AutoSamplingTheory.SALD.unifiedForwardKlSpecializationObligation

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 unifiedForwardKlSpecializationObligation : 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.unified_forward_kl.specialization

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 949-951 and main_body.tex lines 359-395: specialize thm:general-moving-target-SALD by setting c_t=u_t, use prop:guided_path_residual and eq:poisson-eq to obtain the transport velocity v_t=u_t+w_t for pi_t, identify m_t=v_t-c_t=w_t, and preserve the sigma-weighted VA-SALD bound.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldUnifiedForwardKlSource — 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.unifiedForwardKlSpecializationContract
2. sald.unified_forward_kl.transport_bridge_middle
3. sald.unified_forward_kl.transport_velocity_bridge
4. sald.guided_path_residual.identity
5. eq:poisson-eq
6. eq:SALD_Ito
7. sald.general_moving_target.gronwall_application
8. sald.general_moving_target.gronwall_side_conditions
9. sald.general_moving_target.pure_contraction

note : String (explicit)

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

unifiedForwardKlSpecializationContract records the source bridge from the guided residual and correction-field equation to the general theorem specialization. The proof is a specialization only; no new path-space or direct SALD argument should be substituted.

def unifiedForwardKlSpecializationObligation : ProofObligation where
  id := "sald.unified_forward_kl.specialization"
  statement := "Formalize appendix lines 949-951 and main_body.tex lines 359-395: specialize thm:general-moving-target-SALD by setting c_t=u_t, use prop:guided_path_residual and eq:poisson-eq to obtain the transport velocity v_t=u_t+w_t for pi_t, identify m_t=v_t-c_t=w_t, and preserve the sigma-weighted VA-SALD bound."
  source := saldUnifiedForwardKlSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.unifiedForwardKlSpecializationContract",
    "sald.unified_forward_kl.transport_bridge_middle",
    "sald.unified_forward_kl.transport_velocity_bridge",
    "sald.guided_path_residual.identity",
    "eq:poisson-eq",
    "eq:SALD_Ito",
    "sald.general_moving_target.gronwall_application",
    "sald.general_moving_target.gronwall_side_conditions",
    "sald.general_moving_target.pure_contraction"
  ]
  note := "unifiedForwardKlSpecializationContract records the source bridge from the guided residual and correction-field equation to the general theorem specialization.  The proof is a specialization only; no new path-space or direct SALD argument should be substituted."