AutoSamplingTheory.SALD.cycle184GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLawMiddleObligation

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 cycle184GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLawMiddleObligation :
    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.cycle184_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_source_integral_law_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 184 middle dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hBrownianCoordinateGeneratorSourceIntegralDef is no longer primitive once the source supplies hBrownianCoordinateGeneratorNormalizedLawDef, hNormalizedVectorLaw, hCoordinateLawDef, and hVarianceDef. SALD.selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfStdGaussianVectorLaw compiles the bridge: rewrite the coordinate generator as the integral of sourceTaylorIntegrand under normalizedCoordinateLaw, use SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw to identify that law with gaussianReal 0 1, use SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw to show variance phi x i = 1, then rewrite the integral to gaussianReal 0 (variance phi x i). Remaining exact source boundary is hBrownianCoordinateGeneratorNormalizedLawDef plus hSourceTaylorIntegrandPointwise, hRemainderGeneratorLimitDef, and hRemainderMeas/hRemainderBound/hRemainderBoundInt. hSourceHasHessian/hSourceHessianBound remain documented source-contract gaps and are not active proof targets.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteEmSource — 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.selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfStdGaussianVectorLaw
2. SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw
3. SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw
4. hBrownianCoordinateGeneratorSourceIntegralDef
5. hBrownianCoordinateGeneratorNormalizedLawDef
6. hNormalizedVectorLaw
7. hCoordinateLawDef
8. hVarianceDef
9. normalizedVectorLaw
10. normalizedCoordinateLaw
11. sourceTaylorIntegrand
12. brownianCoordinateGenerator
13. variance
14. ProbabilityTheory.stdGaussian
15. ProbabilityTheory.gaussianReal
16. ProbabilityTheory.IsGaussian.map_eq_gaussianReal
17. ProbabilityTheory.integral_strongDual_stdGaussian
18. ProbabilityTheory.variance_dual_stdGaussian
19. ProbabilityTheory.variance_id_gaussianReal
20. eq:SALD_general_EM
21. eq:general_moving_target_SALD_frozen_interp
22. appendix.tex:958-996
23. appendix.tex:1170-1176
24. appendix.tex:1379-1387
25. 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 external SLT file was consulted, imported, ported, or marked formalized. The packet uses only compiled local SALD Gaussian coordinate/variance bridges, which themselves use Mathlib Gaussian APIs. It does not prove hSourceTaylorIntegrandPointwise, hRemainderGeneratorLimitDef, any remainder measurability/domination field, or the selected weak-test Hessian source-contract fields.

def cycle184GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLawMiddleObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle184_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_source_integral_law_middle"
  statement := "Cycle 184 middle dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hBrownianCoordinateGeneratorSourceIntegralDef is no longer primitive once the source supplies hBrownianCoordinateGeneratorNormalizedLawDef, hNormalizedVectorLaw, hCoordinateLawDef, and hVarianceDef. SALD.selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfStdGaussianVectorLaw compiles the bridge: rewrite the coordinate generator as the integral of sourceTaylorIntegrand under normalizedCoordinateLaw, use SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw to identify that law with gaussianReal 0 1, use SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw to show variance phi x i = 1, then rewrite the integral to gaussianReal 0 (variance phi x i). Remaining exact source boundary is hBrownianCoordinateGeneratorNormalizedLawDef plus hSourceTaylorIntegrandPointwise, hRemainderGeneratorLimitDef, and hRemainderMeas/hRemainderBound/hRemainderBoundInt. hSourceHasHessian/hSourceHessianBound remain documented source-contract gaps and are not active proof targets."
  source := saldGeneralMovingTargetDiscreteEmSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfStdGaussianVectorLaw",
    "SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw",
    "SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw",
    "hBrownianCoordinateGeneratorSourceIntegralDef",
    "hBrownianCoordinateGeneratorNormalizedLawDef",
    "hNormalizedVectorLaw",
    "hCoordinateLawDef",
    "hVarianceDef",
    "normalizedVectorLaw",
    "normalizedCoordinateLaw",
    "sourceTaylorIntegrand",
    "brownianCoordinateGenerator",
    "variance",
    "ProbabilityTheory.stdGaussian",
    "ProbabilityTheory.gaussianReal",
    "ProbabilityTheory.IsGaussian.map_eq_gaussianReal",
    "ProbabilityTheory.integral_strongDual_stdGaussian",
    "ProbabilityTheory.variance_dual_stdGaussian",
    "ProbabilityTheory.variance_id_gaussianReal",
    "eq:SALD_general_EM",
    "eq:general_moving_target_SALD_frozen_interp",
    "appendix.tex:958-996",
    "appendix.tex:1170-1176",
    "appendix.tex:1379-1387",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "No external SLT file was consulted, imported, ported, or marked formalized. The packet uses only compiled local SALD Gaussian coordinate/variance bridges, which themselves use Mathlib Gaussian APIs. It does not prove hSourceTaylorIntegrandPointwise, hRemainderGeneratorLimitDef, any remainder measurability/domination field, or the selected weak-test Hessian source-contract fields."

/-- Cycle-184 lower_2 packet for the normalized scalar-coordinate law leaf.

The compiled theorem transports the sample-space expectation of the paper's
source Taylor integrand to the normalized scalar-coordinate law by
`MeasureTheory.integral_map`.  It narrows the law-definition leaf to the
source-facing scalar-coordinate measurability, pushforward-law definition,
integrand measurability, and sample-space generator definition.
-/