AutoSamplingTheory.SALD.cycle205GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorRawSplitLower2Obligation

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 cycle205GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorRawSplitLower2Obligation :
    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.cycle205_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_selected_line_taylor_raw_split_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 205 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact missing theorem boundary narrowed: hSelectedLineTaylorRawSplitDef is the raw selected scalar-line Taylor identity still required below the accepted cycle-204 hSourceTaylorIntegrandSelectedIncrementDef naming gap. Existing compiled bridges SALD.selectedWeakTestSelectedLineTaylorSplitDefOfRawTaylorAndTermDefs, SALD.selectedWeakTestSourceTaylorIntegrandDefOfRawAndLineTaylorSplit, SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralRawTaylorAndTermDefs, SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfScalarPushforwardRawTaylorAndTermDefs, and SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfScalarPushforwardRawTaylorAndDominatedRemainder already reuse this raw Taylor field, so lower_2 must not add another consumer wrapper. Lean inspection shows normalizedRemainder is an abstract parameter of those bridges rather than a reducible residual definition, and a targeted original-source search excluding sald_version_2.tex found no displayed selected-line Taylor expansion or definition of normalizedRemainder as that residual. The needed source-backed shape is testRegular -> forall phi x i z, selectedTest phi (x + z smul stdOrthonormalBasis Real E i) - selectedTest phi x = deriv (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 0 * z + ((2 : Real) * taylorCoeffWithin (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 2 Set.univ 0) * z ^ 2 + normalizedRemainder phi x i z. Source anchors appendix.tex:958-970, appendix.tex:983-996, appendix.tex:1161-1176, and appendix.tex:1379-1387 identify the EM increment, frozen interpolation, normalized Gaussian coordinate, and weak-FP consumer while keeping sigma_eta^2 / 2 outside this scalar Taylor residual field. No endpoint/naming replay, normalized-remainder-bound replay, source-Hessian work, testRegular repackaging, VP score-Hessian substitution, direct SLT dependency, broad route audit, or wrapper churn is introduced.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteEmSource — 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. hSelectedLineTaylorRawSplitDef
2. hSelectedLineTaylorSplitDef
3. hSourceTaylorIntegrandSelectedIncrementDef
4. hSourceTaylorIntegrandRawDef
5. hSourceLinearTermTaylorDef
6. hScalarLineFirstCoeffDef
7. hSourceQuadraticTermTaylorDef
8. hScalarLineTaylorCoeffDef
9. selectedTest
10. normalizedRemainder
11. deriv
12. taylorCoeffWithin
13. Set.univ
14. stdOrthonormalBasis
15. SALD.selectedWeakTestSelectedLineTaylorSplitDefOfRawTaylorAndTermDefs
16. SALD.selectedWeakTestSourceTaylorIntegrandDefOfRawAndLineTaylorSplit
17. SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralRawTaylorAndTermDefs
18. SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfScalarPushforwardRawTaylorAndTermDefs
19. SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfScalarPushforwardRawTaylorAndDominatedRemainder
20. SALD.cycle188GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorSplitLower2Obligation
21. SALD.cycle204GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandSelectedIncrementLower2Obligation
22. runs/20260613-050236-675526-ASTIS-SALD-001-cycle205/lower_1_selected_line_taylor_raw_split_route.md
23. runs/20260613-050236-675526-ASTIS-SALD-001-cycle205/lower_3_selected_line_taylor_api_churn_rejection.md
24. needed_shape=testRegular -> forall phi x i z, selectedTest phi (x + z smul stdOrthonormalBasis Real E i) - selectedTest phi x = deriv (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 0 * z + ((2 : Real) * taylorCoeffWithin (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 2 Set.univ 0) * z ^ 2 + normalizedRemainder phi x i z
25. leaf=hSelectedLineTaylorRawSplitDef
26. error_class=source_contract_gap_missing_selected_line_taylor_raw_split_definition
27. blocked_by=normalizedRemainder is an abstract parameter in the compiled Taylor bridges and no original-paper definition identifies it with the selected scalar Taylor residual
28. eq:SALD_general_EM
29. eq:general_moving_target_SALD_frozen_interp
30. appendix.tex:958-970
31. appendix.tex:983-996
32. appendix.tex:1161-1176
33. appendix.tex:1379-1387
34. source_contract_gap_missing_selected_line_taylor_raw_split_definition
35. sald.general_moving_target_discrete.em_interpolation_fp
36. sald.general_moving_target_discrete.kl_derivative
37. divergence/FI/IBP handoff
38. 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=hSelectedLineTaylorRawSplitDef; error_class=source_contract_gap_missing_selected_line_taylor_raw_split_definition; needed_shape=testRegular -> forall phi x i z, selectedTest phi (x + z smul stdOrthonormalBasis Real E i) - selectedTest phi x = deriv (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 0 * z + ((2 : Real) * taylorCoeffWithin (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 2 Set.univ 0) * z ^ 2 + normalizedRemainder phi x i z; source_lines=appendix.tex:958-970;appendix.tex:983-996;appendix.tex:1161-1176;appendix.tex:1379-1387; blocked_by=normalizedRemainder is an abstract parameter of the compiled cycle-188/189/192/193 bridges and no original-paper or local Lean definition identifies it with the residual of the selected scalar Taylor expansion; lower_1_route_artifact=runs/20260613-050236-675526-ASTIS-SALD-001-cycle205/lower_1_selected_line_taylor_raw_split_route.md; lower_3_api_artifact=runs/20260613-050236-675526-ASTIS-SALD-001-cycle205/lower_3_selected_line_taylor_api_churn_rejection.md; packet_type=dynamic-leaf worker packet.

