AutoSamplingTheory.SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower1Obligation

Data definition / provenance and workflow record

Meaning and type

The result has data type AutoSamplingTheory.ProofObligation. A value of this type stores descriptions; it is not a proof of the statements in those descriptions.

def cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower1Obligation :
    ProofObligation

Construction and field-by-field explanation

Construct a data record from explicit fields and the audited defaults shown below.

This Lean definition constructs provenance or workflow data. It does not prove the mathematical statements stored as text. Status labels, named dependencies and citations are data, not compilation, proof or source certificates.

id : String (explicit)

Stable obligation identifier.

sald.general_moving_target_discrete.cycle172_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_hessian_source_lower1

statement : String (explicit)

Desired mathematical or workflow content as String; it is not a proposition in Prop and the def does not prove it.

Cycle 172 lower_1 proof-scout/illness-area refiner packet. Classification: rejected-wrapper-churn. Exact missing theorem boundary preserved: hHessianOpNorm : forall z : E, norm (fderiv Real (fderiv Real sourceTest) z) <= C1 under sald.general_moving_target_discrete.em_interpolation_fp. The refreshed blueprint and local Lean declarations show that the Brownian coordinate unit direction is already discharged by SALD.gaussianRealStdOrthonormalBasisUnit, and the bridge from hHessianOpNorm to hSecondFDerivOpNorm is already compiled by SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm. Source anchors appendix.tex:984-995, appendix.tex:1026-1072, appendix.tex:1379-1387, and main_body.tex:273-305, plus the original-source search excluding sald_version_2.tex, give the EM frozen interpolation, drift/score Lipschitz assumptions, and weak Fokker-Planck Laplacian display, but no selected weak-test global bounded-Hessian field. Lower_2-ready theorem shape, only after source correspondence supplies that field: selectedWeakTestHessianOpNormOfSourceBoundedHessian (sourceTest : E -> Real) (C1 : Real) (hSourceC2b : SourceSelectedWeakTestC2bBoundedHessian sourceTest C1) : forall z : E, norm (fderiv Real (fderiv Real sourceTest) z) <= C1. Without the source-backed hSourceC2b field, this projection is exactly the wrapper churn rejected in cycle 171 and by cycle 172 middle.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource — audited data reference, not expanded and not a compiled dependency edge

status : AutoSamplingTheory.ProofStatus (explicit)

Stored ProofStatus, default obligation; even an explicitly stored formalized does not independently certify a Lean theorem.

AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification

dependsOn : List String (explicit)

List of declared dependency names as strings; may mix theorem names, obligations, source labels, or descriptions. Not the compiled dependency DAG.

Ordered data items

1. SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceMiddleObligation
2. SALD.cycle171GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation
3. SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm
4. SALD.gaussianRealStdOrthonormalBasisUnit
5. SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis
6. hHessianOpNorm
7. hSecondFDerivOpNorm
8. sourceTest
9. testRegular
10. SourceSelectedWeakTestC2bBoundedHessian proposed only as source-backed field
11. appendix.tex:984-995
12. appendix.tex:1026-1072
13. appendix.tex:1379-1387
14. main_body.tex:273-305
15. paper-wide original-source search excluding sald_version_2.tex
16. source-contract-gap
17. sald.general_moving_target_discrete.em_interpolation_fp

note : String (explicit)

Recorded evidence/caveats; may distinguish a compiled scalar helper from still-open source analysis.

No SLT file was consulted because this is not an SLT theorem: the configured local SLT clone is absent, and the relevant local/Mathlib pieces have already compiled. The packet supplies a proof route only conditional on a real source-backed bounded-Hessian field; it does not add or formalize that field.

def cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower1Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle172_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_hessian_source_lower1"
  statement := "Cycle 172 lower_1 proof-scout/illness-area refiner packet. Classification: rejected-wrapper-churn. Exact missing theorem boundary preserved: hHessianOpNorm : forall z : E, norm (fderiv Real (fderiv Real sourceTest) z) <= C1 under sald.general_moving_target_discrete.em_interpolation_fp. The refreshed blueprint and local Lean declarations show that the Brownian coordinate unit direction is already discharged by SALD.gaussianRealStdOrthonormalBasisUnit, and the bridge from hHessianOpNorm to hSecondFDerivOpNorm is already compiled by SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm. Source anchors appendix.tex:984-995, appendix.tex:1026-1072, appendix.tex:1379-1387, and main_body.tex:273-305, plus the original-source search excluding sald_version_2.tex, give the EM frozen interpolation, drift/score Lipschitz assumptions, and weak Fokker-Planck Laplacian display, but no selected weak-test global bounded-Hessian field. Lower_2-ready theorem shape, only after source correspondence supplies that field: selectedWeakTestHessianOpNormOfSourceBoundedHessian (sourceTest : E -> Real) (C1 : Real) (hSourceC2b : SourceSelectedWeakTestC2bBoundedHessian sourceTest C1) : forall z : E, norm (fderiv Real (fderiv Real sourceTest) z) <= C1. Without the source-backed hSourceC2b field, this projection is exactly the wrapper churn rejected in cycle 171 and by cycle 172 middle."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceMiddleObligation",
    "SALD.cycle171GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation",
    "SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm",
    "SALD.gaussianRealStdOrthonormalBasisUnit",
    "SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis",
    "hHessianOpNorm",
    "hSecondFDerivOpNorm",
    "sourceTest",
    "testRegular",
    "SourceSelectedWeakTestC2bBoundedHessian proposed only as source-backed field",
    "appendix.tex:984-995",
    "appendix.tex:1026-1072",
    "appendix.tex:1379-1387",
    "main_body.tex:273-305",
    "paper-wide original-source search excluding sald_version_2.tex",
    "source-contract-gap",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "No SLT file was consulted because this is not an SLT theorem: the configured local SLT clone is absent, and the relevant local/Mathlib pieces have already compiled. The packet supplies a proof route only conditional on a real source-backed bounded-Hessian field; it does not add or formalize that field."

/-- Cycle-172 lower_2 rejection of the only admissible Hessian projection route
after checking that no source-backed selected weak-test bounded-Hessian field
is available. -/