dirichlet form and its integrated-Gamma representation from stationary and symmetric generator identities",
    note := "All three expanded Gamma terms carry explicit integrability hypotheses; the concrete semigroup supplies stationarity and symmetry separately."
  },
  {
    key := "carre-du-champ.chewi-corollary-1-2-15",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp.negativeGenerator_quadratic_nonneg",
    upstreamDecl := "Chewi Corollary 1.2.15",
    upstreamFile := "Log-Concave Sampling, book page 14 / PDF page 26",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "generator", "nonnegative", "Dirichlet-form", "carre-du-champ"],
    saldUse := "Chewi Corollary 1.2.15: turn pointwise Gamma nonnegativity and stationary integration by parts into nonnegativity of the negative-generator quadratic form",
    note := "Consumes the exact Theorem 1.2.14 leaf; proving Gamma nonnegative from Markov Jensen is Lemma 1.2.13 and remains separate."
  },
  {
    key := "carre-du-champ.chewi-definition-1-2-28",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp.iteratedCarreDuChamp",
    upstreamDecl := "Chewi Definition 1.2.28",
    upstreamFile := "Log-Concave Sampling, book page 18 / PDF page 30",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "carre-du-champ", "Gamma2", "generator"],
    saldUse := "Chewi Definition 1.2.28 root: define the iterated carre du champ consumed by the Bakry-Emery criterion",
    note := "Exact algebraic source definition; concrete Langevin Hessian identification remains downstream."
  },
  {
    key := "carre-du-champ.chewi-definition-1-2-29",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp.SatisfiesBakryEmery",
    upstreamDecl := "Chewi Definition 1.2.29",
    upstreamFile := "Log-Concave Sampling, book page 19 / PDF page 31",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Bakry-Emery", "curvature-dimension", "Gamma2"],
    saldUse := "Chewi Definition 1.2.29 root: state CD(alpha,infinity) with the source positivity requirement on alpha",
    note := "Exact predicate only; the Bakry-Emery implication and concrete Langevin curvature calculation are separate theorem routes."
  },
  {
    key := "markov-semigroup.chewi-lemma-1-2-2",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.chewi_lemma_1_2_2",
    upstreamDecl := "Kernel.lintegral_id' / Kernel.lintegral_comp",
    upstreamFile := "Mathlib.Probability.Kernel.Basic; Mathlib.Probability.Kernel.Composition.Comp; Chewi Lemma 1.2.2",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Markov-semigroup", "transition-kernel", "Chapman-Kolmogorov"],
    saldUse := "Chewi Chapter 1.2 root: derive the observable semigroup laws from identity and Chapman-Kolmogorov transition kernels",
    note := "Compiled kernel-to-operator theorem. It does not construct a concrete diffusion kernel or prove time continuity."
  },
  {
    key := "markov-semigroup.constant-preservation",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.markovOperator_const",
    upstreamDecl := "lintegral_const with probability-kernel mass one",
    upstreamFile := "Mathlib.Probability.Kernel.Basic; Mathlib.MeasureTheory.Integral.Lebesgue.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Markov-operator", "constant-preservation", "probability-kernel"],
    saldUse := "Record the conservative Markov-operator law P_t 1 = 1",
    note := "The ENNReal observable type encodes nonnegativity; the Markov instance supplies total mass one."
  },
  {
    key := "markov-semigroup.monotonicity",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.markovOperator_apply_mono",
    upstreamDecl := "lintegral_mono",
    upstreamFile := "Mathlib.MeasureTheory.Integral.Lebesgue.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Markov-operator", "monotonicity", "lintegral"],
    saldUse := "Order-preserving interface for semigroup comparison and contraction arguments",
    note := "Pointwise monotonicity on measurable ENNReal-valued observables."
  },
  {
    key := "feller-semigroup.kernel-operator-contraction",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.norm_fellerOperator_apply_le",
    upstreamDecl := "norm_integral_le_of_norm_le_const",
    upstreamFile := "Mathlib.MeasureTheory.Integral.Bochner.Basic; Mathlib.Topology.ContinuousMap.Bounded.Normed",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Feller", "Markov-operator", "contraction", "sup-norm"],
    saldUse := "Place transition-kernel operators on the bounded-continuous sup-norm space",
    note := "The norm bound is derived from probability-kernel integration, not postulated as an operator contract."
  },
  {
    key := "feller-semigroup.chewi-display-1-2-11",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.sq_fellerOperator_apply_le",
    upstreamDecl := "ConvexOn.map_integral_le / Even.convexOn_pow",
    upstreamFile := "Mathlib.Analysis.Convex.Integral; Mathlib.Analysis.Convex.Mul; Chewi display (1.2.11)",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Feller", "Markov-operator", "Jensen", "carre-du-champ"],
    saldUse := "Supply the finite-time Jensen gap used to prove non-negativity of the carre du champ",
    note := "The square inequality is proved by Jensen integration under the probability transition kernel; it is not stored as a semigroup-contract assumption."
  },
  {
    key := "feller-semigroup.operator-semigroup-law",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.fellerOperator_add",
    upstreamDecl := "Kernel.integral_comp",
    upstreamFile := "Mathlib.Probability.Kernel.Composition.IntegralCompProd",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Feller", "continuous-linear-map", "Chapman-Kolmogorov", "semigroup"],
    saldUse := "Convert Chapman-Kolmogorov into composition of bounded continuous linear operators",
    note := "This is the concrete kernel-to-operator bridge needed before the norm-topology generator layer."
  },
  {
    key := "feller-semigroup.continuous-linear-semigroup",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.continuousLinearSemigroupOfFeller",
    upstreamDecl := "fellerOperator_zero / fellerOperator_add",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Feller", "continuous-linear-semigroup", "generator-boundary"],
    saldUse := "Supply the exact ContinuousLinearSemigroup consumed by the right-generator definitions",
    note := "Strong continuity in time and concrete diffusion construction remain explicit downstream obligations."
  },
  {
    key := "operator-generator.chewi-definition-1-2-3",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator.HasRightGeneratorAt",
    upstreamDecl := "Chewi Definition 1.2.3",
    upstreamFile := "Log-Concave Sampling, book page 11 / PDF page 23",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "generator", "right-derivative", "Tendsto", "domain"],
    saldUse := "Chewi Definition 1.2.3 root: state the infinitesimal generator as an actual right difference-quotient limit",
    note := "Exact norm-topology relation on the explicit generator domain; concrete differential-operator identification remains separate."
  },
  {
    key := "operator-generator.domain-invariance",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator.generatorDomain_map",
    upstreamDecl := "ContinuousLinearMap.continuous / Tendsto.comp / semigroup commutation",
    upstreamFile := "Mathlib.Analysis.Normed.Operator.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "generator", "domain", "semigroup", "Tendsto"],
    saldUse := "Chewi Definition 1.2.3 root: preserve the right-generator domain under P_t",
    note := "Abstract norm-topology generator theorem; no closed differential Langevin operator is identified."
  },
  {
    key := "operator-generator.kolmogorov-backward-right",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator.kolmogorov_backward_right",
    upstreamDecl := "rightDifferenceQuotient_map / Tendsto.comp",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator; Mathlib.Analysis.Normed.Operator.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Kolmogorov-backward", "right-derivative", "generator", "semigroup"],
    saldUse := "Chewi Proposition 1.2.5 root: prove the right orbit quotient tends to P_t Lf",
    note := "Actual one-sided Tendsto theorem; the forward equation and concrete Langevin generator remain downstream."
  },
  {
    key := "operator-generator.strong-continuity-right-orbit",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.StronglyContinuousSemigroup.tendsto_op_add",
    upstreamDecl := "ContinuousLinearSemigroup.op_add_apply / ContinuousLinearMap.continuous",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain; Mathlib.Analysis.Normed.Operator.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "strong-continuity", "semigroup", "right-orbit", "Tendsto"],
    saldUse := "Chewi Definition 1.2.1 topology leaf: propagate strong continuity at zero to every right-shifted orbit",
    note := "Abstract orbit-continuity theorem. It does not prove that a concrete Langevin or general Feller semigroup is strongly continuous on a selected observable space."
  },
  {
    key := "operator-generator.right-limit-unique",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.hasRightGeneratorAt_unique",
    upstreamDecl := "tendsto_nhds_unique",
    upstreamFile := "Mathlib.Topology.Sequences; Mathlib.Topology.Order.DenselyOrdered",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "generator", "right-derivative", "uniqueness", "Tendsto"],
    saldUse := "Chewi Definition 1.2.3 root: make the norm-topology right-generator relation single-valued",
    note := "Uniqueness uses the nontrivial positive-time filter at zero; generator existence remains encoded by domain membership."
  },
  {
    key := "operator-generator.domain-submodule",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.generatorDomainSubmodule",
    upstreamDecl := "hasRightGeneratorAt_zero / hasRightGeneratorAt_add / hasRightGeneratorAt_smul",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "generator", "domain", "submodule", "linearity"],
    saldUse := "Package the right-generator domain as a real submodule before constructing the infinitesimal operator",
    note := "This is an algebraic domain theorem for an abstract continuous-linear semigroup, not a closedness or smooth-core theorem."
  },
  {
    key := "operator-generator.linear-map",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.rightGenerator",
    upstreamDecl := "hasRightGeneratorAt_unique / generatorDomainSubmodule",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "generator", "linear-map", "domain", "right-derivative"],
    saldUse := "Chewi Definition 1.2.3 root: bundle the unique infinitesimal generator value as a linear map on its domain",
    note := "The codomain is the ambient normed space. Continuity, closedness, and equality with the Langevin differential expression are not asserted."
  },
  {
    key := "operator-generator.bundled-domain-commutation",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.rightGenerator_map",
    upstreamDecl := "generatorDomain_map / hasRightGeneratorAt_unique",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "generator", "semigroup", "commutation", "invariant-domain"],
    saldUse := "Chewi Proposition 1.2.5 root: state L(P_t f)=P_t(Lf) using the canonical bundled generator",
    note := "The theorem applies on the abstract right-generator domain preserved by the semigroup."
  },
  {
    key := "operator-generator.kolmogorov-backward-bundled",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.kolmogorov_backward_right_generator",
    upstreamDecl := "kolmogorov_backward_right / rightGeneratorValue_spec",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Kolmogorov-backward", "generator", "right-derivative", "bundled-domain"],
    saldUse := "Chewi Proposition 1.2.5 root: expose the right backward equation without an existential generator witness",
    note := "Actual one-sided norm-limit theorem on the bundled domain; it does not identify a concrete Langevin generator or prove a forward equation."
  },
  {
    key := "weak-generator.sample-to-law-derivative",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakGenerator.weakGeneratorFromSampleDerivative",
    upstreamDecl := "law-map integral rewrite plus supplied Ito-generator derivative",
    upstreamFile := "Mathlib Measure.map / ASTIS lawIntegralHasDerivAtOfMeasureMapEqAndSample",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["weak-generator", "weak-FP", "law-map", "HasDerivAt", "SDE"],
    saldUse := "rewrite frozen EM/Ito sample-space weak-test derivative into a named law-level weak-generator identity",
    note := "Pro-assimilated leaf: does not prove Ito, density PDE, or conditional drift construction."
  },
  {
    key := "fokker-planck.scalar-divergence-rewrite",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FokkerPlanckAlgebra.fpRewriteScalarAlgebra",
    upstreamDecl := "elementary scalar algebra after supplied FP/divergence identities",
    upstreamFile := "local Mathlib ring proof",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["weak-FP", "Fokker-Planck", "scalar-algebra", "divergence"],
    saldUse := "post-process supplied weak FP and score-split identities before Fisher/IBP handoff",
    note := "Small Mathlib-ready algebra leaf; analytic PDE hypotheses remain explicit contracts."
  },
  {
    key := "fisher.ibp.scalar-algebra",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FokkerPlanckAlgebra.fisherIbpAlgebra",
    upstreamDecl := "elementary scalar algebra after supplied IBP identities",
    upstreamFile := "local Mathlib ring proof",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Fisher-information", "IBP", "scalar-algebra"],
    saldUse := "combine two no-boundary IBP identities into the Fisher/cross-term display",
    note := "Small Mathlib-ready algebra leaf; no-boundary/divergence theorem remains a separate analytic contract."
  },
  {
    key := "langevin.gibbs-weighted-generator-hasDerivAt-1d",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.hasDerivAt_gibbsWeight_mul_testDeriv_eq_langevinGenerator_1d",
    upstreamDecl := "ordinary product rule plus derivative of exp(-V)",
    upstreamFile := "Mathlib.Analysis.SpecialFunctions.ExpDeriv; Mathlib.Analysis.Calculus.Deriv.Mul",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Gibbs-weight", "generator", "one-dimensional", "pointwise-calculus"],
    saldUse := "Chewi SDE/DENS root: pointwise bridge from the 1D overdamped generator `f'' - V' * f'` to the derivative of the Gibbs-weighted test derivative",
    note := "This is only a pointwise ordinary-derivative calculation. Stationarity, reversibility, IBP, boundary terms, generator domains, and invariant-law proofs remain separate red obligations."
  },
  {
    key := "langevin.gibbs-weighted-generator-deriv-1d",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.deriv_gibbsWeight_mul_testDeriv_eq_langevinGenerator_1d",
    upstreamDecl := "hasDerivAt_gibbsWeight_mul_testDeriv_eq_langevinGenerator_1d",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Gibbs-weight", "generator", "one-dimensional", "deriv"],
    saldUse := "Chewi SDE/DENS root: rewrite `(exp(-V) f')'` as the Gibbs weight times the 1D overdamped generator expression",
    note := "Derivative-form wrapper only; it does not assert integration by parts, zero boundary term, stationarity, reversibility, or a normalized Gibbs law."
  },
  {
    key := "langevin.gibbs-weighted-divergence-generator-algebra",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.weightedDivergence_gibbsWeight_langevinGenerator_algebra",
    upstreamDecl := "inner-product algebra after supplied product rule and Gibbs-weight gradient identity",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin; Mathlib.Analysis.InnerProductSpace.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Gibbs-weight", "generator", "divergence-form", "inner-product", "algebra"],
    saldUse := "Chewi SDE/DENS root: assemble `div (rho ∇f) = rho * (lapF - <∇V, ∇f>)` once product-rule and chain-rule facts are supplied",
    note := "Supplied-hypothesis algebra only. It does not define or prove gradient, divergence, Laplacian, product rule, chain rule, IBP, boundary decay, stationarity, reversibility, or invariant Gibbs law."
