AutoSamplingTheory.SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitDag

Data definition / provenance and workflow record

Meaning and type

The result has data type List AutoSamplingTheory.ProofDagBlock. 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 cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitDag :
    List ProofDagBlock

Construction and field-by-field explanation

Construct an ordered list of the following data items. It is not a logical conjunction or proof DAG.

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.

Ordered data items

id : String (explicit)
Stable plan-node identifier.
ASTIS.SALD.cycle92.middle_generator_to_law_source_map
interface : String (explicit)
Text describing intended mathematical interface.
Middle source map: appendix.tex:1379-1387 is the Fokker-Planck invocation for the frozen EM interpolation. Reduce one input to the existing sample-generator law-transport handoff rather than adding another source-sign wrapper; choose the generator split because the source generator is drift plus diffusion before the drift/divergence and Laplacian source actions are identified.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource — audited data reference, not expanded and not a compiled dependency edge
targetLean : String (explicit)
Textual planned implementation destination.
AutoSamplingTheory/SALD.lean
dependsOn : List String (explicit)
Declared input names as strings, default empty.
Ordered data items
1. SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitMiddleObligation
2. SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff
3. SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation
4. SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation
5. sald.general_moving_target_discrete.em_interpolation_fp
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. sald.general_moving_target_discrete.em_interpolation_fp
2. sald.discrete_forward_kl.em_interpolation_fp
3. thm:forward-KL-discrete
4. thm:general-moving-target-SALD-discrete
status : AutoSamplingTheory.ProofStatus (explicit)
Recorded workflow status, default planned.
AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification
id : String (explicit)
Stable plan-node identifier.
ASTIS.SALD.cycle92.lower_sample_split_generator_handoff
interface : String (explicit)
Text describing intended mathematical interface.
Lower proof-producing handoff: SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff removes hgeneratorSplit by making generatorAction the definitional sum driftAction+diffusionAction, then applies the existing sample-space derivative to mapped-law weak derivative transport and source-sign packaging.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource — audited data reference, not expanded and not a compiled dependency edge
targetLean : String (explicit)
Textual planned implementation destination.
AutoSamplingTheory/SALD.lean
dependsOn : List String (explicit)
Declared input names as strings, default empty.
Ordered data items
1. SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation
2. SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff
3. AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample
4. SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff
5. SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. sald.general_moving_target_discrete.em_interpolation_fp
2. sald.discrete_forward_kl.em_interpolation_fp
3. thm:forward-KL-discrete
4. thm:general-moving-target-SALD-discrete
status : AutoSamplingTheory.ProofStatus (explicit)
Recorded workflow status, default planned.
AutoSamplingTheory.ProofStatus.formalized — stored label only; no proof certification
id : String (explicit)
Stable plan-node identifier.
ASTIS.SALD.cycle92.lower_law_derivative_of_sample_split_generator
interface : String (explicit)
Text describing intended mathematical interface.
Lower proof-producing derivative handoff: SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff transports the sample-space split-generator derivative to the mapped-law weak-test integral and rewrites the derivative value to the paper's drift/divergence and diffusion/Laplacian source signs, so the direct weak derivative route no longer needs a separate hlawDerivative premise.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource — audited data reference, not expanded and not a compiled dependency edge
targetLean : String (explicit)
Textual planned implementation destination.
AutoSamplingTheory/SALD.lean
dependsOn : List String (explicit)
Declared input names as strings, default empty.
Ordered data items
1. SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation
2. SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff
3. AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample
4. Mathlib.Analysis.Calculus.ParametricIntegral
5. Mathlib.MeasureTheory.Integral.Bochner.Basic
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. sald.general_moving_target_discrete.em_interpolation_fp
2. sald.discrete_forward_kl.em_interpolation_fp
3. thm:forward-KL-discrete
4. thm:general-moving-target-SALD-discrete
status : AutoSamplingTheory.ProofStatus (explicit)
Recorded workflow status, default planned.
AutoSamplingTheory.ProofStatus.formalized — stored label only; no proof certification
id : String (explicit)
Stable plan-node identifier.
ASTIS.SALD.cycle92.remaining_generator_to_law_boundary
interface : String (explicit)
Text describing intended mathematical interface.
Remaining exact boundary after the split-generator and direct law-derivative handoffs: prove the sample-path/Bochner parametric-integral derivative of the split generator sum, connect driftAction to -div(hat rho_s*bar b_{k,s}) through the conditional drift, and connect diffusionAction to +(sigma_eta^2/2) Delta hat rho_s with boundary hypotheses. A named partialS route can still use a supplied law-derivative witness, but the direct weak-test HasDerivAt route no longer needs hlawDerivative.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource — audited data reference, not expanded and not a compiled dependency edge
targetLean : String (explicit)
Textual planned implementation destination.
AutoSamplingTheory/SALD.lean
dependsOn : List String (explicit)
Declared input names as strings, default empty.
Ordered data items
1. Mathlib.Analysis.Calculus.ParametricIntegral
2. Mathlib.MeasureTheory.Integral.Bochner.Basic
3. SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions
4. SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff
5. SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff
6. appendix.tex:1379-1387
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. sald.general_moving_target_discrete.em_interpolation_fp
2. sald.discrete_forward_kl.em_interpolation_fp
status : AutoSamplingTheory.ProofStatus (explicit)
Recorded workflow status, default planned.
AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification

