AutoSamplingTheory.SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundLowerObligation

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 cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundLowerObligation :
    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.cycle165_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_normalized_remainder_quadratic_bound

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 165 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hBound input to SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero for the concrete selected-test scalar normalized Taylor remainder is no longer primitive once a deterministic scalar Taylor quotient estimate is supplied. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound compiles that bridge: from forall r, the norm of (sourceTest (x + r • e) - taylorWithinEval (fun r => sourceTest (x + r • e)) 2 Set.univ 0 r) / r^2 is at most C, it produces the eventual Gaussian-a.e. domination by fun z => C * z^2. Remaining source-cited work is the deterministic Taylor quotient bound itself, plus hFrozenScalarBrownianItoTaylorMomentDecomposition, hFrozenScalarBrownianItoQuadraticVariationNormalization, and hFrozenScalarBrownianItoEventFieldCoordinateSum. hMeas and hBoundInt remain discharged by the cycle-164 compiled theorems. 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.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound
2. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero
3. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable
4. SALD.gaussianRealSelectedTestLineSecondOrderQuadraticBoundIntegrable
5. hTaylorQuotientBound
6. hBound
7. hBoundInt
8. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
9. hFrozenScalarBrownianItoTaylorMomentDecomposition
10. hFrozenScalarBrownianItoQuadraticVariationNormalization
11. hFrozenScalarBrownianItoEventFieldCoordinateSum
12. taylorWithinEval
13. norm_mul
14. mul_le_mul_of_nonneg_right
15. Mathlib.Analysis.Calculus.Taylor
16. eq:general_moving_target_SALD_frozen_interp
17. appendix.tex:984-995
18. appendix.tex:1379-1387
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.

Dynamic-leaf worker packet inside the scalar Brownian/Ito illness area. Local Mathlib use is only arithmetic/norm closure and the already imported Taylor expression vocabulary; the configured local SLT clone is absent, and no SLT theorem was consulted, imported, ported, or marked formalized.

def cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundLowerObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle165_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_normalized_remainder_quadratic_bound"
  statement := "Cycle 165 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hBound input to SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero for the concrete selected-test scalar normalized Taylor remainder is no longer primitive once a deterministic scalar Taylor quotient estimate is supplied. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound compiles that bridge: from forall r, the norm of (sourceTest (x + r • e) - taylorWithinEval (fun r => sourceTest (x + r • e)) 2 Set.univ 0 r) / r^2 is at most C, it produces the eventual Gaussian-a.e. domination by fun z => C * z^2. Remaining source-cited work is the deterministic Taylor quotient bound itself, plus hFrozenScalarBrownianItoTaylorMomentDecomposition, hFrozenScalarBrownianItoQuadraticVariationNormalization, and hFrozenScalarBrownianItoEventFieldCoordinateSum. hMeas and hBoundInt remain discharged by the cycle-164 compiled theorems. 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.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound",
    "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero",
    "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable",
    "SALD.gaussianRealSelectedTestLineSecondOrderQuadraticBoundIntegrable",
    "hTaylorQuotientBound",
    "hBound",
    "hBoundInt",
    "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes",
    "hFrozenScalarBrownianItoTaylorMomentDecomposition",
    "hFrozenScalarBrownianItoQuadraticVariationNormalization",
    "hFrozenScalarBrownianItoEventFieldCoordinateSum",
    "taylorWithinEval",
    "norm_mul",
    "mul_le_mul_of_nonneg_right",
    "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 worker packet inside the scalar Brownian/Ito illness area. Local Mathlib use is only arithmetic/norm closure and the already imported Taylor expression vocabulary; the configured local SLT clone is absent, and no SLT theorem was consulted, imported, ported, or marked formalized."

/-- Cycle-165 lower_2 packet splitting the deterministic Taylor quotient bound. -/