AutoSamplingTheory.SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorScoutObligation

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 cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorScoutObligation :
    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_scout

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_1 proof-scout packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor should no longer treat the scalar Brownian second moment as open; SALD.gaussianRealZeroSecondMoment compiles the centered real Gaussian second moment needed for one Brownian coordinate. The remaining smaller source-cited theorem is the stochastic Taylor/generator passage hFrozenScalarBrownianItoOneDimTaylorExpansion : testRegular -> forall phi x i, the source-defined brownianCoordinateGenerator for the scalar increment sigma_eta (W_s-W_{s_k}) has derivative-per-unit-time contribution SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest phi) x ((stdOrthonormalBasis Real E) i), using selected-test C^2 regularity, zero first moment for the centered Brownian coordinate, the compiled second moment, and drift separation into the first-order conditional drift field. The coefficient sigma_eta^2/2 remains in the surrounding weak-FP action; 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.gaussianRealZeroSecondMoment
2. hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor
3. hFrozenScalarBrownianItoOneDimTaylorExpansion
4. SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator
5. ProbabilityTheory.integral_id_gaussianReal
6. ProbabilityTheory.variance_id_gaussianReal
7. ProbabilityTheory.variance_eq_integral
8. eq:general_moving_target_SALD_frozen_interp
9. appendix.tex:984-995
10. appendix.tex:1379-1387
11. Mathlib.Probability.Distributions.Gaussian.Real
12. 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_1 proof scout inside the scalar Brownian/Ito illness area. Consulted Mathlib Gaussian.Real for mean/variance of real Gaussian coordinates and the existing local Laplacian bridge; CovarianceBilin remains a future vector-covariance option, but the current one-dimensional leaf only needs the scalar centered second moment. The configured local SLT clone is absent, and no SLT theorem was imported or marked formalized.

def cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorScoutObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle161_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_one_dim_taylor_scout"
  statement := "Cycle 161 lower_1 proof-scout packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor should no longer treat the scalar Brownian second moment as open; SALD.gaussianRealZeroSecondMoment compiles the centered real Gaussian second moment needed for one Brownian coordinate. The remaining smaller source-cited theorem is the stochastic Taylor/generator passage hFrozenScalarBrownianItoOneDimTaylorExpansion : testRegular -> forall phi x i, the source-defined brownianCoordinateGenerator for the scalar increment sigma_eta (W_s-W_{s_k}) has derivative-per-unit-time contribution SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator (selectedTest phi) x ((stdOrthonormalBasis Real E) i), using selected-test C^2 regularity, zero first moment for the centered Brownian coordinate, the compiled second moment, and drift separation into the first-order conditional drift field. The coefficient sigma_eta^2/2 remains in the surrounding weak-FP action; hFrozenScalarBrownianItoEventFieldCoordinateSum remains explicit."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.gaussianRealZeroSecondMoment",
    "hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor",
    "hFrozenScalarBrownianItoOneDimTaylorExpansion",
    "SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator",
    "ProbabilityTheory.integral_id_gaussianReal",
    "ProbabilityTheory.variance_id_gaussianReal",
    "ProbabilityTheory.variance_eq_integral",
    "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_1 proof scout inside the scalar Brownian/Ito illness area. Consulted Mathlib Gaussian.Real for mean/variance of real Gaussian coordinates and the existing local Laplacian bridge; CovarianceBilin remains a future vector-covariance option, but the current one-dimensional leaf only needs the scalar centered second moment. The configured local SLT clone is absent, and no SLT theorem was imported or marked formalized."

/-- Cycle-161 lower_2 moment-algebra narrowing below the one-dimensional Taylor boundary. -/