AutoSamplingTheory.SALD.cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseTaylorLowerObligation

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 cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseTaylorLowerObligation :
    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.cycle163_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_pointwise_taylor

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 163 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the concrete source Taylor pointwise hPoint input below SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT is no longer open for the selected scalar line r |-> sourceTest (x + r • e). SALD.gaussianRealSelectedTestLineSecondOrderTaylorRemainderPointwiseAE compiles the a.e. Gaussian pointwise Tendsto statement from ContDiffOn Real 2 of that line and Mathlib Real.taylor_tendsto, with scalar coordinate scaling r = h*z. Lower_1 also compiles SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfSelectedTestLineEq, which turns an eventual a.e. source equality between the paper's normalizedRemainder and that scaled line remainder into the DCT integral limit. Remaining source-cited inputs are the source equality hSourceEq, the event/measurable domination hypotheses hMeas/hBound/hBoundInt, hFrozenScalarBrownianItoTaylorMomentDecomposition, hFrozenScalarBrownianItoQuadraticVariationNormalization, and the sibling hFrozenScalarBrownianItoEventFieldCoordinateSum. Weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM fallback, wrapper churn, broad audit, theorem-status promotion, fake closure, Lake/toolchain change, SLT import, and sald_version_2.tex routes are rejected.

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.gaussianRealSelectedTestLineSecondOrderTaylorRemainderPointwiseAE
2. SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfSelectedTestLineEq
3. SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT
4. Real.taylor_tendsto
5. Filter.Tendsto.congr'
6. taylorWithinEval
7. hSourceEq
8. hPoint
9. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
10. hFrozenScalarBrownianItoTaylorMomentDecomposition
11. hFrozenScalarBrownianItoQuadraticVariationNormalization
12. hFrozenScalarBrownianItoEventFieldCoordinateSum
13. Mathlib.Analysis.Calculus.Taylor
14. Mathlib.MeasureTheory.Integral.DominatedConvergence
15. eq:general_moving_target_SALD_frozen_interp
16. appendix.tex:984-995
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.

Dynamic-leaf worker packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib Analysis.Calculus.Taylor and the already compiled DCT theorem; the configured local SLT clone is absent, and no SLT theorem was imported, ported, or marked formalized.

def cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseTaylorLowerObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle163_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_pointwise_taylor"
  statement := "Cycle 163 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the concrete source Taylor pointwise hPoint input below SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT is no longer open for the selected scalar line r |-> sourceTest (x + r • e). SALD.gaussianRealSelectedTestLineSecondOrderTaylorRemainderPointwiseAE compiles the a.e. Gaussian pointwise Tendsto statement from ContDiffOn Real 2 of that line and Mathlib Real.taylor_tendsto, with scalar coordinate scaling r = h*z. Lower_1 also compiles SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfSelectedTestLineEq, which turns an eventual a.e. source equality between the paper's normalizedRemainder and that scaled line remainder into the DCT integral limit. Remaining source-cited inputs are the source equality hSourceEq, the event/measurable domination hypotheses hMeas/hBound/hBoundInt, hFrozenScalarBrownianItoTaylorMomentDecomposition, hFrozenScalarBrownianItoQuadraticVariationNormalization, and the sibling hFrozenScalarBrownianItoEventFieldCoordinateSum. Weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM fallback, wrapper churn, broad audit, theorem-status promotion, fake closure, Lake/toolchain change, SLT import, and sald_version_2.tex routes are rejected."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.gaussianRealSelectedTestLineSecondOrderTaylorRemainderPointwiseAE",
    "SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfSelectedTestLineEq",
    "SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT",
    "Real.taylor_tendsto",
    "Filter.Tendsto.congr'",
    "taylorWithinEval",
    "hSourceEq",
    "hPoint",
    "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes",
    "hFrozenScalarBrownianItoTaylorMomentDecomposition",
    "hFrozenScalarBrownianItoQuadraticVariationNormalization",
    "hFrozenScalarBrownianItoEventFieldCoordinateSum",
    "Mathlib.Analysis.Calculus.Taylor",
    "Mathlib.MeasureTheory.Integral.DominatedConvergence",
    "eq:general_moving_target_SALD_frozen_interp",
    "appendix.tex:984-995",
    "appendix.tex:1379-1387",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "Dynamic-leaf worker packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib Analysis.Calculus.Taylor and the already compiled DCT theorem; the configured local SLT clone is absent, and no SLT theorem was imported, ported, or marked formalized."

/-- Cycle-163 lower_2 packet for the normalized-remainder source equality. -/