def cycle205GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorRawSplitLower2Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle205_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_selected_line_taylor_raw_split_lower2"
  statement := "Cycle 205 lower_2 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact missing theorem boundary narrowed: hSelectedLineTaylorRawSplitDef is the raw selected scalar-line Taylor identity still required below the accepted cycle-204 hSourceTaylorIntegrandSelectedIncrementDef naming gap. Existing compiled bridges SALD.selectedWeakTestSelectedLineTaylorSplitDefOfRawTaylorAndTermDefs, SALD.selectedWeakTestSourceTaylorIntegrandDefOfRawAndLineTaylorSplit, SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralRawTaylorAndTermDefs, SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfScalarPushforwardRawTaylorAndTermDefs, and SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfScalarPushforwardRawTaylorAndDominatedRemainder already reuse this raw Taylor field, so lower_2 must not add another consumer wrapper. Lean inspection shows normalizedRemainder is an abstract parameter of those bridges rather than a reducible residual definition, and a targeted original-source search excluding sald_version_2.tex found no displayed selected-line Taylor expansion or definition of normalizedRemainder as that residual. The needed source-backed shape is testRegular -> forall phi x i z, selectedTest phi (x + z smul stdOrthonormalBasis Real E i) - selectedTest phi x = deriv (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 0 * z + ((2 : Real) * taylorCoeffWithin (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 2 Set.univ 0) * z ^ 2 + normalizedRemainder phi x i z. Source anchors appendix.tex:958-970, appendix.tex:983-996, appendix.tex:1161-1176, and appendix.tex:1379-1387 identify the EM increment, frozen interpolation, normalized Gaussian coordinate, and weak-FP consumer while keeping sigma_eta^2 / 2 outside this scalar Taylor residual field. No endpoint/naming replay, normalized-remainder-bound replay, source-Hessian work, testRegular repackaging, VP score-Hessian substitution, direct SLT dependency, broad route audit, or wrapper churn is introduced."
  source := saldGeneralMovingTargetDiscreteEmSource
  status := ProofStatus.obligation
  dependsOn := [
    "hSelectedLineTaylorRawSplitDef",
    "hSelectedLineTaylorSplitDef",
    "hSourceTaylorIntegrandSelectedIncrementDef",
    "hSourceTaylorIntegrandRawDef",
    "hSourceLinearTermTaylorDef",
    "hScalarLineFirstCoeffDef",
    "hSourceQuadraticTermTaylorDef",
    "hScalarLineTaylorCoeffDef",
    "selectedTest",
    "normalizedRemainder",
    "deriv",
    "taylorCoeffWithin",
    "Set.univ",
    "stdOrthonormalBasis",
    "SALD.selectedWeakTestSelectedLineTaylorSplitDefOfRawTaylorAndTermDefs",
    "SALD.selectedWeakTestSourceTaylorIntegrandDefOfRawAndLineTaylorSplit",
    "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralRawTaylorAndTermDefs",
    "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfScalarPushforwardRawTaylorAndTermDefs",
    "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfScalarPushforwardRawTaylorAndDominatedRemainder",
    "SALD.cycle188GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorSplitLower2Obligation",
    "SALD.cycle204GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandSelectedIncrementLower2Obligation",
    "runs/20260613-050236-675526-ASTIS-SALD-001-cycle205/lower_1_selected_line_taylor_raw_split_route.md",
    "runs/20260613-050236-675526-ASTIS-SALD-001-cycle205/lower_3_selected_line_taylor_api_churn_rejection.md",
    "needed_shape=testRegular -> forall phi x i z, selectedTest phi (x + z smul stdOrthonormalBasis Real E i) - selectedTest phi x = deriv (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 0 * z + ((2 : Real) * taylorCoeffWithin (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 2 Set.univ 0) * z ^ 2 + normalizedRemainder phi x i z",
    "leaf=hSelectedLineTaylorRawSplitDef",
    "error_class=source_contract_gap_missing_selected_line_taylor_raw_split_definition",
    "blocked_by=normalizedRemainder is an abstract parameter in the compiled Taylor bridges and no original-paper definition identifies it with the selected scalar Taylor residual",
    "eq:SALD_general_EM",
    "eq:general_moving_target_SALD_frozen_interp",
    "appendix.tex:958-970",
    "appendix.tex:983-996",
    "appendix.tex:1161-1176",
    "appendix.tex:1379-1387",
    "source_contract_gap_missing_selected_line_taylor_raw_split_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=hSelectedLineTaylorRawSplitDef; error_class=source_contract_gap_missing_selected_line_taylor_raw_split_definition; needed_shape=testRegular -> forall phi x i z, selectedTest phi (x + z smul stdOrthonormalBasis Real E i) - selectedTest phi x = deriv (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 0 * z + ((2 : Real) * taylorCoeffWithin (fun q => selectedTest phi (x + q smul stdOrthonormalBasis Real E i)) 2 Set.univ 0) * z ^ 2 + normalizedRemainder phi x i z; source_lines=appendix.tex:958-970;appendix.tex:983-996;appendix.tex:1161-1176;appendix.tex:1379-1387; blocked_by=normalizedRemainder is an abstract parameter of the compiled cycle-188/189/192/193 bridges and no original-paper or local Lean definition identifies it with the residual of the selected scalar Taylor expansion; lower_1_route_artifact=runs/20260613-050236-675526-ASTIS-SALD-001-cycle205/lower_1_selected_line_taylor_raw_split_route.md; lower_3_api_artifact=runs/20260613-050236-675526-ASTIS-SALD-001-cycle205/lower_3_selected_line_taylor_api_churn_rejection.md; packet_type=dynamic-leaf worker packet."

/-- Cycle-205 proof-DAG pane for the selected-line raw Taylor split gap. -/