AutoSamplingTheory.SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoIntervalTaylorLower1Obligation

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 cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoIntervalTaylorLower1Obligation :
    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.cycle166_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_interval_taylor_nonneg

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 166 lower_1 dynamic-leaf proof-scout/worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the non-quotient first-order quadratic Taylor-remainder estimate is no longer wholly primitive on the nonnegative side. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor compiles the r >= 0 case from Mathlib taylor_mean_remainder_bound, assuming ContDiffOn Real 2 on Icc 0 r, a selected-line interval second-derivative bound, and the interval-to-Set.univ Taylor-polynomial compatibility equality. Remaining boundary: prove the negative/reflection side, source the selected-line interval second-derivative bound and compatibility from the paper's bounded-Hessian/selected-test regularity, then combine signed sides to supply hFirstQuadraticRemainder. hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves remain separate.

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.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor
2. SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder
3. Mathlib.Analysis.Calculus.Taylor.taylor_mean_remainder_bound
4. ContDiffOn Real 2
5. iteratedDerivWithin
6. Set.Icc
7. Set.univ
8. hTaylorCompat
9. selected-line interval second-derivative bound
10. negative/reflection sign split
11. eq:general_moving_target_SALD_frozen_interp
12. appendix.tex:984-995
13. appendix.tex:1379-1387
14. 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 packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib Taylor; no SLT theorem was needed because this is deterministic one-dimensional calculus. The nonnegative side is now a compiled local theorem; the remaining named theory/source gap is the signed combination plus the source proof of the interval derivative bound and Taylor-polynomial compatibility.

def cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoIntervalTaylorLower1Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle166_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_interval_taylor_nonneg"
  statement := "Cycle 166 lower_1 dynamic-leaf proof-scout/worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the non-quotient first-order quadratic Taylor-remainder estimate is no longer wholly primitive on the nonnegative side. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor compiles the r >= 0 case from Mathlib taylor_mean_remainder_bound, assuming ContDiffOn Real 2 on Icc 0 r, a selected-line interval second-derivative bound, and the interval-to-Set.univ Taylor-polynomial compatibility equality. Remaining boundary: prove the negative/reflection side, source the selected-line interval second-derivative bound and compatibility from the paper's bounded-Hessian/selected-test regularity, then combine signed sides to supply hFirstQuadraticRemainder. hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves remain separate."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor",
    "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder",
    "Mathlib.Analysis.Calculus.Taylor.taylor_mean_remainder_bound",
    "ContDiffOn Real 2",
    "iteratedDerivWithin",
    "Set.Icc",
    "Set.univ",
    "hTaylorCompat",
    "selected-line interval second-derivative bound",
    "negative/reflection sign split",
    "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 packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib Taylor; no SLT theorem was needed because this is deterministic one-dimensional calculus. The nonnegative side is now a compiled local theorem; the remaining named theory/source gap is the signed combination plus the source proof of the interval derivative bound and Taylor-polynomial compatibility."

/-- Cycle-166 lower_2 signed interval Taylor combination packet. -/