Samplinglib
Lean gate not recorded for this source state main · 0e31a3cda412

AutoSamplingTheory.SALD.cycle39ForwardKlDerivativeMiddleObligation

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 cycle39ForwardKlDerivativeMiddleObligation : 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.forward_kl.cycle39_derivative_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.

Maintain the cycle 39 middle/lower source-to-Lean map for appendix.tex lines 191-228 and the surrounding pre-DV derivative pipeline: SALD.forwardKlInverseScheduleDerivativeScalar, SALD.forwardKlTimeChangeSquareCoefficientRewriteOfProductScalar, SALD.forwardKlVelocitySquareScalingScalar, SALD.forwardKlTimeChangedDerivativeBoundOfProductScalar, SALD.forwardKlPreDvDerivativeBoundOfProductScalar, SALD.forwardKlPreDvDerivativeBoundOfVelocityScalingScalar, SALD.forwardKlLsiDerivativeBoundOfKlFiScalar, and SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar close only the Real algebra after explicit analytic chain-rule, inverse-schedule, velocity-square, KL derivative, target Cauchy, and source KL/FI comparison premises are supplied.
source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldForwardKlDerivativeSource — 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.cycle39ForwardKlDerivativeUpperPacket
  2. SALD.cycle39ForwardKlDerivativeMiddleContract
  3. SALD.cycle39ForwardKlDerivativeUpperObligation
  4. SALD.forwardKlPreDvDerivativeBoundScalar
  5. SALD.forwardKlInverseScheduleDerivativeScalar
  6. SALD.forwardKlTimeChangeSquareCoefficientRewriteOfProductScalar
  7. SALD.forwardKlVelocitySquareScalingScalar
  8. SALD.forwardKlTimeChangedDerivativeBoundOfProductScalar
  9. SALD.forwardKlPreDvDerivativeBoundOfProductScalar
  10. SALD.forwardKlPreDvDerivativeBoundOfVelocityScalingScalar
  11. SALD.forwardKlLsiDerivativeBoundOfKlFiScalar
  12. SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar
  13. SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar
  14. SALD.forwardKlLsiDerivativeBoundScalar
  15. SALD.forwardKlTimeChangedDerivativeBoundScalar
  16. SALD.forwardKlDerivativeSideConditionContract
  17. SALD.forwardKlDensityBoundaryObligation
  18. SALD.forwardKlScheduleTimeChangeObligation
  19. sald.forward_kl.cycle39_derivative_upper
  20. sald.forward_kl.density_boundary_regular
  21. sald.forward_kl.schedule_time_change
  22. probability.lsi_to_kl_fi
  23. sald.forward_kl.kl_derivative
note : String (explicit)

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

Cycle 39 middle translates the source time-change lines into source-shaped scalar lemmas: product inverse identity instead of a pre-rewritten derivative inverse, norm-square scaling instead of an opaque velocity-square equality, and source KL/FI comparison instead of a pre-multiplied LSI half-Fisher premise. It does not prove the analytic inverse-function theorem, L2 transport scaling, KL differentiation, Fokker-Planck identity, integration by parts, LSI-to-KL/FI, DV, Gronwall, EM interpolation, or thm:forward-KL.
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 cycle39ForwardKlDerivativeMiddleObligation : ProofObligation where
  id := "sald.forward_kl.cycle39_derivative_middle"
  statement := "Maintain the cycle 39 middle/lower source-to-Lean map for appendix.tex lines 191-228 and the surrounding pre-DV derivative pipeline: SALD.forwardKlInverseScheduleDerivativeScalar, SALD.forwardKlTimeChangeSquareCoefficientRewriteOfProductScalar, SALD.forwardKlVelocitySquareScalingScalar, SALD.forwardKlTimeChangedDerivativeBoundOfProductScalar, SALD.forwardKlPreDvDerivativeBoundOfProductScalar, SALD.forwardKlPreDvDerivativeBoundOfVelocityScalingScalar, SALD.forwardKlLsiDerivativeBoundOfKlFiScalar, and SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar close only the Real algebra after explicit analytic chain-rule, inverse-schedule, velocity-square, KL derivative, target Cauchy, and source KL/FI comparison premises are supplied."
  source := saldForwardKlDerivativeSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle39ForwardKlDerivativeUpperPacket",
    "SALD.cycle39ForwardKlDerivativeMiddleContract",
    "SALD.cycle39ForwardKlDerivativeUpperObligation",
    "SALD.forwardKlPreDvDerivativeBoundScalar",
    "SALD.forwardKlInverseScheduleDerivativeScalar",
    "SALD.forwardKlTimeChangeSquareCoefficientRewriteOfProductScalar",
    "SALD.forwardKlVelocitySquareScalingScalar",
    "SALD.forwardKlTimeChangedDerivativeBoundOfProductScalar",
    "SALD.forwardKlPreDvDerivativeBoundOfProductScalar",
    "SALD.forwardKlPreDvDerivativeBoundOfVelocityScalingScalar",
    "SALD.forwardKlLsiDerivativeBoundOfKlFiScalar",
    "SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar",
    "SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar",
    "SALD.forwardKlLsiDerivativeBoundScalar",
    "SALD.forwardKlTimeChangedDerivativeBoundScalar",
    "SALD.forwardKlDerivativeSideConditionContract",
    "SALD.forwardKlDensityBoundaryObligation",
    "SALD.forwardKlScheduleTimeChangeObligation",
    "sald.forward_kl.cycle39_derivative_upper",
    "sald.forward_kl.density_boundary_regular",
    "sald.forward_kl.schedule_time_change",
    "probability.lsi_to_kl_fi",
    "sald.forward_kl.kl_derivative"
  ]
  note := "Cycle 39 middle translates the source time-change lines into source-shaped scalar lemmas: product inverse identity instead of a pre-rewritten derivative inverse, norm-square scaling instead of an opaque velocity-square equality, and source KL/FI comparison instead of a pre-multiplied LSI half-Fisher premise. It does not prove the analytic inverse-function theorem, L2 transport scaling, KL differentiation, Fokker-Planck identity, integration by parts, LSI-to-KL/FI, DV, Gronwall, EM interpolation, or thm:forward-KL."

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