AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract

Data definition / provenance and workflow record

Meaning and type

The result has data type AutoSamplingTheory.SALD.GeneralMovingTargetDiscreteKlDerivativeWeakFpHandoffContract. A value of this type stores descriptions; it is not a proof of the statements in those descriptions.

def generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract :
    GeneralMovingTargetDiscreteKlDerivativeWeakFpHandoffContract

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.

sourceBlock : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource — audited data reference, not expanded and not a compiled dependency edge

parentInterface : String (explicit)

Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.

sald.general_moving_target_discrete.em_interpolation_fp / appendix.tex:1358-1387

differentiatedKlFormula : String (explicit)

Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.

eq:general_KL_derivative_0_discrete: d/ds KL(hat rho_s||tilde pi_s)=int partial_s hat rho_s*log(hat rho_s/tilde pi_s) - int (hat rho_s/tilde pi_s)*partial_s tilde pi_s, using int partial_s hat rho_s dx=0.

selectedWeakTest : String (explicit)

Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.

Use the log-density-ratio test phi_s=log(hat rho_s/tilde pi_s), or a justified admissible approximation, in the weak conditional Fokker--Planck identity for hat rho_s.

admissibilityInterface : String (explicit)

Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.

The log-ratio test must belong to the admissible weak-test class, with density positivity, finite KL/FI, local Sobolev/smooth approximation, and boundary behavior sufficient for the divergence and Laplacian dual actions.

weakFpSubstitution : String (explicit)

Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.

Substitute partial_s hat rho_s paired with phi_s by the supplied weak FP action -div(hat rho_s*bar b_{k,s}) paired with phi_s plus (sigma_eta^2/2)*Delta hat rho_s paired with phi_s; SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar compiles only this scalar substitution under explicit hypotheses, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns compiles the same substitution from already-normalized admissible weak-FP source signs, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction records both the log-ratio weak-FP action and the resulting dK display, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns composes through the cycle-72 admissible source-sign theorem, and SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces composes it through the cycle-77 generator-piece source-sign handoff.

sourceSignedDerivative : String (explicit)

Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.

The resulting derivative display preserves the source signs: negative drift-divergence action, positive sigma_eta^2/2 Laplacian action, and the unchanged target-time term -int (hat rho_s/tilde pi_s)*partial_s tilde pi_s.

integrationByPartsHandoff : String (explicit)

Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.

After this handoff, a separate analytic backend must justify integration by parts, the Laplacian split Delta hat rho_s=div(hat rho_s*A_s)+div(hat rho_s*nabla log tilde pi_s), and the Fisher-information identification before the frozen/residual Young and LSI steps.

downstreamDerivativeInterface : String (explicit)

Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.

Feeds SALD.generalMovingTargetDiscreteDerivativeCandidateContract and sald.general_moving_target_discrete.kl_derivative; it is also reusable by the discrete forward-KL EM interpolation backend with its simpler frozen drift.

explicitHypotheses : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. hat rho_s and tilde pi_s live on the same measurable/smooth state space and hat rho_s is absolutely continuous enough for a log-density ratio
2. finite KL and enough time regularity to differentiate KL and drop int partial_s hat rho_s dx by mass conservation
3. the log-density-ratio test is admissible or approximated in the weak-test class
4. the weak FP identity from cycle 72 applies to that test with sigmaCoeff=sigma_eta^2/2
5. boundary/no-flux or decay hypotheses support the later integration-by-parts handoff

downstreamHandoffs : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar
2. SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns
3. SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction
4. SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns
5. SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces
6. SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff
7. SALD.generalMovingTargetDiscreteDerivativeCandidateContract
8. sald.general_moving_target_discrete.kl_derivative
9. sald.discrete_forward_kl.kl_derivative

exclusions : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. Do not prove the weak conditional Fokker--Planck theorem in this handoff.
2. Do not prove admissibility of the log-density-ratio test, density positivity, absolute continuity, finite KL/FI, or integration by parts here.
3. Do not promote LSI/KL/FI, DV, Gronwall, EM interpolation, or either discrete theorem contract above its current status.
4. Do not add the log-ratio admissibility or boundary requirements as hidden assumptions to a theorem statement.

dependencies : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. eq:general_KL_derivative_0_discrete
2. SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract
3. SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff
4. SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff
5. SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar
6. SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns
7. SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction
8. SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns
9. SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces
10. SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract
11. SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract
12. SALD.generalMovingTargetDiscreteDerivativeSideConditionContract
13. SALD.generalMovingTargetDiscreteDerivativeCandidateContract
14. sald.general_moving_target_discrete.em_interpolation_fp
15. sald.general_moving_target_discrete.kl_derivative

sourceGaps : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. appendix.tex:1358-1387 differentiates KL and invokes the associated Fokker--Planck equation without stating the log-ratio weak-test admissibility theorem.
2. The source does not isolate the density/absolute-continuity and boundary hypotheses needed to substitute the weak FP identity into the KL derivative.
3. The following integration-by-parts and Fisher-information identifications remain separate analytic obligations.

status : AutoSamplingTheory.ProofStatus (explicit)

Stored workflow tag; honor the exact default but do not infer mathematical certification.

AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification

def generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract :
    GeneralMovingTargetDiscreteKlDerivativeWeakFpHandoffContract where
  sourceBlock := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource
  parentInterface := "sald.general_moving_target_discrete.em_interpolation_fp / appendix.tex:1358-1387"
  differentiatedKlFormula := "eq:general_KL_derivative_0_discrete: d/ds KL(hat rho_s||tilde pi_s)=int partial_s hat rho_s*log(hat rho_s/tilde pi_s) - int (hat rho_s/tilde pi_s)*partial_s tilde pi_s, using int partial_s hat rho_s dx=0."
  selectedWeakTest := "Use the log-density-ratio test phi_s=log(hat rho_s/tilde pi_s), or a justified admissible approximation, in the weak conditional Fokker--Planck identity for hat rho_s."
  admissibilityInterface := "The log-ratio test must belong to the admissible weak-test class, with density positivity, finite KL/FI, local Sobolev/smooth approximation, and boundary behavior sufficient for the divergence and Laplacian dual actions."
  weakFpSubstitution := "Substitute partial_s hat rho_s paired with phi_s by the supplied weak FP action -div(hat rho_s*bar b_{k,s}) paired with phi_s plus (sigma_eta^2/2)*Delta hat rho_s paired with phi_s; SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar compiles only this scalar substitution under explicit hypotheses, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns compiles the same substitution from already-normalized admissible weak-FP source signs, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction records both the log-ratio weak-FP action and the resulting dK display, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns composes through the cycle-72 admissible source-sign theorem, and SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces composes it through the cycle-77 generator-piece source-sign handoff."
  sourceSignedDerivative := "The resulting derivative display preserves the source signs: negative drift-divergence action, positive sigma_eta^2/2 Laplacian action, and the unchanged target-time term -int (hat rho_s/tilde pi_s)*partial_s tilde pi_s."
  integrationByPartsHandoff := "After this handoff, a separate analytic backend must justify integration by parts, the Laplacian split Delta hat rho_s=div(hat rho_s*A_s)+div(hat rho_s*nabla log tilde pi_s), and the Fisher-information identification before the frozen/residual Young and LSI steps."
  downstreamDerivativeInterface := "Feeds SALD.generalMovingTargetDiscreteDerivativeCandidateContract and sald.general_moving_target_discrete.kl_derivative; it is also reusable by the discrete forward-KL EM interpolation backend with its simpler frozen drift."
  explicitHypotheses := [
    "hat rho_s and tilde pi_s live on the same measurable/smooth state space and hat rho_s is absolutely continuous enough for a log-density ratio",
    "finite KL and enough time regularity to differentiate KL and drop int partial_s hat rho_s dx by mass conservation",
    "the log-density-ratio test is admissible or approximated in the weak-test class",
    "the weak FP identity from cycle 72 applies to that test with sigmaCoeff=sigma_eta^2/2",
    "boundary/no-flux or decay hypotheses support the later integration-by-parts handoff"
  ]
  downstreamHandoffs := [
    "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar",
    "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns",
    "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction",
    "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns",
    "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces",
    "SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff",
    "SALD.generalMovingTargetDiscreteDerivativeCandidateContract",
    "sald.general_moving_target_discrete.kl_derivative",
    "sald.discrete_forward_kl.kl_derivative"
  ]
  exclusions := [
    "Do not prove the weak conditional Fokker--Planck theorem in this handoff.",
    "Do not prove admissibility of the log-density-ratio test, density positivity, absolute continuity, finite KL/FI, or integration by parts here.",
    "Do not promote LSI/KL/FI, DV, Gronwall, EM interpolation, or either discrete theorem contract above its current status.",
    "Do not add the log-ratio admissibility or boundary requirements as hidden assumptions to a theorem statement."
  ]
  dependencies := [
    "eq:general_KL_derivative_0_discrete",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff",
    "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar",
    "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns",
    "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction",
    "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns",
    "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces",
    "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract",
    "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract",
    "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract",
    "SALD.generalMovingTargetDiscreteDerivativeCandidateContract",
    "sald.general_moving_target_discrete.em_interpolation_fp",
    "sald.general_moving_target_discrete.kl_derivative"
  ]
  sourceGaps := [
    "appendix.tex:1358-1387 differentiates KL and invokes the associated Fokker--Planck equation without stating the log-ratio weak-test admissibility theorem.",
    "The source does not isolate the density/absolute-continuity and boundary hypotheses needed to substitute the weak FP identity into the KL derivative.",
    "The following integration-by-parts and Fisher-information identifications remain separate analytic obligations."
  ]
  status := ProofStatus.obligation