AutoSamplingTheory.SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneLower1Obligation

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 cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneLower1Obligation :
    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.cycle176_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_variance_one_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 176 lower_1 dynamic-leaf proof-scout packet. Classification: narrows-source-cited-boundary. Exact missing theorem boundary narrowed: the downstream hVarianceOne : testRegular -> forall phi x i, (variance phi x i : Real) = 1 should be discharged only from the smaller source-facing normalized Brownian variance definition hNormalizedVarianceDef : testRegular -> forall phi x i, variance phi x i = (1 : NNReal). Source route: appendix.tex:958-970 states xi_k iid N(0,I_d) in the EM step; appendix.tex:984-995 gives the frozen interpolation Brownian increment sigma_eta*(W_s-W_{s_k}); appendix.tex:1170-1176 rewrites the increment as sigma_eta(t(s))*sqrt(s-s_k)*xi with xi ~ N(0,I); appendix.tex:1379-1387 keeps sigma_eta^2/2 outside the Brownian event field. Thus the local variance field for the normalized scalar Brownian coordinate is the unit NNReal variance before the external diffusion coefficient is applied. Lower_2-ready theorem shape: selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef (variance : Test -> E -> Fin (Module.finrank Real E) -> NNReal) (testRegular : Prop) (hNormalizedVarianceDef : testRegular -> forall phi x i, variance phi x i = (1 : NNReal)) : testRegular -> forall phi x i, (variance phi x i : Real) = 1. Proof route: intro htests phi x i; rewrite by hNormalizedVarianceDef htests phi x i; close the coercion of (1 : NNReal) to Real by norm_num. This does not derive hNormalizedVarianceDef itself, move sigma_eta^2/2 into the event field, reintroduce selected weak-test Hessian wrappers, or use VP score-Hessian regularity.

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.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneMiddleObligation
2. SALD.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef
3. SALD.selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne
4. hVarianceOne
5. hNormalizedVarianceDef
6. hSecondTaylorCoeffDef
7. hSourceHasHessian
8. hSourceHessianBound
9. hSecondCoeff
10. hFrozenScalarBrownianItoTaylorMomentDecomposition
11. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
12. hFrozenScalarBrownianItoEventFieldCoordinateSum
13. eq:SALD_general_EM
14. eq:general_moving_target_SALD_frozen_interp
15. eq:increment_basic_bound_revised
16. appendix.tex:958-970
17. appendix.tex:984-995
18. appendix.tex:1170-1176
19. appendix.tex:1379-1387
20. paper-wide original-source search excluding sald_version_2.tex
21. 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 theorem was consulted or imported. Mathlib/local input needed by lower_2 is only the existing NNReal-to-Real coercion simplification by norm_num after rewriting by hNormalizedVarianceDef. The remaining source-facing stochastic work is hNormalizedVarianceDef from xi ~ N(0,I).

def cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneLower1Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle176_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_variance_one_lower1"
  statement := "Cycle 176 lower_1 dynamic-leaf proof-scout packet. Classification: narrows-source-cited-boundary. Exact missing theorem boundary narrowed: the downstream hVarianceOne : testRegular -> forall phi x i, (variance phi x i : Real) = 1 should be discharged only from the smaller source-facing normalized Brownian variance definition hNormalizedVarianceDef : testRegular -> forall phi x i, variance phi x i = (1 : NNReal). Source route: appendix.tex:958-970 states xi_k iid N(0,I_d) in the EM step; appendix.tex:984-995 gives the frozen interpolation Brownian increment sigma_eta*(W_s-W_{s_k}); appendix.tex:1170-1176 rewrites the increment as sigma_eta(t(s))*sqrt(s-s_k)*xi with xi ~ N(0,I); appendix.tex:1379-1387 keeps sigma_eta^2/2 outside the Brownian event field. Thus the local variance field for the normalized scalar Brownian coordinate is the unit NNReal variance before the external diffusion coefficient is applied. Lower_2-ready theorem shape: selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef (variance : Test -> E -> Fin (Module.finrank Real E) -> NNReal) (testRegular : Prop) (hNormalizedVarianceDef : testRegular -> forall phi x i, variance phi x i = (1 : NNReal)) : testRegular -> forall phi x i, (variance phi x i : Real) = 1. Proof route: intro htests phi x i; rewrite by hNormalizedVarianceDef htests phi x i; close the coercion of (1 : NNReal) to Real by norm_num. This does not derive hNormalizedVarianceDef itself, move sigma_eta^2/2 into the event field, reintroduce selected weak-test Hessian wrappers, or use VP score-Hessian regularity."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneMiddleObligation",
    "SALD.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef",
    "SALD.selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne",
    "hVarianceOne",
    "hNormalizedVarianceDef",
    "hSecondTaylorCoeffDef",
    "hSourceHasHessian",
    "hSourceHessianBound",
    "hSecondCoeff",
    "hFrozenScalarBrownianItoTaylorMomentDecomposition",
    "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes",
    "hFrozenScalarBrownianItoEventFieldCoordinateSum",
    "eq:SALD_general_EM",
    "eq:general_moving_target_SALD_frozen_interp",
    "eq:increment_basic_bound_revised",
    "appendix.tex:958-970",
    "appendix.tex:984-995",
    "appendix.tex:1170-1176",
    "appendix.tex:1379-1387",
    "paper-wide original-source search excluding sald_version_2.tex",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "No local SLT theorem was consulted or imported. Mathlib/local input needed by lower_2 is only the existing NNReal-to-Real coercion simplification by norm_num after rewriting by hNormalizedVarianceDef. The remaining source-facing stochastic work is hNormalizedVarianceDef from xi ~ N(0,I)."

/-- Cycle-176 lower_2 compiled bridge through the normalized variance field.

The direct variance-one theorem was already available when lower_2 arrived, so
this packet composes it with the cycle-175 second-Taylor coefficient bridge.
The older `hQuadraticCoeffDef`/`hVarianceOne` pair is replaced by the smaller
source-facing `hSecondTaylorCoeffDef`/`hNormalizedVarianceDef` pair for the
quadratic-variation normalization step.
-/