AutoSamplingTheory.SALD.cycle15DiscreteForwardKlConditionalDriftDensityObligation

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 cycle15DiscreteForwardKlConditionalDriftDensityObligation : 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.conditional_drift_density

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 the appendix lines 347-354 input to the conditional Fokker--Planck slice: for the frozen EM pair (X_k^eta, hat X_s), construct the law/density interface and measurable conditional drift bar b_{k,s}(x)=E[nabla log pi_{t_k}(X_k^eta) | hat X_s=x] with enough integrability for div(hat rho_s*bar b_{k,s}).

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldForwardKlDiscreteConditionalFpSource — 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.cycle15DiscreteForwardKlConditionalDriftDensityContract
2. eq:frozen_interp_terminal_disc_prop_additive_final
3. SALD.discreteSaldEulerMaruyamaContract
4. sald.discrete_forward_kl.em_endpoint_laws

note : String (explicit)

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

This lower sub-obligation stops before the Fokker--Planck identity, Laplacian split, KL derivative integration by parts, one-step Gamma/Delta bound, DV, Gronwall, and accumulated-error algebra.

def cycle15DiscreteForwardKlConditionalDriftDensityObligation : ProofObligation where
  id := "sald.discrete_forward_kl.conditional_drift_density"
  statement := "Formalize the appendix lines 347-354 input to the conditional Fokker--Planck slice: for the frozen EM pair (X_k^eta, hat X_s), construct the law/density interface and measurable conditional drift bar b_{k,s}(x)=E[nabla log pi_{t_k}(X_k^eta) | hat X_s=x] with enough integrability for div(hat rho_s*bar b_{k,s})."
  source := saldForwardKlDiscreteConditionalFpSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle15DiscreteForwardKlConditionalDriftDensityContract",
    "eq:frozen_interp_terminal_disc_prop_additive_final",
    "SALD.discreteSaldEulerMaruyamaContract",
    "sald.discrete_forward_kl.em_endpoint_laws"
  ]
  note := "This lower sub-obligation stops before the Fokker--Planck identity, Laplacian split, KL derivative integration by parts, one-step Gamma/Delta bound, DV, Gronwall, and accumulated-error algebra."