AutoSamplingTheory.SALD.cycle43LsiKlFiLowerObligation

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 cycle43LsiKlFiLowerObligation : 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.lsi_kl_fi.cycle43_lower_integral_fisher_chain

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 43 lower compiles a finite-coordinate integral Fisher-chain handoff for main_body.tex lines 208-215: if r>0 almost everywhere and the coordinate derivative identities for sqrt(r) and log(r) hold almost everywhere, then the integral of the squared sqrt-density coordinate derivatives equals the integral of (1/4)*r times the squared log-density coordinate derivatives; after supplied finite-coordinate Dirichlet and Fisher identifications, this yields the exact dirichlet=(1/4)*FI scalar input consumed by the existing LSI/KL/FI bridge. This still leaves vector-gradient equivalence, zero-density Sobolev handling, smooth/admissible sqrt-test approximation, finite KL/FI theorem interfaces, and full probability.lsi_to_kl_fi open.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldKlFiLsiSource — 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.cycle43LsiKlFiUpperPacket
2. SALD.cycle43LsiKlFiMiddleObligation
3. SALD.saldLsiKlFiDensityTestContract
4. SALD.lsiKlFiDensityTestObligation
5. AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar
6. AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainOfDerivativesScalar
7. AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumScalar
8. AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumHandoffScalar
9. AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralFiniteSum
10. AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralHandoffScalar
11. SALD.lsiKlFiDensityTestBridgeScalar
12. SALD.lsiKlFiDensityTestHalfFisherScalar
13. probability.lsi_to_kl_fi

note : String (explicit)

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

This is a proof-producing lower slice for the Fisher side of the source density-test bridge only. eq:LSI-KL-FI remains an obligation because the full Sobolev/vector-gradient backend and sqrt-test admissibility/approximation are not closed.

def cycle43LsiKlFiLowerObligation : ProofObligation where
  id := "sald.lsi_kl_fi.cycle43_lower_integral_fisher_chain"
  statement := "Cycle 43 lower compiles a finite-coordinate integral Fisher-chain handoff for main_body.tex lines 208-215: if r>0 almost everywhere and the coordinate derivative identities for sqrt(r) and log(r) hold almost everywhere, then the integral of the squared sqrt-density coordinate derivatives equals the integral of (1/4)*r times the squared log-density coordinate derivatives; after supplied finite-coordinate Dirichlet and Fisher identifications, this yields the exact dirichlet=(1/4)*FI scalar input consumed by the existing LSI/KL/FI bridge. This still leaves vector-gradient equivalence, zero-density Sobolev handling, smooth/admissible sqrt-test approximation, finite KL/FI theorem interfaces, and full probability.lsi_to_kl_fi open."
  source := saldKlFiLsiSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle43LsiKlFiUpperPacket",
    "SALD.cycle43LsiKlFiMiddleObligation",
    "SALD.saldLsiKlFiDensityTestContract",
    "SALD.lsiKlFiDensityTestObligation",
    "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar",
    "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainOfDerivativesScalar",
    "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumScalar",
    "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumHandoffScalar",
    "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralFiniteSum",
    "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralHandoffScalar",
    "SALD.lsiKlFiDensityTestBridgeScalar",
    "SALD.lsiKlFiDensityTestHalfFisherScalar",
    "probability.lsi_to_kl_fi"
  ]
  note := "This is a proof-producing lower slice for the Fisher side of the source density-test bridge only. eq:LSI-KL-FI remains an obligation because the full Sobolev/vector-gradient backend and sqrt-test admissibility/approximation are not closed."

/-- Cycle-44 upper ledger for the main SALD theorem-skeleton sprint.

The cycle focus is not another isolated scalar lemma.  This packet checks the
five slow analytic interfaces, keeps their unproved backends below formalized
status, and wires them into the theorem route requested by the task contract.
-/