AutoSamplingTheory.SALD.cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLineSecondLower2Obligation

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 cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLineSecondLower2Obligation :
    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.cycle168_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_line_second_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 168 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hNonnegSecond and hNegSecond below SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn are no longer primitive signed interval iteratedDerivWithin 2 domination hypotheses once global selected/reflected scalar-line iteratedDeriv 2 bounds are supplied. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds compiles the bridge: for r=0 it uses accPt_iff_nhds plus derivWithin_zero_of_not_accPt to close the singleton interval by hC1; for 0<r it uses uniqueDiffOn_Icc and iteratedDerivWithin_eq_iteratedDeriv to replace the interval second derivative by the global line derivative, then applies hLineSecond or hNegLineSecond. Remaining source-cited boundary: prove hLineSecond and hNegLineSecond from the selected-test bounded-Hessian regularity, with ambient ContDiffOn and hC1 still explicit. 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.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds
2. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn
3. SALD.gaussianRealSelectedTestLineContDiffOnOfSourceContDiffOn
4. hSource
5. hC1
6. hLineSecond
7. hNegLineSecond
8. hNonnegSecond
9. hNegSecond
10. accPt_iff_nhds
11. derivWithin_zero_of_not_accPt
12. iteratedDerivWithin_succ
13. uniqueDiffOn_Icc
14. iteratedDerivWithin_eq_iteratedDeriv
15. ContDiff.contDiffAt
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_2 packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib IteratedDeriv.Defs, Deriv.Basic, ClusterPt, and TangentCone.Real through the imported Taylor stack; the configured local SLT clone is absent, and no SLT theorem was imported, ported, or marked formalized. This rejects wrapper churn by removing two signed interval second-derivative hypotheses and exposing only global selected/reflected line second-derivative bounds as the smaller source-facing boundary.

def cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLineSecondLower2Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle168_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_line_second_lower2"
  statement := "Cycle 168 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hNonnegSecond and hNegSecond below SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn are no longer primitive signed interval iteratedDerivWithin 2 domination hypotheses once global selected/reflected scalar-line iteratedDeriv 2 bounds are supplied. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds compiles the bridge: for r=0 it uses accPt_iff_nhds plus derivWithin_zero_of_not_accPt to close the singleton interval by hC1; for 0<r it uses uniqueDiffOn_Icc and iteratedDerivWithin_eq_iteratedDeriv to replace the interval second derivative by the global line derivative, then applies hLineSecond or hNegLineSecond. Remaining source-cited boundary: prove hLineSecond and hNegLineSecond from the selected-test bounded-Hessian regularity, with ambient ContDiffOn and hC1 still explicit. hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves remain separate."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds",
    "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn",
    "SALD.gaussianRealSelectedTestLineContDiffOnOfSourceContDiffOn",
    "hSource",
    "hC1",
    "hLineSecond",
    "hNegLineSecond",
    "hNonnegSecond",
    "hNegSecond",
    "accPt_iff_nhds",
    "derivWithin_zero_of_not_accPt",
    "iteratedDerivWithin_succ",
    "uniqueDiffOn_Icc",
    "iteratedDerivWithin_eq_iteratedDeriv",
    "ContDiff.contDiffAt",
    "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 IteratedDeriv.Defs, Deriv.Basic, ClusterPt, and TangentCone.Real through the imported Taylor stack; the configured local SLT clone is absent, and no SLT theorem was imported, ported, or marked formalized. This rejects wrapper churn by removing two signed interval second-derivative hypotheses and exposing only global selected/reflected line second-derivative bounds as the smaller source-facing boundary."

/-- Cycle-169 ambient directional-Hessian narrowing packet. -/