AutoSamplingTheory.SALD.cycle38LsiKlFiMiddleObligation

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 cycle38LsiKlFiMiddleObligation : 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.cycle38_middle_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.

Translate main_body.tex lines 208-215 beyond the cycle-33 sqrt-density scalar helpers: for positive density ratio r, AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar proves the local 1/4 Fisher-integrand coefficient from d sqrt(r) and d log(r), and SALD.lsiKlFiDensityTestHalfFisherScalar turns a normalized LSI density-test bridge into the half-Fisher form C_LSI*KL <= (1/2)*FI used by the forward-KL derivative. The vector-gradient, integral, admissibility/approximation, zero-density, finite KL/FI, and Radon-Nikodym backends remain obligations.

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.cycle38LsiKlFiUpperPacket
2. AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar
3. AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainOfDerivativesScalar
4. SALD.lsiKlFiHalfFisherScalar
5. SALD.lsiKlFiDensityTestHalfFisherScalar
6. SALD.lsiKlFiDensityTestBridgeScalar
7. SALD.lsiKlFiCoefficientAuditScalar
8. SALD.saldLsiKlFiDensityTestContract
9. SALD.lsiKlFiDensityTestObligation
10. probability.lsi_to_kl_fi

note : String (explicit)

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

Middle cycle 38 adds theorem-independent scalar Fisher-chain and half-Fisher coefficient lemmas tied to the source substitution phi=sqrt(rho/pi). It does not prove the full LSI-to-KL/FI comparison, the vector Sobolev chain rule, or any downstream SALD theorem.

def cycle38LsiKlFiMiddleObligation : ProofObligation where
  id := "sald.lsi_kl_fi.cycle38_middle_fisher_chain"
  statement := "Translate main_body.tex lines 208-215 beyond the cycle-33 sqrt-density scalar helpers: for positive density ratio r, AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar proves the local 1/4 Fisher-integrand coefficient from d sqrt(r) and d log(r), and SALD.lsiKlFiDensityTestHalfFisherScalar turns a normalized LSI density-test bridge into the half-Fisher form C_LSI*KL <= (1/2)*FI used by the forward-KL derivative. The vector-gradient, integral, admissibility/approximation, zero-density, finite KL/FI, and Radon-Nikodym backends remain obligations."
  source := saldKlFiLsiSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle38LsiKlFiUpperPacket",
    "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar",
    "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainOfDerivativesScalar",
    "SALD.lsiKlFiHalfFisherScalar",
    "SALD.lsiKlFiDensityTestHalfFisherScalar",
    "SALD.lsiKlFiDensityTestBridgeScalar",
    "SALD.lsiKlFiCoefficientAuditScalar",
    "SALD.saldLsiKlFiDensityTestContract",
    "SALD.lsiKlFiDensityTestObligation",
    "probability.lsi_to_kl_fi"
  ]
  note := "Middle cycle 38 adds theorem-independent scalar Fisher-chain and half-Fisher coefficient lemmas tied to the source substitution phi=sqrt(rho/pi). It does not prove the full LSI-to-KL/FI comparison, the vector Sobolev chain rule, or any downstream SALD theorem."

/-- Cycle-38 lower obligation after compiling a finite-coordinate Fisher-chain handoff. -/