AutoSamplingTheory.SALD.cycle33LsiKlFiDensityTestLowerObligation

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.

Lean statement of this data definition

The part after the colon is the output data type. This declaration takes no mathematical proof inputs.

def cycle33LsiKlFiDensityTestLowerObligation : 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_lower

statement : String (explicit)

Desired mathematical or workflow content as String; it is not a proposition in Prop and the def does not prove it.

Use SALD.lsiKlFiDensityTestBridgeScalar after the analytic backend has supplied the normalized LSI test inequality for phi=sqrt(r), the entropy-to-KL identity, and the Fisher chain-rule identity dirichlet=(1/4)*FI. The Radon-Nikodym density, integral transport, smooth/admissible sqrt test or approximation, finite KL/FI, zero-density convention, and FI chain rule remain open 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

SALD.lsiKlFiDensityTestBridgeScalar
SALD.lsiKlFiCoefficientAuditScalar
AutoSamplingTheory.lsiKlFiSqrtDensitySquareScalar
AutoSamplingTheory.lsiKlFiSqrtDensityEntropyIntegrandScalar
AutoSamplingTheory.lsiKlFiSqrtDensityNormalizationScalar
SALD.cycle33LsiKlFiDensityTestMiddleObligation
SALD.saldLsiKlFiDensityTestContract
SALD.lsiKlFiDensityTestObligation
probability.lsi_to_kl_fi

note : String (explicit)

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

Lower cycle 33 formalizes only the theorem-independent real-order handoff from an already normalized LSI test inequality plus entropy/FI identifications to the displayed KL/FI coefficient. It does not close eq:LSI-KL-FI.

Exact Lean data construction

Each field assignment stores the corresponding value shown above. Omitted fields use the explicitly identified schema defaults. Strings that name theorems remain strings; they do not call those theorems.

def cycle33LsiKlFiDensityTestLowerObligation : ProofObligation where
  id := "sald.lsi_kl_fi.cycle33_density_test_lower"
  statement := "Use SALD.lsiKlFiDensityTestBridgeScalar after the analytic backend has supplied the normalized LSI test inequality for phi=sqrt(r), the entropy-to-KL identity, and the Fisher chain-rule identity dirichlet=(1/4)*FI. The Radon-Nikodym density, integral transport, smooth/admissible sqrt test or approximation, finite KL/FI, zero-density convention, and FI chain rule remain open obligations."
  source := saldKlFiLsiSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.lsiKlFiDensityTestBridgeScalar",
    "SALD.lsiKlFiCoefficientAuditScalar",
    "AutoSamplingTheory.lsiKlFiSqrtDensitySquareScalar",
    "AutoSamplingTheory.lsiKlFiSqrtDensityEntropyIntegrandScalar",
    "AutoSamplingTheory.lsiKlFiSqrtDensityNormalizationScalar",
    "SALD.cycle33LsiKlFiDensityTestMiddleObligation",
    "SALD.saldLsiKlFiDensityTestContract",
    "SALD.lsiKlFiDensityTestObligation",
    "probability.lsi_to_kl_fi"
  ]
  note := "Lower cycle 33 formalizes only the theorem-independent real-order handoff from an already normalized LSI test inequality plus entropy/FI identifications to the displayed KL/FI coefficient. It does not close eq:LSI-KL-FI."

/-- Cycle-38 upper packet for the LSI/KL/FI proof-closure target.

This packet follows the current proof-closure order after cycle 36 advanced
Gronwall assembly and cycle 37 advanced the one-sided Donsker--Varadhan
backend.  It selects `eq:LSI-KL-FI` as item (3) and narrows lower work to the
remaining density-test chain-rule/admissibility bridge in `main_body.tex:202-215`.
-/

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