AutoSamplingTheory.SALD.cycle65ForwardKlDerivativePointwiseLowerObligation

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 cycle65ForwardKlDerivativePointwiseLowerObligation : 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.cycle65_derivative_pointwise_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.

Cycle 65 lower compiles the pointwise wrapper for appendix.tex:168-228: after the analytic backend supplies the raw KL derivative split, mass conservation, SALD Fokker-Planck/integration-by-parts first-term identity, target Cauchy estimate, LSI/KL/FI comparison, slowed-velocity scaling, and inverse-schedule product identity at each t, SALD.forwardKlPointwisePreDvDerivativeBoundOfRawKlFiVelocityScaling derives the pointwise t-time pre-DV inequality K'(t) <= -dot{s}(t)*C_LSI(t)*K(t)+(1/2)*dot{s}(t)^(-1)*||v_t||^2.

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.cycle65ForwardKlSkeletonMiddleContract
2. SALD.cycle65ForwardKlSkeletonMiddleObligation
3. SALD.cycle60ForwardKlDerivativeRawLowerObligation
4. SALD.forwardKlPointwisePreDvDerivativeBoundOfRawKlFiVelocityScaling
5. SALD.forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar
6. SALD.forwardKlDerivativeCandidateContract
7. SALD.forwardKlDerivativeSideConditionContract
8. SALD.forwardKlDerivativeObligation
9. SALD.forwardKlDensityBoundaryObligation
10. SALD.forwardKlScheduleTimeChangeObligation
11. SALD.saldLsiKlFiDensityTestContract
12. probability.lsi_to_kl_fi
13. sald.forward_kl.density_boundary_regular
14. sald.forward_kl.schedule_time_change
15. sald.forward_kl.kl_derivative

note : String (explicit)

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

The new Lean theorem is a pointwise Real/order wrapper only. It does not prove mass conservation, differentiation under the KL integral, Fokker-Planck, integration by parts, target transport, Cauchy-Schwarz, LSI/KL/FI, inverse-function calculus, DV, Gronwall, or thm:forward-KL.

def cycle65ForwardKlDerivativePointwiseLowerObligation : ProofObligation where
  id := "sald.forward_kl.cycle65_derivative_pointwise_lower"
  statement := "Cycle 65 lower compiles the pointwise wrapper for appendix.tex:168-228: after the analytic backend supplies the raw KL derivative split, mass conservation, SALD Fokker-Planck/integration-by-parts first-term identity, target Cauchy estimate, LSI/KL/FI comparison, slowed-velocity scaling, and inverse-schedule product identity at each t, SALD.forwardKlPointwisePreDvDerivativeBoundOfRawKlFiVelocityScaling derives the pointwise t-time pre-DV inequality K'(t) <= -dot{s}(t)*C_LSI(t)*K(t)+(1/2)*dot{s}(t)^(-1)*||v_t||^2."
  source := saldForwardKlDerivativeSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle65ForwardKlSkeletonMiddleContract",
    "SALD.cycle65ForwardKlSkeletonMiddleObligation",
    "SALD.cycle60ForwardKlDerivativeRawLowerObligation",
    "SALD.forwardKlPointwisePreDvDerivativeBoundOfRawKlFiVelocityScaling",
    "SALD.forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar",
    "SALD.forwardKlDerivativeCandidateContract",
    "SALD.forwardKlDerivativeSideConditionContract",
    "SALD.forwardKlDerivativeObligation",
    "SALD.forwardKlDensityBoundaryObligation",
    "SALD.forwardKlScheduleTimeChangeObligation",
    "SALD.saldLsiKlFiDensityTestContract",
    "probability.lsi_to_kl_fi",
    "sald.forward_kl.density_boundary_regular",
    "sald.forward_kl.schedule_time_change",
    "sald.forward_kl.kl_derivative"
  ]
  note := "The new Lean theorem is a pointwise Real/order wrapper only. It does not prove mass conservation, differentiation under the KL integral, Fokker-Planck, integration by parts, target transport, Cauchy-Schwarz, LSI/KL/FI, inverse-function calculus, DV, Gronwall, or thm:forward-KL."

/-- Cycle-65 proof-DAG pane for the post-cycle-64 continuous forward-KL
skeleton route. -/