AutoSamplingTheory.SALD.cycle45ForwardKlSkeletonObligation

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 cycle45ForwardKlSkeletonObligation : 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.cycle45_theorem_skeleton_route

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 45 upper wires the source-cited analytic interfaces into thm:forward-KL: the continuous KL derivative/Fokker-Planck backend yields the pre-DV K'(t) inequality; eq:LSI-KL-FI supplies the LSI contraction coefficient; lem:dv_variation with the alpha-complexity finite-log-mgf witness supplies the velocity-energy bound; lem:gronwall with endpoint schedule identities and coefficient side conditions yields exactly the main_body.tex:243-246 display. The EM interpolation Fokker-Planck interface remains checked as the downstream discrete slow backend but is not promoted or used to alter the continuous theorem.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldForwardKlProofSource — 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.cycle44MainSkeletonAnalyticInterfaceObligation
2. SALD.cycle45ForwardKlSkeletonUpperPacket
3. SALD.continuousForwardKlStatementContract
4. SALD.forwardKlMiddleSourceToLeanMapObligation
5. SALD.forwardKlDerivativeCandidateContract
6. SALD.forwardKlDerivativeSideConditionContract
7. sald.forward_kl.density_boundary_regular
8. sald.forward_kl.schedule_time_change
9. sald.forward_kl.kl_derivative
10. SALD.saldLsiKlFiDensityTestContract
11. probability.lsi_to_kl_fi
12. SALD.forwardKlDvFiniteLogMgfWitnessContract
13. sald.forward_kl.dv_finite_log_mgf_witness
14. sald.forward_kl.dv_energy_bound
15. SALD.forwardKlGronwallInstantiationContract
16. sald.forward_kl.gronwall_application
17. sald.forward_kl.endpoint_schedule_identities
18. sald.forward_kl.gronwall_side_conditions
19. SALD.discreteForwardKlEmInterpolationSideConditionContract

note : String (explicit)

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

This is a theorem-route wrapper for main skeleton sprint 2. It keeps Gronwall, DV, LSI/KL/FI, continuous KL derivative, and EM interpolation backends below formalized status and does not change the paper theorem display.

def cycle45ForwardKlSkeletonObligation : ProofObligation where
  id := "sald.forward_kl.cycle45_theorem_skeleton_route"
  statement := "Cycle 45 upper wires the source-cited analytic interfaces into thm:forward-KL: the continuous KL derivative/Fokker-Planck backend yields the pre-DV K'(t) inequality; eq:LSI-KL-FI supplies the LSI contraction coefficient; lem:dv_variation with the alpha-complexity finite-log-mgf witness supplies the velocity-energy bound; lem:gronwall with endpoint schedule identities and coefficient side conditions yields exactly the main_body.tex:243-246 display. The EM interpolation Fokker-Planck interface remains checked as the downstream discrete slow backend but is not promoted or used to alter the continuous theorem."
  source := saldForwardKlProofSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle44MainSkeletonAnalyticInterfaceObligation",
    "SALD.cycle45ForwardKlSkeletonUpperPacket",
    "SALD.continuousForwardKlStatementContract",
    "SALD.forwardKlMiddleSourceToLeanMapObligation",
    "SALD.forwardKlDerivativeCandidateContract",
    "SALD.forwardKlDerivativeSideConditionContract",
    "sald.forward_kl.density_boundary_regular",
    "sald.forward_kl.schedule_time_change",
    "sald.forward_kl.kl_derivative",
    "SALD.saldLsiKlFiDensityTestContract",
    "probability.lsi_to_kl_fi",
    "SALD.forwardKlDvFiniteLogMgfWitnessContract",
    "sald.forward_kl.dv_finite_log_mgf_witness",
    "sald.forward_kl.dv_energy_bound",
    "SALD.forwardKlGronwallInstantiationContract",
    "sald.forward_kl.gronwall_application",
    "sald.forward_kl.endpoint_schedule_identities",
    "sald.forward_kl.gronwall_side_conditions",
    "SALD.discreteForwardKlEmInterpolationSideConditionContract"
  ]
  note := "This is a theorem-route wrapper for main skeleton sprint 2. It keeps Gronwall, DV, LSI/KL/FI, continuous KL derivative, and EM interpolation backends below formalized status and does not change the paper theorem display."

/-- Cycle-45 proof-DAG pane for the continuous forward-KL theorem route. -/