  },
  {
    key := "langevin.exp-neg-weighted-divergence-generator-algebra",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.expNeg_weightedDivergence_langevinGenerator_algebra",
    upstreamDecl := "weightedDivergence_gibbsWeight_langevinGenerator_algebra with `rho = exp (-Vx)`",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Gibbs-weight", "exp-neg-potential", "divergence-form", "algebra"],
    saldUse := "Chewi SDE/DENS root: source-facing `exp(-V)` weighted-divergence algebra before the invariant Gibbs and reversibility proof branches",
    note := "Thin source-facing wrapper only; all analytic facts about `∇ exp(-V)`, divergence theorem, no-boundary terms, domains, and invariant laws remain red obligations."
  },
  {
    key := "langevin.finite-coordinate-weighted-divergence-generator-algebra",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteCoord_weightedDivergence_langevinGenerator_algebra",
    upstreamDecl := "finite-coordinate summation algebra after supplied coordinate product and chain rules",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin; Mathlib.Algebra.BigOperators.Fin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "finite-coordinate", "Gibbs-weight", "divergence-form", "Laplacian", "algebra"],
    saldUse := "Chewi SDE/DENS root: aggregate coordinate identities `∂ᵢ(rho ∂ᵢf)` into the finite-sum Langevin divergence-form expression",
    note := "Supplied-hypothesis finite-sum algebra only. It does not define/prove partial derivatives, gradients, divergence, Laplacian, product rule, chain rule, IBP, boundary decay, stationarity, reversibility, or invariant Gibbs law."
  },
  {
    key := "langevin.finite-coordinate-named-weighted-divergence-generator-algebra",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteCoord_named_weightedDivergence_langevinGenerator_algebra",
    upstreamDecl := "finiteCoord_weightedDivergence_langevinGenerator_algebra plus supplied names for divergence, Laplacian, and gradient inner-product sums",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "finite-coordinate", "Laplacian", "gradient-inner-product", "divergence-form", "algebra"],
    saldUse := "Chewi SDE/DENS root: source-facing finite-coordinate handoff to named `divWeighted = rho * (lapF - innerGradVGradF)`",
    note := "Names the coordinate sums only after they are supplied as hypotheses; true Euclidean `grad/div/laplace` API wrappers and analytic regularity remain red obligations."
  },
  {
    key := "langevin.finite-coordinate-toLp-inner-weighted-divergence-generator-algebra",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteCoord_toLpInner_weightedDivergence_langevinGenerator_algebra",
    upstreamDecl := "finite-coordinate Langevin algebra plus EuclideanSpace `WithLp.toLp 2` inner-product coordinate bridge",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin; AutoSamplingTheory.TechnicalLemmas.Geometry.EuclideanSpaceCoordinates",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "finite-coordinate", "EuclideanSpace", "WithLp.toLp", "gradient-inner-product", "algebra"],
    saldUse := "Chewi SDE/DENS root: rewrite finite-coordinate Langevin divergence algebra with Mathlib `EuclideanSpace` inner-product notation after coordinate representatives are supplied",
    note := "Coordinate-to-inner-product notation bridge only; it does not define/prove gradients, divergence, Laplacian, IBP, stationarity, reversibility, or invariant Gibbs law."
  },
  {
    key := "langevin.finite-coordinate-euclidean-inner-weighted-divergence-generator-algebra",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteCoord_euclideanInner_weightedDivergence_langevinGenerator_algebra",
    upstreamDecl := "finite-coordinate Langevin algebra plus direct `EuclideanSpace` inner-product coordinate bridge",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin; AutoSamplingTheory.TechnicalLemmas.Geometry.EuclideanSpaceCoordinates",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "finite-coordinate", "EuclideanSpace", "gradient-inner-product", "algebra"],
    saldUse := "Chewi SDE/DENS root: source-facing finite-dimensional handoff to `divWeighted = rho * (lapF - inner ℝ gradV gradF)`",
    note := "Direct Euclidean inner-product wrapper after supplied coordinate product-rule, chain-rule, and Laplacian identities; analytic generator and invariant-law theorems remain red."
  },
  {
    key := "langevin.finite-euclidean-generator-basis-display",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteEuclidean_langevinGenerator_basisDisplay",
    upstreamDecl := "InnerProductSpace.laplacian_eq_iteratedFDeriv_orthonormalBasis plus EuclideanSpace inner-product coordinate bridge",
    upstreamFile := "Mathlib.Analysis.InnerProductSpace.Laplacian; AutoSamplingTheory.TechnicalLemmas.Geometry.EuclideanSpaceCoordinates",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "finite-dimensional", "EuclideanSpace", "generator-display", "Laplacian", "gradient-inner-product", "basisFun"],
    saldUse := "Chewi SDE/ANALYSIS root: pointwise finite-dimensional display of the formal differential expression `Laplacian.laplacian f x - inner ℝ (gradient V x) (gradient f x)` using Mathlib's `EuclideanSpace.basisFun` coordinate basis",
    note := "Pointwise display leaf only using Mathlib's total `gradient` and `Laplacian.laplacian` definitions. It does not prove divergence, product rules for `rho ∇f`, IBP, boundary decay, generator domains, stationarity, reversibility, invariant Gibbs law, or KL/FI dissipation."
  },
  {
    key := "langevin.finite-euclidean-generator-coordinate-display",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteEuclidean_langevinGenerator_coordinateDisplay",
    upstreamDecl := "finiteEuclidean_langevinGenerator_basisDisplay plus EuclideanSpace.basisFun_apply",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin; Mathlib.Analysis.InnerProductSpace.PiL2",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "finite-dimensional", "EuclideanSpace", "generator-display", "Laplacian", "gradient-inner-product", "coordinate-unit"],
    saldUse := "Chewi SDE/ANALYSIS root: pointwise coordinate-unit display of the formal Langevin differential expression `Δ f - <∇V, ∇f>` as diagonal second derivatives minus the gradient-coordinate inner-product sum",
    note := "Explicit coordinate-unit display only. It requires `[DecidableEq ι]` to unfold `EuclideanSpace.basisFun` to `EuclideanSpace.single i 1`; it does not assert divergence, IBP, stationarity, reversibility, invariant Gibbs law, or KL/FI dissipation."
  },
  {
    key := "langevin.finite-euclidean-weighted-divergence-basis-handoff",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteEuclidean_weightedDivergence_langevinGenerator_basisHandoff",
    upstreamDecl := "finiteCoord_weightedDivergence_langevinGenerator_algebra plus finiteEuclidean_langevinGenerator_basisDisplay",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "finite-dimensional", "EuclideanSpace", "weighted-divergence", "supplied-hypothesis", "basisFun", "handoff"],
    saldUse := "Chewi SDE/DENS root: after coordinate product-rule and Gibbs-weight chain-rule facts are supplied, rewrite the finite weighted-divergence sum as `rho * (Laplacian.laplacian f x - inner ℝ (gradient V x) (gradient f x))` using the basis display",
    note := "Supplied-hypothesis handoff only. It does not prove coordinate product rules, the a.e. bridge or box-integrability assumptions needed by the compiled box divergence wrapper, IBP, boundary decay, semigroup generator theorem, stationarity, reversibility, invariant Gibbs law, or KL/FI dissipation."
  },
  {
    key := "langevin.finite-euclidean-weighted-divergence-coordinate-handoff",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteEuclidean_weightedDivergence_langevinGenerator_coordinateHandoff",
    upstreamDecl := "finiteCoord_weightedDivergence_langevinGenerator_algebra plus finiteEuclidean_langevinGenerator_coordinateDisplay",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "finite-dimensional", "EuclideanSpace", "weighted-divergence", "supplied-hypothesis", "coordinate-unit", "handoff"],
    saldUse := "Chewi SDE/DENS root: explicit coordinate-unit handoff from supplied coordinate divergence/product-rule and Gibbs-weight chain-rule facts to the Mathlib expression `rho * (Δ f - <∇V, ∇f>)`",
    note := "Coordinate-unit supplied-hypothesis handoff only. It does not prove divergence theorem, product rule, IBP, no-boundary term, semigroup/Ito generator result, stationarity, reversibility, invariant Gibbs law, or KL/FI dissipation."
  },
  {
    key := "langevin.finite-euclidean-exp-neg-weighted-divergence-basis-handoff",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteEuclidean_expNeg_weightedDivergence_langevinGenerator_basisHandoff",
    upstreamDecl := "finiteEuclidean_weightedDivergence_langevinGenerator_basisHandoff plus gradient_expNegPotential_coordinate_eq_of_differentiableAt",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "finite-dimensional", "EuclideanSpace", "Gibbs-weight", "weighted-divergence", "DifferentiableAt", "basisFun", "handoff"],
    saldUse := "Chewi SDE/DENS root: basis-coordinate handoff from supplied divergence sum and coordinate product-rule facts to `exp(-V x) * (Laplacian.laplacian f x - inner ℝ (gradient V x) (gradient f x))`, with the Gibbs-weight chain-rule coordinate equality discharged from `DifferentiableAt ℝ V x`",
    note := "Discharges only the Gibbs-weight chain-rule hypothesis used by the earlier supplied-hypothesis handoff. Coordinate product rules, the a.e. bridge and box-integrability assumptions needed by the compiled box divergence wrapper, IBP, boundary decay, semigroup/Ito generator theorem, stationarity, reversibility, invariant Gibbs law, and KL/FI remain red."
  },
  {
    key := "langevin.finite-euclidean-exp-neg-weighted-divergence-coordinate-handoff",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteEuclidean_expNeg_weightedDivergence_langevinGenerator_coordinateHandoff",
    upstreamDecl := "finiteEuclidean_weightedDivergence_langevinGenerator_coordinateHandoff plus gradient_expNegPotential_coordinate_eq_of_differentiableAt",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "finite-dimensional", "EuclideanSpace", "Gibbs-weight", "weighted-divergence", "DifferentiableAt", "coordinate-unit", "handoff"],
    saldUse := "Chewi SDE/DENS root: coordinate-unit handoff from supplied divergence sum and coordinate product-rule facts to `exp(-V x) * (Δ f - <∇V, ∇f>)`, with the Gibbs-weight chain-rule coordinate equality discharged from `DifferentiableAt ℝ V x`",
    note := "Discharges only the Gibbs-weight chain-rule hypothesis for the coordinate-unit display. It does not prove divergence theorem, coordinate product rule, IBP/no-boundary term, semigroup/Ito generator result, stationarity, reversibility, invariant Gibbs law, or KL/FI dissipation."
  },
  {
    key := "langevin.finite-euclidean-exp-neg-lineDeriv-fderiv-coordinate-sum-display",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteEuclidean_expNeg_lineDeriv_fderiv_coordinateSum_langevinGenerator_display",
    upstreamDecl := "lineDeriv_expNegPotential_mul_fderiv_coordinate_eq plus finiteEuclidean_expNeg_weightedDivergence_langevinGenerator_coordinateHandoff",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "finite-dimensional", "EuclideanSpace", "Gibbs-weight", "lineDeriv", "fderiv", "coordinate-sum", "generator-display", "coordinate-unit"],
    saldUse := "Chewi Ch.1 Langevin root: aggregate the compiled coordinate line-derivative calculation for `exp(-V) * fderiv f eᵢ` into the finite Euclidean `exp(-V x) * (Δ f - <∇V, ∇f>)` display",
    note := "Pointwise finite-coordinate sum display only. It discharges the coordinate product-rule/diagonal second-derivative branch for the explicit field `exp(-V) * fderiv f eᵢ`, but keeps the gradient-coordinate bridge `fderiv ℝ f x eᵢ = (gradient f x) i` as a hypothesis. It does not define a divergence operator, assert that the sum is a divergence, prove IBP/no-boundary terms, semigroup/Ito generator domains, stationarity, reversibility, invariant Gibbs law, or KL/FI."
  },
  {
    key := "langevin.finite-euclidean-exp-neg-lineDeriv-fderiv-coordinate-sum-display-differentiable",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteEuclidean_expNeg_lineDeriv_fderiv_coordinateSum_langevinGenerator_display_of_differentiableAt",
    upstreamDecl := "finiteEuclidean_expNeg_lineDeriv_fderiv_coordinateSum_langevinGenerator_display plus fderiv_apply_coordinate_eq_gradient_coordinate_of_differentiableAt",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "finite-dimensional", "EuclideanSpace", "Gibbs-weight", "lineDeriv", "fderiv", "coordinate-sum", "generator-display", "DifferentiableAt"],
    saldUse := "Chewi Ch.1 Langevin root: coordinate-sum display for `exp(-V) * fderiv f eᵢ` with the local `fderiv`-coordinate-to-`gradient`-coordinate bridge discharged from `DifferentiableAt ℝ f x`",
    note := "Pointwise finite-coordinate sum display only. It removes the earlier supplied `hgradF` hypothesis by adding the explicit assumption `DifferentiableAt ℝ f x`. It does not assert that differentiability of `fun y => fderiv ℝ f y` implies differentiability of `f`; it does not define divergence, prove that the sum is divergence, prove IBP/no-boundary terms, semigroup/Ito generator domains, stationarity, reversibility, invariant Gibbs law, or KL/FI."
  },
  {
    key := "langevin.coordinate-divergence-exp-neg-fderiv-display",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.coordinateDivergence_expNeg_fderivCoordinateField_langevinGenerator_display_of_differentiableAt",
    upstreamDecl := "coordinateDivergence_eq_sum_lineDeriv plus finiteEuclidean_expNeg_lineDeriv_fderiv_coordinateSum_langevinGenerator_display_of_differentiableAt",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "coordinate-divergence", "finite-dimensional", "EuclideanSpace", "Gibbs-weight", "fderiv", "generator-display", "DifferentiableAt"],
    saldUse := "Chewi Ch.1 Langevin root: rewrite the explicit Gibbs-weighted first-derivative coordinate field through ASTIS `coordinateDivergence` and recover `exp(-V x) * (Delta f - <grad V, grad f>)` pointwise",
    note := "Named coordinate-divergence display only. It packages the compiled pointwise coordinate sum; it does not prove Mathlib's divergence theorem, weighted IBP, boundary decay, semigroup/Ito generator domains, invariant Gibbs law, reversibility, or KL/FI."
