AutoSamplingTheory.SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneMiddleObligation

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 cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneMiddleObligation :
    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_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 176 middle dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the downstream hVarianceOne field under SALD.selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne is no longer primitive once the source correspondence supplies hNormalizedVarianceDef : testRegular -> forall phi x i, variance phi x i = (1 : NNReal), expressing that the scalar Brownian coordinate has been normalized to a standard Gaussian coordinate. SALD.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef compiles the local NNReal-to-Real coercion bridge to recover hVarianceOne. Source anchors: appendix.tex:958-970 gives xi_k iid N(0,I_d) in the EM step, appendix.tex:984-995 gives the frozen interpolation Brownian increment sigma_eta*(W_s-W_{s_k}), appendix.tex:1170-1176 rewrites the increment as sigma_eta(t(s))*sqrt(s-s_k)*xi with xi ~ N(0,I), and appendix.tex:1379-1387 keeps sigma_eta^2/2 outside the Brownian event field. Remaining source-facing gaps are hNormalizedVarianceDef, hSecondTaylorCoeffDef, hSourceHasHessian/hSourceHessianBound, hSecondCoeff, Taylor moment decomposition, normalized-remainder DCT data, and coordinate-sum; no source-Hessian wrapper, VP score-Hessian substitution, SLT import, theorem-status promotion, or sald_version_2 use.

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.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef
2. SALD.selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne
3. hVarianceOne
4. hNormalizedVarianceDef
5. hSecondTaylorCoeffDef
6. hSourceHasHessian
7. hSourceHessianBound
8. hSecondCoeff
9. hFrozenScalarBrownianItoTaylorMomentDecomposition
10. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
11. hFrozenScalarBrownianItoEventFieldCoordinateSum
12. eq:SALD_general_EM
13. eq:general_moving_target_SALD_frozen_interp
14. eq:increment_basic_bound_revised
15. appendix.tex:958-970
16. appendix.tex:984-995
17. appendix.tex:1170-1176
18. appendix.tex:1379-1387
19. 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 only compiled proof ingredient is the local NNReal-to-Real coercion bridge for the normalized Brownian variance field; the stochastic source correspondence still has to supply hNormalizedVarianceDef from the standard Gaussian coordinate normalization.

def cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneMiddleObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle176_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_variance_one_middle"
  statement := "Cycle 176 middle dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the downstream hVarianceOne field under SALD.selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne is no longer primitive once the source correspondence supplies hNormalizedVarianceDef : testRegular -> forall phi x i, variance phi x i = (1 : NNReal), expressing that the scalar Brownian coordinate has been normalized to a standard Gaussian coordinate. SALD.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef compiles the local NNReal-to-Real coercion bridge to recover hVarianceOne. Source anchors: appendix.tex:958-970 gives xi_k iid N(0,I_d) in the EM step, appendix.tex:984-995 gives the frozen interpolation Brownian increment sigma_eta*(W_s-W_{s_k}), appendix.tex:1170-1176 rewrites the increment as sigma_eta(t(s))*sqrt(s-s_k)*xi with xi ~ N(0,I), and appendix.tex:1379-1387 keeps sigma_eta^2/2 outside the Brownian event field. Remaining source-facing gaps are hNormalizedVarianceDef, hSecondTaylorCoeffDef, hSourceHasHessian/hSourceHessianBound, hSecondCoeff, Taylor moment decomposition, normalized-remainder DCT data, and coordinate-sum; no source-Hessian wrapper, VP score-Hessian substitution, SLT import, theorem-status promotion, or sald_version_2 use."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef",
    "SALD.selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne",
    "hVarianceOne",
    "hNormalizedVarianceDef",
    "hSecondTaylorCoeffDef",
    "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 only compiled proof ingredient is the local NNReal-to-Real coercion bridge for the normalized Brownian variance field; the stochastic source correspondence still has to supply hNormalizedVarianceDef from the standard Gaussian coordinate normalization."

/-- Cycle-176 lower_1 proof-scout route for the normalized scalar Brownian
variance field. -/