AutoSamplingTheory.SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoScalarLineCoeffLower2Obligation

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 cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoScalarLineCoeffLower2Obligation :
    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.cycle179_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_scalar_line_coeff_lower2

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 179 lower_2 dynamic-leaf Lean implementer packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hScalarLineSecondCoeffDef is no longer primitive once the source correspondence supplies hScalarLineTaylorCoeffDef : testRegular -> forall phi x i, quadraticCoeff phi x i = (2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest phi (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0. SALD.selectedWeakTestScalarLineSecondCoeffDefOfTaylorCoeffWithin compiles the local Mathlib bridge by rewriting hScalarLineTaylorCoeffDef, unfolding taylorCoeffWithin over Set.univ, rewriting iteratedDerivWithin_univ through simp, and cancelling (2 : Real) * (2 ! : Real)^-1. Remaining exact source-facing coefficient boundary is hScalarLineTaylorCoeffDef from appendix.tex:984-995, appendix.tex:1170-1176, and appendix.tex:1379-1387. The theorem keeps hSource, hNormalizedVectorLaw, hCoordinateLawDef, hVarianceDef, hSecondCoeff, Taylor moment decomposition, DCT domination, coordinate-sum, and hSourceHasHessian/hSourceHessianBound explicit; it does not define sourceHessian from the desired conclusion, use VP score-Hessian regularity, move sigma_eta^2/2 into the event field, or use sald_version_2.tex.

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.selectedWeakTestScalarLineSecondCoeffDefOfTaylorCoeffWithin
2. SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoScalarLineCoeffLower1Obligation
3. SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianAuditScalarLineCoeffMiddleObligation
4. hScalarLineTaylorCoeffDef
5. hScalarLineSecondCoeffDef
6. taylorCoeffWithin
7. iteratedDerivWithin_univ
8. Nat.factorial
9. hSource
10. hNormalizedVectorLaw
11. hCoordinateLawDef
12. hVarianceDef
13. hSecondCoeff
14. hFrozenScalarBrownianItoTaylorMomentDecomposition
15. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
16. hFrozenScalarBrownianItoEventFieldCoordinateSum
17. hSourceHasHessian
18. hSourceHessianBound
19. appendix.tex:984-995
20. appendix.tex:1170-1176
21. appendix.tex:1379-1387
22. 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.

No local SLT file was consulted or imported. Mathlib consultation was limited to the existing imports Mathlib.Analysis.Calculus.Taylor and Mathlib.Analysis.Calculus.IteratedDeriv.Defs/Lemmas through taylorCoeffWithin and iteratedDerivWithin_univ. This packet is one compiled theorem plus synchronized obligation metadata, not a wrapper around hScalarLineSecondCoeffDef.

def cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoScalarLineCoeffLower2Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle179_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_scalar_line_coeff_lower2"
  statement := "Cycle 179 lower_2 dynamic-leaf Lean implementer packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hScalarLineSecondCoeffDef is no longer primitive once the source correspondence supplies hScalarLineTaylorCoeffDef : testRegular -> forall phi x i, quadraticCoeff phi x i = (2 : Real) * taylorCoeffWithin (fun q : Real => selectedTest phi (x + q • (stdOrthonormalBasis Real E i))) 2 Set.univ 0. SALD.selectedWeakTestScalarLineSecondCoeffDefOfTaylorCoeffWithin compiles the local Mathlib bridge by rewriting hScalarLineTaylorCoeffDef, unfolding taylorCoeffWithin over Set.univ, rewriting iteratedDerivWithin_univ through simp, and cancelling (2 : Real) * (2 ! : Real)^-1. Remaining exact source-facing coefficient boundary is hScalarLineTaylorCoeffDef from appendix.tex:984-995, appendix.tex:1170-1176, and appendix.tex:1379-1387. The theorem keeps hSource, hNormalizedVectorLaw, hCoordinateLawDef, hVarianceDef, hSecondCoeff, Taylor moment decomposition, DCT domination, coordinate-sum, and hSourceHasHessian/hSourceHessianBound explicit; it does not define sourceHessian from the desired conclusion, use VP score-Hessian regularity, move sigma_eta^2/2 into the event field, or use sald_version_2.tex."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.selectedWeakTestScalarLineSecondCoeffDefOfTaylorCoeffWithin",
    "SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoScalarLineCoeffLower1Obligation",
    "SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianAuditScalarLineCoeffMiddleObligation",
    "hScalarLineTaylorCoeffDef",
    "hScalarLineSecondCoeffDef",
    "taylorCoeffWithin",
    "iteratedDerivWithin_univ",
    "Nat.factorial",
    "hSource",
    "hNormalizedVectorLaw",
    "hCoordinateLawDef",
    "hVarianceDef",
    "hSecondCoeff",
    "hFrozenScalarBrownianItoTaylorMomentDecomposition",
    "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes",
    "hFrozenScalarBrownianItoEventFieldCoordinateSum",
    "hSourceHasHessian",
    "hSourceHessianBound",
    "appendix.tex:984-995",
    "appendix.tex:1170-1176",
    "appendix.tex:1379-1387",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "No local SLT file was consulted or imported. Mathlib consultation was limited to the existing imports Mathlib.Analysis.Calculus.Taylor and Mathlib.Analysis.Calculus.IteratedDeriv.Defs/Lemmas through taylorCoeffWithin and iteratedDerivWithin_univ. This packet is one compiled theorem plus synchronized obligation metadata, not a wrapper around hScalarLineSecondCoeffDef."

/-! ### Cycle 180 middle: Taylor moment decomposition integral split -/

/-- Cycle-180 middle packet for the Taylor moment decomposition leaf.

The source-Hessian fields remain source-contract gaps.  This packet follows the
cycle-180 upper assignment and narrows the sibling Brownian/Ito Taylor moment
decomposition to the source coordinate-generator integral definition, summand
integrability, and the normalized-remainder generator definition.
-/