AutoSamplingTheory.SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryMiddleObligation

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 cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryMiddleObligation :
    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.cycle85_conditional_kernel_boundary_middle

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 85 middle narrows the appendix.tex:1368-1377 conditional-kernel theorem boundary by compiling local Mathlib-backed helpers AutoSamplingTheory.condDistribAeEqCondExpKernelMap, AutoSamplingTheory.condDistribIntegralAEStronglyMeasurable, and AutoSamplingTheory.condDistribIntegralIntegrable. These orient X_k^eta | hat X_s=x as condDistrib X_k^eta hatX_s mu (hatX_s omega), identify it a.e. with condExpKernel mu (mState.comap hatX_s) mapped by X_k^eta, and supply vector-valued conditional-integral measurability/integrability from Mathlib finite-measure, standard-Borel, a.e.-measurability, and integrability hypotheses. Classification: narrows-source-cited-boundary. The remaining exact lower theorem is the source-specific state-field versioning theorem: from these sample-space a.e. condDistrib/condExpKernel integral facts and hatRhoS=Law(hatX_s), construct the named component fields condC_{k,s}, condScore_{k,s} as hat-rho_s-a.e. fields and feed SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents or SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff without supplied component-regularity hypotheses.

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.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.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryUpperObligation
2. AutoSamplingTheory.condDistribAeEqCondExpKernelMap
3. AutoSamplingTheory.condDistribIntegralAEStronglyMeasurable
4. AutoSamplingTheory.condDistribIntegralIntegrable
5. ProbabilityTheory.condDistrib_apply_ae_eq_condExpKernel_map
6. MeasureTheory.AEStronglyMeasurable.integral_condDistrib
7. MeasureTheory.Integrable.integral_condDistrib
8. Mathlib.Probability.Kernel.Condexp
9. Mathlib.Probability.Kernel.CondDistrib
10. SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface
11. SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation
12. SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation
13. SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityMiddleObligation
14. SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation
15. SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents
16. SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff
17. 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.

Middle boundary narrowing only. The orientation and conditional-integral regularity facts now compile locally, but the SALD-specific field-version theorem still has to identify hatRhoS as the conditioning marginal and turn sample-space a.e. integral formulas into named hat-rho_s-a.e. component fields. No weak FP, KL derivative, theorem contract, SLT status, or Lake dependency is promoted.

def cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryMiddleObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle85_conditional_kernel_boundary_middle"
  statement := "Cycle 85 middle narrows the appendix.tex:1368-1377 conditional-kernel theorem boundary by compiling local Mathlib-backed helpers AutoSamplingTheory.condDistribAeEqCondExpKernelMap, AutoSamplingTheory.condDistribIntegralAEStronglyMeasurable, and AutoSamplingTheory.condDistribIntegralIntegrable. These orient X_k^eta | hat X_s=x as condDistrib X_k^eta hatX_s mu (hatX_s omega), identify it a.e. with condExpKernel mu (mState.comap hatX_s) mapped by X_k^eta, and supply vector-valued conditional-integral measurability/integrability from Mathlib finite-measure, standard-Borel, a.e.-measurability, and integrability hypotheses. Classification: narrows-source-cited-boundary. The remaining exact lower theorem is the source-specific state-field versioning theorem: from these sample-space a.e. condDistrib/condExpKernel integral facts and hatRhoS=Law(hatX_s), construct the named component fields condC_{k,s}, condScore_{k,s} as hat-rho_s-a.e. fields and feed SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents or SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff without supplied component-regularity hypotheses."
  source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryUpperObligation",
    "AutoSamplingTheory.condDistribAeEqCondExpKernelMap",
    "AutoSamplingTheory.condDistribIntegralAEStronglyMeasurable",
    "AutoSamplingTheory.condDistribIntegralIntegrable",
    "ProbabilityTheory.condDistrib_apply_ae_eq_condExpKernel_map",
    "MeasureTheory.AEStronglyMeasurable.integral_condDistrib",
    "MeasureTheory.Integrable.integral_condDistrib",
    "Mathlib.Probability.Kernel.Condexp",
    "Mathlib.Probability.Kernel.CondDistrib",
    "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface",
    "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation",
    "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation",
    "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityMiddleObligation",
    "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation",
    "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents",
    "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "Middle boundary narrowing only. The orientation and conditional-integral regularity facts now compile locally, but the SALD-specific field-version theorem still has to identify hatRhoS as the conditioning marginal and turn sample-space a.e. integral formulas into named hat-rho_s-a.e. component fields. No weak FP, KL derivative, theorem contract, SLT status, or Lake dependency is promoted."

/-- Cycle-85 lower boundary reduction for named conditional-integral fields.

The lower packet compiles the law-space conditional-integral regularity facts
needed after the middle sample-space orientation work.  The theorem
`AutoSamplingTheory.condDistribIntegralNamedFieldRegularity` removes the
generic supplied component measurability/integrability hypotheses for any
component field that is chosen as a `hatRhoS`-a.e. version of the canonical
`condDistrib` integral.  The remaining analytic work is now the
source-specific conditional-expectation/version choice and the kernel
compatibility/disintegration theorem, not the Mathlib law-space regularity
of those integrals.
-/