AutoSamplingTheory.SALD.cycle33LsiKlFiDensityTestMiddleObligation

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 cycle33LsiKlFiDensityTestMiddleObligation : 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.cycle33_density_test_middle

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 into proof-producing scalar density-test code for phi=sqrt(r): nonnegative r gives phi^2=r, hence the pointwise entropy integrand phi^2*log(phi^2)=r*log(r), and once an integral backend identifies testMass with densityMass, density normalization gives the LSI test normalization. The full Radon-Nikodym backend, smooth/admissible sqrt test, finite KL/FI, zero-density convention, and FI chain rule 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. AutoSamplingTheory.lsiKlFiSqrtDensitySquareScalar
2. AutoSamplingTheory.lsiKlFiSqrtDensityEntropyIntegrandScalar
3. AutoSamplingTheory.lsiKlFiSqrtDensityNormalizationScalar
4. SALD.saldLsiKlFiDensityTestContract
5. SALD.lsiKlFiDensityTestObligation
6. probability.lsi_to_kl_fi

note : String (explicit)

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

Cycle 33 closes only theorem-independent scalar pieces of the source substitution phi=sqrt(r). It does not prove LSI, the KL/FI integral identities, or the Fisher-information chain rule.

def cycle33LsiKlFiDensityTestMiddleObligation : ProofObligation where
  id := "sald.lsi_kl_fi.cycle33_density_test_middle"
  statement := "Translate main_body.tex lines 208-215 into proof-producing scalar density-test code for phi=sqrt(r): nonnegative r gives phi^2=r, hence the pointwise entropy integrand phi^2*log(phi^2)=r*log(r), and once an integral backend identifies testMass with densityMass, density normalization gives the LSI test normalization. The full Radon-Nikodym backend, smooth/admissible sqrt test, finite KL/FI, zero-density convention, and FI chain rule remain obligations."
  source := saldKlFiLsiSource
  status := ProofStatus.obligation
  dependsOn := [
    "AutoSamplingTheory.lsiKlFiSqrtDensitySquareScalar",
    "AutoSamplingTheory.lsiKlFiSqrtDensityEntropyIntegrandScalar",
    "AutoSamplingTheory.lsiKlFiSqrtDensityNormalizationScalar",
    "SALD.saldLsiKlFiDensityTestContract",
    "SALD.lsiKlFiDensityTestObligation",
    "probability.lsi_to_kl_fi"
  ]
  note := "Cycle 33 closes only theorem-independent scalar pieces of the source substitution phi=sqrt(r). It does not prove LSI, the KL/FI integral identities, or the Fisher-information chain rule."

/-- Cycle-33 lower obligation for the normalized LSI-test scalar bridge. -/