AutoSamplingTheory.SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneLower2Obligation

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 cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneLower2Obligation :
    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.cycle176_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_variance_one_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 176 lower_2 dynamic-leaf Lean implementer packet. Classification: narrows-source-cited-boundary. Lower_1's exact variance theorem SALD.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef was already present, so lower_2 implemented the next non-duplicative compiled bridge in the same Brownian/Ito chain: SALD.selectedWeakTestQuadraticVariationNormalizationOfSecondTaylorCoeffAndNormalizedVarianceDef. Exact boundary narrowed: the quadratic-variation normalization no longer needs the older intermediate fields hQuadraticCoeffDef and hVarianceOne once the source correspondence supplies hSecondTaylorCoeffDef : testRegular -> forall phi x i, quadraticCoeff phi x i = iteratedFDeriv Real 2 (selectedTest phi) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] and hNormalizedVarianceDef : testRegular -> forall phi x i, variance phi x i = (1 : NNReal). The proof composes SALD.selectedWeakTestQuadraticCoeffDefOfSecondTaylorCoeffDef, SALD.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef, and SALD.selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne. Remaining source-facing gaps are exactly hSecondTaylorCoeffDef and hNormalizedVarianceDef for this normalization step; hSourceHasHessian/hSourceHessianBound, hSecondCoeff, Taylor moment decomposition, normalized-remainder DCT 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.selectedWeakTestQuadraticVariationNormalizationOfSecondTaylorCoeffAndNormalizedVarianceDef
2. SALD.selectedWeakTestQuadraticCoeffDefOfSecondTaylorCoeffDef
3. SALD.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef
4. SALD.selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne
5. SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneLower1Obligation
6. SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneMiddleObligation
7. hSecondTaylorCoeffDef
8. hNormalizedVarianceDef
9. hQuadraticCoeffDef
10. hVarianceOne
11. hSourceHasHessian
12. hSourceHessianBound
13. hSecondCoeff
14. hFrozenScalarBrownianItoTaylorMomentDecomposition
15. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
16. hFrozenScalarBrownianItoEventFieldCoordinateSum
17. eq:SALD_general_EM
18. eq:general_moving_target_SALD_frozen_interp
19. eq:increment_basic_bound_revised
20. appendix.tex:958-970
21. appendix.tex:984-995
22. appendix.tex:1170-1176
23. appendix.tex:1379-1387
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. The compiled proof only composes local bridges already in the Brownian/Ito chain; it does not prove the stochastic source fields hSecondTaylorCoeffDef or hNormalizedVarianceDef, does not move sigma_eta^2/2 inside the event field, and does not add a source-Hessian wrapper.

def cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneLower2Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle176_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_variance_one_lower2"
  statement := "Cycle 176 lower_2 dynamic-leaf Lean implementer packet. Classification: narrows-source-cited-boundary. Lower_1's exact variance theorem SALD.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef was already present, so lower_2 implemented the next non-duplicative compiled bridge in the same Brownian/Ito chain: SALD.selectedWeakTestQuadraticVariationNormalizationOfSecondTaylorCoeffAndNormalizedVarianceDef. Exact boundary narrowed: the quadratic-variation normalization no longer needs the older intermediate fields hQuadraticCoeffDef and hVarianceOne once the source correspondence supplies hSecondTaylorCoeffDef : testRegular -> forall phi x i, quadraticCoeff phi x i = iteratedFDeriv Real 2 (selectedTest phi) x ![(stdOrthonormalBasis Real E) i, (stdOrthonormalBasis Real E) i] and hNormalizedVarianceDef : testRegular -> forall phi x i, variance phi x i = (1 : NNReal). The proof composes SALD.selectedWeakTestQuadraticCoeffDefOfSecondTaylorCoeffDef, SALD.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef, and SALD.selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne. Remaining source-facing gaps are exactly hSecondTaylorCoeffDef and hNormalizedVarianceDef for this normalization step; hSourceHasHessian/hSourceHessianBound, hSecondCoeff, Taylor moment decomposition, normalized-remainder DCT data, and coordinate-sum remain separate."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.selectedWeakTestQuadraticVariationNormalizationOfSecondTaylorCoeffAndNormalizedVarianceDef",
    "SALD.selectedWeakTestQuadraticCoeffDefOfSecondTaylorCoeffDef",
    "SALD.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef",
    "SALD.selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne",
    "SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneLower1Obligation",
    "SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneMiddleObligation",
    "hSecondTaylorCoeffDef",
    "hNormalizedVarianceDef",
    "hQuadraticCoeffDef",
    "hVarianceOne",
    "hSourceHasHessian",
    "hSourceHessianBound",
    "hSecondCoeff",
    "hFrozenScalarBrownianItoTaylorMomentDecomposition",
    "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes",
    "hFrozenScalarBrownianItoEventFieldCoordinateSum",
    "eq:SALD_general_EM",
    "eq:general_moving_target_SALD_frozen_interp",
    "eq:increment_basic_bound_revised",
    "appendix.tex:958-970",
    "appendix.tex:984-995",
    "appendix.tex:1170-1176",
    "appendix.tex:1379-1387",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "No local SLT file was consulted or imported. The compiled proof only composes local bridges already in the Brownian/Ito chain; it does not prove the stochastic source fields hSecondTaylorCoeffDef or hNormalizedVarianceDef, does not move sigma_eta^2/2 inside the event field, and does not add a source-Hessian wrapper."

/-- Cycle-176 proof-DAG pane for the normalized Brownian variance field. -/