AutoSamplingTheory.SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceMiddleObligation

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 cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceMiddleObligation :
    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.cycle178_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_normalized_variance_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 178 middle dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hNormalizedVarianceDef is no longer primitive once the source correspondence supplies a normalized scalar Brownian coordinate law normalizedCoordinateLaw with hNormalizedCoordinateLaw : testRegular -> forall phi x i, normalizedCoordinateLaw phi x i = ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal), and the local variance-field definition hVarianceDef : testRegular -> forall phi x i, (variance phi x i : Real) = ProbabilityTheory.variance (id : Real -> Real) (normalizedCoordinateLaw phi x i). SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw compiles the local Mathlib bridge using ProbabilityTheory.variance_id_gaussianReal and NNReal.coe_injective. Source anchors: appendix.tex:958-970 gives xi_k iid N(0,I_d), appendix.tex:984-995 gives the frozen Brownian increment sigma_eta*(W_s-W_{s_k}), appendix.tex:1170-1176 rewrites the increment with xi ~ N(0,I), and appendix.tex:1379-1387 keeps sigma_eta^2/2 outside the Brownian event field. Remaining exact source boundary after this packet is hNormalizedCoordinateLaw plus hVarianceDef on the variance side, together with the separate hScalarLineSecondCoeffDef coefficient boundary; hSourceHasHessian/hSourceHessianBound, hSecondCoeff, Taylor moment decomposition, normalized-remainder DCT data, and coordinate-sum stay 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.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw
2. ProbabilityTheory.variance_id_gaussianReal
3. NNReal.coe_injective
4. hNormalizedVarianceDef
5. hNormalizedCoordinateLaw
6. hVarianceDef
7. hScalarLineSecondCoeffDef
8. SALD.selectedWeakTestQuadraticVariationNormalizationOfScalarLineSecondCoeffAndNormalizedVarianceDef
9. SALD.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef
10. hSourceHasHessian
11. hSourceHessianBound
12. hSecondCoeff
13. hFrozenScalarBrownianItoTaylorMomentDecomposition
14. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
15. hFrozenScalarBrownianItoEventFieldCoordinateSum
16. eq:SALD_general_EM
17. eq:general_moving_target_SALD_frozen_interp
18. eq:increment_basic_bound_revised
19. appendix.tex:958-970
20. appendix.tex:984-995
21. appendix.tex:1170-1176
22. appendix.tex:1379-1387
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. The compiled proof uses only Mathlib's real Gaussian variance theorem and NNReal coercion injectivity. It does not reprove the stochastic source law hNormalizedCoordinateLaw, does not move sigma_eta^2/2 into the event field, does not derive source-Hessian fields, and does not use sald_version_2.tex.

def cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceMiddleObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle178_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_normalized_variance_middle"
  statement := "Cycle 178 middle dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hNormalizedVarianceDef is no longer primitive once the source correspondence supplies a normalized scalar Brownian coordinate law normalizedCoordinateLaw with hNormalizedCoordinateLaw : testRegular -> forall phi x i, normalizedCoordinateLaw phi x i = ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal), and the local variance-field definition hVarianceDef : testRegular -> forall phi x i, (variance phi x i : Real) = ProbabilityTheory.variance (id : Real -> Real) (normalizedCoordinateLaw phi x i). SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw compiles the local Mathlib bridge using ProbabilityTheory.variance_id_gaussianReal and NNReal.coe_injective. Source anchors: appendix.tex:958-970 gives xi_k iid N(0,I_d), appendix.tex:984-995 gives the frozen Brownian increment sigma_eta*(W_s-W_{s_k}), appendix.tex:1170-1176 rewrites the increment with xi ~ N(0,I), and appendix.tex:1379-1387 keeps sigma_eta^2/2 outside the Brownian event field. Remaining exact source boundary after this packet is hNormalizedCoordinateLaw plus hVarianceDef on the variance side, together with the separate hScalarLineSecondCoeffDef coefficient boundary; hSourceHasHessian/hSourceHessianBound, hSecondCoeff, Taylor moment decomposition, normalized-remainder DCT data, and coordinate-sum stay explicit."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw",
    "ProbabilityTheory.variance_id_gaussianReal",
    "NNReal.coe_injective",
    "hNormalizedVarianceDef",
    "hNormalizedCoordinateLaw",
    "hVarianceDef",
    "hScalarLineSecondCoeffDef",
    "SALD.selectedWeakTestQuadraticVariationNormalizationOfScalarLineSecondCoeffAndNormalizedVarianceDef",
    "SALD.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef",
    "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 uses only Mathlib's real Gaussian variance theorem and NNReal coercion injectivity. It does not reprove the stochastic source law hNormalizedCoordinateLaw, does not move sigma_eta^2/2 into the event field, does not derive source-Hessian fields, and does not use sald_version_2.tex."

/-- Cycle-178 lower_1 proof-scout packet for the normalized coordinate law. -/