AutoSamplingTheory.SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract

Data definition / provenance and workflow record

Meaning and type

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

def generalMovingTargetDiscreteConditionalLawMeasurabilityContract :
    GeneralMovingTargetDiscreteConditionalLawMeasurabilityContract

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.saldGeneralMovingTargetDiscreteDerivativeSource — 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

commonSpace : String (explicit)

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

A fixed EM interval [s_k,s_{k+1}] uses one probability space carrying X_k^eta, hat X_s, c_{t_k}(X_k^eta), nabla log pi_{t_k}(X_k^eta), and the Brownian increment in eq:general_moving_target_SALD_frozen_interp.

interpolationLaw : String (explicit)

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

hat rho_s is Law(hat X_s), with endpoint bookkeeping supplied separately by SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation and the paired endpoint helpers.

conditionalKernel : String (explicit)

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

A regular conditional kernel for X_k^eta given hat X_s=x, or an equivalent kernel for the joint law of (X_k^eta,hat X_s), must be supplied before forming the conditional expectations.

kernelCompatibility : String (explicit)

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

The kernel must disintegrate the joint law and be compatible with the hat rho_s marginal, so conditional expectations are defined hat-rho_s-a.e. on the same state space as the weak FP equation.

componentConditionalFields : String (explicit)

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

Name condC_{k,s}(x)=E[c_{t_k}(X_k^eta)|hat X_s=x] and condScore_{k,s}(x)=E[nabla log pi_{t_k}(X_k^eta)|hat X_s=x] with the same conditional-expectation version.

selectedDriftField : String (explicit)

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

Name bar b_{k,s}(x)=E[dot t_k*c_{t_k}(X_k^eta)+(sigma_eta^2/2)*nabla log pi_{t_k}(X_k^eta)|hat X_s=x]; SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents compiles only the algebraic rewrite to dot t_k*condC_{k,s}+(sigma_eta^2/2)*condScore_{k,s} after linearity is supplied.

measurabilitySideConditions : String (explicit)

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

The kernel must be measurable in x and the named fields condC_{k,s}, condScore_{k,s}, and bar b_{k,s} must be measurable; SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents derives regularity of bar b_{k,s} from supplied component regularity and add/smul closure, while SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff transfers any supplied congruence-stable measurability predicate from the named component combination to bar b_{k,s}.

integrabilitySideConditions : String (explicit)

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

Both source summands must be conditionally integrable under the kernel, and bar b_{k,s} must be integrable enough against hat rho_s to define div(hat rho_s*bar b_{k,s}) in the chosen weak-test class.

weakFpHandoff : String (explicit)

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

This contract stops before the weak conditional Fokker-Planck theorem; SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract records the required weak-test statement, while cycle 72, cycle 69, cycle 54, and cycle 73 wrappers consume supplied weak FP and Laplacian/KL-derivative identities without constructing them. The cycle-72 admissible-test wrapper keeps the chosen test predicate explicit, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract keeps the log-ratio test admissibility separate, and SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns composes the two supplied handoffs.

exclusions : List String (explicit)

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

Ordered data items

1. Do not mark the regular conditional kernel or disintegration theorem as formalized here.
2. Do not infer density or absolute-continuity of hat rho_s or tilde pi_s from this conditional-law interface.
3. Do not prove the weak Fokker-Planck equation, KL differentiation, integration by parts, LSI, DV, Gronwall, or theorem closure in this packet.
4. Do not add the conditional-law requirements as hidden assumptions to thm:forward-KL-discrete or thm:general-moving-target-SALD-discrete.

dependencies : List String (explicit)

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

Ordered data items

1. eq:general_moving_target_SALD_frozen_interp
2. SALD.generalMovingTargetDiscreteConditionalDriftContract
3. SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents
4. SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff
5. SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents
6. SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination
7. SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination
8. SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation
9. SALD.generalMovingTargetDiscreteDerivativeSideConditionContract
10. SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract
11. SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff
12. SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff
13. SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract
14. SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar
15. SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns
16. SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff
17. SALD.cycle69GeneralMovingTargetDiscreteEmFpSourceSignsLowerObligation
18. sald.general_moving_target_discrete.em_interpolation_fp

sourceGaps : List String (explicit)

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

Ordered data items

1. appendix.tex defines bar b_{k,s} by conditional expectation but does not state a regular conditional probability theorem.
2. appendix.tex immediately passes from the conditional drift definition to the associated Fokker-Planck equation, so Lean needs separate measurability and integrability hypotheses.
3. No local Mathlib or SLT theorem has yet been ported for this disintegration/conditional-expectation backend.

