AutoSamplingTheory.SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSignedIntervalTaylorLower2Obligation

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 cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSignedIntervalTaylorLower2Obligation :
    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_signed_interval_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 166 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the source-facing all-r non-quotient first-order quadratic Taylor-remainder estimate hFirstQuadraticRemainder is no longer primitive once signed interval Taylor data are supplied. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor compiles the signed combination: for r >= 0 it uses SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor on the selected line q |-> sourceTest (x + q • e), and for r < 0 it applies the same theorem to the reflected line q |-> sourceTest (x + q • (-e)) at -r, with an explicit Set.univ Taylor-polynomial reflection compatibility. Remaining source-cited work is to prove the signed interval ContDiffOn/second-derivative domination, interval-to-Set.univ Taylor compatibility, and reflected Taylor-polynomial compatibility from the paper's selected-test bounded-Hessian/regularity hypothesis. 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.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor
2. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor
3. SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder
4. hFirstQuadraticRemainder
5. hNonnegCont
6. hNonnegSecond
7. hNonnegTaylorCompat
8. hNegCont
9. hNegSecond
10. hNegTaylorCompat
11. hNegTaylorReflect
12. Mathlib.Analysis.Calculus.Taylor.taylor_mean_remainder_bound
13. selected-test second-derivative or bounded-Hessian source hypothesis
14. eq:general_moving_target_SALD_frozen_interp
15. appendix.tex:984-995
16. appendix.tex:1379-1387
17. 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_2 packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib Taylor through the lower_1 theorem and basic order/reflection arithmetic; no SLT theorem was needed because this is deterministic one-dimensional calculus. The remaining named source/theory gap is now the signed interval regularity/second-derivative domination and Taylor-compatibility data, not a primitive all-r quadratic remainder.

def cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSignedIntervalTaylorLower2Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle166_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_signed_interval_taylor"
  statement := "Cycle 166 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the source-facing all-r non-quotient first-order quadratic Taylor-remainder estimate hFirstQuadraticRemainder is no longer primitive once signed interval Taylor data are supplied. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor compiles the signed combination: for r >= 0 it uses SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor on the selected line q |-> sourceTest (x + q • e), and for r < 0 it applies the same theorem to the reflected line q |-> sourceTest (x + q • (-e)) at -r, with an explicit Set.univ Taylor-polynomial reflection compatibility. Remaining source-cited work is to prove the signed interval ContDiffOn/second-derivative domination, interval-to-Set.univ Taylor compatibility, and reflected Taylor-polynomial compatibility from the paper's selected-test bounded-Hessian/regularity hypothesis. hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves remain separate."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor",
    "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor",
    "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder",
    "hFirstQuadraticRemainder",
    "hNonnegCont",
    "hNonnegSecond",
    "hNonnegTaylorCompat",
    "hNegCont",
    "hNegSecond",
    "hNegTaylorCompat",
    "hNegTaylorReflect",
    "Mathlib.Analysis.Calculus.Taylor.taylor_mean_remainder_bound",
    "selected-test second-derivative or bounded-Hessian source hypothesis",
    "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_2 packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib Taylor through the lower_1 theorem and basic order/reflection arithmetic; no SLT theorem was needed because this is deterministic one-dimensional calculus. The remaining named source/theory gap is now the signed interval regularity/second-derivative domination and Taylor-compatibility data, not a primitive all-r quadratic remainder."

/-- Cycle-166 proof-DAG pane for the first-order selected-line remainder split. -/