AutoSamplingTheory.SALD.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryDag

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 cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryDag :
    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.cycle98.middle_barB_divergence_no_boundary_source_map
interface : String (explicit)
Text describing intended mathematical interface.
Middle source map: appendix.tex:1379-1387 uses the source sign -div(hat rho_s*bar b_{k,s}). After cycles 94-97, the remaining divergence half should be proved as an integration-by-parts/no-boundary identity for the law integral of the test-gradient contraction with barB.
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.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryMiddleObligation
2. ASTIS.SALD.cycle94.remaining_barB_divergence_boundary
3. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction
4. Mathlib.MeasureTheory.Integral.DivergenceTheorem
5. 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
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.cycle98.lower_packet.barB_no_boundary_integral
interface : String (explicit)
Text describing intended mathematical interface.
Compiled local handoff: SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral proves the old hbarBWeakDivergence shape from the law-integral weak-pairing definition and the no-boundary divergence identity; SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral feeds it into the cycle-94 drift-source route. The bounded-pairing variants additionally remove hpairIntegrable by proving it from Integrable barB hatRhoS and an a.e. test-gradient contraction bound.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource — 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.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryLowerObligation
2. SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound
3. SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral
4. SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing
5. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral
6. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral
7. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff
2. SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff
3. sald.general_moving_target_discrete.em_interpolation_fp
4. sald.discrete_forward_kl.em_interpolation_fp
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.cycle98.remaining_exact_no_boundary_theorem
interface : String (explicit)
Text describing intended mathematical interface.
Remaining exact theorem: for admissible weak tests, prove the concrete a.e. norm bound for the test-gradient contraction from barB regularity/test admissibility, align weakGradPairing with that Bochner integral, and prove driftDiv phi equals its negative by the Mathlib/local divergence theorem under the paper boundary/no-flux hypotheses. The abstract hpairIntegrable premise has been discharged by SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound once the bound is supplied.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource — 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.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound
2. SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing
3. SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral
4. Mathlib.MeasureTheory.Integral.DivergenceTheorem
5. Mathlib.MeasureTheory.Integral.Bochner.Basic
6. appendix.tex:1379-1387
7. appendix.tex:1368-1377
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 cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle98.middle_barB_divergence_no_boundary_source_map"
      interface := "Middle source map: appendix.tex:1379-1387 uses the source sign -div(hat rho_s*bar b_{k,s}).  After cycles 94-97, the remaining divergence half should be proved as an integration-by-parts/no-boundary identity for the law integral of the test-gradient contraction with barB."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryMiddleObligation",
        "ASTIS.SALD.cycle94.remaining_barB_divergence_boundary",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction",
        "Mathlib.MeasureTheory.Integral.DivergenceTheorem",
        "appendix.tex:1379-1387"
      ]
      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.cycle98.lower_packet.barB_no_boundary_integral"
      interface := "Compiled local handoff: SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral proves the old hbarBWeakDivergence shape from the law-integral weak-pairing definition and the no-boundary divergence identity; SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral feeds it into the cycle-94 drift-source route. The bounded-pairing variants additionally remove hpairIntegrable by proving it from Integrable barB hatRhoS and an a.e. test-gradient contraction bound."
      source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryLowerObligation",
        "SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound",
        "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral",
        "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction"
      ]
      reusedBy := [
        "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff",
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle98.remaining_exact_no_boundary_theorem"
      interface := "Remaining exact theorem: for admissible weak tests, prove the concrete a.e. norm bound for the test-gradient contraction from barB regularity/test admissibility, align weakGradPairing with that Bochner integral, and prove driftDiv phi equals its negative by the Mathlib/local divergence theorem under the paper boundary/no-flux hypotheses. The abstract hpairIntegrable premise has been discharged by SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound once the bound is supplied."
      source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound",
        "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing",
        "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral",
        "Mathlib.MeasureTheory.Integral.DivergenceTheorem",
        "Mathlib.MeasureTheory.Integral.Bochner.Basic",
        "appendix.tex:1379-1387",
        "appendix.tex:1368-1377"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp"
      ]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 99: raw KL derivative finite-KL `llr` boundary -/

/-- Cycle-99 middle obligation for the KL/log-ratio analytic boundary.

This returns to `appendix.tex:1358-1366` and narrows the remaining primitive
`hklRaw` display to a source-cited raw-KL theorem at the exact Mathlib
`llr hatRho tildePi` weak test.  It reuses cycle-87 finite-KL log-ratio
regularity, cycle-88 admissibility closure, and cycle-93 mapped-law mass
handoffs instead of adding another broad wrapper.
-/

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