AutoSamplingTheory.SALD.cycle55ForwardKlDerivativeMassLowerObligation

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 cycle55ForwardKlDerivativeMassLowerObligation : 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.forward_kl.cycle55_derivative_mass_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 55 lower compiles the first source-shaped scalar handoff in appendix.tex:168-185: after the KL derivative backend supplies the raw derivative split with the mass term, mass conservation supplies that term as zero, and the SALD Fokker-Planck/integration-by-parts backend supplies the first term as -FI, SALD.forwardKlMassConservationFirstTermFisherScalar derives the reduced derivative display dK = -FI + targetTerm. The analytic mass conservation, differentiation-under-integral, Fokker-Planck, boundary, and FI-identification facts remain obligations.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldForwardKlDerivativeSource — 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.cycle55ForwardKlSkeletonMiddleContract
SALD.cycle55ForwardKlSkeletonMiddleObligation
SALD.forwardKlDerivativeCandidateContract
SALD.forwardKlDerivativeSideConditionContract
SALD.forwardKlDerivativeObligation
SALD.forwardKlDensityBoundaryObligation
SALD.forwardKlMassConservationDropScalar
SALD.forwardKlMassConservationFirstTermFisherScalar
SALD.forwardKlFirstTermFisherSubstitutionScalar
sald.forward_kl.kl_derivative
sald.forward_kl.density_boundary_regular
eq:SALD
eq:FP-eq

note : String (explicit)

Recorded evidence/caveats; may distinguish a compiled scalar helper from still-open source analysis.

This is only theorem-independent Real equality bookkeeping for the first derivative slice. It does not close sald.forward_kl.kl_derivative, prove the Fokker-Planck equation, or promote thm:forward-KL.

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 cycle55ForwardKlDerivativeMassLowerObligation : ProofObligation where
  id := "sald.forward_kl.cycle55_derivative_mass_lower"
  statement := "Cycle 55 lower compiles the first source-shaped scalar handoff in appendix.tex:168-185: after the KL derivative backend supplies the raw derivative split with the mass term, mass conservation supplies that term as zero, and the SALD Fokker-Planck/integration-by-parts backend supplies the first term as -FI, SALD.forwardKlMassConservationFirstTermFisherScalar derives the reduced derivative display dK = -FI + targetTerm. The analytic mass conservation, differentiation-under-integral, Fokker-Planck, boundary, and FI-identification facts remain obligations."
  source := saldForwardKlDerivativeSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle55ForwardKlSkeletonMiddleContract",
    "SALD.cycle55ForwardKlSkeletonMiddleObligation",
    "SALD.forwardKlDerivativeCandidateContract",
    "SALD.forwardKlDerivativeSideConditionContract",
    "SALD.forwardKlDerivativeObligation",
    "SALD.forwardKlDensityBoundaryObligation",
    "SALD.forwardKlMassConservationDropScalar",
    "SALD.forwardKlMassConservationFirstTermFisherScalar",
    "SALD.forwardKlFirstTermFisherSubstitutionScalar",
    "sald.forward_kl.kl_derivative",
    "sald.forward_kl.density_boundary_regular",
    "eq:SALD",
    "eq:FP-eq"
  ]
  note := "This is only theorem-independent Real equality bookkeeping for the first derivative slice. It does not close sald.forward_kl.kl_derivative, prove the Fokker-Planck equation, or promote thm:forward-KL."

/-- Cycle-55 proof-DAG pane for the focused continuous forward-KL route. -/

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