  },
  {
    key := "langevin.trace-exp-neg-fderiv-coordinate-field-display",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.trace_expNeg_fderivCoordinateField_langevinGenerator_display_of_hasFDerivAt",
    upstreamDecl := "coordinateDivergence_wrapped_toPi_trace_of_hasFDerivAt plus coordinateDivergence_expNeg_fderivCoordinateField_langevinGenerator_display_of_differentiableAt",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "trace-style", "Pi-space", "Gibbs-weight", "fderiv", "generator-display", "HasFDerivAt"],
    saldUse := "Chewi Ch.1 Langevin root: pointwise handoff from Mathlib's Pi-space finite-box trace summand for the explicit field `exp(-V) * fderiv f eᵢ` to the scalar display `exp(-V) * (Delta f - <grad V, grad f>)`",
    note := "Pointwise trace-display handoff only. It assumes the explicit Pi-space field has the supplied Frechet derivative and the needed pointwise differentiability hypotheses. It does not prove differentiability on a box, continuity, integrability, a divergence theorem, boundary cancellation, weighted IBP, generator domains, invariant Gibbs law, reversibility, stationarity, or KL/FI."
  },
  {
    key := "langevin.integrableOn-trace-exp-neg-fderiv-coordinate-field-continuous",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integrableOn_trace_expNeg_fderivCoordinateField_of_continuousOn",
    upstreamDecl := "ContinuousOn.integrableOn_compact on Set.Icc plus trace_expNeg_fderivCoordinateField_langevinGenerator_display_of_hasFDerivAt and IntegrableOn.congr_fun",
    upstreamFile := "Mathlib.MeasureTheory.Function.LocallyIntegrable; Mathlib.MeasureTheory.Integral.IntegrableOn; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "IntegrableOn", "trace-style", "closed-box", "ContinuousOn", "Gibbs-weight", "generator-display"],
    saldUse := "Chewi Ch.1 Langevin root: finite closed-box handoff proving Mathlib trace-summand `IntegrableOn` for the explicit Gibbs-weighted fderiv-coordinate field when the scalar Langevin display is already continuous on the box",
    note := "Finite closed-box regularity handoff only. It assumes the trace/display equality hypotheses and continuity of the displayed scalar RHS; it does not prove RHS continuity, field differentiability, whole-space integrability, boundary cancellation, weighted IBP, generator domains, invariant Gibbs law, reversibility, stationarity, or KL/FI."
  },
  {
    key := "langevin.continuousOn-exp-neg-generator-rhs-components",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.continuousOn_expNeg_langevinGenerator_rhs_of_components",
    upstreamDecl := "ContinuousOn.rexp / ContinuousOn.mul / ContinuousOn.sub / ContinuousOn.inner",
    upstreamFile := "Mathlib.Analysis.SpecialFunctions.Exp; Mathlib.Analysis.InnerProductSpace.Continuous; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "ContinuousOn", "closed-box", "Gibbs-weight", "generator-display", "components"],
    saldUse := "Chewi Ch.1 Langevin root: assemble component continuity of `V`, `Delta f`, `grad V`, and `grad f` into continuity of the scalar display `exp(-V) * (Delta f - <grad V, grad f>)` on a finite Pi-box",
    note := "Component-continuity assembly only. It does not derive those component continuity hypotheses from a ContDiff/test-function class, does not prove field differentiability, integrability, IBP, boundary cancellation, domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "langevin.integrableOn-trace-exp-neg-fderiv-coordinate-field-components",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integrableOn_trace_expNeg_fderivCoordinateField_of_component_continuousOn",
    upstreamDecl := "continuousOn_expNeg_langevinGenerator_rhs_of_components plus integrableOn_trace_expNeg_fderivCoordinateField_of_continuousOn",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "IntegrableOn", "trace-style", "closed-box", "ContinuousOn", "components", "Gibbs-weight"],
    saldUse := "Chewi Ch.1 Langevin root: finite closed-box trace `IntegrableOn` handoff for the explicit Gibbs-weighted fderiv-coordinate field under component continuity plus the existing trace/display differentiability hypotheses",
    note := "Finite closed-box handoff under component continuity only. It still assumes the explicit Pi-space field derivative and pointwise differentiability hypotheses; it does not derive them from a test-function class and does not prove whole-space integrability, IBP, boundary cancellation, domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "langevin.continuousOn-exp-neg-generator-rhs-contDiff",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.continuousOn_expNeg_langevinGenerator_rhs_of_contDiff",
    upstreamDecl := "continuousOn_expNeg_langevinGenerator_rhs_of_components plus continuous_gradient_of_contDiff_one and continuous_laplacian_of_contDiff_two",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "ContinuousOn", "ContDiff", "closed-box", "Gibbs-weight", "generator-display", "components"],
    saldUse := "Chewi Ch.1 Langevin root: derive closed-box continuity of `exp(-V) * (Delta f - <grad V, grad f>)` from global `C¹/C²` hypotheses",
    note := "Component continuity is now discharged from global `ContDiff` regularity. The explicit Pi-space field derivative, trace integrability without that derivative, whole-space integrability, weighted IBP, boundary cancellation, domains, invariant law, reversibility, and KL/FI remain separate obligations."
  },
  {
    key := "langevin.integrableOn-trace-exp-neg-fderiv-coordinate-field-contDiff",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integrableOn_trace_expNeg_fderivCoordinateField_of_contDiff",
    upstreamDecl := "integrableOn_trace_expNeg_fderivCoordinateField_of_component_continuousOn plus global `ContDiff` component regularity",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "IntegrableOn", "ContDiff", "trace-style", "closed-box", "Gibbs-weight"],
    saldUse := "Chewi Ch.1 Langevin root: close finite-box trace `IntegrableOn` from global `C¹/C²` regularity once the explicit Pi-space field derivative is supplied",
    note := "Finite-box trace handoff under `ContDiff ℝ 1 V`, `ContDiff ℝ 2 f`, and supplied field derivative only. It does not prove that field derivative, whole-space integrability, weighted IBP, boundary cancellation, domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "langevin.hasFDerivAt-exp-neg-fderiv-coordinate-field-contDiff",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.hasFDerivAt_expNeg_fderivCoordinateField_of_contDiff",
    upstreamDecl := "DifferentiableAt.hasFDerivAt, differentiableAt_pi, PiLp.hasFDerivAt_toLp, product/chain differentiability",
    upstreamFile := "Mathlib.Analysis.Calculus.FDeriv.Prod; Mathlib.Analysis.Calculus.FDeriv.WithLp; Mathlib.Analysis.SpecialFunctions.ExpDeriv; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "HasFDerivAt", "ContDiff", "Pi-space", "Gibbs-weight", "fderiv", "regularity"],
    saldUse := "Chewi Ch.1 Langevin root: discharge the explicit Pi-space field differentiability input for `z ↦ exp(-V(toLp z)) * fderiv f (toLp z) eᵢ` from global `C¹/C²` regularity",
    note := "The derivative representative is Mathlib's `fderiv` of the field, not a closed-form Jacobian. It does not prove a divergence theorem, whole-space integrability, weighted IBP, boundary cancellation, domains, invariant law, reversibility, or KL/FI."
  },
  {
    key := "langevin.integrableOn-trace-exp-neg-fderiv-coordinate-field-contDiff-fderiv",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integrableOn_trace_expNeg_fderivCoordinateField_of_contDiff_fderiv",
    upstreamDecl := "integrableOn_trace_expNeg_fderivCoordinateField_of_contDiff plus hasFDerivAt_expNeg_fderivCoordinateField_of_contDiff",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "IntegrableOn", "ContDiff", "HasFDerivAt", "fderiv", "trace-style", "closed-box"],
    saldUse := "Chewi Ch.1 Langevin root: finite-box trace `IntegrableOn` for the canonical `fderiv` trace of the explicit Gibbs-weighted first-derivative field under global `C¹/C²` regularity",
    note := "This removes the supplied field-derivative hypothesis only for the canonical `fderiv` representative. It remains finite-box regularity, not boundary cancellation, weighted IBP, generator-domain semantics, invariant law, reversibility, or KL/FI."
  },
  {
    key := "langevin.integrable-exp-neg-fderiv-coordinate-source-field",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integrable_expNeg_fderivCoordinateField_of_lintegral_expNeg_ne_top_of_fderiv_norm_le",
    upstreamDecl := "lintegral_ofReal_ne_top_iff_integrable, PiLp.volume_preserving_toLp, Integrable.smul_bdd, ContinuousLinearMap.le_opNorm",
    upstreamFile := "Mathlib.MeasureTheory.Function.L1Space.Integrable; Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Integrable", "whole-space", "Gibbs-weight", "fderiv", "coordinate-field", "bounded-derivative"],
    saldUse := "Chewi Ch.1 Example 1.2.8 cutoff route: discharge the concrete `Integrable G volume` premise for `G = exp(-V) * fderiv f` in raw Pi coordinates from finite Gibbs mass and a bounded first derivative",
    note := "Whole-space source-field integrability only. It uses genuine `C¹` regularity for `fderiv`, transports the scalar Gibbs weight through the volume-preserving PiLp bridge, and bounds the raw Pi sup norm coordinatewise. It does not prove a Gibbs tail, cutoff main-term convergence, weighted IBP, generator/semigroup domains, stationarity, invariant law, reversibility, or any second-order cutoff estimate."
  },
  {
    key := "langevin.integrable-exp-neg-generator-rhs-compact-test",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integrable_expNeg_langevinGenerator_rhs_of_contDiff_of_hasCompactSupport",
    upstreamDecl := "Continuous.integrable_of_hasCompactSupport, notMem_tsupport_iff_eventuallyEq, InnerProductSpace.laplacian_congr_nhds",
    upstreamFile := "Mathlib.MeasureTheory.Function.LocallyIntegrable; Mathlib.Analysis.Calculus.FDeriv.Const; Mathlib.Analysis.InnerProductSpace.Laplacian; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Integrable", "whole-space", "Gibbs-weight", "generator-display", "compact-support", "test-function"],
    saldUse := "Chewi Ch.1 Example 1.2.8 cutoff route: prove whole-space integrability of the exact Euclidean display `exp(-V) * (Delta f - <grad V, grad f>)` for a C_c^2 test function and a C^1 potential",
    note := "Concrete generator-display integrability on EuclideanSpace. Compact support is imposed on the test function; no bounded gradient of the potential and no finite Gibbs-mass premise is needed. The theorem does not prove a cutoff limit, Gibbs tail, weighted IBP, generator/semigroup domains, stationarity, invariance, reversibility, or KL/FI."
  },
  {
    key := "langevin.integrable-exp-neg-generator-rhs-compact-test-raw-pi",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integrable_expNeg_langevinGenerator_rhs_comp_toLp_of_contDiff_of_hasCompactSupport",
    upstreamDecl := "integrable_expNeg_langevinGenerator_rhs_of_contDiff_of_hasCompactSupport plus PiLp.volume_preserving_toLp",
    upstreamFile := "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Integrable", "whole-space", "Gibbs-weight", "generator-display", "compact-support", "PiLp"],
    saldUse := "Chewi Ch.1 Example 1.2.8 cutoff route: expose concrete generator-display integrability in the raw finite-Pi coordinate shape consumed by radial-cutoff dominated convergence",
    note := "Volume-preserving coordinate transport of the Euclidean compact-test theorem. It does not itself take the radial-cutoff limit or prove Gibbs tails, weighted IBP, generator/semigroup domains, stationarity, invariance, reversibility, or KL/FI."
  },
  {
    key := "langevin.integrable-exp-neg-weight-raw-pi",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integrable_expNeg_comp_toLp_of_lintegral_expNeg_ne_top",
    upstreamDecl := "lintegral_ofReal_ne_top_iff_integrable plus PiLp.volume_preserving_toLp",
    upstreamFile := "Mathlib.MeasureTheory.Function.L1Space.Integrable; Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Gibbs", "Integrable", "whole-space", "PiLp", "weight"],
    saldUse := "Chewi Ch.1 Gibbs-tail route: transport finite unnormalized Gibbs mass to an Integrable raw-Pi scalar weight shared by cutoff and tail consumers",
    note := "Integrability transport only. It does not prove tail convergence, generator-display integrability, weighted IBP, domains, stationarity, invariance, reversibility, or KL/FI."
  },
  {
    key := "langevin.gibbs-weight-tail-tendsto-zero-raw-pi",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.tendsto_setIntegral_expNeg_norm_ge_comp_toLp_of_lintegral_expNeg_ne_top",
    upstreamDecl := "integrable_expNeg_comp_toLp_of_lintegral_expNeg_ne_top plus tendsto_setIntegral_norm_norm_ge_comp_toLp",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin; AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Gibbs", "tail", "whole-space", "PiLp", "set-integral", "Tendsto"],
    saldUse := "Chewi Ch.1 Example 1.2.8 cutoff route: prove that finite unnormalized Gibbs mass outside expanding Euclidean balls tends to zero in raw-Pi coordinates",
    note := "Concrete Gibbs-tail convergence only. It does not assemble the cutoff divergence identity, prove whole-space weighted IBP, supply generator/semigroup domains, or establish stationarity, invariance, reversibility, or KL/FI."
  },
  {
    key := "langevin.whole-space-gibbs-weighted-generator-ibp-compact-test",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integral_expNeg_langevinGenerator_rhs_eq_zero_of_contDiff_of_hasCompactSupport",
    upstreamDecl := "integral_coordinateDivergence_wrapped_eq_zero_of_contDiff_of_hasCompactSupport plus trace_expNeg_fderivCoordinateField_langevinGenerator_display_of_hasFDerivAt and PiLp.volume_preserving_toLp",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin; Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Gibbs", "weighted-IBP", "whole-space", "compact-support", "generator", "EuclideanSpace"],
    saldUse := "Chewi Ch.1 Example 1.2.8: justify the omitted whole-space integration-by-parts identity integral exp(-V) * (Delta f - inner (grad V) (grad f)) = 0 for C_c^2 tests",
    note := "Compiled analytic core identity. Compact test support removes the boundary without a finite Gibbs-mass hypothesis. Closed-generator domains, semigroup differentiation, invariant probability law, reversibility, and KL/FI dissipation remain separate."
  },
  {
    key := "analysis.calculus.laplacian-product-rule",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCarreDuChamp.laplacian_mul",
    upstreamDecl := "fderiv_mul / iteratedFDeriv_two_apply / OrthonormalBasis.sum_inner_mul_inner",
    upstreamFile := "Mathlib.Analysis.Calculus.FDeriv.Mul; Mathlib.Analysis.InnerProductSpace.Laplacian",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Laplacian", "product-rule", "gradient", "C2"],
    saldUse := "Chewi Example 1.2.17 root: expand Delta(fg) with the two cross-gradient terms",
    note := "Generic finite-dimensional C2 product rule; no Langevin potential or measure is required."
