AutoSamplingTheory.SALD.cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderScoutObligation

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 cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderScoutObligation :
    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.cycle162_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_normalized_remainder_scout

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 162 lower_1 proof-scout packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes should be implemented as a dominated-convergence theorem for the normalized scalar Taylor remainder integral, separate from hFrozenScalarBrownianItoTaylorMomentDecomposition and hFrozenScalarBrownianItoQuadraticVariationNormalization. Lower_2-ready theorem shape: SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT proves Tendsto (fun h => integral z, normalizedRemainder h z d gaussianReal 0 v) (nhdsWithin 0 (Set.Ioi 0)) (nhds 0) from eventual AEStronglyMeasurable, eventual ae domination by an Integrable bound, and ae pointwise convergence of normalizedRemainder h z to 0. The follow-up source Taylor step should supply that pointwise convergence for r |-> selectedTest phi (x + r • e_i) using Real.taylor_tendsto or taylor_isLittleO under selected-test C^2/ContDiffOn hypotheses. The sibling hFrozenScalarBrownianItoEventFieldCoordinateSum remains explicit; sigma_eta^2/2 stays outside the event field.

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. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
2. hFrozenScalarBrownianItoTaylorMomentDecomposition
3. hFrozenScalarBrownianItoQuadraticVariationNormalization
4. SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder
5. MeasureTheory.tendsto_integral_filter_of_dominated_convergence
6. Real.taylor_tendsto
7. taylor_isLittleO
8. Mathlib.MeasureTheory.Integral.DominatedConvergence
9. Mathlib.Analysis.Calculus.Taylor
10. eq:general_moving_target_SALD_frozen_interp
11. appendix.tex:984-995
12. appendix.tex:1379-1387
13. 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 lower_1 proof scout inside the scalar Brownian/Ito illness area. Consulted Mathlib Taylor for the scalar path little-o theorem and Mathlib dominated convergence for the integral-limit handoff; the configured local SLT clone is absent, and no SLT theorem was imported or marked formalized.

def cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderScoutObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle162_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_normalized_remainder_scout"
  statement := "Cycle 162 lower_1 proof-scout packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes should be implemented as a dominated-convergence theorem for the normalized scalar Taylor remainder integral, separate from hFrozenScalarBrownianItoTaylorMomentDecomposition and hFrozenScalarBrownianItoQuadraticVariationNormalization. Lower_2-ready theorem shape: SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT proves Tendsto (fun h => integral z, normalizedRemainder h z d gaussianReal 0 v) (nhdsWithin 0 (Set.Ioi 0)) (nhds 0) from eventual AEStronglyMeasurable, eventual ae domination by an Integrable bound, and ae pointwise convergence of normalizedRemainder h z to 0. The follow-up source Taylor step should supply that pointwise convergence for r |-> selectedTest phi (x + r • e_i) using Real.taylor_tendsto or taylor_isLittleO under selected-test C^2/ContDiffOn hypotheses. The sibling hFrozenScalarBrownianItoEventFieldCoordinateSum remains explicit; sigma_eta^2/2 stays outside the event field."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.obligation
  dependsOn := [
    "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes",
    "hFrozenScalarBrownianItoTaylorMomentDecomposition",
    "hFrozenScalarBrownianItoQuadraticVariationNormalization",
    "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder",
    "MeasureTheory.tendsto_integral_filter_of_dominated_convergence",
    "Real.taylor_tendsto",
    "taylor_isLittleO",
    "Mathlib.MeasureTheory.Integral.DominatedConvergence",
    "Mathlib.Analysis.Calculus.Taylor",
    "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 lower_1 proof scout inside the scalar Brownian/Ito illness area. Consulted Mathlib Taylor for the scalar path little-o theorem and Mathlib dominated convergence for the integral-limit handoff; the configured local SLT clone is absent, and no SLT theorem was imported or marked formalized."

/-- Cycle-162 lower_2 compiled DCT theorem for the normalized scalar Taylor remainder. -/