AutoSamplingTheory.SALD.cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPolynomialIntegrabilityLower1Obligation

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 cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPolynomialIntegrabilityLower1Obligation :
    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_polynomial_integrability_lower1

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_1 dynamic-leaf proof-scout/compiled packet. Classification: discharges-supplied-hypothesis. Exact supplied hypotheses discharged: hLinearInt and hQuadraticInt in the Taylor moment integral split below hFrozenScalarBrownianItoTaylorMomentDecomposition. SALD.gaussianRealLinearQuadraticTaylorSummandsIntegrable compiles Gaussian integrability of z and z^2 summands with arbitrary scalar coefficients from ProbabilityTheory.integrable_exp_mul_gaussianReal and ProbabilityTheory.integrable_pow_of_integrable_exp_mul. SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndGaussianPolynomialIntegrability then composes that helper with SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefs, so the downstream Taylor moment decomposition no longer needs hLinearInt or hQuadraticInt as supplied fields. Remaining exact Taylor-integral source boundary: hBrownianCoordinateGeneratorTaylorIntegralDef, hRemainderInt, and hRemainderGeneratorLimitDef, with hScalarLineTaylorCoeffDef, hNormalizedVectorLaw/hCoordinateLawDef/hVarianceDef, normalized-remainder DCT data, coordinate-sum, and hSourceHasHessian/hSourceHessianBound still 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.gaussianRealLinearQuadraticTaylorSummandsIntegrable
2. SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndGaussianPolynomialIntegrability
3. SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefs
4. hLinearInt
5. hQuadraticInt
6. hBrownianCoordinateGeneratorTaylorIntegralDef
7. hRemainderInt
8. hRemainderGeneratorLimitDef
9. ProbabilityTheory.integrable_exp_mul_gaussianReal
10. ProbabilityTheory.integrable_pow_of_integrable_exp_mul
11. MeasureTheory.Integrable.const_mul
12. appendix.tex:984-995
13. appendix.tex:1170-1176
14. appendix.tex:1379-1387
15. hScalarLineTaylorCoeffDef
16. hNormalizedVectorLaw
17. hCoordinateLawDef
18. hVarianceDef
19. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
20. hFrozenScalarBrownianItoEventFieldCoordinateSum
21. hSourceHasHessian
22. hSourceHessianBound
23. 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 ProbabilityTheory.integrable_exp_mul_gaussianReal, ProbabilityTheory.integrable_pow_of_integrable_exp_mul, and MeasureTheory.Integrable.const_mul already available through the current imports. This is not wrapper churn: it removes hLinearInt and hQuadraticInt from the newer Taylor moment bridge while preserving the source Taylor integral/remainder boundary.

def cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPolynomialIntegrabilityLower1Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle180_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_polynomial_integrability_lower1"
  statement := "Cycle 180 lower_1 dynamic-leaf proof-scout/compiled packet. Classification: discharges-supplied-hypothesis. Exact supplied hypotheses discharged: hLinearInt and hQuadraticInt in the Taylor moment integral split below hFrozenScalarBrownianItoTaylorMomentDecomposition. SALD.gaussianRealLinearQuadraticTaylorSummandsIntegrable compiles Gaussian integrability of z and z^2 summands with arbitrary scalar coefficients from ProbabilityTheory.integrable_exp_mul_gaussianReal and ProbabilityTheory.integrable_pow_of_integrable_exp_mul. SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndGaussianPolynomialIntegrability then composes that helper with SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefs, so the downstream Taylor moment decomposition no longer needs hLinearInt or hQuadraticInt as supplied fields. Remaining exact Taylor-integral source boundary: hBrownianCoordinateGeneratorTaylorIntegralDef, hRemainderInt, and hRemainderGeneratorLimitDef, with hScalarLineTaylorCoeffDef, hNormalizedVectorLaw/hCoordinateLawDef/hVarianceDef, normalized-remainder DCT data, coordinate-sum, and hSourceHasHessian/hSourceHessianBound still separate."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.gaussianRealLinearQuadraticTaylorSummandsIntegrable",
    "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndGaussianPolynomialIntegrability",
    "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefs",
    "hLinearInt",
    "hQuadraticInt",
    "hBrownianCoordinateGeneratorTaylorIntegralDef",
    "hRemainderInt",
    "hRemainderGeneratorLimitDef",
    "ProbabilityTheory.integrable_exp_mul_gaussianReal",
    "ProbabilityTheory.integrable_pow_of_integrable_exp_mul",
    "MeasureTheory.Integrable.const_mul",
    "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 ProbabilityTheory.integrable_exp_mul_gaussianReal, ProbabilityTheory.integrable_pow_of_integrable_exp_mul, and MeasureTheory.Integrable.const_mul already available through the current imports. This is not wrapper churn: it removes hLinearInt and hQuadraticInt from the newer Taylor moment bridge while preserving the source Taylor integral/remainder boundary."

/-- Cycle-180 lower_2 dominated-remainder integrability bridge.

This worker packet discharges the `hRemainderInt` input in the Taylor moment
split from the normalized-remainder measurability/domination package already
tracked in the Brownian/Ito scalar branch.  It leaves the source Taylor
integral definition and the remainder-generator definition explicit.
-/