  },
  {
    key := "analysis.calculus.gradient-product-rule",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCarreDuChamp.gradient_mul",
    upstreamDecl := "fderiv_mul / InnerProductSpace.gradient",
    upstreamFile := "Mathlib.Analysis.Calculus.FDeriv.Mul; Mathlib.Analysis.Calculus.Gradient.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "gradient", "product-rule", "differentiable"],
    saldUse := "Chewi Example 1.2.17 root: expand the drift action on a product before cancellation",
    note := "Generic differentiable product rule expressed in Mathlib's gradient API."
  },
  {
    key := "langevin.chewi-example-1-2-17",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCarreDuChamp.langevinCarreDuChamp_eq_inner",
    upstreamDecl := "Chewi Example 1.2.17",
    upstreamFile := "Log-Concave Sampling, book page 15 / PDF page 27",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "carre-du-champ", "gradient", "Dirichlet-form"],
    saldUse := "Chewi Example 1.2.17: identify the formal Langevin carre du champ with the gradient inner product",
    note := "Actual C2 product-rule proof for the displayed differential operator; closed semigroup-generator identification remains separate."
  },
  {
    key := "langevin-generator.compactly-supported-c2-core",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator.CompactlySupportedC2",
    upstreamDecl := "ContDiff and HasCompactSupport",
    upstreamFile := "Mathlib.Analysis.Calculus.ContDiff.Defs; Mathlib.Topology.Algebra.Support; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "generator", "domain", "core", "C2", "compact-support"],
    saldUse := "Chewi Ch.1 generator-domain branch: name the C_c^2 test core on which the whole-space Gibbs-weighted IBP theorem is proved",
    note := "Domain predicate only. It does not claim closability, closed-generator membership beyond the core, semigroup generation, invariance, or reversibility."
  },
  {
    key := "langevin-generator.displayed-operator",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator.operator",
    upstreamDecl := "Laplacian.laplacian and gradient",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.{Gradient,Laplacian}; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "generator", "operator", "Laplacian", "gradient"],
    saldUse := "Chewi Ch.1 generator-domain branch: name the displayed differential action Delta f - inner (grad V) (grad f) independently of any operator-domain claim",
    note := "Formal differential operator only. Genuine generator semantics are supplied by CoreContract rather than inferred from this definition."
  },
  {
    key := "langevin-generator.core-domain-contract",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator.CoreContract",
    upstreamDecl := "CompactlySupportedC2 and LangevinGenerator.operator",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "generator", "domain", "contract", "core", "operator-agreement"],
    saldUse := "Chewi Ch.1 generator-domain branch: require a candidate generator domain to contain C_c^2 and its action to agree there with the displayed Langevin operator",
    note := "Honest core-domain interface. It deliberately does not construct a closed operator or prove that a Markov semigroup satisfies the contract; the semigroup bridge remains separate."
  },
  {
    key := "langevin-generator.normalized-gibbs-annihilates-compact-c2-core",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator.integral_operator_normalizedGibbs_eq_zero_on_compactlySupportedC2",
    upstreamDecl := "integral_expNeg_langevinGenerator_rhs_eq_zero_of_contDiff_of_hasCompactSupport plus GibbsIntegral integral_withDensity rewrite",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.{Langevin,LangevinGenerator}; AutoSamplingTheory.TechnicalLemmas.Measure.GibbsIntegral",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "generator", "Gibbs", "normalized", "core", "infinitesimal-stationarity"],
    saldUse := "Chewi Ch.1 Example 1.2.8 to Corollary 1.2.9: prove normalized Gibbs expectation of the displayed generator is zero on the C_c^2 test core",
    note := "Compiled core-level infinitesimal stationarity. It is not semigroup invariance: Langevin evolution generally does not preserve compact support, so a domain-extension/core theorem or martingale-problem uniqueness remains necessary."
  },
  {
    key := "weak-generator.invariance-on-tests",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakGenerator.IsInvariantOn",
    upstreamDecl := "MeasureTheory.integral",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakGenerator",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "semigroup", "invariance", "measure", "test-class"],
    saldUse := "Chewi Ch.1 invariant-law branch: state invariance for nonnegative times on an explicit test class before any measure-extensionality upgrade",
    note := "Definition only. It does not assert that a test class determines measures or that a supplied operator family is Markov."
  },
  {
    key := "weak-generator.integrated-semigroup-generator-contract",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakGenerator.IntegratedSemigroupGeneratorContract",
    upstreamDecl := "HasDerivWithinAt on Ici and ContinuousOn on finite intervals",
    upstreamFile := "Mathlib.Analysis.Calculus.MeanValue; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakGenerator",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "semigroup", "generator", "domain", "right-derivative", "integral-pairing"],
    saldUse := "Chewi Ch.1 operator branch: package identity, semigroup law, domain preservation, pairing continuity, and right-generator derivative without extending time to a group",
    note := "Explicit semigroup/domain contract. It is not constructed for the Langevin SDE here; positivity, mass preservation, strong continuity, and the concrete SDE realization remain separate when needed."
  },
  {
    key := "weak-generator.semigroup-domain-to-invariance",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakGenerator.isInvariantOn_of_integral_generator_eq_zero",
    upstreamDecl := "constant_of_has_deriv_right_zero",
    upstreamFile := "Mathlib.Analysis.Calculus.MeanValue; AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakGenerator",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "semigroup", "generator", "domain", "invariance", "mean-zero"],
    saldUse := "Chewi Ch.1 Corollary 1.2.9 route: turn domain preservation, the integrated generator derivative, and zero generator mean into semigroup invariance on the domain",
    note := "Compiled abstract bridge. A concrete Langevin invariant law still requires constructing the semigroup contract and extending the Gibbs mean-zero identity from C_c^2 to its semigroup-stable domain."
  },
  {
    key := "langevin-generator.conditional-normalized-gibbs-core-invariance",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator.isInvariantOn_normalizedGibbs_on_compactlySupportedC2",
    upstreamDecl := "integral_operator_normalizedGibbs_eq_zero_on_compactlySupportedC2 plus isInvariantOn_of_integral_generator_eq_zero",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.{LangevinGenerator,WeakGenerator}",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Langevin", "Gibbs", "semigroup", "generator", "core", "conditional-invariance"],
    saldUse := "Chewi Ch.1 Corollary 1.2.9 route: compose Gibbs core annihilation with an explicit integrated-semigroup contract on C_c^2",
    note := "Compiled conditional core-level bridge. It does not construct the Langevin semigroup and does not extend equality from C_c^2 to a measure-determining or semigroup-stable domain."
  },
  {
    key := "girsanov.finite-gaussian-cylinder-integral",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Girsanov.finiteGaussianGirsanovCylinderIntegral",
    upstreamDecl := "stdGaussian_shift_integral_map_toLp / finite-dimensional Gaussian Esscher",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian; Chewi finite-dimensional Girsanov cylinder route",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Girsanov", "Gaussian", "cylindrical", "path-space", "change-of-measure"],
    saldUse := "Chewi PATH root: package the finite-dimensional Gaussian change-of-measure as a cylindrical Girsanov likelihood ratio before full Brownian path-space RN derivatives",
    note := "Compiled finite-dimensional PATH-facing wrapper; does not assert filtrations, martingales, Novikov, Ito, or continuous-time Brownian path-space Girsanov."
  },
  {
    key := "girsanov.finite-gaussian-cylinder-rn-density",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Girsanov.finiteGaussianGirsanovCylinderMeasure_eq_withDensity",
    upstreamDecl := "measurableEquiv_map_withDensity / pi_gaussianReal_withDensity_exp_shift / map_pi_eq_stdGaussian",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym; AutoSamplingTheory.TechnicalLemmas.ProbabilityDistributions.Gaussian",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Girsanov", "Radon-Nikodym", "withDensity", "Gaussian", "cylindrical", "path-space"],
    saldUse := "Chewi PATH root: identify the finite-dimensional cylindrical shifted Gaussian path measure as a withDensity tilt of centered stdGaussian",
    note := "Measure-level RN-style finite-dimensional Girsanov identity; full Brownian path-space RN derivative remains a separate analytic theorem."
  }
]

def probabilityMemory : List LemmaMemoryEntry := [
  {
    key := "gaussian-polynomial-l2-differentiation",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.GaussianPolynomialDifferentiation.actual_gaussian_polynomial_ridge_L2_frechet_derivative",
    upstreamDecl := "Original generic feature differentiation and transport supplement",
    upstreamFile := "Pinned Mathlib Gaussian moments, Frechet calculus, Hilbert adjoints, Bochner integrals and transport infimum",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["feature", "derivative", "transport", "shared"],
    saldUse := "Generic probability, optimisation and sphere-feature prerequisites; concrete model compositions remain separate",
    note := "Actual ambient Gaussian polynomial feature and directional-derivative memberships, represented bounded-linear derivative and full Frechet limit produced internally. Fresh independent source and compiler audits accepted. Final root/reader gates and postmerge purification remain separate; no whole-model or convergence credit."
  },
  {
    key := "gaussian-polynomial-derivative-bounds",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.GaussianPolynomialDifferentiation.actual_gaussian_polynomial_ridge_L2_derivative_bounds",
    upstreamDecl := "Original generic feature differentiation and transport supplement",
    upstreamFile := "Pinned Mathlib Gaussian moments, Frechet calculus, Hilbert adjoints, Bochner integrals and transport infimum",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["feature", "derivative", "transport", "shared"],
    saldUse := "Generic probability, optimisation and sphere-feature prerequisites; concrete model compositions remain separate",
    note := "Same genuine Gaussian derivative with positive uniform operator and operator-Lipschitz constants on the closed unit ball. Constants may depend on the polynomial and ambient space. Fresh independent source and compiler audits accepted. Final root/reader gates and postmerge purification remain separate; no whole-model or convergence credit."
  },
  {
    key := "gaussian-polynomial-l2-continuity",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.GaussianPolynomialAveraging.continuous_gaussian_polynomial_ridge_L2",
    upstreamDecl := "Original actual probability and Gaussian feature regularity supplement",
    upstreamFile := "Pinned Mathlib probability, Gaussian moments, L2 representatives and Bochner calculus",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["probability", "L2", "risk", "shared"],
    saldUse := "Generic probability-law calculus and feature averaging; no model completion",
    note := "Actual Gaussian polynomial L2 membership and continuity, with radial domination derived internally. Independent fresh compilation and whole-module current-context semantic review accepted. Final root/reader gates and postmerge purification separate. No spatial gradient regularity, SDE or stochastic convergence claim."
  },
  {
    key := "gaussian-polynomial-parameter-average",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.GaussianPolynomialAveraging.gaussian_polynomial_parameter_average_L2",
    upstreamDecl := "Original actual probability and Gaussian feature regularity supplement",
    upstreamFile := "Pinned Mathlib probability, Gaussian moments, L2 representatives and Bochner calculus",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["probability", "L2", "risk", "shared"],
    saldUse := "Generic probability-law calculus and feature averaging; no model completion",
    note := "Actual scalar parameter mean membership and its L2 class identified with the genuine Bochner feature mean. Independent fresh compilation and whole-module current-context semantic review accepted. Final root/reader gates and postmerge purification separate. No spatial gradient regularity, SDE or stochastic convergence claim."
  },
  {
    key := "gaussian-polynomial-joint-risk",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.GaussianPolynomialRisk.gaussian_polynomial_joint_risk_first_variation",
    upstreamDecl := "Original actual probability and Gaussian feature regularity supplement",
    upstreamFile := "Pinned Mathlib probability, Gaussian moments, L2 representatives and Bochner calculus",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["probability", "L2", "risk", "shared"],
    saldUse := "Generic probability-law calculus and feature averaging; no model completion",
    note := "Actual joint-Gaussian polynomial risk first variation and linear convexity, with finite label second moment explicit and no independence hypothesis. Independent fresh compilation and whole-module current-context semantic review accepted. Final root/reader gates and postmerge purification separate. No spatial gradient regularity, SDE or stochastic convergence claim."
  },
  {
    key := "probability.gaussian.polynomial-ridge-uniform-l2",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.GaussianPolynomialRidge.standard_gaussian_polynomial_ridge_uniform_L2",
    upstreamDecl := "Original Gaussian polynomial ridge L2 moment and isometry supplement",
    upstreamFile := "Pinned Mathlib Gaussian moments, actual linear projection and L2 pullback",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Gaussian", "polynomial", "L2", "moments", "shared"],
    saldUse := "Generic Gaussian observables and planned quadratic-feature inputs; no SALD completion",
    note := "One actual standard Gaussian polynomial moment producer. All ridge L2 membership and the common positive dimension-independent bound are derived, never supplied. Non-even and zero polynomials and dimension0 allowed. Independent fresh module/test and current-context7-slot source review accepted; exact final root/reader gates and postmerge purification remain pending. No label moment, global parameter averaging, statistical first variation or convergence claim."
  },
  {
    key := "probability.kernel.retained-coordinate-reversibility",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.KernelRetainedReversibility.retained_coordinate_isReversible",
    upstreamDecl := "Almost-everywhere conditional detailed balance lifts through a retained coordinate; ASTIS standard derived interface",
    upstreamFile := "Mathlib db584cd6d46c92f209a44c0f1c829460d327499d Kernel.IsReversible, deterministic product and Measure.compProd",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["KERN", "MEAS", "reversibility", "disintegration", "PBPS", "MCMC", "shared"],
    saldUse := "Planned actual PBPS half-turn A.2/A.9 and general component-wise Gibbs adapters; no SALD completion",
    note := "For arbitrary measurable A,B, s-finite mu and s-finite measurable kernels R,F, lift mu-a.e. fiber reversibility under R(a) to detailed balance of deterministic-fst product F under mu compProd R. General measurable joint-set sections and nonnegative Tonelli avoid rectangle-only restrictions and finite-flux assumptions. Zero/empty/infinite-flux cases allowed; no Markov premise or invariance conclusion. Independent source/compiler review at63408b2af; accepted anonymous seven-slot audit ASTIS-RT-20261017-RetainedKernelReversibility, standard three axioms only. Actual SAME-H/R disintegration, swap transport, all-L2 half-turn, full-K operator, semigroup/restart, hypocoercivity, implementation/error/cap/cost, initialization/composition and full papers remain separate."
