AutoSamplingTheory.SALD.cycle183GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralMiddleObligation

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 cycle183GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralMiddleObligation :
    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.cycle183_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_source_integral_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 183 middle dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hBrownianCoordinateGeneratorTaylorIntegralDef is no longer primitive once the source supplies hBrownianCoordinateGeneratorSourceIntegralDef and hBrownianCoordinateGeneratorTaylorIntegrandAE. SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralAndAE compiles the MeasureTheory.integral_congr_ae bridge: rewrite brownianCoordinateGenerator as the integral of the paper's sourceTaylorIntegrand under ProbabilityTheory.gaussianReal 0 (variance phi x i), then replace the integrand a.e. by linearCoeff*z + quadraticCoeff*z^2 + normalizedRemainder phi x i z. Remaining exact source boundary is hBrownianCoordinateGeneratorSourceIntegralDef plus hBrownianCoordinateGeneratorTaylorIntegrandAE, with hRemainderGeneratorLimitDef and hRemainderMeas/hRemainderBound/hRemainderBoundInt still explicit. 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.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralAndAE
2. MeasureTheory.integral_congr_ae
3. hBrownianCoordinateGeneratorTaylorIntegralDef
4. hBrownianCoordinateGeneratorSourceIntegralDef
5. hBrownianCoordinateGeneratorTaylorIntegrandAE
6. sourceTaylorIntegrand
7. brownianCoordinateGenerator
8. linearCoeff
9. quadraticCoeff
10. normalizedRemainder
11. variance
12. ProbabilityTheory.gaussianReal
13. hRemainderGeneratorLimitDef
14. hRemainderMeas
15. hRemainderBound
16. hRemainderBoundInt
17. hScalarLineTaylorCoeffDef
18. hNormalizedVectorLaw
19. hCoordinateLawDef
20. hVarianceDef
21. hSourceHasHessian
22. hSourceHessianBound
23. eq:general_moving_target_SALD_frozen_interp
24. appendix.tex:984-995
25. appendix.tex:1170-1176
26. appendix.tex:1379-1387
27. 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, or marked formalized. Mathlib use is exactly MeasureTheory.integral_congr_ae. The lower-ready target is the a.e. scalar Taylor integrand identity hBrownianCoordinateGeneratorTaylorIntegrandAE, or a strictly smaller pointwise source Taylor identity feeding it; do not reprove the integral-congruence bridge.

def cycle183GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralMiddleObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle183_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_source_integral_middle"
  statement := "Cycle 183 middle dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hBrownianCoordinateGeneratorTaylorIntegralDef is no longer primitive once the source supplies hBrownianCoordinateGeneratorSourceIntegralDef and hBrownianCoordinateGeneratorTaylorIntegrandAE. SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralAndAE compiles the MeasureTheory.integral_congr_ae bridge: rewrite brownianCoordinateGenerator as the integral of the paper's sourceTaylorIntegrand under ProbabilityTheory.gaussianReal 0 (variance phi x i), then replace the integrand a.e. by linearCoeff*z + quadraticCoeff*z^2 + normalizedRemainder phi x i z. Remaining exact source boundary is hBrownianCoordinateGeneratorSourceIntegralDef plus hBrownianCoordinateGeneratorTaylorIntegrandAE, with hRemainderGeneratorLimitDef and hRemainderMeas/hRemainderBound/hRemainderBoundInt still explicit. hSourceHasHessian/hSourceHessianBound remain documented source-contract gaps and are not active proof targets."
  source := saldGeneralMovingTargetDiscreteEmSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralAndAE",
    "MeasureTheory.integral_congr_ae",
    "hBrownianCoordinateGeneratorTaylorIntegralDef",
    "hBrownianCoordinateGeneratorSourceIntegralDef",
    "hBrownianCoordinateGeneratorTaylorIntegrandAE",
    "sourceTaylorIntegrand",
    "brownianCoordinateGenerator",
    "linearCoeff",
    "quadraticCoeff",
    "normalizedRemainder",
    "variance",
    "ProbabilityTheory.gaussianReal",
    "hRemainderGeneratorLimitDef",
    "hRemainderMeas",
    "hRemainderBound",
    "hRemainderBoundInt",
    "hScalarLineTaylorCoeffDef",
    "hNormalizedVectorLaw",
    "hCoordinateLawDef",
    "hVarianceDef",
    "hSourceHasHessian",
    "hSourceHessianBound",
    "eq:general_moving_target_SALD_frozen_interp",
    "appendix.tex:984-995",
    "appendix.tex:1170-1176",
    "appendix.tex:1379-1387",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "No external SLT file was consulted, imported, or marked formalized. Mathlib use is exactly MeasureTheory.integral_congr_ae. The lower-ready target is the a.e. scalar Taylor integrand identity hBrownianCoordinateGeneratorTaylorIntegrandAE, or a strictly smaller pointwise source Taylor identity feeding it; do not reprove the integral-congruence bridge."

/-- Cycle-183 lower_2 packet for the Brownian source Taylor integrand leaf.

The compiled theorem turns the source-facing pointwise scalar Taylor identity
into the a.e. equality consumed by the middle integral-congruence bridge.  It
does not prove the analytic Taylor identity itself.
-/