AutoSamplingTheory.SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianAuditScalarLineCoeffMiddleObligation

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 cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianAuditScalarLineCoeffMiddleObligation :
    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.cycle179_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_hessian_audit_scalar_line_coeff_middle

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 179 middle dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact source decision: hSourceHasHessian : forall z, HasFDerivAt (fderiv Real sourceTest) (sourceHessian z) z and hSourceHessianBound : forall z, norm (sourceHessian z) <= C1 remain source-contract gaps. The checked original-source anchors appendix.tex:984-995, appendix.tex:1026-1072, appendix.tex:1170-1176, appendix.tex:1379-1387, main_body.tex:273-305, and iteration_complexity.tex:309-321 do not supply selected weak-test Hessian regularity; iteration_complexity.tex:309-321 is a VP score-Hessian result, not a Hessian bound for sourceTest. Middle therefore rejects testRegular projection, sourceHessian-by-conclusion, SourceSelectedWeakTestC2bBoundedHessian or SelectedWeakTestC2bBoundedHessian without source backing, VP score-Hessian substitution, SLT import, theorem-status promotion, and wrapper churn. Exact next lower boundary: hScalarLineSecondCoeffDef, the scalar Brownian coordinate Taylor coefficient before the external sigma_eta^2/2 diffusion coefficient in appendix.tex:1379-1387. Lower_1 should give the natural-language source/Mathlib route from appendix.tex:984-995, appendix.tex:1170-1176, and appendix.tex:1379-1387; lower_2 should implement exactly one compiled theorem or record one strictly smaller source-cited obligation narrowing hScalarLineSecondCoeffDef. Keep hNormalizedVectorLaw, hCoordinateLawDef, hVarianceDef, hSecondCoeff, Taylor moment decomposition, DCT domination, coordinate-sum, and the Hessian gaps explicit.

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.selectedWeakTestHessianOpNormOfSourceHessianField
2. SALD.selectedWeakTestSecondTaylorCoeffDefOfScalarLineSecondCoeffDef
3. SALD.selectedWeakTestQuadraticVariationNormalizationOfScalarLineSecondCoeffAndNormalizedVarianceDef
4. SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw
5. SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw
6. hSourceHasHessian
7. hSourceHessianBound
8. sourceHessian
9. hScalarLineSecondCoeffDef
10. hNormalizedVectorLaw
11. hCoordinateLawDef
12. hVarianceDef
13. hSecondCoeff
14. hFrozenScalarBrownianItoTaylorMomentDecomposition
15. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
16. hFrozenScalarBrownianItoEventFieldCoordinateSum
17. eq:SALD_general_EM
18. eq:general_moving_target_SALD_frozen_interp
19. eq:increment_basic_bound_revised
20. appendix.tex:958-970
21. appendix.tex:984-995
22. appendix.tex:1026-1072
23. appendix.tex:1170-1176
24. appendix.tex:1379-1387
25. main_body.tex:273-305
26. iteration_complexity.tex:309-321 score Hessian bound rejected for sourceTest
27. paper-wide original-source search excluding sald_version_2.tex
28. 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 local SLT file was consulted or imported; this is a source-dependency audit plus lower packet assignment. Mathlib lookup was not needed beyond existing compiled local declarations. The packet does not define sourceHessian from the desired conclusion, does not move sigma_eta^2/2 into the event field, does not use VP score-Hessian regularity for the selected weak test, and does not use sald_version_2.tex.

def cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianAuditScalarLineCoeffMiddleObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle179_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_hessian_audit_scalar_line_coeff_middle"
  statement := "Cycle 179 middle dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact source decision: hSourceHasHessian : forall z, HasFDerivAt (fderiv Real sourceTest) (sourceHessian z) z and hSourceHessianBound : forall z, norm (sourceHessian z) <= C1 remain source-contract gaps. The checked original-source anchors appendix.tex:984-995, appendix.tex:1026-1072, appendix.tex:1170-1176, appendix.tex:1379-1387, main_body.tex:273-305, and iteration_complexity.tex:309-321 do not supply selected weak-test Hessian regularity; iteration_complexity.tex:309-321 is a VP score-Hessian result, not a Hessian bound for sourceTest. Middle therefore rejects testRegular projection, sourceHessian-by-conclusion, SourceSelectedWeakTestC2bBoundedHessian or SelectedWeakTestC2bBoundedHessian without source backing, VP score-Hessian substitution, SLT import, theorem-status promotion, and wrapper churn. Exact next lower boundary: hScalarLineSecondCoeffDef, the scalar Brownian coordinate Taylor coefficient before the external sigma_eta^2/2 diffusion coefficient in appendix.tex:1379-1387. Lower_1 should give the natural-language source/Mathlib route from appendix.tex:984-995, appendix.tex:1170-1176, and appendix.tex:1379-1387; lower_2 should implement exactly one compiled theorem or record one strictly smaller source-cited obligation narrowing hScalarLineSecondCoeffDef. Keep hNormalizedVectorLaw, hCoordinateLawDef, hVarianceDef, hSecondCoeff, Taylor moment decomposition, DCT domination, coordinate-sum, and the Hessian gaps explicit."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.selectedWeakTestHessianOpNormOfSourceHessianField",
    "SALD.selectedWeakTestSecondTaylorCoeffDefOfScalarLineSecondCoeffDef",
    "SALD.selectedWeakTestQuadraticVariationNormalizationOfScalarLineSecondCoeffAndNormalizedVarianceDef",
    "SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw",
    "SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw",
    "hSourceHasHessian",
    "hSourceHessianBound",
    "sourceHessian",
    "hScalarLineSecondCoeffDef",
    "hNormalizedVectorLaw",
    "hCoordinateLawDef",
    "hVarianceDef",
    "hSecondCoeff",
    "hFrozenScalarBrownianItoTaylorMomentDecomposition",
    "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes",
    "hFrozenScalarBrownianItoEventFieldCoordinateSum",
    "eq:SALD_general_EM",
    "eq:general_moving_target_SALD_frozen_interp",
    "eq:increment_basic_bound_revised",
    "appendix.tex:958-970",
    "appendix.tex:984-995",
    "appendix.tex:1026-1072",
    "appendix.tex:1170-1176",
    "appendix.tex:1379-1387",
    "main_body.tex:273-305",
    "iteration_complexity.tex:309-321 score Hessian bound rejected for sourceTest",
    "paper-wide original-source search excluding sald_version_2.tex",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "No local SLT file was consulted or imported; this is a source-dependency audit plus lower packet assignment. Mathlib lookup was not needed beyond existing compiled local declarations. The packet does not define sourceHessian from the desired conclusion, does not move sigma_eta^2/2 into the event field, does not use VP score-Hessian regularity for the selected weak test, and does not use sald_version_2.tex."

/-- Cycle-179 proof-DAG pane for the source-Hessian audit decision and next
scalar-line coefficient packet. -/