AutoSamplingTheory.SALD.cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondMiddleObligation

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 cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondMiddleObligation :
    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.cycle169_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_directional_second_middle

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 169 middle dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hLineSecond and hNegLineSecond below SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds are no longer primitive global selected/reflected scalar-line iteratedDeriv 2 bounds once the single ambient diagonal directional-Hessian bound forall z, norm (iteratedFDeriv Real 2 sourceTest z (fun _ : Fin 2 => e)) <= C1 is supplied. SALD.gaussianRealSelectedTestLineSecondDerivEqDirectionalFDeriv compiles the Mathlib iteratedDeriv_vcomp_two chain rule for the affine line q |-> x + q • e; SALD.gaussianRealSelectedTestLineSecondBoundsOfDirectionalSecondBound uses ContinuousMultilinearMap.map_smul_univ to handle the reflected direction -e; SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound feeds those bounds into the accepted cycle-168 bridge and derives hC1 from the ambient norm bound. Remaining source-cited boundary: prove the ambient diagonal directional-Hessian bound from the paper's selected-test bounded-Hessian/regularity hypothesis at appendix.tex:984-995 and appendix.tex:1379-1387. hSource, 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.gaussianRealSelectedTestLineSecondDerivEqDirectionalFDeriv
2. SALD.gaussianRealSelectedTestLineSecondBoundsOfDirectionalSecondBound
3. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound
4. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds
5. hDirectionalSecond
6. hLineSecond
7. hNegLineSecond
8. hC1
9. iteratedDeriv_vcomp_two
10. iteratedDeriv_const_add
11. iteratedDeriv_smul_const
12. iteratedDeriv_fun_id
13. deriv_smul_const
14. ContinuousMultilinearMap.map_smul_univ
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 middle packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib IteratedDeriv.FaaDiBruno and IteratedDeriv.Lemmas; the configured local SLT clone is absent and not needed. This rejects wrapper churn by removing the two scalar-line second-derivative hypotheses and exposing only the ambient selected-test directional Hessian bound as the smaller source-facing boundary.

def cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondMiddleObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle169_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_directional_second_middle"
  statement := "Cycle 169 middle dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hLineSecond and hNegLineSecond below SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds are no longer primitive global selected/reflected scalar-line iteratedDeriv 2 bounds once the single ambient diagonal directional-Hessian bound forall z, norm (iteratedFDeriv Real 2 sourceTest z (fun _ : Fin 2 => e)) <= C1 is supplied. SALD.gaussianRealSelectedTestLineSecondDerivEqDirectionalFDeriv compiles the Mathlib iteratedDeriv_vcomp_two chain rule for the affine line q |-> x + q • e; SALD.gaussianRealSelectedTestLineSecondBoundsOfDirectionalSecondBound uses ContinuousMultilinearMap.map_smul_univ to handle the reflected direction -e; SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound feeds those bounds into the accepted cycle-168 bridge and derives hC1 from the ambient norm bound. Remaining source-cited boundary: prove the ambient diagonal directional-Hessian bound from the paper's selected-test bounded-Hessian/regularity hypothesis at appendix.tex:984-995 and appendix.tex:1379-1387. hSource, hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves remain separate."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.gaussianRealSelectedTestLineSecondDerivEqDirectionalFDeriv",
    "SALD.gaussianRealSelectedTestLineSecondBoundsOfDirectionalSecondBound",
    "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound",
    "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds",
    "hDirectionalSecond",
    "hLineSecond",
    "hNegLineSecond",
    "hC1",
    "iteratedDeriv_vcomp_two",
    "iteratedDeriv_const_add",
    "iteratedDeriv_smul_const",
    "iteratedDeriv_fun_id",
    "deriv_smul_const",
    "ContinuousMultilinearMap.map_smul_univ",
    "appendix.tex:984-995",
    "appendix.tex:1379-1387",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "Dynamic-leaf middle packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib IteratedDeriv.FaaDiBruno and IteratedDeriv.Lemmas; the configured local SLT clone is absent and not needed. This rejects wrapper churn by removing the two scalar-line second-derivative hypotheses and exposing only the ambient selected-test directional Hessian bound as the smaller source-facing boundary."

/-- Cycle-169 lower_2 second-Frechet-derivative operator-norm narrowing packet. -/