AutoSamplingTheory.SALD.cycle170GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianOpNormLower2Obligation

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 cycle170GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianOpNormLower2Obligation :
    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.cycle170_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_hessian_op_norm_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 170 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hSecondFDerivOpNorm : forall z, norm (iteratedFDeriv Real 2 sourceTest z) <= C1 below SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis is no longer the primitive source-facing shape once a uniform Hessian-as-derivative-of-Frechet-derivative operator-norm bound forall z, norm (fderiv Real (fderiv Real sourceTest) z) <= C1 is supplied. SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm compiles this Mathlib bridge using norm_iteratedFDeriv_fderiv and norm_iteratedFDeriv_one. The remaining source-contract gap is to expose that selected weak-test C2_b/bounded-Hessian operator-norm bound from the paper/testRegular interface at appendix.tex:984-995 and appendix.tex:1379-1387. The Brownian coordinate unit-direction side condition was already discharged by SALD.gaussianRealStdOrthonormalBasisUnit; hSource, hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, coordinate-sum leaves, and EM weak-FP leaves remain separate.

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.formalized — 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.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm
2. SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis
3. SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm
4. SALD.gaussianRealStdOrthonormalBasisUnit
5. hSecondFDerivOpNorm
6. hHessianOpNorm
7. norm_iteratedFDeriv_fderiv
8. norm_iteratedFDeriv_one
9. appendix.tex:984-995
10. appendix.tex:1379-1387
11. 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.

Dynamic-leaf lower_2 packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib Analysis.Calculus.ContDiff.FTaylorSeries through the already imported Taylor stack; no SLT theorem was needed or imported. This rejects opaque testRegular-to-hSecondFDerivOpNorm wrapper churn by only translating a real bounded-Hessian source field into the Lean iteratedFDeriv operator-norm shape. If the paper/testRegular interface does not contain such a field, the remaining item is a source-contract gap, not a Mathlib or SLT gap.

def cycle170GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianOpNormLower2Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle170_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_hessian_op_norm_lower2"
  statement := "Cycle 170 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hSecondFDerivOpNorm : forall z, norm (iteratedFDeriv Real 2 sourceTest z) <= C1 below SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis is no longer the primitive source-facing shape once a uniform Hessian-as-derivative-of-Frechet-derivative operator-norm bound forall z, norm (fderiv Real (fderiv Real sourceTest) z) <= C1 is supplied. SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm compiles this Mathlib bridge using norm_iteratedFDeriv_fderiv and norm_iteratedFDeriv_one. The remaining source-contract gap is to expose that selected weak-test C2_b/bounded-Hessian operator-norm bound from the paper/testRegular interface at appendix.tex:984-995 and appendix.tex:1379-1387. The Brownian coordinate unit-direction side condition was already discharged by SALD.gaussianRealStdOrthonormalBasisUnit; hSource, hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, coordinate-sum leaves, and EM weak-FP leaves remain separate."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm",
    "SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis",
    "SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm",
    "SALD.gaussianRealStdOrthonormalBasisUnit",
    "hSecondFDerivOpNorm",
    "hHessianOpNorm",
    "norm_iteratedFDeriv_fderiv",
    "norm_iteratedFDeriv_one",
    "appendix.tex:984-995",
    "appendix.tex:1379-1387",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "Dynamic-leaf lower_2 packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib Analysis.Calculus.ContDiff.FTaylorSeries through the already imported Taylor stack; no SLT theorem was needed or imported. This rejects opaque testRegular-to-hSecondFDerivOpNorm wrapper churn by only translating a real bounded-Hessian source field into the Lean iteratedFDeriv operator-norm shape. If the paper/testRegular interface does not contain such a field, the remaining item is a source-contract gap, not a Mathlib or SLT gap."

/-- Cycle-170 lower_2 proof-DAG pane for the selected-test Hessian source
interface. -/