AutoSamplingTheory.SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationMiddleObligation

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 cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationMiddleObligation :
    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.cycle175_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_taylor_domination_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 175 middle dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the standard-basis selected-line first-order quadratic Taylor remainder below SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff is no longer primitive once the source correspondence supplies ambient selected-test ContDiffOn Real 2, a Hessian representative sourceHessian, hSourceHasHessian : forall z, HasFDerivAt (fderiv Real sourceTest) (sourceHessian z) z, and hSourceHessianBound : forall z, norm (sourceHessian z) <= C1. SALD.gaussianRealSelectedTestStdOrthonormalFirstOrderQuadraticRemainderBoundOfSourceHessianField compiles this bridge for e = (stdOrthonormalBasis Real E) i by composing SALD.selectedWeakTestHessianOpNormOfSourceHessianField, SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm, SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis, and SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound. The remaining source-facing boundary is still hSource plus hSourceHasHessian/hSourceHessianBound for the selected weak test; hSecondCoeff, Taylor moment decomposition, hQuadraticCoeffDef/hVarianceOne, normalized-remainder DCT data, and coordinate-sum leaves remain 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.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.gaussianRealSelectedTestStdOrthonormalFirstOrderQuadraticRemainderBoundOfSourceHessianField
2. SALD.selectedWeakTestHessianOpNormOfSourceHessianField
3. SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm
4. SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis
5. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound
6. hSource
7. hSourceHasHessian
8. hSourceHessianBound
9. hSecondCoeff
10. hFrozenScalarBrownianItoTaylorMomentDecomposition
11. hQuadraticCoeffDef
12. hVarianceOne
13. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
14. hFrozenScalarBrownianItoEventFieldCoordinateSum
15. eq:general_moving_target_SALD_frozen_interp
16. appendix.tex:984-995
17. appendix.tex:1026-1072
18. appendix.tex:1379-1387
19. main_body.tex:273-305
20. 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. The compiled proof uses only existing local Mathlib/Lean bridges for HasFDerivAt.fderiv uniqueness, iterated-Frechet norm conversion, standard-basis unit directions, directional-Hessian control, and the selected-line Taylor remainder. It rejects deriving hSourceHasHessian or hSourceHessianBound from unexpanded testRegular, a sourceHessian defined from the desired conclusion, an unsourced SelectedWeakTestC2bBoundedHessian predicate, or VP score-Hessian regularity.

def cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationMiddleObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle175_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_taylor_domination_middle"
  statement := "Cycle 175 middle dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the standard-basis selected-line first-order quadratic Taylor remainder below SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff is no longer primitive once the source correspondence supplies ambient selected-test ContDiffOn Real 2, a Hessian representative sourceHessian, hSourceHasHessian : forall z, HasFDerivAt (fderiv Real sourceTest) (sourceHessian z) z, and hSourceHessianBound : forall z, norm (sourceHessian z) <= C1. SALD.gaussianRealSelectedTestStdOrthonormalFirstOrderQuadraticRemainderBoundOfSourceHessianField compiles this bridge for e = (stdOrthonormalBasis Real E) i by composing SALD.selectedWeakTestHessianOpNormOfSourceHessianField, SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm, SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis, and SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound. The remaining source-facing boundary is still hSource plus hSourceHasHessian/hSourceHessianBound for the selected weak test; hSecondCoeff, Taylor moment decomposition, hQuadraticCoeffDef/hVarianceOne, normalized-remainder DCT data, and coordinate-sum leaves remain explicit."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.gaussianRealSelectedTestStdOrthonormalFirstOrderQuadraticRemainderBoundOfSourceHessianField",
    "SALD.selectedWeakTestHessianOpNormOfSourceHessianField",
    "SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm",
    "SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis",
    "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound",
    "hSource",
    "hSourceHasHessian",
    "hSourceHessianBound",
    "hSecondCoeff",
    "hFrozenScalarBrownianItoTaylorMomentDecomposition",
    "hQuadraticCoeffDef",
    "hVarianceOne",
    "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes",
    "hFrozenScalarBrownianItoEventFieldCoordinateSum",
    "eq:general_moving_target_SALD_frozen_interp",
    "appendix.tex:984-995",
    "appendix.tex:1026-1072",
    "appendix.tex:1379-1387",
    "main_body.tex:273-305",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "No local SLT file was consulted or imported. The compiled proof uses only existing local Mathlib/Lean bridges for HasFDerivAt.fderiv uniqueness, iterated-Frechet norm conversion, standard-basis unit directions, directional-Hessian control, and the selected-line Taylor remainder. It rejects deriving hSourceHasHessian or hSourceHessianBound from unexpanded testRegular, a sourceHessian defined from the desired conclusion, an unsourced SelectedWeakTestC2bBoundedHessian predicate, or VP score-Hessian regularity."

/-- Cycle-175 lower_1 proof-scout packet for the quadratic-coefficient source
boundary.

After the selected-line Taylor-domination bridge is compiled, the connected
Brownian/Ito coefficient leaf should not be closed by a broad test-regularity
wrapper.  This packet narrows `hQuadraticCoeffDef` to the exact source-facing
second-Taylor-coefficient identity for the standard Brownian coordinate.
-/