AutoSamplingTheory.SALD.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityDag

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 cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityDag :
    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.cycle136.middle_packet.weak_fp_std_basis_source_density
interface : String (explicit)
Text describing intended mathematical interface.
Compiled dynamic-leaf worker theorem: SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula derives hweakFpStdBasisDef from hweakFpDensityLaplacianAction plus hdensityLaplacianStdBasisDef. It keeps htestLaplacianStdBasisDef and the second-Green, box-divergence, trace, diffusion, and first-Green leaves explicit, while avoiding the older htestLaplacianOperator, htestLaplacianActionDef, and hweakFpLaplacianDef boundaries.
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.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation
2. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula
3. SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula
4. appendix.tex:1379-1387
5. appendix.tex:1392-1427
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.cycle136.lower_1_packet.density_laplacian_pointwise_source_formula
interface : String (explicit)
Text describing intended mathematical interface.
Compiled lower_1 proof-scout theorem: SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula derives hdensityLaplacianStdBasisDef from hdensityLaplacianActionDef, the source functional representation of the density-Laplacian weak action, plus hsourceDensityLaplacianStdBasis, the pointwise standard-basis second-derivative source formula for the selected weak test.
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.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation
2. SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula
3. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula
4. appendix.tex:1379-1387
5. appendix.tex:1392-1427
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.cycle136.lower_2_packet.source_density_laplacian_laplacian_source_field
interface : String (explicit)
Text describing intended mathematical interface.
Compiled lower_2 Lean theorem: SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField derives hsourceDensityLaplacianStdBasis from hsourceDensityLaplacianEqLaplacian, the source-field equality identifying the named density-Laplacian source field with Mathlib's Laplacian of the selected weak test. The standard-basis second-derivative formula is supplied by SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv.
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.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation
2. SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField
3. SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula
4. SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv
5. Mathlib.Analysis.InnerProductSpace.Laplacian
6. appendix.tex:1379-1387
7. appendix.tex:1392-1427
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.cycle136.remaining_weak_fp_source_density_formula
interface : String (explicit)
Text describing intended mathematical interface.
Remaining exact theorem boundary after cycle 136 lower_2: prove hweakFpDensityLaplacianAction, selecting the source density-Laplacian weak action as the weak-FP abstract Laplacian; prove hdensityLaplacianActionDef, representing the density-Laplacian action as the source functional applied to a named pointwise source field; and prove hsourceDensityLaplacianEqLaplacian, identifying that pointwise source field with Mathlib's Laplacian of the selected weak test. The sibling htestLaplacianStdBasisDef and all second-Green, box-divergence, trace, diffusion, and first-Green leaves remain separate.
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. appendix.tex:1379-1387
2. appendix.tex:1392-1427
3. SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField
4. SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula
5. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula
6. SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula
7. SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula
8. SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv
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
id : String (explicit)
Stable plan-node identifier.
ASTIS.SALD.cycle136.reviewer_weak_fp_std_basis_source_density_check
interface : String (explicit)
Text describing intended mathematical interface.
Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hweakFpStdBasisDef is narrowed to the density-Laplacian standard-basis source formula through SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula, the old htestLaplacianOperator/htestLaplacianActionDef/hweakFpLaplacianDef leaves are not reintroduced, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, or sald_version_2 use.
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.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation
2. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula
3. SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula
4. appendix.tex:1379-1427
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. cycle 136 reviewer
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 cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle136.middle_packet.weak_fp_std_basis_source_density"
      interface := "Compiled dynamic-leaf worker theorem: SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula derives hweakFpStdBasisDef from hweakFpDensityLaplacianAction plus hdensityLaplacianStdBasisDef. It keeps htestLaplacianStdBasisDef and the second-Green, box-divergence, trace, diffusion, and first-Green leaves explicit, while avoiding the older htestLaplacianOperator, htestLaplacianActionDef, and hweakFpLaplacianDef boundaries."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation",
        "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula",
        "appendix.tex:1379-1387",
        "appendix.tex:1392-1427"
      ]
      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.cycle136.lower_1_packet.density_laplacian_pointwise_source_formula"
      interface := "Compiled lower_1 proof-scout theorem: SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula derives hdensityLaplacianStdBasisDef from hdensityLaplacianActionDef, the source functional representation of the density-Laplacian weak action, plus hsourceDensityLaplacianStdBasis, the pointwise standard-basis second-derivative source formula for the selected weak test."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation",
        "SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula",
        "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula",
        "appendix.tex:1379-1387",
        "appendix.tex:1392-1427"
      ]
      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.cycle136.lower_2_packet.source_density_laplacian_laplacian_source_field"
      interface := "Compiled lower_2 Lean theorem: SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField derives hsourceDensityLaplacianStdBasis from hsourceDensityLaplacianEqLaplacian, the source-field equality identifying the named density-Laplacian source field with Mathlib's Laplacian of the selected weak test. The standard-basis second-derivative formula is supplied by SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation",
        "SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField",
        "SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula",
        "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv",
        "Mathlib.Analysis.InnerProductSpace.Laplacian",
        "appendix.tex:1379-1387",
        "appendix.tex:1392-1427"
      ]
      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.cycle136.remaining_weak_fp_source_density_formula"
      interface := "Remaining exact theorem boundary after cycle 136 lower_2: prove hweakFpDensityLaplacianAction, selecting the source density-Laplacian weak action as the weak-FP abstract Laplacian; prove hdensityLaplacianActionDef, representing the density-Laplacian action as the source functional applied to a named pointwise source field; and prove hsourceDensityLaplacianEqLaplacian, identifying that pointwise source field with Mathlib's Laplacian of the selected weak test. The sibling htestLaplacianStdBasisDef and all second-Green, box-divergence, trace, diffusion, and first-Green leaves remain separate."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "appendix.tex:1379-1387",
        "appendix.tex:1392-1427",
        "SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField",
        "SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula",
        "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula",
        "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula",
        "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle136.reviewer_weak_fp_std_basis_source_density_check"
      interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hweakFpStdBasisDef is narrowed to the density-Laplacian standard-basis source formula through SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula, the old htestLaplacianOperator/htestLaplacianActionDef/hweakFpLaplacianDef leaves are not reintroduced, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, or sald_version_2 use."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation",
        "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula",
        "appendix.tex:1379-1427"
      ]
      reusedBy := ["cycle 136 reviewer"]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 137: EM weak-FP density-Laplacian pointwise IBP -/

/-- Cycle-137 lower obligation narrowing the weak-FP density-Laplacian action
boundary to pointwise weak Laplacian integration by parts. -/

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