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 generalMovingTargetDiscreteConditionalLawMeasurabilityContract :
    GeneralMovingTargetDiscreteConditionalLawMeasurabilityContract where
  sourceBlock := saldGeneralMovingTargetDiscreteDerivativeSource
  parentInterface := "sald.general_moving_target_discrete.em_interpolation_fp / appendix.tex:1358-1387"
  commonSpace := "A fixed EM interval [s_k,s_{k+1}] uses one probability space carrying X_k^eta, hat X_s, c_{t_k}(X_k^eta), nabla log pi_{t_k}(X_k^eta), and the Brownian increment in eq:general_moving_target_SALD_frozen_interp."
  interpolationLaw := "hat rho_s is Law(hat X_s), with endpoint bookkeeping supplied separately by SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation and the paired endpoint helpers."
  conditionalKernel := "A regular conditional kernel for X_k^eta given hat X_s=x, or an equivalent kernel for the joint law of (X_k^eta,hat X_s), must be supplied before forming the conditional expectations."
  kernelCompatibility := "The kernel must disintegrate the joint law and be compatible with the hat rho_s marginal, so conditional expectations are defined hat-rho_s-a.e. on the same state space as the weak FP equation."
  componentConditionalFields := "Name condC_{k,s}(x)=E[c_{t_k}(X_k^eta)|hat X_s=x] and condScore_{k,s}(x)=E[nabla log pi_{t_k}(X_k^eta)|hat X_s=x] with the same conditional-expectation version."
  selectedDriftField := "Name bar b_{k,s}(x)=E[dot t_k*c_{t_k}(X_k^eta)+(sigma_eta^2/2)*nabla log pi_{t_k}(X_k^eta)|hat X_s=x]; SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents compiles only the algebraic rewrite to dot t_k*condC_{k,s}+(sigma_eta^2/2)*condScore_{k,s} after linearity is supplied."
  measurabilitySideConditions := "The kernel must be measurable in x and the named fields condC_{k,s}, condScore_{k,s}, and bar b_{k,s} must be measurable; SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents derives regularity of bar b_{k,s} from supplied component regularity and add/smul closure, while SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff transfers any supplied congruence-stable measurability predicate from the named component combination to bar b_{k,s}."
  integrabilitySideConditions := "Both source summands must be conditionally integrable under the kernel, and bar b_{k,s} must be integrable enough against hat rho_s to define div(hat rho_s*bar b_{k,s}) in the chosen weak-test class."
  weakFpHandoff := "This contract stops before the weak conditional Fokker-Planck theorem; SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract records the required weak-test statement, while cycle 72, cycle 69, cycle 54, and cycle 73 wrappers consume supplied weak FP and Laplacian/KL-derivative identities without constructing them. The cycle-72 admissible-test wrapper keeps the chosen test predicate explicit, SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract keeps the log-ratio test admissibility separate, and SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns composes the two supplied handoffs."
  exclusions := [
    "Do not mark the regular conditional kernel or disintegration theorem as formalized here.",
    "Do not infer density or absolute-continuity of hat rho_s or tilde pi_s from this conditional-law interface.",
    "Do not prove the weak Fokker-Planck equation, KL differentiation, integration by parts, LSI, DV, Gronwall, or theorem closure in this packet.",
    "Do not add the conditional-law requirements as hidden assumptions to thm:forward-KL-discrete or thm:general-moving-target-SALD-discrete."
  ]
  dependencies := [
    "eq:general_moving_target_SALD_frozen_interp",
    "SALD.generalMovingTargetDiscreteConditionalDriftContract",
    "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents",
    "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff",
    "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents",
    "SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination",
    "SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination",
    "SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation",
    "SALD.generalMovingTargetDiscreteDerivativeSideConditionContract",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff",
    "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract",
    "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar",
    "SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns",
    "SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff",
    "SALD.cycle69GeneralMovingTargetDiscreteEmFpSourceSignsLowerObligation",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  sourceGaps := [
    "appendix.tex defines bar b_{k,s} by conditional expectation but does not state a regular conditional probability theorem.",
    "appendix.tex immediately passes from the conditional drift definition to the associated Fokker-Planck equation, so Lean needs separate measurability and integrability hypotheses.",
    "No local Mathlib or SLT theorem has yet been ported for this disintegration/conditional-expectation backend."
  ]
  status := ProofStatus.obligation