  },
  {
    key := "probability.kernel.invariant-l2-operator",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.KernelL2.invariant_kernel_l2_operator",
    upstreamDecl := "Finite invariant Markov kernel real-L2 contraction; ASTIS standard derived interface",
    upstreamFile := "Mathlib db584cd6d46c92f209a44c0f1c829460d327499d Kernel/compProd, L2Space and Adjoint",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["KERN", "L2", "invariance", "reversibility", "PBPS", "shared"],
    saldUse := "Planned actual retained-coordinate half-turn A.2/A.9 and one-step observable B.1 adapters; no SALD completion",
    note := "For arbitrary measurable X and finite invariant mu, construct ONE real-L2 contraction realizing EVERY MemLp representative with genuine AE fiber integrability, AE order positivity, fixed one and preserved mean. Reversibility additionally yields self-adjointness of the SAME operator; order positivity is not Hilbert PSD. Zero measure and empty X allowed. Independent source/compiler and anonymous reconstruction accepted at14cebf6c; standard three axioms only. Shared prerequisite, not completion of actual PBPS A.2/B.1, semigroup, density evolution, hypocoercivity, rate, implementation/error/cap/expected cost, initialization/composition or any paper."
  },
  {
    key := "probability.balanced-count.actual-clt"
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.BalancedRademacherCLT.balanced_count_sum_tendsto_gaussian"
    upstreamDecl := "Mathlib tendsto_charFun_inv_sqrt_mul_pow / ProbabilityMeasure.tendsto_iff_tendsto_charFun"
    upstreamFile := "Mathlib db584cd6d46c92f209a44c0f1c829460d327499d; ASTIS authored actual Bool-count adapter"
    status := LemmaMemoryStatus.formalizedLocal
    tags := ["probability", "CLT", "Rademacher", "Gaussian-background", "SPHMC"]
    saldUse := "Actual SPHMC omitted Gaussian LSI background; no SALD or second-route admission"
    note := "Actual allN normalized counting pushforward of signed sum divided by sqrtN; N0 Dirac0 and successor weak convergence to mean0 variance1 real Gaussian. Actual coordinate moments and finite CF factorization internally produced. Pinned Mathlib scalar CLT and Levy reused; no entropy, full flip-energy, GaussianLSI/T2 or cost upgrade. Independently verified 4d9e71a6b835b54453a5cfd2d132f9a51f66f69c; paper main and actual cost/composition remain open."
  },
  {
    key := "probability.empirical-covariance.preconcentration",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.EmpiricalCovariance.empiricalSecondMoment_preconcentration",
    upstreamDecl := "rank-one empirical covariance decomposition before matrix Bernstein",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Probability.EmpiricalCovariance",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["empirical-covariance", "rank-one", "matrix-concentration", "operator-norm", "finite-sample"],
    saldUse := "prepare bounded symmetric centered summands for a later matrix Bernstein or covariance concentration theorem",
    note := "Proves the exact normalized decomposition, symmetry, and the 2 B^2 summand bound. It does not assume or prove a matrix concentration tail inequality."
  },
  {
    key := "probability.empirical-covariance.unit-rank-one-moments",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.EmpiricalCovariance.unitRankOneCovariance_momentPackage",
    upstreamDecl := "unit-vector rank-one covariance first and centered second operator moments",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Probability.EmpiricalCovariance",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["empirical-covariance", "rank-one", "matrix-concentration", "operator-moment", "Bochner-integral"],
    saldUse := "derive self-adjointness, contraction, the exact C-C^2 centered second moment, and its norm bound before invoking self-adjoint matrix Bernstein",
    note := "Compiles the model-independent expectation calculation for unit-vector samples. Independence, noncommutative MGF estimates, trace bounds, probability tails, and sample complexity remain outside the theorem."
  },
  {
    key := "probability.kernel-hybrid.mean-event-error",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.KernelHybridTelescope.abs_real_comp_sub_le_integral_eventBound",
    upstreamDecl := "Chen--Chewi--Lu--Zhang arXiv:2609.38710v1, Lemma 6.3 mean one-call replacement step",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Probability.KernelHybridTelescope; arXiv:2609.38710v1 Section 6.1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Gaussian-cloud", "Markov-kernel", "total-variation", "adaptive", "integrability"],
    saldUse := "average a history-dependent eventwise kernel discrepancy under the ideal history law without replacing expectation by a uniform bound",
    note := "Compiled reusable core only. It does not construct the paper's dummy-filled histories or prove the local conditional-error estimates."
  },
  {
    key := "probability.kernel-hybrid.common-suffix",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.KernelHybridTelescope.abs_real_comp_commonSuffix_sub_le_integral_eventBound",
    upstreamDecl := "Lemma 6.3 adjacent-hybrid replacement plus Markov-kernel data processing",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Probability.{KernelHybridTelescope,KernelTotalVariation}; arXiv:2609.38710v1 Section 6.1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Gaussian-cloud", "Markov-kernel", "total-variation", "adaptive", "data-processing"],
    saldUse := "absorb every later adaptive call into one common Markov suffix while preserving the mean one-slot error",
    note := "The downstream suffix must be the same Markov kernel on both adjacent hybrids; history construction and measurability stay separate."
  },
  {
    key := "probability.kernel-hybrid.finite-adjacent-sum",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Probability.KernelHybridTelescope.abs_sub_zero_le_sum_range_of_adjacent",
    upstreamDecl := "Lemma 6.3 finite adjacent-hybrid triangle telescope",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Probability.KernelHybridTelescope; arXiv:2609.38710v1 Section 6.1",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Gaussian-cloud", "hybrid-argument", "error-budget", "finite-sum"],
    saldUse := "sum the event discrepancy of adjacent ideal/implemented hybrids across a fixed finite call cap",
    note := "Real-valued telescope only; the capped/uncapped comparison epsilon_cap is an independent source obligation."
  }
]

def klDensityMemory : List LemmaMemoryEntry := [
  {
    key := "kl-density.pointwise-derivative-simplify",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.InformationTheory.KLDensity.klPointwiseDerivSimplify",
    upstreamDecl := "pointwise real-field algebra for d/ds q log(q/p)",
    upstreamFile := "local Mathlib field_simp/ring proof",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["KL", "density", "log", "pointwise-algebra"],
    saldUse := "separate KL density derivative algebra from dominated differentiation and density regularity assumptions",
    note := "Pro-assimilated leaf; positivity/nonzero and domination stay explicit."
  },
  {
    key := "kl-density.remove-mass-term",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.InformationTheory.KLDensity.klDerivativeRemoveMassTerm",
    upstreamDecl := "mass-conservation derivative simplification",
    upstreamFile := "local Mathlib HasDerivAt congruence/simp proof",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["KL", "density", "mass-conservation", "HasDerivAt"],
    saldUse := "remove the integral qdot term in KL differentiation after mass conservation is supplied",
    note := "Small derivative-target rewrite; mass conservation itself remains a separate theorem or hypothesis."
  },
  {
    key := "information-theory.tilted-log-ratio-ae-representative",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.InformationTheory.TiltedLogRatio.llr_tilted_tilted_ae",
    upstreamDecl := "MeasureTheory.llr_tilted_left / llr_tilted_right / llr_self",
    upstreamFile := "Mathlib.MeasureTheory.Measure.Tilted; Lee--Shen--Tian arXiv:2010.03106v4 equation (11)",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["SPHMC", "RGO", "log-density-ratio", "tilted-measure", "representative", "almost-everywhere"],
    saldUse := "SPHMC Lemma 4.17: identify the canonical measurable llr almost everywhere with the explicit normalized tilt representative",
    note := "The equality is under the left tilted law. It does not identify derivatives of arbitrary a.e.-equal representatives or prove Fisher, LSI, T2, or Wasserstein bounds."
  },
  {
    key := "information-theory.quadratic-tilt-representative-gradient",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.InformationTheory.TiltedLogRatio.gradient_quadratic_representative",
    upstreamDecl := "HasFDerivAt.norm_sq / InnerProductSpace.toDual / Lee--Shen--Tian equation (11)",
    upstreamFile := "Mathlib.Analysis.InnerProductSpace.Calculus; Lee--Shen--Tian arXiv:2010.03106v4 equation (11)",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["SPHMC", "RGO", "quadratic-tilt", "score", "gradient", "representative"],
    saldUse := "SPHMC Lemma 4.17: compute the constant relative score (y-y')/eta for restricted-Gaussian fibers",
    note := "Differentiates only the explicit smooth representative, not Mathlib's canonical measurable llr. The eta != 0 reusable statement is consumed with source eta > 0."
  },
  {
    key := "information-theory.tilted-kl-explicit-integral",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.InformationTheory.TiltedKL.finite_klDiv_and_toReal_eq_integral_normalizedLogRatio",
    upstreamDecl := "InformationTheory.klDiv_ne_top / toReal_klDiv_of_measure_eq / TiltedLogRatio.llr_tilted_tilted_ae",
    upstreamFile := "Mathlib.InformationTheory.KullbackLeibler.Basic; AutoSamplingTheory.TechnicalLemmas.InformationTheory.TiltedLogRatio; Lee--Shen--Tian arXiv:2010.03106v4 Lemma 2",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["SPHMC", "RGO", "KL", "tilted-measure", "log-density-ratio", "representative", "integrability"],
    saldUse := "SPHMC Lemma 4.17 route: evaluate canonical KL through the explicit normalized tilt representative without differentiating the canonical llr",
    note := "Assumes integrability of the displayed representative under the left tilt, proves finite canonical KL and its exact real integral formula. LSI, Fisher, T2 and W2 contraction remain separate."
  },
  {
    key := "information-theory.quadratic-tilt-relative-fisher",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.InformationTheory.QuadraticTiltFisher.information_quadratic_representative",
    upstreamDecl := "TiltedLogRatio.gradient_quadratic_representative / RelativeFisher.information / Lee--Shen--Tian Lemma 2",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.InformationTheory.TiltedLogRatio; AutoSamplingTheory.TechnicalLemmas.InformationTheory.RelativeFisher; Lee--Shen--Tian arXiv:2010.03106v4 Lemma 2",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["SPHMC", "RGO", "quadratic-tilt", "relative-Fisher", "score", "probability-integral"],
    saldUse := "SPHMC Lemma 4.17: integrate the exact restricted-Gaussian relative score under the left probability fiber",
    note := "The common normalizer base and the probability integration law are separate. This evaluates only the explicit smooth representative; canonical score domains, LSI, T2 and W2 contraction remain separate."
  }
]

def renyiDensityMemory : List LemmaMemoryEntry := [
  {
    key := "renyi-density.integrand-positivity",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi.renyiIntegrand_pos",
    upstreamDecl := "Real.rpow_pos_of_pos / Real.rpow_nonneg",
    upstreamFile := "Mathlib.Analysis.SpecialFunctions.Pow.Real",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Renyi", "density", "positivity", "rpow"],
    saldUse := "Chewi DENS/FI root: expose Renyi integrand positivity before integral and derivative contracts",
    note := "Companion nonnegative theorem is `renyiIntegrand_nonneg`; full Renyi divergence remains separate."
  },
  {
    key := "renyi-density.integrand-measurable",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi.measurable_renyiIntegrandENNReal",
    upstreamDecl := "Real.continuous_rpow_const / Measurable.ennreal_ofReal",
    upstreamFile := "Mathlib.Analysis.SpecialFunctions.Pow.Continuity; Mathlib.MeasureTheory.Constructions.BorelSpace.Real",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Renyi", "density", "measurability", "ENNReal"],
    saldUse := "Chewi DENS/FI root: turn measurable density representatives into a measurable Renyi lintegrand",
    note := "Requires order `a ∈ [0,1]`; representative choice and absolute continuity stay explicit."
  },
  {
    key := "renyi-density.integral-finite-envelope",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi.lintegral_renyiIntegrandENNReal_ne_top_of_ae_le",
    upstreamDecl := "lintegral_mono_ae / ne_top_of_le_ne_top",
    upstreamFile := "Mathlib.MeasureTheory.Integral.Lebesgue.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Renyi", "density", "lintegral", "envelope", "finite"],
    saldUse := "Chewi DENS/FI root: reduce Renyi integrability to a finite ENNReal envelope",
    note := "This mirrors the Gibbs finite-envelope contract and keeps tail/domination estimates outside the algebra leaf."
  },
  {
    key := "renyi-density.pointwise-derivative",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi.hasDerivAt_renyiIntegrand",
    upstreamDecl := "HasDerivAt.rpow_const / HasDerivAt.mul",
    upstreamFile := "Mathlib.Analysis.SpecialFunctions.Pow.Deriv",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Renyi", "density", "pointwise-derivative", "rpow"],
    saldUse := "Chewi chapter 6 root: isolate pointwise Renyi derivative algebra before dominated differentiation under the integral",
    note := "Positivity/nonzero, domination, and path-space regularity remain explicit source contracts."
  }
]

def variationalMemory : List LemmaMemoryEntry := [
  {
    key := "dv.scaled-test.energy-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.InformationTheory.DonskerVaradhan.dvVariationalScaledTestEnergyBound",
    upstreamDecl := "Donsker--Varadhan one-sided variational consequence",
    upstreamFile := "Boucheron-style cited result / future SLT entropy-duality port",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["DV", "KL", "energy"],
    saldUse := "convert finite log-mgf and KL hypotheses into residual energy bounds",
    note := "Small compiled consequence; full DV theorem remains a cited-result obligation."
  },
  {
    key := "lsi.sqrt-density.fisher-chain",
    localDecl := "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralHandoffScalar",
    upstreamDecl := "LSI density and Fisher-information bookkeeping",
    upstreamFile := "Mathlib/SLT-inspired entropy and LSI proof shape",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["LSI", "FI", "density"],
    saldUse := "bookkeeping for LSI-to-KL/FI handoff after density assumptions are supplied",
    note := "Compiled scalar/integral algebra; full LSI analytic theorem remains an obligation."
  },
  {
    key := "poincare.variance",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance",
    upstreamDecl := "MeasureTheory.integral",
    upstreamFile := "Mathlib.MeasureTheory.Integral.Bochner.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "chapter-2", "Poincare", "variance", "definition"],
    saldUse := "shared Chapter 2 variance interface with the centering convention visible in Lean",
    note := "The integral is totalized; consumers must carry the separate admissibility hypotheses."
