AutoSamplingTheory.SALD.generalMovingTargetDerivativeObligation

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 generalMovingTargetDerivativeObligation : 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.general_moving_target.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 765-884: differentiate KL(rho_s||pi_{t(s)}), insert the general VA-SALD Fokker--Planck equation, combine c_t with the target transport velocity v_t to expose m_t=v_t-c_t, apply Young's inequality with epsilon=2*dot{t}(s)/sigma_{t(s)}^2, change from s to t, and apply LSI.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDerivativeSource — 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. eq:general_moving_target_SALD
2. TransportVelocityContract
3. FokkerPlanckContract
4. KLContract
5. FIContract
6. probability.lsi_to_kl_fi
7. SALD.generalMovingTargetKlDerivativeResidualSplitScalar
8. SALD.generalMovingTargetKlDerivativeScaledResidualDisplayScalar
9. SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar
10. SALD.generalMovingTargetPostYoungDerivativeBoundScalar
11. SALD.generalMovingTargetLsiDerivativeBoundScalar
12. SALD.generalMovingTargetTimeChangedDerivativeBoundScalar
13. SALD.generalMovingTargetPreDvDerivativeBoundScalar
14. sald.forward_kl.density_boundary_regular
15. sald.forward_kl.schedule_time_change

note : String (explicit)

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

generalMovingTargetDerivativeCandidateContract records the sigma-weighted derivative route and its implicit regularity gaps before proof search. Cycle 47 lower adds compiled scalar/order handoffs for residual Young, LSI substitution, and inverse-schedule coefficient rewriting; cycle 57 lower adds the raw KL-derivative residual split before those handoffs; cycle 62 lower adds the dot{t}(s)-scaled residual sign handoff for the target-transport term. The Fokker-Planck, mass conservation, integration-by-parts, target transport, LSI density-test, and schedule backends remain obligations.

def generalMovingTargetDerivativeObligation : ProofObligation where
  id := "sald.general_moving_target.kl_derivative"
  statement := "Formalize appendix lines 765-884: differentiate KL(rho_s||pi_{t(s)}), insert the general VA-SALD Fokker--Planck equation, combine c_t with the target transport velocity v_t to expose m_t=v_t-c_t, apply Young's inequality with epsilon=2*dot{t}(s)/sigma_{t(s)}^2, change from s to t, and apply LSI."
  source := saldGeneralMovingTargetDerivativeSource
  status := ProofStatus.obligation
  dependsOn := [
    "eq:general_moving_target_SALD",
    "TransportVelocityContract",
    "FokkerPlanckContract",
    "KLContract",
      "FIContract",
      "probability.lsi_to_kl_fi",
      "SALD.generalMovingTargetKlDerivativeResidualSplitScalar",
      "SALD.generalMovingTargetKlDerivativeScaledResidualDisplayScalar",
      "SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar",
      "SALD.generalMovingTargetPostYoungDerivativeBoundScalar",
      "SALD.generalMovingTargetLsiDerivativeBoundScalar",
      "SALD.generalMovingTargetTimeChangedDerivativeBoundScalar",
      "SALD.generalMovingTargetPreDvDerivativeBoundScalar",
      "sald.forward_kl.density_boundary_regular",
      "sald.forward_kl.schedule_time_change"
    ]
  note := "generalMovingTargetDerivativeCandidateContract records the sigma-weighted derivative route and its implicit regularity gaps before proof search. Cycle 47 lower adds compiled scalar/order handoffs for residual Young, LSI substitution, and inverse-schedule coefficient rewriting; cycle 57 lower adds the raw KL-derivative residual split before those handoffs; cycle 62 lower adds the dot{t}(s)-scaled residual sign handoff for the target-transport term. The Fokker-Planck, mass conservation, integration-by-parts, target transport, LSI density-test, and schedule backends remain obligations."