AutoSamplingTheory.SALD.discreteForwardKlDerivativeObligation

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 discreteForwardKlDerivativeObligation : 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.discrete_forward_kl.kl_derivative

statement : String (explicit)

Desired mathematical or workflow content as String; it is not a proposition in Prop and the def does not prove it.

Formalize appendix lines 334-491: differentiate KL(hat rho_s||tilde pi_s), use the EM interpolation Fokker--Planck equation, transport identity for tilde pi_s, the frozen-defect bound, Young, and LSI to obtain the source pre-DV inequality with the remaining ||tilde v_s||^2 term.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldForwardKlDiscreteDerivativeSource — 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.discrete_forward_kl.em_interpolation_fp
2. sald.discrete_forward_kl.em_conditional_fokker_planck
3. sald.discrete_forward_kl.frozen_delta_cross_lip
4. sald.forward_kl.density_boundary_regular
5. sald.forward_kl.schedule_time_change
6. probability.lsi_to_kl_fi
7. SALD.discreteForwardKlPostLsiDerivativeBoundScalar
8. SALD.discreteForwardKlPostLsiDerivativeBoundOfKlFiScalar

note : String (explicit)

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

This is the discrete analogue of the continuous KL derivative backend up to the moving-velocity norm. Cycle 51 lower compiles only the scalar handoff from supplied EM-FP derivative, frozen-cross, moving Young, and LSI inputs to the source pre-DV inequality; the separate discreteForwardKlDvFiniteLogMgfWitnessObligation and discreteForwardKlDvVelocityObligation cover appendix lines 493-523.

def discreteForwardKlDerivativeObligation : ProofObligation where
  id := "sald.discrete_forward_kl.kl_derivative"
  statement := "Formalize appendix lines 334-491: differentiate KL(hat rho_s||tilde pi_s), use the EM interpolation Fokker--Planck equation, transport identity for tilde pi_s, the frozen-defect bound, Young, and LSI to obtain the source pre-DV inequality with the remaining ||tilde v_s||^2 term."
  source := saldForwardKlDiscreteDerivativeSource
  status := ProofStatus.obligation
  dependsOn := ["sald.discrete_forward_kl.em_interpolation_fp", "sald.discrete_forward_kl.em_conditional_fokker_planck", "sald.discrete_forward_kl.frozen_delta_cross_lip", "sald.forward_kl.density_boundary_regular", "sald.forward_kl.schedule_time_change", "probability.lsi_to_kl_fi", "SALD.discreteForwardKlPostLsiDerivativeBoundScalar", "SALD.discreteForwardKlPostLsiDerivativeBoundOfKlFiScalar"]
  note := "This is the discrete analogue of the continuous KL derivative backend up to the moving-velocity norm. Cycle 51 lower compiles only the scalar handoff from supplied EM-FP derivative, frozen-cross, moving Young, and LSI inputs to the source pre-DV inequality; the separate discreteForwardKlDvFiniteLogMgfWitnessObligation and discreteForwardKlDvVelocityObligation cover appendix lines 493-523."