AutoSamplingTheory.SALD.cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondLower2Obligation

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 cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondLower2Obligation :
    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.cycle169_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_directional_second_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 169 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hDirectionalSecond below SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound is no longer primitive once the selected-test second Frechet derivative has a uniform operator-norm bound forall z, norm (iteratedFDeriv Real 2 sourceTest z) <= C1 and the selected direction satisfies norm e <= 1. SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm compiles this local bounded-Hessian leaf using ContinuousMultilinearMap.le_mul_prod_of_opNorm_le_of_le and the product over Fin 2 of unit bounds. Remaining source-cited boundary: expose that selected weak-test C^2_b/bounded-Hessian operator-norm hypothesis, and the intended Brownian coordinate unit-direction fact, from the paper/testRegular interface at appendix.tex:984-995 and appendix.tex:1379-1387. hSource, hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum 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.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm
2. hDirectionalSecond
3. hSecondFDerivOpNorm
4. heUnit
5. ContinuousMultilinearMap.le_mul_prod_of_opNorm_le_of_le
6. Finset.prod_const
7. appendix.tex:984-995
8. appendix.tex:1379-1387
9. 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.Normed.Module.Multilinear.Basic; local SLT clone is absent and not needed. This rejects wrapper churn by removing the diagonal directional-Hessian hypothesis in favor of the smaller selected-test bounded-Hessian operator-norm contract plus a unit-direction side condition.

def cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondLower2Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle169_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_directional_second_lower2"
  statement := "Cycle 169 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hDirectionalSecond below SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound is no longer primitive once the selected-test second Frechet derivative has a uniform operator-norm bound forall z, norm (iteratedFDeriv Real 2 sourceTest z) <= C1 and the selected direction satisfies norm e <= 1. SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm compiles this local bounded-Hessian leaf using ContinuousMultilinearMap.le_mul_prod_of_opNorm_le_of_le and the product over Fin 2 of unit bounds. Remaining source-cited boundary: expose that selected weak-test C^2_b/bounded-Hessian operator-norm hypothesis, and the intended Brownian coordinate unit-direction fact, from the paper/testRegular interface at appendix.tex:984-995 and appendix.tex:1379-1387. hSource, hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves remain separate."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm",
    "hDirectionalSecond",
    "hSecondFDerivOpNorm",
    "heUnit",
    "ContinuousMultilinearMap.le_mul_prod_of_opNorm_le_of_le",
    "Finset.prod_const",
    "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.Normed.Module.Multilinear.Basic; local SLT clone is absent and not needed. This rejects wrapper churn by removing the diagonal directional-Hessian hypothesis in favor of the smaller selected-test bounded-Hessian operator-norm contract plus a unit-direction side condition."

/-- Cycle-167 proof-DAG pane for the Taylor-data narrowing. -/