AutoSamplingTheory.SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation

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.

Lean statement of this data definition

The part after the colon is the output data type. This declaration takes no mathematical proof inputs.

def cycle76GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation :
    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.cycle76_endpoint_conditional_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 76 middle synchronizes appendix.tex:1358-1387 by connecting the endpoint law statements hat rho_{s_k}=rho_k^eta and hat rho_{s_{k+1}}=rho_{k+1}^eta with the conditional drift definition of bar b_{k,s}. The lower wrapper should consume existing endpoint Measure.map handoffs, the named hat rho_s=Law(hat X_s) representation for the interior time s, and the cycle-75 swapped condDistrib orientation, producing original-orientation kernel compatibility for the weak Fokker-Planck statement without constructing the conditional law.

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.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

SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalUpperObligation
SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation
SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap
SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents
SALD.generalMovingTargetDiscreteEndpointMeasureMapToSwappedConditionalCompatibility
SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract
SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract
SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface
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 synchronization only. Endpoint laws, swapped marginal compatibility, and original-orientation kernel compatibility are aligned; standard-Borel, finite/probability, AEMeasurable, Integrable, conditional expectation, density/AC, weak FP, and KL differentiation remain explicit obligations.

Exact Lean data construction

Each field assignment stores the corresponding value shown above. Omitted fields use the explicitly identified schema defaults. Strings that name theorems remain strings; they do not call those theorems.

def cycle76GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle76_endpoint_conditional_middle"
  statement := "Cycle 76 middle synchronizes appendix.tex:1358-1387 by connecting the endpoint law statements hat rho_{s_k}=rho_k^eta and hat rho_{s_{k+1}}=rho_{k+1}^eta with the conditional drift definition of bar b_{k,s}. The lower wrapper should consume existing endpoint Measure.map handoffs, the named hat rho_s=Law(hat X_s) representation for the interior time s, and the cycle-75 swapped condDistrib orientation, producing original-orientation kernel compatibility for the weak Fokker-Planck statement without constructing the conditional law."
  source := saldGeneralMovingTargetDiscreteDerivativeSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalUpperObligation",
    "SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation",
    "SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap",
    "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents",
    "SALD.generalMovingTargetDiscreteEndpointMeasureMapToSwappedConditionalCompatibility",
    "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract",
    "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract",
    "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "Middle synchronization only. Endpoint laws, swapped marginal compatibility, and original-orientation kernel compatibility are aligned; standard-Borel, finite/probability, AEMeasurable, Integrable, conditional expectation, density/AC, weak FP, and KL differentiation remain explicit obligations."

/-- Cycle-76 lower obligation for the endpoint Measure.map to swapped
conditional-kernel wrapper. -/

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