  },
  {
    key := "poincare.dirichlet-energy",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.dirichletEnergy",
    upstreamDecl := "Analysis.Calculus.gradient / MeasureTheory.integral",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient; Mathlib.MeasureTheory.Integral.Bochner.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "chapter-2", "Poincare", "Dirichlet-energy", "definition"],
    saldUse := "shared gradient-energy term for Poincare statements and later semigroup dissipation leaves",
    note := "This is the Euclidean/inner-product energy interface, not a closed Dirichlet-form construction."
  },
  {
    key := "poincare.admissible",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.Admissible",
    upstreamDecl := "MeasureTheory.Integrable",
    upstreamFile := "Mathlib.MeasureTheory.Integral.Bochner.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "chapter-2", "Poincare", "integrability", "domain"],
    saldUse := "prevent totalized integrals from hiding the variance and gradient-energy domain",
    note := "Function, centered square, and squared gradient integrability are all explicit."
  },
  {
    key := "poincare.satisfies",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.Satisfies",
    upstreamDecl := "Poincare inequality, Chewi section 2.1",
    upstreamFile := "Log-Concave Sampling, Chapter 2",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "chapter-2", "Poincare", "test-class", "interface"],
    saldUse := "typed replacement for string-only PI task contracts on an explicit test class",
    note := "Probability normalization is explicit; no concrete target measure is asserted to satisfy PI here."
  },
  {
    key := "poincare.variance-nonnegative",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance_nonneg",
    upstreamDecl := "integral_nonneg_of_ae / sq_nonneg",
    upstreamFile := "Mathlib.MeasureTheory.Integral.Bochner.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "chapter-2", "Poincare", "variance", "nonnegative"],
    saldUse := "basic order leaf for variance estimates",
    note := "Proves nonnegativity of the explicit centered-square integral."
  },
  {
    key := "poincare.dirichlet-energy-nonnegative",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.dirichletEnergy_nonneg",
    upstreamDecl := "integral_nonneg_of_ae / sq_nonneg",
    upstreamFile := "Mathlib.MeasureTheory.Integral.Bochner.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "chapter-2", "Poincare", "Dirichlet-energy", "nonnegative"],
    saldUse := "order-theoretic leaf used when changing Poincare constants",
    note := "No differentiability claim is needed because Mathlib derivatives are totalized."
  },
  {
    key := "poincare.constant-monotonicity",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.mono_constant",
    upstreamDecl := "mul_le_mul_of_nonneg_right",
    upstreamFile := "Mathlib algebra/order API",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "chapter-2", "Poincare", "constant", "monotonicity"],
    saldUse := "reuse a proved PI bound at any larger nonnegative constant",
    note := "Preserves the same test class and exact admissibility domain."
  },
  {
    key := "poincare.variance-bound-elimination",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance_le",
    upstreamDecl := "Poincare.Satisfies",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "chapter-2", "Poincare", "consumer", "interface"],
    saldUse := "stable consumer API for extracting the PI inequality",
    note := "Requires explicit test membership and admissibility."
  },
  {
    key := "poincare.test-class-monotonicity",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.mono_tests",
    upstreamDecl := "Set inclusion",
    upstreamFile := "Mathlib.Data.Set.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "chapter-2", "Poincare", "test-class", "monotonicity"],
    saldUse := "restrict a PI package to a smaller core or downstream test class",
    note := "Keeps probability normalization, constant, and admissibility unchanged."
  }
]

def geometryMemory : List LemmaMemoryEntry := [
  {
    key := "unit-sphere-angular-cost",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.UnitSphereAngularCost.unit_sphere_angular_cost_properties",
    upstreamDecl := "Original generic sphere geometry and feature-law supplement",
    upstreamFile := "Pinned Mathlib vector-angle, Frechet calculus, Hilbert adjoints and actual Bochner/coupling interfaces",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["sphere", "gradient", "transport", "shared"],
    saldUse := "Generic geometric optimisation and law-dependent feature prerequisites; actual model compositions remain separate",
    note := "Actual unit-subtype arccos cost is continuous/measurable, bounded, chord-dominating, separating, symmetric and triangular. No installed metric or intrinsic path-length identity. Fresh independent source/compiler and exact candidate audits accepted. Final root/reader gates and postmerge purification separate; no whole-model claim."
  },
  {
    key := "quadratic-feature-law-gradient",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.QuadraticFeatureLawGradient.quadratic_feature_law_gradient_angular_transport_control",
    upstreamDecl := "Original generic sphere geometry and feature-law supplement",
    upstreamFile := "Pinned Mathlib vector-angle, Frechet calculus, Hilbert adjoints and actual Bochner/coupling interfaces",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["sphere", "gradient", "transport", "shared"],
    saldUse := "Generic geometric optimisation and law-dependent feature prerequisites; actual model compositions remain separate",
    note := "Actual ambient feature derivatives and visible operator bounds produce genuine mean integrability, potential gradient, tangent uniform bound, finite angular coupling cost and joint law/point control. Concrete feature-model inputs must still be produced. Fresh independent source/compiler and exact candidate audits accepted. Final root/reader gates and postmerge purification separate; no whole-model claim."
  },
  {
    key := "quadratic-feature-tangent-gradient",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.QuadraticFeatureTangentGradient.quadratic_feature_potential_tangent_gradient",
    upstreamDecl := "Original generic feature differentiation and transport supplement",
    upstreamFile := "Pinned Mathlib Gaussian moments, Frechet calculus, Hilbert adjoints, Bochner integrals and transport infimum",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["feature", "derivative", "transport", "shared"],
    saldUse := "Generic probability, optimisation and sphere-feature prerequisites; concrete model compositions remain separate",
    note := "Generic actual Hilbert feature derivative and visible ball bounds yield scalar potential derivative, tangent Riesz representation and joint residual-point field control. No manifold API identity is asserted. Fresh independent source and compiler audits accepted. Final root/reader gates and postmerge purification remain separate; no whole-model or convergence credit."
  },
  {
    key := "geometry.sphere-rayleigh-gradient",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.SphereRayleigh.sphere_rayleigh_characterization",
    upstreamDecl := "SphereGradient.sphere_gradient_characterization; LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf; Submodule.starProjection_orthogonal_val; Submodule.starProjection_unit_singleton",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.SphereGradient; Mathlib.Analysis.InnerProductSpace.Rayleigh; Mathlib.Analysis.InnerProductSpace.Projection.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Boumal", "Example3.62", "sphere", "Rayleigh", "induced-gradient", "symmetric-quadratic"],
    saldUse := "Concrete textbook consumer of the canonical true sphere differential and quadratic derivative",
    note := "Unit induced-sphere gradient representative2(Ax-<x,Ax>x), uniquely characterized against the actual inclusion differential. No general manifold/global field/algorithm completion."
  },
  {
    key := "geometry.sphere-rayleigh-critical-eigenvectors",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.SphereRayleigh.sphere_rayleigh_stationary_iff",
    upstreamDecl := "SphereRayleigh.sphere_rayleigh_characterization; Module.End.mem_eigenspace_iff; Module.End.HasEigenvector.apply_eq_smul",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.SphereRayleigh; Mathlib.LinearAlgebra.Eigenspace.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Boumal", "Example3.62", "sphere", "stationary", "eigenvector", "Rayleigh"],
    saldUse := "Actual unit-sphere differential-zero/eigenvector equivalences with both directions and exact eigenvalue coefficient",
    note := "Derives nonzero vectorhood and the Rayleigh eigenvalue from unit membership. Negative and zero eigenvalues allowed; no extremal spectral or convergence theorem."
  },
  {
    key := "geometry.sphere-gradient-characterization",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.SphereGradient.sphere_gradient_characterization",
    upstreamDecl := "range_mvfderiv_subtypeVal; mvfderiv_comp; HasGradientAt.fderiv_apply; Submodule.inner_orthogonalProjectionOnto_eq_of_mem_left; ext_inner_left",
    upstreamFile := "Mathlib.Geometry.Manifold.Instances.Sphere; Mathlib.Geometry.Manifold.MFDeriv.NormedSpace; Mathlib.Analysis.Calculus.Gradient.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Boumal", "sphere", "gradient", "manifold-differential", "induced-metric", "orthogonal-projection"],
    saldUse := "Shared actual nonlinear sphere differential and unique induced tangent-gradient representative for first-order geometry",
    note := "The actual inclusion derivative identifies the ambient tangent plane; its pairing yields the unique projected gradient. No default chart metric, general embedded-manifold theorem, global metric or smooth gradient-field completion is claimed."
  },
  {
    key := "geometry.curve-extension-independence",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.CurveExtension.fderiv_eq_on_curve_velocity",
    upstreamDecl := "HasFDerivAt.comp_hasDerivAt; HasDerivAt.congr_of_eventuallyEq; HasDerivAt.unique",
    upstreamFile := "Mathlib.Analysis.Calculus.Deriv.Comp; Mathlib.Analysis.Calculus.Deriv.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Boumal", "extension-independence", "curve", "relative-germ", "Frechet-derivative"],
    saldUse := "Shared derivative equality on actual nonlinear feasible velocities for embedded first-order geometry",
    note := "Derives the extension-choice proof edge from genuine pointwise derivatives and relative local agreement. No intrinsic differential or tangent adapter and no full Proposition 3.35/3.61 completion."
  },
  {
    key := "geometry.affine-restriction-gradient",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.AffineGradient.hasGradientAt_affine_restriction",
    upstreamDecl := "HasFDerivAt.comp; Submodule.inner_orthogonalProjectionOnto_eq_of_mem_right",
    upstreamFile := "Mathlib.Analysis.Calculus.FDeriv.Comp; Mathlib.Analysis.InnerProductSpace.Projection.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Boumal", "gradient", "affine-subspace", "Hilbert", "orthogonal-projection"],
    saldUse := "Shared inherited-metric restriction gradient for affine constrained objectives and the first-order embedded-geometry route",
    note := "Genuine pointwise gradient derived by the chain rule, with explicit completeness. Canonical affine prerequisite only; not the nonlinear Boumal Proposition 3.61 or a smooth gradient-field theorem."
  },
  {
    key := "geometry.chewi-definition-1-3-26",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.GeodesicConvexity.IsAlphaGeodesicallyConvex",
    upstreamDecl := "Chewi Definition 1.3.26, condition 1",
    upstreamFile := "Log-Concave Sampling, book page 31 / PDF page 43",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "geodesic-convexity", "metric-space", "interpolation", "definition"],
    saldUse := "Chewi Definition 1.3.26 root: state alpha-geodesic convexity through the endpoint interpolation inequality along every selected geodesic",
    note := "The geodesic predicate remains an explicit parameter so Riemannian and Wasserstein realizations can supply their own constant-speed/domain conditions."
  },
  {
    key := "geometry.chewi-display-1-4-7",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.GeodesicConvexity.firstOrder_geodesicConvexity",
    upstreamDecl := "HasDerivAt.tendsto_slope / le_of_tendsto_of_tendsto",
    upstreamFile := "Mathlib.Analysis.Calculus.Deriv.Slope; Chewi display (1.4.7)",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "geodesic-convexity", "first-order-condition", "derivative", "Wasserstein"],
    saldUse := "Derive the first-order alpha-convexity inequality from the endpoint chord condition along a differentiable geodesic",
    note := "The proof takes the positive-time secant-slope limit. Concrete Riemannian or Wasserstein geometry must identify the path derivative with its gradient pairing."
  },
  {
    key := "geometry.chewi-definition-1-3-16",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.MetricCurve.IsAbsolutelyContinuousMetricCurve",
    upstreamDecl := "Chewi Definition 1.3.16 (informal)",
    upstreamFile := "Log-Concave Sampling, book page 26 / PDF page 38",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "metric-derivative", "absolutely-continuous-curve", "Wasserstein", "definition"],
    saldUse := "Chewi Definition 1.3.16 root: require a finite nonnegative punctured-neighborhood metric derivative at almost every time",
    note := "Matches the source's explicitly informal definition. The standard upper-gradient characterization and Wasserstein specialization remain theorem routes."
  },
  {
    key := "geometry.euclidean-space.inner-toLp-toLp-sum",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.EuclideanSpaceCoordinates.euclideanSpace_inner_toLp_toLp_eq_sum_mul",
    upstreamDecl := "PiLp.inner_apply",
    upstreamFile := "Mathlib.Analysis.InnerProductSpace.PiL2",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "EuclideanSpace", "inner-product", "coordinates", "WithLp.toLp", "finite-dimensional"],
    saldUse := "Chewi GAUSS/SDE root: bridge coordinate gradient representatives to Mathlib `EuclideanSpace` inner-product notation",
    note := "Pure finite-dimensional coordinate identity; it does not define gradients, divergence, Laplacian, or analytic regularity."
  },
  {
    key := "geometry.euclidean-space.inner-sum",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.EuclideanSpaceCoordinates.euclideanSpace_inner_eq_sum_mul",
    upstreamDecl := "PiLp.inner_apply",
    upstreamFile := "Mathlib.Analysis.InnerProductSpace.PiL2",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "EuclideanSpace", "inner-product", "coordinates", "finite-dimensional"],
    saldUse := "Chewi GAUSS/SDE root: expose the direct finite-coordinate formula for `inner ℝ u v` in `EuclideanSpace ℝ ι`",
    note := "Reusable notation bridge for Gaussian and Langevin finite-dimensional leaves; no calculus or measure statement is hidden here."
  },
  {
    key := "geometry.log-concavity.def",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn",
    upstreamDecl := "ConcaveOn / strictConcaveOn_log_Ioi",
    upstreamFile := "Mathlib.Analysis.Convex.Function; Mathlib.Analysis.Convex.SpecificFunctions.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "log-concavity", "convex-analysis", "density"],
    saldUse := "Chewi CONV/DENS root: shared definition for positive density log-concavity leaves",
    note := "Small ASTIS-owned wrapper around Mathlib ConcaveOn with explicit positivity."
  },
  {
    key := "geometry.log-concavity.negative-log-potential-convex",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.convexOn_neg_log",
    upstreamDecl := "ConcaveOn.neg",
    upstreamFile := "Mathlib.Analysis.Convex.Function",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "log-concavity", "negative-log", "convex-potential", "Gibbs", "density"],
    saldUse := "Chewi DENS/CONV/SDE root: extract the convex potential `-log f` from a positive log-concave density",
    note := "Converse interface to the existing `exp (-V)` log-concavity leaf; useful for Gibbs potentials, score/FI, and Langevin generator subtrees."
