AutoSamplingTheory.SALD.cycle174GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticVariationMiddleObligation

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 cycle174GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticVariationMiddleObligation :
    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.cycle174_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_quadratic_variation_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 174 middle dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Upper decision accepted by middle: the cycle-173 hSourceHasHessian and hSourceHessianBound fields are not derivable from the original source anchors and remain a source-contract gap; deriving them from unexpanded testRegular, defining sourceHessian from hHessianOpNorm, adding an unsourced SourceSelectedWeakTestC2bBoundedHessian, or using the VP score-Hessian result is rejected wrapper churn. Exact boundary now narrowed for lower work: the next connected Brownian/Ito leaf is hFrozenScalarBrownianItoQuadraticVariationNormalization under SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder. Lower_1 should write the source route from eq:general_moving_target_SALD_frozen_interp: after drift separation, the Brownian coordinate increment is sigma_eta times a centered scalar coordinate with variance one in the normalized variable, the paper's sigma_eta^2/2 coefficient remains outside the event field, and the normalized quadratic coefficient is the diagonal Hessian generator SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator. Lower_2 should implement exactly one compiled algebraic bridge only if the source supplies hQuadraticCoeffDef : testRegular -> forall phi x i, quadraticCoeff phi x i = SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest phi) x ((stdOrthonormalBasis Real E) i) and hVarianceOne : testRegular -> forall phi x i, (variance phi x i : Real) = 1; otherwise record the smaller source-cited gap hQuadraticCoeffDef/hVarianceOne. The remaining sibling leaves hFrozenScalarBrownianItoTaylorMomentDecomposition, hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes, and hFrozenScalarBrownianItoEventFieldCoordinateSum stay 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.cycle173GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation
2. SALD.selectedWeakTestHessianOpNormOfSourceHessianField
3. hSourceHasHessian
4. hSourceHessianBound
5. source-contract-gap for selected weak-test Hessian fields
6. SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder
7. SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator
8. hFrozenScalarBrownianItoQuadraticVariationNormalization
9. hQuadraticCoeffDef
10. hVarianceOne
11. hFrozenScalarBrownianItoTaylorMomentDecomposition
12. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
13. hFrozenScalarBrownianItoEventFieldCoordinateSum
14. eq:general_moving_target_SALD_frozen_interp
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.

No SLT theorem was consulted or imported. The source labels are already indexed under SALD_original.jsonl. Local Mathlib/Lean context consulted only through the existing Gaussian moment and Taylor bridge chain in AutoSamplingTheory/SALD.lean; the new lower packet is source-bookkeeping algebra for the Brownian variance/coefficient normalization, not a probability-library import or theorem-status promotion.

def cycle174GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticVariationMiddleObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle174_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_quadratic_variation_middle"
  statement := "Cycle 174 middle dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Upper decision accepted by middle: the cycle-173 hSourceHasHessian and hSourceHessianBound fields are not derivable from the original source anchors and remain a source-contract gap; deriving them from unexpanded testRegular, defining sourceHessian from hHessianOpNorm, adding an unsourced SourceSelectedWeakTestC2bBoundedHessian, or using the VP score-Hessian result is rejected wrapper churn. Exact boundary now narrowed for lower work: the next connected Brownian/Ito leaf is hFrozenScalarBrownianItoQuadraticVariationNormalization under SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder. Lower_1 should write the source route from eq:general_moving_target_SALD_frozen_interp: after drift separation, the Brownian coordinate increment is sigma_eta times a centered scalar coordinate with variance one in the normalized variable, the paper's sigma_eta^2/2 coefficient remains outside the event field, and the normalized quadratic coefficient is the diagonal Hessian generator SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator. Lower_2 should implement exactly one compiled algebraic bridge only if the source supplies hQuadraticCoeffDef : testRegular -> forall phi x i, quadraticCoeff phi x i = SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest phi) x ((stdOrthonormalBasis Real E) i) and hVarianceOne : testRegular -> forall phi x i, (variance phi x i : Real) = 1; otherwise record the smaller source-cited gap hQuadraticCoeffDef/hVarianceOne. The remaining sibling leaves hFrozenScalarBrownianItoTaylorMomentDecomposition, hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes, and hFrozenScalarBrownianItoEventFieldCoordinateSum stay explicit."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle173GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation",
    "SALD.selectedWeakTestHessianOpNormOfSourceHessianField",
    "hSourceHasHessian",
    "hSourceHessianBound",
    "source-contract-gap for selected weak-test Hessian fields",
    "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder",
    "SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator",
    "hFrozenScalarBrownianItoQuadraticVariationNormalization",
    "hQuadraticCoeffDef",
    "hVarianceOne",
    "hFrozenScalarBrownianItoTaylorMomentDecomposition",
    "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes",
    "hFrozenScalarBrownianItoEventFieldCoordinateSum",
    "eq:general_moving_target_SALD_frozen_interp",
    "appendix.tex:984-995",
    "appendix.tex:1379-1387",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "No SLT theorem was consulted or imported. The source labels are already indexed under SALD_original.jsonl. Local Mathlib/Lean context consulted only through the existing Gaussian moment and Taylor bridge chain in AutoSamplingTheory/SALD.lean; the new lower packet is source-bookkeeping algebra for the Brownian variance/coefficient normalization, not a probability-library import or theorem-status promotion."

/-- Cycle-174 lower_1 proof-scout packet for the Brownian
quadratic-variation normalization leaf.

This records the source route and the exact lower_2 algebraic theorem shape.
It does not close the normalization theorem or promote the coefficient and
variance fields to formalized facts.
-/