AutoSamplingTheory.SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation

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 cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation :
    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_lower2

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_2 illness-area refiner packet. Classification: rejected-wrapper-churn. Exact attempted theorem rejected: selectedWeakTestHessianOpNormOfSourceBoundedHessian from SourceSelectedWeakTestC2bBoundedHessian sourceTest C1 to hHessianOpNorm. This theorem would be valid only after the source correspondence independently defines a selected weak-test C2_b/bounded-Hessian field; implementing it from unexpanded testRegular or from an unsourced SelectedWeakTestC2bBoundedHessian predicate would only rename the missing supplied hypothesis. Lower_2 rechecked the lower_1 route and the original source anchors appendix.tex:984-995, appendix.tex:1026-1072, appendix.tex:1379-1387, main_body.tex:273-305, plus the original SALD TeX search excluding sald_version_2.tex. These sources give the EM Brownian/Laplacian display and nearby drift/score Lipschitz/integrability assumptions, but no selected weak-test global bounded-Hessian field. The VP score Hessian bound in iteration_complexity.tex:309-321 was also rejected for this boundary because it controls the score Hessian, not the selected weak test sourceTest. The remaining exact dependency is hHessianOpNorm : forall z : E, norm (fderiv Real (fderiv Real sourceTest) z) <= C1, classified as a source-contract gap under sald.general_moving_target_discrete.em_interpolation_fp. Brownian coordinate unit direction and the hHessianOpNorm-to-hSecondFDerivOpNorm bridge remain compiled by SALD.gaussianRealStdOrthonormalBasisUnit and SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm.

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.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower1Obligation
2. SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceMiddleObligation
3. SALD.cycle171GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation
4. SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm
5. SALD.gaussianRealStdOrthonormalBasisUnit
6. SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis
7. hHessianOpNorm
8. sourceTest
9. testRegular
10. SourceSelectedWeakTestC2bBoundedHessian proposed only as source-backed field
11. SelectedWeakTestC2bBoundedHessian projection rejected without source backing
12. iteration_complexity.tex:309-321 score Hessian bound rejected for sourceTest
13. appendix.tex:984-995
14. appendix.tex:1026-1072
15. appendix.tex:1379-1387
16. main_body.tex:273-305
17. paper-wide original-source search excluding sald_version_2.tex
18. source-contract-gap
19. 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 or imported. The only candidate outside the named source anchors was the VP score Hessian result in iteration_complexity.tex, and lower_2 rejected it because it is a score regularity theorem rather than a selected weak-test Hessian theorem. This packet keeps hHessianOpNorm as the exact source-contract gap and does not introduce a projection wrapper.

def cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle172_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_hessian_source_lower2"
  statement := "Cycle 172 lower_2 illness-area refiner packet. Classification: rejected-wrapper-churn. Exact attempted theorem rejected: selectedWeakTestHessianOpNormOfSourceBoundedHessian from SourceSelectedWeakTestC2bBoundedHessian sourceTest C1 to hHessianOpNorm. This theorem would be valid only after the source correspondence independently defines a selected weak-test C2_b/bounded-Hessian field; implementing it from unexpanded testRegular or from an unsourced SelectedWeakTestC2bBoundedHessian predicate would only rename the missing supplied hypothesis. Lower_2 rechecked the lower_1 route and the original source anchors appendix.tex:984-995, appendix.tex:1026-1072, appendix.tex:1379-1387, main_body.tex:273-305, plus the original SALD TeX search excluding sald_version_2.tex. These sources give the EM Brownian/Laplacian display and nearby drift/score Lipschitz/integrability assumptions, but no selected weak-test global bounded-Hessian field. The VP score Hessian bound in iteration_complexity.tex:309-321 was also rejected for this boundary because it controls the score Hessian, not the selected weak test sourceTest. The remaining exact dependency is hHessianOpNorm : forall z : E, norm (fderiv Real (fderiv Real sourceTest) z) <= C1, classified as a source-contract gap under sald.general_moving_target_discrete.em_interpolation_fp. Brownian coordinate unit direction and the hHessianOpNorm-to-hSecondFDerivOpNorm bridge remain compiled by SALD.gaussianRealStdOrthonormalBasisUnit and SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower1Obligation",
    "SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceMiddleObligation",
    "SALD.cycle171GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation",
    "SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm",
    "SALD.gaussianRealStdOrthonormalBasisUnit",
    "SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis",
    "hHessianOpNorm",
    "sourceTest",
    "testRegular",
    "SourceSelectedWeakTestC2bBoundedHessian proposed only as source-backed field",
    "SelectedWeakTestC2bBoundedHessian projection rejected without source backing",
    "iteration_complexity.tex:309-321 score Hessian bound rejected for sourceTest",
    "appendix.tex:984-995",
    "appendix.tex:1026-1072",
    "appendix.tex:1379-1387",
    "main_body.tex:273-305",
    "paper-wide original-source search excluding sald_version_2.tex",
    "source-contract-gap",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "No SLT file was consulted or imported. The only candidate outside the named source anchors was the VP score Hessian result in iteration_complexity.tex, and lower_2 rejected it because it is a score regularity theorem rather than a selected weak-test Hessian theorem. This packet keeps hHessianOpNorm as the exact source-contract gap and does not introduce a projection wrapper."

/-! ### Cycle 173 middle: Hessian source-contract recovery packet -/

/-- Cycle-173 middle refiner: blueprint-guided source-contract recovery for
the selected-test Hessian operator norm, with wrapper churn rejected. -/