AutoSamplingTheory.SALD.cycle43LsiKlFiMiddleObligation

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 cycle43LsiKlFiMiddleObligation : 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_middle_density_entropy

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 middle compiles the Radon-Nikodym density normalization and entropy-transport sublemmas for main_body.tex lines 208-215: rho << pi gives unit mass for d rho/d pi, the sqrt density test has unit LSI mass, and the sqrt-test entropy integral is transported to the KL log-likelihood integral. This does not prove smooth/admissible sqrt-test approximation, the vector/integral Fisher chain rule, or the full probability.lsi_to_kl_fi theorem.

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.saldLsiKlFiDensityTestContract
3. SALD.lsiKlFiDensityTestObligation
4. AutoSamplingTheory.lsiKlFiRnDerivLIntegralMassOne
5. AutoSamplingTheory.lsiKlFiRnDerivDensityMassOne
6. AutoSamplingTheory.lsiKlFiSqrtRnDerivTestMassOne
7. AutoSamplingTheory.lsiKlFiRnDerivEntropyIntegral
8. AutoSamplingTheory.lsiKlFiSqrtRnDerivEntropyIntegral
9. AutoSamplingTheory.lsiKlFiSqrtDensitySquareScalar
10. AutoSamplingTheory.lsiKlFiSqrtDensityEntropyIntegrandScalar
11. probability.lsi_to_kl_fi

note : String (explicit)

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

This is formalized measure-level density and entropy plumbing only. eq:LSI-KL-FI remains an obligation because the LSI admissibility/approximation step and the vector/integral Fisher-information chain rule have not been closed.

def cycle43LsiKlFiMiddleObligation : ProofObligation where
  id := "sald.lsi_kl_fi.cycle43_middle_density_entropy"
  statement := "Cycle 43 middle compiles the Radon-Nikodym density normalization and entropy-transport sublemmas for main_body.tex lines 208-215: rho << pi gives unit mass for d rho/d pi, the sqrt density test has unit LSI mass, and the sqrt-test entropy integral is transported to the KL log-likelihood integral. This does not prove smooth/admissible sqrt-test approximation, the vector/integral Fisher chain rule, or the full probability.lsi_to_kl_fi theorem."
  source := saldKlFiLsiSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle43LsiKlFiUpperPacket",
    "SALD.saldLsiKlFiDensityTestContract",
    "SALD.lsiKlFiDensityTestObligation",
    "AutoSamplingTheory.lsiKlFiRnDerivLIntegralMassOne",
    "AutoSamplingTheory.lsiKlFiRnDerivDensityMassOne",
    "AutoSamplingTheory.lsiKlFiSqrtRnDerivTestMassOne",
    "AutoSamplingTheory.lsiKlFiRnDerivEntropyIntegral",
    "AutoSamplingTheory.lsiKlFiSqrtRnDerivEntropyIntegral",
    "AutoSamplingTheory.lsiKlFiSqrtDensitySquareScalar",
    "AutoSamplingTheory.lsiKlFiSqrtDensityEntropyIntegrandScalar",
    "probability.lsi_to_kl_fi"
  ]
  note := "This is formalized measure-level density and entropy plumbing only. eq:LSI-KL-FI remains an obligation because the LSI admissibility/approximation step and the vector/integral Fisher-information chain rule have not been closed."

/-- Cycle-43 lower finite-coordinate integral Fisher-chain obligation.

The accompanying declarations in `AutoSamplingTheory/Probability.lean` push
the cycle-38 finite-coordinate Fisher-chain identity through an arbitrary
measure under a.e. positivity and supplied coordinate derivative identities.
This narrows the source Fisher-chain backend, but it still does not identify
the finite-coordinate integrals with the paper's full vector-gradient Fisher
information.
-/