AutoSamplingTheory.SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoScalarLineCoeffLower1Obligation

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 cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoScalarLineCoeffLower1Obligation :
    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_lower1

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_1 dynamic-leaf proof-scout packet. Classification: narrows-source-cited-boundary. Exact missing theorem boundary narrowed: hScalarLineSecondCoeffDef should not remain a primitive field. The smaller source-facing boundary is 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, expressing the paper convention that the Brownian event-field coefficient is twice the order-two scalar Taylor coefficient, because appendix.tex:1379-1387 keeps the sigma_eta^2/2 diffusion factor outside the event field. Lower_2-ready theorem shape: selectedWeakTestScalarLineSecondCoeffDefOfTaylorCoeffWithin (selectedTest) (quadraticCoeff) (testRegular) (hScalarLineTaylorCoeffDef) : testRegular -> forall phi x i, quadraticCoeff phi x i = iteratedDeriv 2 (fun q : Real => selectedTest phi (x + q • (stdOrthonormalBasis Real E i))) 0. Mathlib route: intro htests phi x i; rewrite hScalarLineTaylorCoeffDef; unfold taylorCoeffWithin; rewrite iteratedDerivWithin_univ; norm_num/simp to cancel (2 : Real) * (2 ! : Real)^-1. Source route: appendix.tex:984-995 gives the frozen Brownian interpolation, appendix.tex:1170-1176 rewrites the increment as sigma_eta(t(s))*sqrt(s-s_k)*xi with xi ~ N(0,I), and appendix.tex:1379-1387 places sigma_eta^2/2 in the weak-FP diffusion term, so the normalized scalar coefficient inside quadraticCoeff is the full second derivative, not a half coefficient and not a place to absorb sigma_eta^2/2. Keep hSource, hNormalizedVectorLaw, hCoordinateLawDef, hVarianceDef, hSecondCoeff, Taylor moment decomposition, DCT domination, coordinate-sum, and hSourceHasHessian/hSourceHessianBound explicit.

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.obligation — 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.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianAuditScalarLineCoeffMiddleObligation
2. SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianAuditScalarLineCoeffDag
3. SALD.selectedWeakTestSecondTaylorCoeffDefOfScalarLineSecondCoeffDef
4. SALD.selectedWeakTestQuadraticVariationNormalizationOfScalarLineSecondCoeffAndNormalizedVarianceDef
5. hScalarLineSecondCoeffDef
6. hScalarLineTaylorCoeffDef
7. taylorCoeffWithin
8. iteratedDerivWithin_univ
9. Nat.factorial
10. hSource
11. hNormalizedVectorLaw
12. hCoordinateLawDef
13. hVarianceDef
14. hSecondCoeff
15. hFrozenScalarBrownianItoTaylorMomentDecomposition
16. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
17. hFrozenScalarBrownianItoEventFieldCoordinateSum
18. hSourceHasHessian
19. hSourceHessianBound
20. appendix.tex:984-995
21. appendix.tex:1170-1176
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.

No local SLT file was consulted or imported. Mathlib consultation was limited to Mathlib.Analysis.Calculus.Taylor.taylorCoeffWithin and Mathlib.Analysis.Calculus.IteratedDeriv.Defs.iteratedDerivWithin_univ. This route does not define sourceHessian from the desired conclusion, does not use VP score-Hessian regularity, does not move sigma_eta^2/2 into the event field, and does not use sald_version_2.tex.

def cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoScalarLineCoeffLower1Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle179_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_scalar_line_coeff_lower1"
  statement := "Cycle 179 lower_1 dynamic-leaf proof-scout packet. Classification: narrows-source-cited-boundary. Exact missing theorem boundary narrowed: hScalarLineSecondCoeffDef should not remain a primitive field. The smaller source-facing boundary is 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, expressing the paper convention that the Brownian event-field coefficient is twice the order-two scalar Taylor coefficient, because appendix.tex:1379-1387 keeps the sigma_eta^2/2 diffusion factor outside the event field. Lower_2-ready theorem shape: selectedWeakTestScalarLineSecondCoeffDefOfTaylorCoeffWithin (selectedTest) (quadraticCoeff) (testRegular) (hScalarLineTaylorCoeffDef) : testRegular -> forall phi x i, quadraticCoeff phi x i = iteratedDeriv 2 (fun q : Real => selectedTest phi (x + q • (stdOrthonormalBasis Real E i))) 0. Mathlib route: intro htests phi x i; rewrite hScalarLineTaylorCoeffDef; unfold taylorCoeffWithin; rewrite iteratedDerivWithin_univ; norm_num/simp to cancel (2 : Real) * (2 ! : Real)^-1. Source route: appendix.tex:984-995 gives the frozen Brownian interpolation, appendix.tex:1170-1176 rewrites the increment as sigma_eta(t(s))*sqrt(s-s_k)*xi with xi ~ N(0,I), and appendix.tex:1379-1387 places sigma_eta^2/2 in the weak-FP diffusion term, so the normalized scalar coefficient inside quadraticCoeff is the full second derivative, not a half coefficient and not a place to absorb sigma_eta^2/2. Keep hSource, hNormalizedVectorLaw, hCoordinateLawDef, hVarianceDef, hSecondCoeff, Taylor moment decomposition, DCT domination, coordinate-sum, and hSourceHasHessian/hSourceHessianBound explicit."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianAuditScalarLineCoeffMiddleObligation",
    "SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianAuditScalarLineCoeffDag",
    "SALD.selectedWeakTestSecondTaylorCoeffDefOfScalarLineSecondCoeffDef",
    "SALD.selectedWeakTestQuadraticVariationNormalizationOfScalarLineSecondCoeffAndNormalizedVarianceDef",
    "hScalarLineSecondCoeffDef",
    "hScalarLineTaylorCoeffDef",
    "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 Mathlib.Analysis.Calculus.Taylor.taylorCoeffWithin and Mathlib.Analysis.Calculus.IteratedDeriv.Defs.iteratedDerivWithin_univ. This route does not define sourceHessian from the desired conclusion, does not use VP score-Hessian regularity, does not move sigma_eta^2/2 into the event field, and does not use sald_version_2.tex."

/-- Cycle-179 lower_2 compiled bridge from the scalar Taylor coefficient
convention to the scalar-line second-derivative coefficient boundary. -/