AutoSamplingTheory.SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefBoundary

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 cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefBoundary :
    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.cycle109_named_barB_source_def_boundary

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 109 records the exact missing theorem behind appendix.tex:1368-1377. Classification: narrows-source-cited-boundary. Proposed theorem boundary: prove the source-definition version of SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpKernelSourceDef from Mathlib conditional-kernel ingredients, with imports Mathlib.Probability.Kernel.CondDistrib, Mathlib.Probability.Kernel.Condexp, Mathlib.MeasureTheory.Integral.Bochner.Basic, AutoSamplingTheory.Probability, and AutoSamplingTheory.SALD. Hypotheses: a finite/probability common law P on a standard-Borel sample space, standard-Borel state space, measurable or a.e.-measurable hatXAtS and XkEta, hhatRhoS : hatRhoS = Measure.map hatXAtS P, the guide and score integrands for dot t_k c_{t_k} and (sigma_eta^2/2) nabla log pi_{t_k}, equality-set measurability for the canonical guide-plus-score field, the measure-valued a.e. kernel alignment condDistrib XkEta hatXAtS P (hatXAtS omega) = condExpKernel P (mState.comap hatXAtS).map XkEta omega, and the paper source definition of barB as the corresponding condExpKernel.map conditional expectation. Conclusion: barB is hatRhoS-a.e. equal to the canonical condDistrib guide-plus-score field consumed by SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource — 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.sourceCited — 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.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpKernelSourceDef
2. SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq
3. SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity
4. AutoSamplingTheory.condDistribAeEqCondExpKernelMap
5. AutoSamplingTheory.condDistribIntegralSampleAeEqOfCondExpKernelMap
6. ProbabilityTheory.condDistrib_apply_ae_eq_condExpKernel_map
7. ProbabilityTheory.condExp_ae_eq_integral_condExpKernel
8. ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib
9. Mathlib.Probability.Kernel.CondDistrib
10. Mathlib.Probability.Kernel.Condexp
11. Mathlib.MeasureTheory.Integral.Bochner.Basic
12. appendix.tex:1368-1377
13. 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.

This is a source-cited theorem boundary, not a broad wrapper. It does not promote the conditional law, weak FP, KL/log-ratio, no-boundary, LSI, DV, Gronwall, theorem status, SLT reuse status, or Lake dependencies.

def cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefBoundary :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle109_named_barB_source_def_boundary"
  statement := "Cycle 109 records the exact missing theorem behind appendix.tex:1368-1377. Classification: narrows-source-cited-boundary. Proposed theorem boundary: prove the source-definition version of SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpKernelSourceDef from Mathlib conditional-kernel ingredients, with imports Mathlib.Probability.Kernel.CondDistrib, Mathlib.Probability.Kernel.Condexp, Mathlib.MeasureTheory.Integral.Bochner.Basic, AutoSamplingTheory.Probability, and AutoSamplingTheory.SALD. Hypotheses: a finite/probability common law P on a standard-Borel sample space, standard-Borel state space, measurable or a.e.-measurable hatXAtS and XkEta, hhatRhoS : hatRhoS = Measure.map hatXAtS P, the guide and score integrands for dot t_k c_{t_k} and (sigma_eta^2/2) nabla log pi_{t_k}, equality-set measurability for the canonical guide-plus-score field, the measure-valued a.e. kernel alignment condDistrib XkEta hatXAtS P (hatXAtS omega) = condExpKernel P (mState.comap hatXAtS).map XkEta omega, and the paper source definition of barB as the corresponding condExpKernel.map conditional expectation. Conclusion: barB is hatRhoS-a.e. equal to the canonical condDistrib guide-plus-score field consumed by SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq."
  source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource
  status := ProofStatus.sourceCited
  dependsOn := [
    "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpKernelSourceDef",
    "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq",
    "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity",
    "AutoSamplingTheory.condDistribAeEqCondExpKernelMap",
    "AutoSamplingTheory.condDistribIntegralSampleAeEqOfCondExpKernelMap",
    "ProbabilityTheory.condDistrib_apply_ae_eq_condExpKernel_map",
    "ProbabilityTheory.condExp_ae_eq_integral_condExpKernel",
    "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib",
    "Mathlib.Probability.Kernel.CondDistrib",
    "Mathlib.Probability.Kernel.Condexp",
    "Mathlib.MeasureTheory.Integral.Bochner.Basic",
    "appendix.tex:1368-1377",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "This is a source-cited theorem boundary, not a broad wrapper. It does not promote the conditional law, weak FP, KL/log-ratio, no-boundary, LSI, DV, Gronwall, theorem status, SLT reuse status, or Lake dependencies."

/-- Cycle-109 middle packet for the named `barB` source-definition bridge. -/