AutoSamplingTheory.SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower1Obligation

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 cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower1Obligation :
    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_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 178 lower_1 dynamic-leaf proof-scout packet. Classification: narrows-source-cited-boundary. Exact missing theorem boundary narrowed: hNormalizedCoordinateLaw should not remain a primitive scalar-law field; it should follow from the paper's vector standard-Gaussian normalized increment law hNormalizedVectorLaw : testRegular -> forall phi x, normalizedVectorLaw phi x = ProbabilityTheory.stdGaussian E, plus the coordinate-law definition hCoordinateLawDef : testRegular -> forall phi x i, normalizedCoordinateLaw phi x i = (normalizedVectorLaw phi x).map (fun y => inner (stdOrthonormalBasis Real E i) y). Source route: appendix.tex:958-970 states xi_k iid N(0,I_d), appendix.tex:984-995 identifies the Brownian increment in the frozen interpolation, appendix.tex:1170-1176 rewrites it 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. Mathlib route for lower_2: import or use Mathlib.Probability.Distributions.Gaussian.Multivariate, apply ProbabilityTheory.IsGaussian.map_eq_gaussianReal to the strong dual InnerProductSpace.toDual Real E ((stdOrthonormalBasis Real E) i), use ProbabilityTheory.integral_strongDual_stdGaussian for mean zero, ProbabilityTheory.variance_dual_stdGaussian and the orthonormal-basis norm simplifier for variance one, and then rewrite hCoordinateLawDef and hNormalizedVectorLaw. The local variance-field definition hVarianceDef remains a separate packaging boundary; composing this coordinate-law theorem with SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw will discharge hNormalizedVarianceDef without moving sigma_eta^2/2 into the event field. Keep hScalarLineSecondCoeffDef and hSourceHasHessian/hSourceHessianBound 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.obligation — 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. hNormalizedCoordinateLaw
2. hNormalizedVectorLaw
3. hCoordinateLawDef
4. hVarianceDef
5. SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw
6. ProbabilityTheory.stdGaussian
7. ProbabilityTheory.IsGaussian.map_eq_gaussianReal
8. ProbabilityTheory.integral_strongDual_stdGaussian
9. ProbabilityTheory.variance_dual_stdGaussian
10. InnerProductSpace.toDual
11. stdOrthonormalBasis
12. appendix.tex:958-970
13. appendix.tex:984-995
14. appendix.tex:1170-1176
15. appendix.tex:1379-1387
16. hScalarLineSecondCoeffDef
17. hSourceHasHessian
18. hSourceHessianBound
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. A scratch Lean probe showed the Mathlib multivariate Gaussian coordinate theorem is available from stdGaussian, IsGaussian.map_eq_gaussianReal, integral_strongDual_stdGaussian, and variance_dual_stdGaussian; lower_2 should implement only this coordinate-law projection or a direct theorem using the same ingredients, not a restatement of hNormalizedCoordinateLaw.

def cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower1Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle178_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_normalized_coordinate_law_lower1"
  statement := "Cycle 178 lower_1 dynamic-leaf proof-scout packet. Classification: narrows-source-cited-boundary. Exact missing theorem boundary narrowed: hNormalizedCoordinateLaw should not remain a primitive scalar-law field; it should follow from the paper's vector standard-Gaussian normalized increment law hNormalizedVectorLaw : testRegular -> forall phi x, normalizedVectorLaw phi x = ProbabilityTheory.stdGaussian E, plus the coordinate-law definition hCoordinateLawDef : testRegular -> forall phi x i, normalizedCoordinateLaw phi x i = (normalizedVectorLaw phi x).map (fun y => inner (stdOrthonormalBasis Real E i) y). Source route: appendix.tex:958-970 states xi_k iid N(0,I_d), appendix.tex:984-995 identifies the Brownian increment in the frozen interpolation, appendix.tex:1170-1176 rewrites it 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. Mathlib route for lower_2: import or use Mathlib.Probability.Distributions.Gaussian.Multivariate, apply ProbabilityTheory.IsGaussian.map_eq_gaussianReal to the strong dual InnerProductSpace.toDual Real E ((stdOrthonormalBasis Real E) i), use ProbabilityTheory.integral_strongDual_stdGaussian for mean zero, ProbabilityTheory.variance_dual_stdGaussian and the orthonormal-basis norm simplifier for variance one, and then rewrite hCoordinateLawDef and hNormalizedVectorLaw. The local variance-field definition hVarianceDef remains a separate packaging boundary; composing this coordinate-law theorem with SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw will discharge hNormalizedVarianceDef without moving sigma_eta^2/2 into the event field. Keep hScalarLineSecondCoeffDef and hSourceHasHessian/hSourceHessianBound separate."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.obligation
  dependsOn := [
    "hNormalizedCoordinateLaw",
    "hNormalizedVectorLaw",
    "hCoordinateLawDef",
    "hVarianceDef",
    "SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw",
    "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",
    "hScalarLineSecondCoeffDef",
    "hSourceHasHessian",
    "hSourceHessianBound",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "No local SLT file was consulted or imported. A scratch Lean probe showed the Mathlib multivariate Gaussian coordinate theorem is available from stdGaussian, IsGaussian.map_eq_gaussianReal, integral_strongDual_stdGaussian, and variance_dual_stdGaussian; lower_2 should implement only this coordinate-law projection or a direct theorem using the same ingredients, not a restatement of hNormalizedCoordinateLaw."

/-- Cycle-178 lower_2 compiled bridge for the normalized Brownian coordinate law. -/