AutoSamplingTheory.SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation

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 cycle63UnifiedDiscreteGeneralMeasureBackfillObligation :
    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.cycle63_joint_endpoint_law_backfill

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 63 backfills one narrow measure-theory detail for appendix.tex:1354-1387: AutoSamplingTheory.lawMapProdEqOfAEEq proves that componentwise almost-everywhere equality of two endpoint representatives implies equality of their paired Measure.map pushforward laws, AutoSamplingTheory.lawMapProdFst/Snd project the paired pushforward back to the two endpoint laws under explicit measurability hypotheses, and SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation plus SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation package the named EM endpoint identities into those helpers. This supports later common-space endpoint stitching for the general EM interpolation but does not construct conditional laws, densities, absolute continuity, weak Fokker-Planck, KL differentiation, or Gronwall regularity.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteDerivativeSource — 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.formalized — 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. AutoSamplingTheory.lawMapEqOfAEEq
2. AutoSamplingTheory.lawMapProdEqOfAEEq
3. AutoSamplingTheory.lawMapProdFst
4. AutoSamplingTheory.lawMapProdSnd
5. SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation
6. SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation
7. SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation
8. SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation
9. SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation
10. 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.

Formalized local Measure.map/a.e.-equality and measurable projection bookkeeping only. SLT Measure.map and a.e. patterns were used as reference style; no SLT theorem is imported or marked as a SALD analytic backend.

def cycle63UnifiedDiscreteGeneralMeasureBackfillObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle63_joint_endpoint_law_backfill"
  statement := "Cycle 63 backfills one narrow measure-theory detail for appendix.tex:1354-1387: AutoSamplingTheory.lawMapProdEqOfAEEq proves that componentwise almost-everywhere equality of two endpoint representatives implies equality of their paired Measure.map pushforward laws, AutoSamplingTheory.lawMapProdFst/Snd project the paired pushforward back to the two endpoint laws under explicit measurability hypotheses, and SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation plus SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation package the named EM endpoint identities into those helpers. This supports later common-space endpoint stitching for the general EM interpolation but does not construct conditional laws, densities, absolute continuity, weak Fokker-Planck, KL differentiation, or Gronwall regularity."
  source := saldGeneralMovingTargetDiscreteDerivativeSource
  status := ProofStatus.formalized
  dependsOn := [
    "AutoSamplingTheory.lawMapEqOfAEEq",
    "AutoSamplingTheory.lawMapProdEqOfAEEq",
    "AutoSamplingTheory.lawMapProdFst",
    "AutoSamplingTheory.lawMapProdSnd",
    "SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation",
    "SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation",
    "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation",
    "SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation",
    "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "Formalized local Measure.map/a.e.-equality and measurable projection bookkeeping only. SLT Measure.map and a.e. patterns were used as reference style; no SLT theorem is imported or marked as a SALD analytic backend."

/-- Cycle-63 middle audit for the unified/discrete general route.

This source-to-Lean synchronization layer checks the upper route and the local
paired endpoint-law backfill against the paper order, then narrows lower work
to the first conditional-law/Fokker--Planck interface that the discrete general
theorem still needs.
-/