  },
  {
    key := "geometry.log-concavity.negative-log-potential-sublevel-convex",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.convex_sublevel_neg_log",
    upstreamDecl := "ConvexOn.quasiconvexOn / LogConcaveOn.convexOn_neg_log",
    upstreamFile := "Mathlib.Analysis.Convex.Quasiconvex; AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "log-concavity", "negative-log", "sublevel", "convex-set", "localization"],
    saldUse := "Chewi CONV/DENS root: turn log-concave density energy sublevels into convex sets",
    note := "Energy-sublevel counterpart to positive-density superlevel convexity; supports localization and warm-start geometry."
  },
  {
    key := "geometry.log-concavity.positive-ray-id",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_id_Ioi",
    upstreamDecl := "strictConcaveOn_log_Ioi",
    upstreamFile := "Mathlib.Analysis.Convex.SpecificFunctions.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "log-concavity", "positive-ray", "convex-analysis"],
    saldUse := "Chewi CONV sanity leaf before density/Gibbs and Prekopa--Leindler ports",
    note := "Compiled one-dimensional leaf reusing Mathlib's strict concavity of log."
  },
  {
    key := "geometry.log-concavity.linear-precomposition",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.comp_linearMap",
    upstreamDecl := "ConcaveOn.comp_linearMap",
    upstreamFile := "Mathlib.Analysis.Convex.Function",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "log-concavity", "linear-map", "preimage", "density", "coordinate"],
    saldUse := "Chewi CONV/DENS root: pull log-concave density factors through linear coordinate maps",
    note := "Preimage-domain transport leaf for coordinate projections, linear observations, and product-density factors."
  },
  {
    key := "geometry.log-concavity.affine-precomposition",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.comp_affineMap",
    upstreamDecl := "ConcaveOn.comp_affineMap",
    upstreamFile := "Mathlib.Analysis.Convex.Function",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "log-concavity", "affine-map", "preimage", "density", "shift"],
    saldUse := "Chewi CONV/DENS root: preserve log-concavity under shifts and affine coordinate changes",
    note := "Vector-space affine transport leaf; useful for shifted Gaussian/Gibbs densities and restricted affine charts."
  },
  {
    key := "geometry.log-concavity.convex-superlevel",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.convex_superlevel",
    upstreamDecl := "ConcaveOn.quasiconcaveOn / Real.log_le_log_iff",
    upstreamFile := "Mathlib.Analysis.Convex.Quasiconvex; Mathlib.Analysis.SpecialFunctions.Log.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "log-concavity", "superlevel", "convex-set", "restricted-oracle"],
    saldUse := "Chewi CONV/DENS root: identify positive-density superlevel sets as convex bodies for restrictions and localization",
    note := "Handles all real thresholds: nonpositive thresholds use positivity, positive thresholds use log monotonicity and concavity."
  },
  {
    key := "geometry.log-concavity.quasiconcave",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.quasiconcaveOn",
    upstreamDecl := "QuasiconcaveOn / convex_superlevel",
    upstreamFile := "Mathlib.Analysis.Convex.Quasiconvex; AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "log-concavity", "quasiconcavity", "superlevel", "convex-set"],
    saldUse := "Chewi CONV root: expose log-concavity as quasiconcavity so all superlevel-set APIs become callable",
    note := "Small wrapper around `convex_superlevel`; useful for convex-body and restricted-Gaussian subtrees."
  },
  {
    key := "geometry.log-concavity.restrict-superlevel",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.restrict_superlevel",
    upstreamDecl := "LogConcaveOn.subset / LogConcaveOn.convex_superlevel",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "log-concavity", "superlevel", "restriction", "restricted-oracle"],
    saldUse := "Chewi CONV/DENS root: preserve log-concavity when restricting a density to a convex superlevel body",
    note := "Consumer-facing restricted-density wrapper for localization, warm starts, restricted Gaussian oracles, and proximal subtrees."
  },
  {
    key := "geometry.log-concavity.positive-rescale",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.const_mul",
    upstreamDecl := "ConcaveOn.add / Real.log_mul",
    upstreamFile := "Mathlib.Analysis.Convex.Function; Mathlib.Analysis.SpecialFunctions.Log.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "log-concavity", "normalization", "density"],
    saldUse := "Chewi DENS/CONV root: keep normalized-density positive constants out of later convexity proofs",
    note := "Separates positive scalar normalization from measure/integral normalization."
  },
  {
    key := "geometry.log-concavity.pointwise-product",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.mul",
    upstreamDecl := "ConcaveOn.add / Real.log_mul",
    upstreamFile := "Mathlib.Analysis.Convex.Function; Mathlib.Analysis.SpecialFunctions.Log.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "log-concavity", "product", "density", "tensorization"],
    saldUse := "Chewi CONV/DENS root: multiply positive log-concave density factors on a shared domain",
    note := "Used before product-density, tilted-density, and normalized-density tensorization leaves."
  },
  {
    key := "geometry.log-concavity.nonnegative-rpow",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.rpow",
    upstreamDecl := "ConcaveOn.smul / Real.log_rpow",
    upstreamFile := "Mathlib.Analysis.Convex.Function; Mathlib.Analysis.SpecialFunctions.Pow.Real",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "log-concavity", "rpow", "density", "positive-function"],
    saldUse := "Chewi CONV/DENS root: nonnegative powers preserve positive log-concavity for density algebra",
    note := "Keeps exponent bookkeeping local before Renyi/Hellinger-style powered density consumers."
  },
  {
    key := "geometry.log-concavity.product-domain-product",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.prod",
    upstreamDecl := "Convex.prod / ConcaveOn.add / Real.log_mul",
    upstreamFile := "Mathlib.Analysis.Convex.Basic; Mathlib.Analysis.Convex.Function; Mathlib.Analysis.SpecialFunctions.Log.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "log-concavity", "product-domain", "density", "tensorization"],
    saldUse := "Chewi CONV/DENS root: tensorize log-concave density factors over Cartesian product domains",
    note := "Small product-domain API used by future Prekopa, Gaussian product-density, and tensorization subtrees."
  },
  {
    key := "geometry.gibbs-density.convex-potential",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_const_mul_exp_neg_of_convexOn",
    upstreamDecl := "ConvexOn.neg / Real.log_exp",
    upstreamFile := "Mathlib.Analysis.Convex.Function; Mathlib.Analysis.SpecialFunctions.Log.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "convex-potential", "log-concavity", "density"],
    saldUse := "Chewi DENS/CONV root for Gibbs target densities before withDensity normalization",
    note := "Compiled convex-analytic Gibbs leaf; no measure normalization or integrability is claimed."
  },
  {
    key := "geometry.convexity.absolute-value",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.convexOn_univ_abs",
    upstreamDecl := "convexOn_univ_norm / Real.norm_eq_abs",
    upstreamFile := "Mathlib.Analysis.Normed.Module.Convex",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "convexity", "absolute-value", "Laplace-tail", "one-dimensional"],
    saldUse := "Chewi CONV root: expose convexity of `|x|` for the one-dimensional Laplace density example",
    note := "Geometry counterpart of the exact one-dimensional Laplace normalizer; this is not a multidimensional norm-tail theorem."
  },
  {
    key := "geometry.convexity.absolute-linear-potential",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.convexOn_univ_const_mul_abs_add",
    upstreamDecl := "convexOn_univ_abs / ConvexOn.smul / ConvexOn.add_const",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "convexity", "absolute-linear-potential", "Laplace-tail", "Gibbs"],
    saldUse := "Chewi CONV/DENS root: package `a|x|+b` with `0 ≤ a` as a convex Gibbs potential",
    note := "Keeps the convexity assumption for Laplace examples explicit before density normalization."
  },
  {
    key := "geometry.gibbs-density.absolute-linear-potential-logconcave",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_exp_neg_abs_linear",
    upstreamDecl := "logConcaveOn_exp_neg_of_convexOn / convexOn_univ_const_mul_abs_add",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "Laplace-tail", "log-concavity", "density", "one-dimensional"],
    saldUse := "Chewi CONV/DENS root: prove the unnormalized one-dimensional Laplace Gibbs shape is log-concave",
    note := "Geometry-only result; measure normalization is supplied by the exact Laplace normalizer leaves."
  },
  {
    key := "geometry.gibbs-density.absolute-linear-positive-rescale-logconcave",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_const_mul_exp_neg_abs_linear",
    upstreamDecl := "logConcaveOn_const_mul_exp_neg_of_convexOn / convexOn_univ_const_mul_abs_add",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "Laplace-tail", "log-concavity", "normalization", "density"],
    saldUse := "Chewi DENS/CONV root: preserve log-concavity after multiplying the Laplace shape by a positive normalizing constant",
    note := "Use with `analysis.integrability.laplace-absolute-linear-tail-ennreal-normalizer`; it does not construct a probability measure by itself."
  },
  {
    key := "geometry.gibbs-density.explicit-laplace-normalized-logconcave",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_explicit_abs_linear_normalized_density",
    upstreamDecl := "logConcaveOn_const_mul_exp_neg_abs_linear / positivity of `2*exp(-b)/a`",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "Laplace-tail", "log-concavity", "normalizer", "density"],
    saldUse := "Chewi DENS/CONV root: expose the explicitly normalized one-dimensional Laplace-type real density as log-concave",
    note := "Real-density geometry companion to the exact ENNReal normalizer and withDensity probability theorem."
  },
  {
    key := "geometry.strong-convexity.convex-potential-nonnegative-modulus",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity.convexOn_of_strongConvexOn_nonneg",
    upstreamDecl := "StrongConvexOn.mono / strongConvexOn_zero",
    upstreamFile := "Mathlib.Analysis.Convex.Strong",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "strong-convexity", "convexity", "convex-potential", "density"],
    saldUse := "Chewi CONV/DENS root: translate strongly convex potential assumptions into ordinary convex-potential geometry",
    note := "Uses Mathlib's strong-convexity monotonicity to lower the modulus to zero; requires `0 ≤ k`."
  },
  {
    key := "geometry.strong-convexity.gibbs-shape-logconcave",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity.logConcaveOn_exp_neg_of_strongConvexOn",
    upstreamDecl := "convexOn_of_strongConvexOn_nonneg / logConcaveOn_exp_neg_of_convexOn",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity; AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "strong-convexity", "Gibbs", "log-concavity", "density"],
    saldUse := "Chewi DENS/CONV root: expose the unnormalized Gibbs density shape of a strongly convex potential as log-concave",
    note := "Geometry-only bridge; it does not claim a finite normalizer, invariant law, or functional inequality."
  },
  {
    key := "geometry.strong-convexity.normalized-gibbs-shape-logconcave",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity.logConcaveOn_const_mul_exp_neg_of_strongConvexOn",
    upstreamDecl := "logConcaveOn_const_mul_exp_neg_of_convexOn / convexOn_of_strongConvexOn_nonneg",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity; AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "strong-convexity", "Gibbs", "log-concavity", "normalization", "density"],
    saldUse := "Chewi DENS/CONV root: keep positive scalar normalizers separate while preserving strong-convex Gibbs log-concavity",
    note := "Pairs with the normalized Gibbs probability branch after a positive normalizing constant has been constructed separately."
  },
  {
    key := "geometry.strong-convexity.minimizer-centered-quadratic-lower-bound",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity.centered_quadratic_lower_bound_of_strongConvexOn_minimizer",
    upstreamDecl := "StrongConvexOn / IsMinOn / UniformConvexOn midpoint inequality",
    upstreamFile := "Mathlib.Analysis.Convex.Strong; Mathlib.Order.Filter.Extr",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "strong-convexity", "minimizer", "quadratic-lower-bound", "Gibbs", "coercivity"],
    saldUse := "Chewi CONV/DENS root: expose the quadratic tail envelope used by strong log-concavity and Gibbs normalization",
    note := "Mathlib-native `StrongConvexOn` bridge with an explicit global minimizer; proves the robust midpoint `k/4` envelope."
  },
  {
    key := "geometry.convexity.norm-square",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.convexOn_univ_norm_sq",
    upstreamDecl := "norm_add_le / norm_smul / sq_le_sq₀",
    upstreamFile := "Mathlib.Analysis.Normed.Group.Basic; Mathlib.Analysis.Normed.MulAction; Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "convexity", "norm-square", "quadratic", "normed-space"],
    saldUse := "Chewi CONV root: reusable convexity of the quadratic norm energy `‖x‖^2`",
    note := "Works in any real normed vector space; uses the triangle inequality and scalar Jensen algebra."
  },
  {
    key := "geometry.convexity.quadratic-norm-potential",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.convexOn_univ_const_mul_norm_sq_add",
    upstreamDecl := "convexOn_univ_norm_sq / ConvexOn.smul / ConvexOn.add_const",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity; Mathlib.Analysis.Convex.Function",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "convexity", "quadratic-potential", "Gibbs", "density"],
    saldUse := "Chewi DENS/CONV root: package nonnegative quadratic norm potentials `a‖x‖^2+b` as convex",
    note := "This is the convex-geometry counterpart of the exact quadratic normalizer in Analysis.Integrability."
  },
  {
    key := "geometry.gibbs-density.quadratic-potential-logconcave",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_exp_neg_quadratic_norm",
    upstreamDecl := "logConcaveOn_exp_neg_of_convexOn / convexOn_univ_const_mul_norm_sq_add",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "quadratic-potential", "log-concavity", "density"],
    saldUse := "Chewi DENS/CONV root: prove `exp (-(a‖x‖^2+b))` is log-concave before measure normalization",
    note := "Keeps log-concavity separate from Lebesgue integrability and probability-measure construction."
  },
  {
    key := "geometry.gibbs-density.quadratic-positive-rescale-logconcave",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_const_mul_exp_neg_quadratic_norm",
    upstreamDecl := "logConcaveOn_const_mul_exp_neg_of_convexOn / convexOn_univ_const_mul_norm_sq_add",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "quadratic-potential", "normalization", "log-concavity", "density"],
    saldUse := "Chewi DENS/CONV root: positive scalar normalizers preserve quadratic Gibbs log-concavity",
    note := "This removes scalar-normalizer bookkeeping from later invariant-law and Prekopa-style leaves."
  },
  {
    key := "geometry.gibbs-density.explicit-quadratic-normalized-logconcave",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_explicit_quadratic_normalized_density",
    upstreamDecl := "logConcaveOn_const_mul_exp_neg_quadratic_norm / Real.rpow_pos_of_pos",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity; Mathlib.Analysis.SpecialFunctions.Pow.Real",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "Lebesgue", "quadratic", "normalizer", "log-concavity", "finite-dimensional"],
    saldUse := "Chewi DENS/CONV root: connect the explicit finite-dimensional quadratic normalizer to log-concavity of the normalized density",
    note := "Geometric companion to `isProbabilityMeasure_withDensity_exp_neg_add_mul_norm_sq`; it claims density log-concavity, not a measure-level theorem."
