AutoSamplingTheory.SALD.cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeLowerObligation

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 cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeLowerObligation :
    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.cycle112_named_barB_condExp_representative_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 112 lower compiles SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq, SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef, SALD.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents, and SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef for appendix.tex:1368-1377. Classification: narrows-source-cited-boundary. The first theorem derives the old hbarBCondExp premise from the conditional-expectation uniqueness criterion: guide/score integrability gives integrability of the frozen drift, while the selected barB representative supplies a.e. measurability, integrability, and matching set integrals on all hatXAtS-measurable finite-measure sets. The state-event bridge narrows that set-integral condition to measurable state events {omega | hatXAtS omega in t}, matching the source conditioning on hat X_s=x. The downstream theorems plug the derived hbarBCondExp into the existing cycle-109 product-condExp source bridge, so the canonical condDistrib equality no longer needs hbarBCondExp as a primitive premise. Remaining exact boundary: prove candidate regularity plus the source state-event set-integral characterization for the selected paper representative barB(hatXAtS omega).

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteConditionalDriftSource — 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.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq
2. SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef
3. SALD.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents
4. SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef
5. SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef
6. SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable
7. MeasureTheory.ae_eq_condExp_of_forall_setIntegral_eq
8. MeasurableSpace.measurableSet_comap
9. MeasureTheory.Integrable.comp_aemeasurable
10. ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib
11. appendix.tex:1368-1377
12. sald.general_moving_target_discrete.em_interpolation_fp
13. sald.discrete_forward_kl.em_interpolation_fp

note : String (explicit)

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

Proof-producing boundary narrowing only. It does not prove the state-event set-integral characterization, conditional law construction, weak Fokker-Planck equation, no-boundary theorem, diffusion source action, KL/log-ratio derivative, theorem closure, SLT reuse, or any Lake dependency.

def cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeLowerObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle112_named_barB_condExp_representative_lower"
  statement := "Cycle 112 lower compiles SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq, SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef, SALD.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents, and SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef for appendix.tex:1368-1377. Classification: narrows-source-cited-boundary. The first theorem derives the old hbarBCondExp premise from the conditional-expectation uniqueness criterion: guide/score integrability gives integrability of the frozen drift, while the selected barB representative supplies a.e. measurability, integrability, and matching set integrals on all hatXAtS-measurable finite-measure sets. The state-event bridge narrows that set-integral condition to measurable state events {omega | hatXAtS omega in t}, matching the source conditioning on hat X_s=x. The downstream theorems plug the derived hbarBCondExp into the existing cycle-109 product-condExp source bridge, so the canonical condDistrib equality no longer needs hbarBCondExp as a primitive premise. Remaining exact boundary: prove candidate regularity plus the source state-event set-integral characterization for the selected paper representative barB(hatXAtS omega)."
  source := saldGeneralMovingTargetDiscreteConditionalDriftSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq",
    "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef",
    "SALD.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents",
    "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef",
    "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef",
    "SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable",
    "MeasureTheory.ae_eq_condExp_of_forall_setIntegral_eq",
    "MeasurableSpace.measurableSet_comap",
    "MeasureTheory.Integrable.comp_aemeasurable",
    "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib",
    "appendix.tex:1368-1377",
    "sald.general_moving_target_discrete.em_interpolation_fp",
    "sald.discrete_forward_kl.em_interpolation_fp"
  ]
  note := "Proof-producing boundary narrowing only. It does not prove the state-event set-integral characterization, conditional law construction, weak Fokker-Planck equation, no-boundary theorem, diffusion source action, KL/log-ratio derivative, theorem closure, SLT reuse, or any Lake dependency."

/-- Cycle-112 proof-DAG pane for the named `barB` conditional-expectation
representative boundary. -/