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 cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle92.middle_generator_to_law_source_map"
      interface := "Middle source map: appendix.tex:1379-1387 is the Fokker-Planck invocation for the frozen EM interpolation. Reduce one input to the existing sample-generator law-transport handoff rather than adding another source-sign wrapper; choose the generator split because the source generator is drift plus diffusion before the drift/divergence and Laplacian source actions are identified."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitMiddleObligation",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff",
        "SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation",
        "SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation",
        "sald.general_moving_target_discrete.em_interpolation_fp"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp",
        "thm:forward-KL-discrete",
        "thm:general-moving-target-SALD-discrete"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle92.lower_sample_split_generator_handoff"
      interface := "Lower proof-producing handoff: SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff removes hgeneratorSplit by making generatorAction the definitional sum driftAction+diffusionAction, then applies the existing sample-space derivative to mapped-law weak derivative transport and source-sign packaging."
      source := saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff",
        "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp",
        "thm:forward-KL-discrete",
        "thm:general-moving-target-SALD-discrete"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle92.lower_law_derivative_of_sample_split_generator"
      interface := "Lower proof-producing derivative handoff: SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff transports the sample-space split-generator derivative to the mapped-law weak-test integral and rewrites the derivative value to the paper's drift/divergence and diffusion/Laplacian source signs, so the direct weak derivative route no longer needs a separate hlawDerivative premise."
      source := saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff",
        "AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample",
        "Mathlib.Analysis.Calculus.ParametricIntegral",
        "Mathlib.MeasureTheory.Integral.Bochner.Basic"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp",
        "thm:forward-KL-discrete",
        "thm:general-moving-target-SALD-discrete"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle92.remaining_generator_to_law_boundary"
      interface := "Remaining exact boundary after the split-generator and direct law-derivative handoffs: prove the sample-path/Bochner parametric-integral derivative of the split generator sum, connect driftAction to -div(hat rho_s*bar b_{k,s}) through the conditional drift, and connect diffusionAction to +(sigma_eta^2/2) Delta hat rho_s with boundary hypotheses. A named partialS route can still use a supplied law-derivative witness, but the direct weak-test HasDerivAt route no longer needs hlawDerivative."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "Mathlib.Analysis.Calculus.ParametricIntegral",
        "Mathlib.MeasureTheory.Integral.Bochner.Basic",
        "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp"
      ]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 93: KL/log-ratio mass-derivative backfill -/

/-- Cycle-93 middle obligation for the KL/log-ratio mass derivative boundary.

This returns to `appendix.tex:1358-1366` after cycle 92's accepted weak-FP
split-generator handoff.  The selected middle packet removes one supplied input
behind the raw KL-to-weak-FP handoff: the primitive scalar mass-conservation
premise is replaced by the mapped-law constant weak-test derivative for
`hat rho_s = Law(hat X_s)`.
-/

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