  },
  {
    key := "geometry.convexity.shifted-quadratic-norm-potential",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.convexOn_univ_const_mul_norm_sub_sq_add",
    upstreamDecl := "ConvexOn.comp_affineMap / convexOn_univ_const_mul_norm_sq_add",
    upstreamFile := "Mathlib.Analysis.Convex.Function; AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "convexity", "shifted-quadratic", "Gibbs", "density", "affine-map"],
    saldUse := "Chewi DENS/CONV/GAUSS root: package shifted quadratic potentials `a‖x-m‖^2+b` as convex",
    note := "Uses affine precomposition of the centered quadratic norm potential; this is the geometry leaf behind shifted Gaussian/Gibbs densities."
  },
  {
    key := "geometry.gibbs-density.shifted-quadratic-potential-logconcave",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_exp_neg_shifted_quadratic_norm",
    upstreamDecl := "logConcaveOn_exp_neg_of_convexOn / convexOn_univ_const_mul_norm_sub_sq_add",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "shifted-quadratic", "log-concavity", "density", "Gaussian"],
    saldUse := "Chewi DENS/CONV/GAUSS root: prove `exp (-(a‖x-m‖^2+b))` is log-concave before measure normalization",
    note := "Shifted counterpart of the centered quadratic Gibbs shape; measure-level translation invariance remains separate."
  },
  {
    key := "geometry.gibbs-density.shifted-quadratic-positive-rescale-logconcave",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_const_mul_exp_neg_shifted_quadratic_norm",
    upstreamDecl := "logConcaveOn_const_mul_exp_neg_of_convexOn / convexOn_univ_const_mul_norm_sub_sq_add",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "shifted-quadratic", "normalization", "log-concavity", "density"],
    saldUse := "Chewi DENS/CONV/GAUSS root: positive scalar normalizers preserve shifted quadratic Gibbs log-concavity",
    note := "Keeps shifted-density scalar normalization separate from exact Lebesgue normalizer and withDensity probability leaves."
  },
  {
    key := "geometry.gibbs-density.explicit-shifted-quadratic-normalized-logconcave",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_explicit_shifted_quadratic_normalized_density",
    upstreamDecl := "logConcaveOn_const_mul_exp_neg_shifted_quadratic_norm / Real.rpow_pos_of_pos",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity; Mathlib.Analysis.SpecialFunctions.Pow.Real",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "Gaussian", "shifted-quadratic", "normalizer", "log-concavity", "finite-dimensional"],
    saldUse := "Chewi DENS/CONV/GAUSS root: connect the explicit finite-dimensional quadratic normalizer to shifted density log-concavity",
    note := "Geometric shifted-density companion to exact quadratic normalizer leaves; it does not assert the translated Lebesgue integral identity."
  },
  {
    key := "geometry.convexity.pair-sub-quadratic-kernel-potential",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.convexOn_univ_const_mul_norm_fst_sub_snd_sq_add",
    upstreamDecl := "ConvexOn.comp_linearMap / convexOn_univ_const_mul_norm_sq_add",
    upstreamFile := "Mathlib.Analysis.Convex.Function; AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "convexity", "pair-difference", "quadratic-kernel", "proximal", "coupling"],
    saldUse := "Chewi DENS/CONV/GAUSS/DISC root: package two-point potentials `a‖x-y‖^2+b` as convex on product space",
    note := "Uses the linear map `(x,y) ↦ x-y`; this is the geometry leaf behind Gaussian transition and proximal-kernel log-concavity."
  },
  {
    key := "geometry.gibbs-density.pair-sub-quadratic-kernel-logconcave",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_exp_neg_pair_sub_quadratic_norm",
    upstreamDecl := "logConcaveOn_exp_neg_of_convexOn / convexOn_univ_const_mul_norm_fst_sub_snd_sq_add",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "Gaussian-kernel", "pair-difference", "log-concavity", "proximal"],
    saldUse := "Chewi DENS/CONV/GAUSS/DISC root: prove `exp (-(a‖x-y‖^2+b))` is log-concave as a kernel shape",
    note := "Product-space kernel geometry only; transition-kernel measurability and conditional normalization remain separate leaves."
  },
  {
    key := "geometry.gibbs-density.pair-sub-quadratic-kernel-positive-rescale-logconcave",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_const_mul_exp_neg_pair_sub_quadratic_norm",
    upstreamDecl := "logConcaveOn_const_mul_exp_neg_of_convexOn / convexOn_univ_const_mul_norm_fst_sub_snd_sq_add",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "Gaussian-kernel", "normalization", "pair-difference", "log-concavity"],
    saldUse := "Chewi DENS/CONV/GAUSS/DISC root: positive constants preserve two-point Gaussian-kernel log-concavity",
    note := "Keeps scalar kernel constants separate from Markov kernel mass, detailed balance, and Lebesgue integral identities."
  },
  {
    key := "geometry.gibbs-density.explicit-pair-sub-quadratic-kernel-logconcave",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_explicit_pair_sub_quadratic_kernel",
    upstreamDecl := "logConcaveOn_const_mul_exp_neg_pair_sub_quadratic_norm / Real.rpow_pos_of_pos",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity; Mathlib.Analysis.SpecialFunctions.Pow.Real",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Chewi", "Gibbs", "Gaussian-kernel", "proximal", "pair-difference", "normalizer", "log-concavity"],
    saldUse := "Chewi GAUSS/DISC root: log-concavity of finite-dimensional Gaussian-kernel shapes with the usual conditional normalizing constant",
    note := "This is not a probability-density theorem on product volume; it is the reusable convex-geometric kernel-shape leaf."
  },
  {
    key := "geometry.unit-sphere.geodesic-derivatives",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.UnitSphere.unitSphereGeodesic_derivatives_and_bounds",
    upstreamDecl := "explicit great-circle geodesic and elementary trigonometric derivatives",
    upstreamFile := "Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc; Mathlib.Analysis.InnerProductSpace.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Riemannian-geometry", "sphere", "geodesic", "C4", "retraction"],
    saldUse := "Fourth-order local analysis for sphere-valued optimization and geometric integrators.",
    note := "Exact derivatives through order four and uniform norm bounds for a unit base point; no finite-dimensional or probabilistic hypothesis."
  },
  {
    key := "geometry.unit-sphere.exponential-map-norm",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.UnitSphere.unitSphereExp_norm_eq_one",
    upstreamDecl := "sin_sq_add_cos_sq / orthogonal norm identity",
    upstreamFile := "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic; Mathlib.Analysis.InnerProductSpace.Basic",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Riemannian-geometry", "sphere", "exponential-map", "constraint-preservation"],
    saldUse := "Exact feasibility certificate for tangent exponential-map updates on the unit sphere.",
    note := "The tangent hypothesis is the ambient inner-product equation; the result works in arbitrary real inner-product spaces."
  },
  {
    key := "analysis.second-order-extremum.necessary-sign",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.SphereSecondOrder.IsLocalMin.deriv_deriv_nonneg_of_continuousAt",
    upstreamDecl := "isLocalMax_of_deriv_deriv_neg / isLocalMin_of_deriv_deriv_pos",
    upstreamFile := "Mathlib.Analysis.Calculus.DerivativeTest",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["calculus", "second-derivative", "local-extremum", "necessary-condition"],
    saldUse := "Reusable second-order sign condition for optimization and maximum-principle arguments.",
    note := "Complements Mathlib's strict sufficient tests with the non-strict necessary direction for the total double derivative."
  },
  {
    key := "geometry.unit-sphere.laplacian-extremum-sign",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.Geometry.SphereSecondOrder.unitSphere_isMinOn_laplacian_nonneg",
    upstreamDecl := "unitSphereExp_norm_eq_one / second-order necessary condition / Finset.sum_nonneg",
    upstreamFile := "AutoSamplingTheory.TechnicalLemmas.Geometry.UnitSphere; Mathlib.Analysis.Calculus.DerivativeTest",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Riemannian-geometry", "sphere", "Hessian", "Laplacian", "maximum-principle"],
    saldUse := "Sign of a supplied finite Hessian trace at a unit-sphere extremum.",
    note := "Exposes tangent directions, curve continuity, diagonal identities, and the finite trace identity; no PDE or model coefficient is included."
  }
]

def saldExtractedMemory : List LemmaMemoryEntry := [
  {
    key := "sald.gronwall.scalar-rewrites",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.SALDExtracted.gronwallExpProductRewriteScalar",
    upstreamDecl := "SALD appendix Gronwall proof plus elementary exponential algebra",
    upstreamFile := "AutoSamplingTheory/SALD.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Gronwall", "scalar-algebra", "SALD-extracted"],
    saldUse := "forward-KL and discrete forward-KL Gronwall display algebra",
    note := "Compiled in SALD and exposed through TechnicalLemmas.SALDExtracted."
  },
  {
    key := "sald.em-endpoint-law-handoff",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.SALDExtracted.discreteForwardKlEmEndpointLawPairHandoff",
    upstreamDecl := "SALD discrete EM endpoint law bookkeeping",
    upstreamFile := "AutoSamplingTheory/SALD.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Euler-Maruyama", "endpoint-law", "SALD-extracted"],
    saldUse := "endpoint-law pair handoff for discrete SALD/VA-SALD proofs",
    note := "SALD-derived local theorem exposed as a searchable memory item."
  },
  {
    key := "sald.brownian-normalization-bridges",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.SALDExtracted.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw",
    upstreamDecl := "SALD Brownian/Ito normalized-coordinate law bridge",
    upstreamFile := "AutoSamplingTheory/SALD.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Brownian", "Ito", "Gaussian", "SALD-extracted"],
    saldUse := "active Brownian/Ito scalar generator backend and coordinate variance leaves",
    note := "Domain-specific SALD bridge; can be generalized later if reused."
  },
  {
    key := "sald.remainder-meas-gaussian-law",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.SALDExtracted.selectedWeakTestRemainderMeasOfStdGaussianVectorLaw",
    upstreamDecl := "SALD Brownian/Ito normalized-remainder measurability bridge",
    upstreamFile := "AutoSamplingTheory/SALD.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Brownian", "Ito", "Gaussian", "measurability", "SALD-extracted"],
    saldUse := "discharge hRemainderMeas in the active Brownian/Ito Taylor moment backend",
    note := "Cycle 194 bridge transporting AEStronglyMeasurable across the normalized coordinate-law and variance equalities."
  },
  {
    key := "sald.remainder-bound-gaussian-law",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.SALDExtracted.selectedWeakTestRemainderBoundOfStdGaussianVectorLaw",
    upstreamDecl := "SALD Brownian/Ito normalized-remainder domination bridge",
    upstreamFile := "AutoSamplingTheory/SALD.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Brownian", "Ito", "Gaussian", "domination", "SALD-extracted"],
    saldUse := "narrow hRemainderBound to normalized-coordinate-law domination in the active Brownian/Ito Taylor moment backend",
    note := "Cycle 194 bridge transporting ae domination across the normalized coordinate-law and variance equalities."
  },
  {
    key := "sald.remainder-bound-integrable-gaussian-law",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.SALDExtracted.selectedWeakTestRemainderBoundIntegrableOfStdGaussianVectorLaw",
    upstreamDecl := "SALD Brownian/Ito normalized-remainder dominating-bound integrability bridge",
    upstreamFile := "AutoSamplingTheory/SALD.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Brownian", "Ito", "Gaussian", "integrability", "SALD-extracted"],
    saldUse := "narrow hRemainderBoundInt to normalized-coordinate-law integrability in the active Brownian/Ito Taylor moment backend",
    note := "Cycle 195 bridge transporting MeasureTheory.Integrable across the normalized coordinate-law and variance equalities."
  },
  {
    key := "sald.normalized-remainder-bound-int-quadratic",
    localDecl := "AutoSamplingTheory.TechnicalLemmas.SALDExtracted.selectedWeakTestNormalizedRemainderBoundIntOfQuadraticBound",
    upstreamDecl := "SALD Brownian/Ito normalized-remainder quadratic domination integrability bridge",
    upstreamFile := "AutoSamplingTheory/SALD.lean",
    status := LemmaMemoryStatus.formalizedLocal,
    tags := ["Brownian", "Ito", "Gaussian", "integrability", "quadratic-bound", "SALD-extracted"],
    saldUse := "narrow hNormalizedRemainderBoundInt to hNormalizedRemainderBoundDef plus normalized-coordinate-law integrability",
    note := "Cycle 196 bridge from source-cited remainderBound = C * z ^ 2 to normalized-coordinate-law integrability."
  }
]

def portQueueMemory : List LemmaMemoryEntry := [
  {
    key := "dv.entropy-duality",
    localDecl := "",
    upstreamDecl := "entropy_duality",
    upstreamFile := "SLT/GaussianLSI/DualityEntropy.lean",
    status := LemmaMemoryStatus.portCandidate,
    tags := ["DV", "entropy", "KL"],
    saldUse := "Donsker--Varadhan variational backend for KL energy bounds",
    note := "Port only when DV becomes the active dynamic leaf."
  },
  {
    key := "lsi.product-gaussian",
    localDecl := "",
    upstreamDecl := "gaussian_logSobolev_W12_pi",
    upstreamFile := "SLT/GaussianLSI/TensorizedGLSI.lean",
    status := LemmaMemoryStatus.portCandidate,
    tags := ["LSI", "Gaussian", "product"],
    saldUse := "LSI-to-KL/FI backend when the faithful proof reaches this cited-result boundary",
    note := "Large theorem; keep as queue entry until a local ASTIS declaration compiles."
  }
]

def technicalLemmaMemory : List LemmaMemoryEntry :=
  analysisMemory ++ gaussianMemory ++ taylorMemory ++ calculusMemory ++ measureMemory ++ functionalInequalityMemory ++ stochasticProcessMemory ++
    probabilityMemory ++ klDensityMemory ++ renyiDensityMemory ++ variationalMemory ++ geometryMemory ++
    saldExtractedMemory ++ portQueueMemory

def formalizedTechnicalLemmaCount : Nat :=
  (technicalLemmaMemory.filter fun entry =>
    entry.status == LemmaMemoryStatus.formalizedLocal).length

end TechnicalLemmas
end AutoSamplingTheory