AutoSamplingTheory.SALD.lsiKlFiDensityTestObligation

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 lsiKlFiDensityTestObligation : 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.density_test_interface

statement : String (explicit)

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

Formalize the source step main_body.tex lines 208-215: for rho << pi, set phi=sqrt(rho/pi), verify the LSI normalization and admissibility conditions, identify the entropy term with KL(rho||pi), and use the Fisher-information chain rule to obtain KL(rho||pi) <= FI(rho||pi)/(2*C_LSI).

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. KLContract
2. FIContract
3. LSIContract
4. SALD.saldLsiKlFiDensityTestContract
5. SALD.cycle29FirstAppendixMiddleAuditContract
6. SALD.cycle29LsiKlFiDensityTestMiddleObligation
7. SALD.cycle29LsiKlFiDensityTestLowerObligation
8. SALD.cycle33LsiKlFiDensityTestMiddleObligation
9. SALD.cycle33LsiKlFiDensityTestLowerObligation
10. SALD.cycle38LsiKlFiUpperPacket
11. SALD.cycle38LsiKlFiUpperObligation
12. SALD.cycle38LsiKlFiMiddleObligation
13. SALD.cycle38LsiKlFiLowerObligation
14. SALD.cycle43LsiKlFiUpperPacket
15. SALD.cycle43LsiKlFiUpperObligation
16. SALD.cycle43LsiKlFiMiddleObligation
17. SALD.cycle43LsiKlFiLowerObligation
18. AutoSamplingTheory.lsiKlFiSqrtDensitySquareScalar
19. AutoSamplingTheory.lsiKlFiSqrtDensityEntropyIntegrandScalar
20. AutoSamplingTheory.lsiKlFiSqrtDensityNormalizationScalar
21. AutoSamplingTheory.lsiKlFiRnDerivLIntegralMassOne
22. AutoSamplingTheory.lsiKlFiRnDerivDensityMassOne
23. AutoSamplingTheory.lsiKlFiSqrtRnDerivTestMassOne
24. AutoSamplingTheory.lsiKlFiRnDerivEntropyIntegral
25. AutoSamplingTheory.lsiKlFiSqrtRnDerivEntropyIntegral
26. AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar
27. AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainOfDerivativesScalar
28. AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumScalar
29. AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumHandoffScalar
30. AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralFiniteSum
31. AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralHandoffScalar
32. SALD.lsiKlFiCoefficientAuditScalar
33. SALD.lsiKlFiDensityTestBridgeScalar
34. probability.lsi_to_kl_fi

note : String (explicit)

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

saldLsiKlFiDensityTestContract records the smooth-density, absolute-continuity, finite-KL/FI, test-function, coefficient, and approximation interfaces; cycle 29 formalized only the final real-algebra coefficient audit, cycle 33 formalizes scalar sqrt-density square, entropy-integrand, normalization handoff, and normalized-test scalar bridge lemmas, cycle 38 adds pointwise plus finite-coordinate Fisher-chain handoffs, cycle 43 middle formalizes the Mathlib Radon-Nikodym mass and entropy-transport sublemmas, and cycle 43 lower formalizes a finite-coordinate a.e. integral Fisher-chain handoff while leaving sqrt-test admissibility/approximation, zero-density Sobolev handling, vector-gradient equivalence, finite theorem-level KL/FI interfaces, and the full LSI-to-KL/FI theorem open. Do not replace the source LSI route with PI, Pinsker, or Talagrand.

def lsiKlFiDensityTestObligation : ProofObligation where
  id := "sald.lsi_kl_fi.density_test_interface"
  statement := "Formalize the source step main_body.tex lines 208-215: for rho << pi, set phi=sqrt(rho/pi), verify the LSI normalization and admissibility conditions, identify the entropy term with KL(rho||pi), and use the Fisher-information chain rule to obtain KL(rho||pi) <= FI(rho||pi)/(2*C_LSI)."
  source := saldKlFiLsiSource
  status := ProofStatus.obligation
  dependsOn := [
    "KLContract",
    "FIContract",
    "LSIContract",
    "SALD.saldLsiKlFiDensityTestContract",
    "SALD.cycle29FirstAppendixMiddleAuditContract",
    "SALD.cycle29LsiKlFiDensityTestMiddleObligation",
    "SALD.cycle29LsiKlFiDensityTestLowerObligation",
    "SALD.cycle33LsiKlFiDensityTestMiddleObligation",
    "SALD.cycle33LsiKlFiDensityTestLowerObligation",
    "SALD.cycle38LsiKlFiUpperPacket",
    "SALD.cycle38LsiKlFiUpperObligation",
    "SALD.cycle38LsiKlFiMiddleObligation",
    "SALD.cycle38LsiKlFiLowerObligation",
    "SALD.cycle43LsiKlFiUpperPacket",
    "SALD.cycle43LsiKlFiUpperObligation",
    "SALD.cycle43LsiKlFiMiddleObligation",
    "SALD.cycle43LsiKlFiLowerObligation",
    "AutoSamplingTheory.lsiKlFiSqrtDensitySquareScalar",
    "AutoSamplingTheory.lsiKlFiSqrtDensityEntropyIntegrandScalar",
    "AutoSamplingTheory.lsiKlFiSqrtDensityNormalizationScalar",
    "AutoSamplingTheory.lsiKlFiRnDerivLIntegralMassOne",
    "AutoSamplingTheory.lsiKlFiRnDerivDensityMassOne",
    "AutoSamplingTheory.lsiKlFiSqrtRnDerivTestMassOne",
    "AutoSamplingTheory.lsiKlFiRnDerivEntropyIntegral",
    "AutoSamplingTheory.lsiKlFiSqrtRnDerivEntropyIntegral",
    "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar",
    "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainOfDerivativesScalar",
    "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumScalar",
    "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumHandoffScalar",
    "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralFiniteSum",
    "AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralHandoffScalar",
    "SALD.lsiKlFiCoefficientAuditScalar",
    "SALD.lsiKlFiDensityTestBridgeScalar",
    "probability.lsi_to_kl_fi"
  ]
  note := "saldLsiKlFiDensityTestContract records the smooth-density, absolute-continuity, finite-KL/FI, test-function, coefficient, and approximation interfaces; cycle 29 formalized only the final real-algebra coefficient audit, cycle 33 formalizes scalar sqrt-density square, entropy-integrand, normalization handoff, and normalized-test scalar bridge lemmas, cycle 38 adds pointwise plus finite-coordinate Fisher-chain handoffs, cycle 43 middle formalizes the Mathlib Radon-Nikodym mass and entropy-transport sublemmas, and cycle 43 lower formalizes a finite-coordinate a.e. integral Fisher-chain handoff while leaving sqrt-test admissibility/approximation, zero-density Sobolev handling, vector-gradient equivalence, finite theorem-level KL/FI interfaces, and the full LSI-to-KL/FI theorem open. Do not replace the source LSI route with PI, Pinsker, or Talagrand."