AutoSamplingTheory.SALD.cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftLowerObligation

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 cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftLowerObligation :
    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.cycle106_canonical_condDistrib_drift_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 106 lower compiles SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity for appendix.tex lines 1368-1377. The theorem proves AEStronglyMeasurable and Integrable for the canonical conditional drift field x |-> dotTk • ∫ guideIntegrand (x,y) d condDistrib Xk hatXAtS P x + sigmaCoeff • ∫ scoreIntegrand (x,y) d condDistrib Xk hatXAtS P x, using hatRhoS = Measure.map hatXAtS P and the local Mathlib wrappers AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable and AutoSamplingTheory.condDistribIntegralNamedLawIntegrable. The companion SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq proves that any named barB representative a.e. equal to this canonical field inherits the same regularity, so the remaining named-representative boundary is only the source-specific hatRhoS-a.e. equality. Classification: discharges-supplied-hypothesis for the old canonical component conditional-integral regularity premise and narrows-source-cited-boundary for the named barB versioning side condition. Weak FP, no-boundary, diffusion, KL/log-ratio, LSI, DV, and Gronwall remain separate.

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.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity
2. SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq
3. AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable
4. AutoSamplingTheory.condDistribIntegralNamedLawIntegrable
5. appendix.tex:1368-1377
6. 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.

Proof-producing local theorem only. It does not construct the EM process, prove the full regular conditional-law theorem for an arbitrary named representative, prove weak Fokker-Planck source signs, close no-boundary identities, promote theorem status, import SLT, or change Lake dependencies.

def cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftLowerObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle106_canonical_condDistrib_drift_lower"
  statement := "Cycle 106 lower compiles SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity for appendix.tex lines 1368-1377. The theorem proves AEStronglyMeasurable and Integrable for the canonical conditional drift field x |-> dotTk • ∫ guideIntegrand (x,y) d condDistrib Xk hatXAtS P x + sigmaCoeff • ∫ scoreIntegrand (x,y) d condDistrib Xk hatXAtS P x, using hatRhoS = Measure.map hatXAtS P and the local Mathlib wrappers AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable and AutoSamplingTheory.condDistribIntegralNamedLawIntegrable. The companion SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq proves that any named barB representative a.e. equal to this canonical field inherits the same regularity, so the remaining named-representative boundary is only the source-specific hatRhoS-a.e. equality. Classification: discharges-supplied-hypothesis for the old canonical component conditional-integral regularity premise and narrows-source-cited-boundary for the named barB versioning side condition. Weak FP, no-boundary, diffusion, KL/log-ratio, LSI, DV, and Gronwall remain separate."
  source := saldGeneralMovingTargetDiscreteConditionalDriftSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity",
    "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq",
    "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable",
    "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable",
    "appendix.tex:1368-1377",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "Proof-producing local theorem only. It does not construct the EM process, prove the full regular conditional-law theorem for an arbitrary named representative, prove weak Fokker-Planck source signs, close no-boundary identities, promote theorem status, import SLT, or change Lake dependencies."

/-- Cycle-106 proof-DAG pane for canonical conditional-integral drift
regularity. -/