AutoSamplingTheory.SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower2Obligation

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 cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower2Obligation :
    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_coordinate_law_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 178 lower_2 dynamic-leaf Lean implementer packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hNormalizedCoordinateLaw is no longer primitive once the source correspondence supplies the normalized vector standard-Gaussian law hNormalizedVectorLaw : testRegular -> forall phi x, normalizedVectorLaw phi x = ProbabilityTheory.stdGaussian E and the coordinate-law definition hCoordinateLawDef : testRegular -> forall phi x i, normalizedCoordinateLaw phi x i = (normalizedVectorLaw phi x).map (fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y). SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw compiles the Mathlib coordinate projection using ProbabilityTheory.IsGaussian.map_eq_gaussianReal, ProbabilityTheory.integral_strongDual_stdGaussian, ProbabilityTheory.variance_dual_stdGaussian, InnerProductSpace.toDual, and the standard-orthonormal-basis norm simplifier. Remaining exact variance-side source boundary is hNormalizedVectorLaw plus hCoordinateLawDef plus the separate hVarianceDef packaging field; hScalarLineSecondCoeffDef and hSourceHasHessian/hSourceHessianBound remain separate source gaps.

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.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw
2. SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower1Obligation
3. SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw
4. hNormalizedCoordinateLaw
5. hNormalizedVectorLaw
6. hCoordinateLawDef
7. hVarianceDef
8. ProbabilityTheory.stdGaussian
9. ProbabilityTheory.IsGaussian.map_eq_gaussianReal
10. ProbabilityTheory.integral_strongDual_stdGaussian
11. ProbabilityTheory.variance_dual_stdGaussian
12. InnerProductSpace.toDual
13. stdOrthonormalBasis
14. appendix.tex:958-970
15. appendix.tex:984-995
16. appendix.tex:1170-1176
17. appendix.tex:1379-1387
18. 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 bridge uses Mathlib's multivariate Gaussian coordinate projection for stdGaussian and source-law rewrites; it does not move sigma_eta^2/2 into the event field, derive source-Hessian fields, promote hVarianceDef, or use sald_version_2.tex.

def cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower2Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle178_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_normalized_coordinate_law_lower2"
  statement := "Cycle 178 lower_2 dynamic-leaf Lean implementer packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hNormalizedCoordinateLaw is no longer primitive once the source correspondence supplies the normalized vector standard-Gaussian law hNormalizedVectorLaw : testRegular -> forall phi x, normalizedVectorLaw phi x = ProbabilityTheory.stdGaussian E and the coordinate-law definition hCoordinateLawDef : testRegular -> forall phi x i, normalizedCoordinateLaw phi x i = (normalizedVectorLaw phi x).map (fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y). SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw compiles the Mathlib coordinate projection using ProbabilityTheory.IsGaussian.map_eq_gaussianReal, ProbabilityTheory.integral_strongDual_stdGaussian, ProbabilityTheory.variance_dual_stdGaussian, InnerProductSpace.toDual, and the standard-orthonormal-basis norm simplifier. Remaining exact variance-side source boundary is hNormalizedVectorLaw plus hCoordinateLawDef plus the separate hVarianceDef packaging field; hScalarLineSecondCoeffDef and hSourceHasHessian/hSourceHessianBound remain separate source gaps."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw",
    "SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower1Obligation",
    "SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw",
    "hNormalizedCoordinateLaw",
    "hNormalizedVectorLaw",
    "hCoordinateLawDef",
    "hVarianceDef",
    "ProbabilityTheory.stdGaussian",
    "ProbabilityTheory.IsGaussian.map_eq_gaussianReal",
    "ProbabilityTheory.integral_strongDual_stdGaussian",
    "ProbabilityTheory.variance_dual_stdGaussian",
    "InnerProductSpace.toDual",
    "stdOrthonormalBasis",
    "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 bridge uses Mathlib's multivariate Gaussian coordinate projection for stdGaussian and source-law rewrites; it does not move sigma_eta^2/2 into the event field, derive source-Hessian fields, promote hVarianceDef, or use sald_version_2.tex."

/-- Cycle-178 proof-DAG pane for normalized Brownian variance law narrowing. -/