AutoSamplingTheory.SALD.cycle164GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderMeasLowerObligation

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 cycle164GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderMeasLowerObligation :
    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.cycle164_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_normalized_remainder_meas

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 164 dynamic-leaf worker packet. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: the hMeas input to SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero for the concrete selected-test scalar normalized Taylor remainder. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable compiles eventual AEStronglyMeasurable under ProbabilityTheory.gaussianReal 0 v from the selected scalar line regularity ContDiffOn Real 2 (fun r => sourceTest (x + r • e)) Set.univ, using ContDiffOn.continuousOn, continuity of taylorWithinEval in the evaluation variable via hasDerivAt_taylorWithinEval_succ, Continuous.comp_aestronglyMeasurable, and AEStronglyMeasurable arithmetic closure. Remaining source-cited inputs are hBound and hBoundInt for the concrete normalized remainder, plus hFrozenScalarBrownianItoTaylorMomentDecomposition, hFrozenScalarBrownianItoQuadraticVariationNormalization, and 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.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable
2. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero
3. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainder
4. ContDiffOn.continuousOn
5. hasDerivAt_taylorWithinEval_succ
6. Continuous.comp_aestronglyMeasurable
7. MeasureTheory.AEStronglyMeasurable.sub
8. MeasureTheory.AEStronglyMeasurable.div₀
9. MeasureTheory.AEStronglyMeasurable.mul
10. MeasureTheory.AEStronglyMeasurable.pow
11. hMeas
12. hBound
13. hBoundInt
14. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
15. hFrozenScalarBrownianItoTaylorMomentDecomposition
16. hFrozenScalarBrownianItoQuadraticVariationNormalization
17. hFrozenScalarBrownianItoEventFieldCoordinateSum
18. Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable
19. Mathlib.Analysis.Calculus.Taylor
20. eq:general_moving_target_SALD_frozen_interp
21. appendix.tex:984-995
22. appendix.tex:1379-1387
23. 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 consulted through existing imports: Analysis.Calculus.Taylor for taylorWithinEval/hasDerivAt_taylorWithinEval_succ and MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable for composition/arithmetic closure. The configured local SLT clone is absent, and no SLT theorem was imported, ported, or marked formalized.

def cycle164GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderMeasLowerObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle164_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_normalized_remainder_meas"
  statement := "Cycle 164 dynamic-leaf worker packet. Classification: discharges-supplied-hypothesis. Exact supplied hypothesis discharged: the hMeas input to SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero for the concrete selected-test scalar normalized Taylor remainder. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable compiles eventual AEStronglyMeasurable under ProbabilityTheory.gaussianReal 0 v from the selected scalar line regularity ContDiffOn Real 2 (fun r => sourceTest (x + r • e)) Set.univ, using ContDiffOn.continuousOn, continuity of taylorWithinEval in the evaluation variable via hasDerivAt_taylorWithinEval_succ, Continuous.comp_aestronglyMeasurable, and AEStronglyMeasurable arithmetic closure. Remaining source-cited inputs are hBound and hBoundInt for the concrete normalized remainder, plus hFrozenScalarBrownianItoTaylorMomentDecomposition, hFrozenScalarBrownianItoQuadraticVariationNormalization, and 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.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable",
    "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero",
    "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainder",
    "ContDiffOn.continuousOn",
    "hasDerivAt_taylorWithinEval_succ",
    "Continuous.comp_aestronglyMeasurable",
    "MeasureTheory.AEStronglyMeasurable.sub",
    "MeasureTheory.AEStronglyMeasurable.div₀",
    "MeasureTheory.AEStronglyMeasurable.mul",
    "MeasureTheory.AEStronglyMeasurable.pow",
    "hMeas",
    "hBound",
    "hBoundInt",
    "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes",
    "hFrozenScalarBrownianItoTaylorMomentDecomposition",
    "hFrozenScalarBrownianItoQuadraticVariationNormalization",
    "hFrozenScalarBrownianItoEventFieldCoordinateSum",
    "Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable",
    "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 consulted through existing imports: Analysis.Calculus.Taylor for taylorWithinEval/hasDerivAt_taylorWithinEval_succ and MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable for composition/arithmetic closure. The configured local SLT clone is absent, and no SLT theorem was imported, ported, or marked formalized."

/-- Cycle-164 lower_2 packet for the concrete selected-test quadratic bound
`hBoundInt`.

Once the source Taylor domination leaf is stated with the quadratic Gaussian
bound `fun z => C * z ^ 2`, this packet discharges the DCT integrability input
using Mathlib's Gaussian exponential-moment theorem.  The pointwise domination
`hBound` remains the next source Taylor estimate.
-/