AutoSamplingTheory.SALD.cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDominatedRemainderLower2Obligation

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 cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDominatedRemainderLower2Obligation :
    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.cycle180_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_dominated_remainder_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 180 lower_2 dynamic-leaf worker packet. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: hRemainderInt in the Taylor moment integral split below hFrozenScalarBrownianItoTaylorMomentDecomposition. SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndDominatedRemainder compiles the dominated-integrability bridge: from hRemainderMeas, hRemainderBound, and hRemainderBoundInt it derives Integrable (fun z => normalizedRemainder phi x i z) by MeasureTheory.Integrable.mono', then reuses the cycle-180 Taylor moment split with Gaussian polynomial integrability. Remaining exact Taylor-integral source boundary: hBrownianCoordinateGeneratorTaylorIntegralDef and hRemainderGeneratorLimitDef, plus the concrete normalized-remainder measurability/domination package hRemainderMeas/hRemainderBound/hRemainderBoundInt. hSourceHasHessian/hSourceHessianBound remain source-contract gaps; hScalarLineTaylorCoeffDef, normalized Gaussian law fields, DCT pointwise data, and coordinate-sum 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.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndDominatedRemainder
2. SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndGaussianPolynomialIntegrability
3. SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefs
4. MeasureTheory.Integrable.mono'
5. hRemainderInt
6. hRemainderMeas
7. hRemainderBound
8. hRemainderBoundInt
9. hBrownianCoordinateGeneratorTaylorIntegralDef
10. hRemainderGeneratorLimitDef
11. normalizedRemainder
12. remainderBound
13. appendix.tex:984-995
14. appendix.tex:1170-1176
15. appendix.tex:1379-1387
16. hScalarLineTaylorCoeffDef
17. hNormalizedVectorLaw
18. hCoordinateLawDef
19. hVarianceDef
20. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
21. hFrozenScalarBrownianItoEventFieldCoordinateSum
22. hSourceHasHessian
23. hSourceHessianBound
24. 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. Mathlib consultation was limited to MeasureTheory.Integrable.mono' and the already compiled Gaussian/Taylor moment bridge. This is not wrapper churn: it removes hRemainderInt from the Taylor moment split and exposes the smaller normalized-remainder measurability/domination package without changing the source theorem target.

def cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDominatedRemainderLower2Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle180_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_dominated_remainder_lower2"
  statement := "Cycle 180 lower_2 dynamic-leaf worker packet. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: hRemainderInt in the Taylor moment integral split below hFrozenScalarBrownianItoTaylorMomentDecomposition. SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndDominatedRemainder compiles the dominated-integrability bridge: from hRemainderMeas, hRemainderBound, and hRemainderBoundInt it derives Integrable (fun z => normalizedRemainder phi x i z) by MeasureTheory.Integrable.mono', then reuses the cycle-180 Taylor moment split with Gaussian polynomial integrability. Remaining exact Taylor-integral source boundary: hBrownianCoordinateGeneratorTaylorIntegralDef and hRemainderGeneratorLimitDef, plus the concrete normalized-remainder measurability/domination package hRemainderMeas/hRemainderBound/hRemainderBoundInt. hSourceHasHessian/hSourceHessianBound remain source-contract gaps; hScalarLineTaylorCoeffDef, normalized Gaussian law fields, DCT pointwise data, and coordinate-sum remain separate."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndDominatedRemainder",
    "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndGaussianPolynomialIntegrability",
    "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefs",
    "MeasureTheory.Integrable.mono'",
    "hRemainderInt",
    "hRemainderMeas",
    "hRemainderBound",
    "hRemainderBoundInt",
    "hBrownianCoordinateGeneratorTaylorIntegralDef",
    "hRemainderGeneratorLimitDef",
    "normalizedRemainder",
    "remainderBound",
    "appendix.tex:984-995",
    "appendix.tex:1170-1176",
    "appendix.tex:1379-1387",
    "hScalarLineTaylorCoeffDef",
    "hNormalizedVectorLaw",
    "hCoordinateLawDef",
    "hVarianceDef",
    "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes",
    "hFrozenScalarBrownianItoEventFieldCoordinateSum",
    "hSourceHasHessian",
    "hSourceHessianBound",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "No local SLT file was consulted or imported. Mathlib consultation was limited to MeasureTheory.Integrable.mono' and the already compiled Gaussian/Taylor moment bridge. This is not wrapper churn: it removes hRemainderInt from the Taylor moment split and exposes the smaller normalized-remainder measurability/domination package without changing the source theorem target."

/-- Cycle-180 proof-DAG pane for the Taylor moment decomposition split. -/