AutoSamplingTheory.SALD.cycle30ForwardKlDensityBoundaryLowerObligation

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 cycle30ForwardKlDensityBoundaryLowerObligation : 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.cycle30_density_boundary_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.

Record the cycle 30 lower scalar substitution for appendix.tex lines 168-185: once mass conservation and differentiation under the KL integral give dK/ds as the first SALD term plus the target term, and the SALD Fokker-Planck/integration-by-parts backend identifies the first term as -FI(rho_s||tilde_pi_s), SALD.forwardKlFirstTermFisherSubstitutionScalar rewrites the derivative display as -FI plus the target term. Smooth densities, positivity, domination, mass conservation, Fokker-Planck, boundary/no-flux, and the FI identification 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

1. SALD.cycle30ForwardKlUpperPacket
2. SALD.cycle30ForwardKlMiddleContract
3. sald.forward_kl.cycle30_derivative_side_upper
4. sald.forward_kl.cycle30_derivative_side_middle
5. SALD.forwardKlDerivativeSideConditionContract
6. SALD.forwardKlDensityBoundaryObligation
7. SALD.forwardKlFirstTermFisherSubstitutionScalar
8. sald.forward_kl.density_boundary_regular
9. eq:FP-eq
10. KLContract
11. FIContract
12. FokkerPlanckContract

note : String (explicit)

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

Lower cycle 30 closes only theorem-independent Real equality substitution after the analytic derivative and first-term identities have been supplied. It does not prove the KL derivative, SALD Fokker-Planck equation, integration by parts, Fisher-information identity, target-side transport, LSI, DV, time change, or thm:forward-KL.

def cycle30ForwardKlDensityBoundaryLowerObligation : ProofObligation where
  id := "sald.forward_kl.cycle30_density_boundary_lower"
  statement := "Record the cycle 30 lower scalar substitution for appendix.tex lines 168-185: once mass conservation and differentiation under the KL integral give dK/ds as the first SALD term plus the target term, and the SALD Fokker-Planck/integration-by-parts backend identifies the first term as -FI(rho_s||tilde_pi_s), SALD.forwardKlFirstTermFisherSubstitutionScalar rewrites the derivative display as -FI plus the target term. Smooth densities, positivity, domination, mass conservation, Fokker-Planck, boundary/no-flux, and the FI identification remain obligations."
  source := saldForwardKlDerivativeSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle30ForwardKlUpperPacket",
    "SALD.cycle30ForwardKlMiddleContract",
    "sald.forward_kl.cycle30_derivative_side_upper",
    "sald.forward_kl.cycle30_derivative_side_middle",
    "SALD.forwardKlDerivativeSideConditionContract",
    "SALD.forwardKlDensityBoundaryObligation",
    "SALD.forwardKlFirstTermFisherSubstitutionScalar",
    "sald.forward_kl.density_boundary_regular",
    "eq:FP-eq",
    "KLContract",
    "FIContract",
    "FokkerPlanckContract"
  ]
  note := "Lower cycle 30 closes only theorem-independent Real equality substitution after the analytic derivative and first-term identities have been supplied. It does not prove the KL derivative, SALD Fokker-Planck equation, integration by parts, Fisher-information identity, target-side transport, LSI, DV, time change, or thm:forward-KL."

/-- Cycle-34 upper/lower scalar obligation for the derivative closure sprint. -/