AutoSamplingTheory.SALD.cycle198GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateSumLower2Obligation

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 cycle198GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateSumLower2Obligation :
    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.cycle198_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_coordinate_sum_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 198 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact missing theorem boundary narrowed: hFrozenScalarBrownianItoEventFieldCoordinateSum would need the source-backed definition emGeneratorLaplacianEventField phi x = Finset.univ.sum (fun i : Fin (Module.finrank Real E) => brownianCoordinateGenerator phi x i). Lean inspection shows SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator consumes this equality as an input, while SALD.emFrozenScalarBrownianItoGeneratorEventField unfolds only to the finite sum of diagonal iteratedFDeriv terms after the coordinate generators have already been identified. No local definition connects the abstract event field to the coordinate-generator sum. Source anchors appendix.tex:983-996 and appendix.tex:1379-1387 motivate the frozen Brownian coordinate decomposition with sigma_eta^2/2 kept outside the event field, but the original source does not spell out the Lean-facing definition of emGeneratorLaplacianEventField as that finite coordinate sum. No source-Hessian repackaging, hNormalizedRemainderBoundDef closure claim, direct SLT dependency, broad route audit, or same-shape wrapper theorem is introduced.

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. hFrozenScalarBrownianItoEventFieldCoordinateSum
2. emGeneratorLaplacianEventField
3. brownianCoordinateGenerator
4. SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator
5. SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor
6. SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfScalarPushforwardRawTaylorAndDominatedRemainder
7. AutoSamplingTheory.TechnicalLemmas.SALDExtracted.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw
8. SALD.emFrozenScalarBrownianItoGeneratorEventField
9. eq:general_moving_target_SALD_frozen_interp
10. appendix.tex:983-996
11. appendix.tex:1379-1387
12. source_contract_gap_missing_event_field_coordinate_sum_definition
13. sald.general_moving_target_discrete.em_interpolation_fp
14. sald.general_moving_target_discrete.kl_derivative
15. divergence/FI/IBP handoff
16. thm:general-moving-target-SALD-discrete

note : String (explicit)

Recorded evidence/caveats; may distinguish a compiled scalar helper from still-open source analysis.

typed_verifier_feedback: leaf=hFrozenScalarBrownianItoEventFieldCoordinateSum; error_class=source_contract_gap_missing_event_field_coordinate_sum_definition; needed_shape=emGeneratorLaplacianEventField phi x = Finset.univ.sum (fun i : Fin (Module.finrank Real E) => brownianCoordinateGenerator phi x i); source_lines=appendix.tex:983-996; appendix.tex:1379-1387; blocked_by=no original-paper/Lean definition connecting emGeneratorLaplacianEventField to the finite sum of brownianCoordinateGenerator; lower_1_route_artifact=runs/20260613-024704-572927-ASTIS-SALD-001-cycle198/lower_1_coordinate_sum_route.md; packet_type=dynamic-leaf worker packet.

def cycle198GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateSumLower2Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle198_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_coordinate_sum_lower2"
  statement := "Cycle 198 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact missing theorem boundary narrowed: hFrozenScalarBrownianItoEventFieldCoordinateSum would need the source-backed definition emGeneratorLaplacianEventField phi x = Finset.univ.sum (fun i : Fin (Module.finrank Real E) => brownianCoordinateGenerator phi x i). Lean inspection shows SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator consumes this equality as an input, while SALD.emFrozenScalarBrownianItoGeneratorEventField unfolds only to the finite sum of diagonal iteratedFDeriv terms after the coordinate generators have already been identified. No local definition connects the abstract event field to the coordinate-generator sum. Source anchors appendix.tex:983-996 and appendix.tex:1379-1387 motivate the frozen Brownian coordinate decomposition with sigma_eta^2/2 kept outside the event field, but the original source does not spell out the Lean-facing definition of emGeneratorLaplacianEventField as that finite coordinate sum. No source-Hessian repackaging, hNormalizedRemainderBoundDef closure claim, direct SLT dependency, broad route audit, or same-shape wrapper theorem is introduced."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.obligation
  dependsOn := [
    "hFrozenScalarBrownianItoEventFieldCoordinateSum",
    "emGeneratorLaplacianEventField",
    "brownianCoordinateGenerator",
    "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator",
    "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor",
    "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfScalarPushforwardRawTaylorAndDominatedRemainder",
    "AutoSamplingTheory.TechnicalLemmas.SALDExtracted.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw",
    "SALD.emFrozenScalarBrownianItoGeneratorEventField",
    "eq:general_moving_target_SALD_frozen_interp",
    "appendix.tex:983-996",
    "appendix.tex:1379-1387",
    "source_contract_gap_missing_event_field_coordinate_sum_definition",
    "sald.general_moving_target_discrete.em_interpolation_fp",
    "sald.general_moving_target_discrete.kl_derivative",
    "divergence/FI/IBP handoff",
    "thm:general-moving-target-SALD-discrete"
  ]
  note := "typed_verifier_feedback: leaf=hFrozenScalarBrownianItoEventFieldCoordinateSum; error_class=source_contract_gap_missing_event_field_coordinate_sum_definition; needed_shape=emGeneratorLaplacianEventField phi x = Finset.univ.sum (fun i : Fin (Module.finrank Real E) => brownianCoordinateGenerator phi x i); source_lines=appendix.tex:983-996; appendix.tex:1379-1387; blocked_by=no original-paper/Lean definition connecting emGeneratorLaplacianEventField to the finite sum of brownianCoordinateGenerator; lower_1_route_artifact=runs/20260613-024704-572927-ASTIS-SALD-001-cycle198/lower_1_coordinate_sum_route.md; packet_type=dynamic-leaf worker packet."

/-- Cycle-198 proof-DAG pane for the frozen Brownian coordinate-sum gap. -/