AutoSamplingTheory.SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorMomentLowerObligation

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 cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorMomentLowerObligation :
    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.cycle161_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_one_dim_taylor_moment_lower

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 161 lower_2 packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hFrozenScalarBrownianItoOneDimTaylorExpansion should no longer treat the centered scalar Gaussian first/second moment algebra as open; SALD.gaussianRealZeroOneDimTaylorMomentContribution compiles the local fact that a linear Taylor coefficient times E[Z] plus a quadratic coefficient times E[Z^2] collapses to the quadratic coefficient times the variance for Z ~ gaussianReal 0 v. The remaining smaller source-cited theorem is hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit: under selected-test C^2 regularity, integrability, and dominated Taylor remainder hypotheses, the source-defined brownianCoordinateGenerator for sigma_eta (W_s-W_{s_k}) has derivative-per-unit-time contribution SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator after drift separation. The coefficient sigma_eta^2/2 remains outside this event field, and hFrozenScalarBrownianItoEventFieldCoordinateSum remains 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.gaussianRealZeroOneDimTaylorMomentContribution
2. SALD.gaussianRealZeroSecondMoment
3. ProbabilityTheory.integral_id_gaussianReal
4. hFrozenScalarBrownianItoOneDimTaylorExpansion
5. hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit
6. SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator
7. eq:general_moving_target_SALD_frozen_interp
8. appendix.tex:984-995
9. appendix.tex:1379-1387
10. Mathlib.Probability.Distributions.Gaussian.Real
11. 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 lower_2 worker packet inside the scalar Brownian/Ito illness area. It uses only Mathlib Gaussian.Real moment facts already imported in SALD, with no local SLT clone or SLT import. This is not a weak-FP source-field, trace-field, total-event/source-functional, non-EM, route-audit, or wrapper-churn packet; it leaves the analytic dominated Taylor/generator limit explicit.

def cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorMomentLowerObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle161_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_one_dim_taylor_moment_lower"
  statement := "Cycle 161 lower_2 packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: hFrozenScalarBrownianItoOneDimTaylorExpansion should no longer treat the centered scalar Gaussian first/second moment algebra as open; SALD.gaussianRealZeroOneDimTaylorMomentContribution compiles the local fact that a linear Taylor coefficient times E[Z] plus a quadratic coefficient times E[Z^2] collapses to the quadratic coefficient times the variance for Z ~ gaussianReal 0 v. The remaining smaller source-cited theorem is hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit: under selected-test C^2 regularity, integrability, and dominated Taylor remainder hypotheses, the source-defined brownianCoordinateGenerator for sigma_eta (W_s-W_{s_k}) has derivative-per-unit-time contribution SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator after drift separation. The coefficient sigma_eta^2/2 remains outside this event field, and hFrozenScalarBrownianItoEventFieldCoordinateSum remains explicit."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.gaussianRealZeroOneDimTaylorMomentContribution",
    "SALD.gaussianRealZeroSecondMoment",
    "ProbabilityTheory.integral_id_gaussianReal",
    "hFrozenScalarBrownianItoOneDimTaylorExpansion",
    "hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit",
    "SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator",
    "eq:general_moving_target_SALD_frozen_interp",
    "appendix.tex:984-995",
    "appendix.tex:1379-1387",
    "Mathlib.Probability.Distributions.Gaussian.Real",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "Dynamic-leaf lower_2 worker packet inside the scalar Brownian/Ito illness area. It uses only Mathlib Gaussian.Real moment facts already imported in SALD, with no local SLT clone or SLT import. This is not a weak-FP source-field, trace-field, total-event/source-functional, non-EM, route-audit, or wrapper-churn packet; it leaves the analytic dominated Taylor/generator limit explicit."

/-- Cycle-161 proof-DAG pane for the one-dimensional coordinate Taylor split. -/