AutoSamplingTheory.SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorCompatLower1Obligation

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 cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorCompatLower1Obligation :
    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.cycle167_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_taylor_compat_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 167 lower_1 dynamic-leaf worker/scout packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hNonnegTaylorCompat and hNegTaylorCompat below SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect are no longer primitive compatibility assumptions once the selected and reflected scalar lines are differentiable at 0. SALD.gaussianRealSelectedTestLineFirstOrderTaylorCompatOfDifferentiableAt compiles the interval-to-Set.univ first-order Taylor-polynomial compatibility for any r >= 0 from DifferentiableAt at the expansion point, using the r = 0 algebraic case and Mathlib uniqueDiffOn_Icc/DifferentiableAt.derivWithin for 0 < r. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff then supplies the all-r first-order quadratic remainder from signed interval ContDiffOn, signed interval second-derivative domination, and base differentiability of the selected/reflected scalar lines. Remaining source-cited boundary: derive hNonnegCont, hNonnegSecond, hNegCont, hNegSecond, hNonnegBaseDiff, and hNegBaseDiff from the paper's selected-test bounded-Hessian/regularity hypotheses. 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.gaussianRealSelectedTestLineFirstOrderTaylorCompatOfDifferentiableAt
2. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff
3. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect
4. hNonnegTaylorCompat
5. hNegTaylorCompat
6. hNonnegBaseDiff
7. hNegBaseDiff
8. hNonnegCont
9. hNonnegSecond
10. hNegCont
11. hNegSecond
12. uniqueDiffOn_Icc
13. DifferentiableAt.derivWithin
14. derivWithin_univ
15. taylorWithinEval_succ
16. appendix.tex:984-995
17. appendix.tex:1379-1387
18. 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 plus Deriv/FDeriv endpoint derivative-within APIs and TangentCone.Real uniqueDiffOn_Icc. The configured local SLT clone is absent; no SLT theorem was imported, ported, or marked formalized. This rejects wrapper churn by removing two explicit Taylor-compatibility hypotheses and exposing base differentiability as the smaller regularity leaf.

def cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorCompatLower1Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle167_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_taylor_compat_lower1"
  statement := "Cycle 167 lower_1 dynamic-leaf worker/scout packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hNonnegTaylorCompat and hNegTaylorCompat below SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect are no longer primitive compatibility assumptions once the selected and reflected scalar lines are differentiable at 0. SALD.gaussianRealSelectedTestLineFirstOrderTaylorCompatOfDifferentiableAt compiles the interval-to-Set.univ first-order Taylor-polynomial compatibility for any r >= 0 from DifferentiableAt at the expansion point, using the r = 0 algebraic case and Mathlib uniqueDiffOn_Icc/DifferentiableAt.derivWithin for 0 < r. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff then supplies the all-r first-order quadratic remainder from signed interval ContDiffOn, signed interval second-derivative domination, and base differentiability of the selected/reflected scalar lines. Remaining source-cited boundary: derive hNonnegCont, hNonnegSecond, hNegCont, hNegSecond, hNonnegBaseDiff, and hNegBaseDiff from the paper's selected-test bounded-Hessian/regularity hypotheses. hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves remain separate."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.gaussianRealSelectedTestLineFirstOrderTaylorCompatOfDifferentiableAt",
    "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff",
    "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect",
    "hNonnegTaylorCompat",
    "hNegTaylorCompat",
    "hNonnegBaseDiff",
    "hNegBaseDiff",
    "hNonnegCont",
    "hNonnegSecond",
    "hNegCont",
    "hNegSecond",
    "uniqueDiffOn_Icc",
    "DifferentiableAt.derivWithin",
    "derivWithin_univ",
    "taylorWithinEval_succ",
    "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 plus Deriv/FDeriv endpoint derivative-within APIs and TangentCone.Real uniqueDiffOn_Icc. The configured local SLT clone is absent; no SLT theorem was imported, ported, or marked formalized. This rejects wrapper churn by removing two explicit Taylor-compatibility hypotheses and exposing base differentiability as the smaller regularity leaf."

/-- Cycle-167 lower_2 global line regularity narrowing packet. -/