Samplinglib
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AutoSamplingTheory.SALD.cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisDag

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 cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisDag :
    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

  1. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle135.lower_packet.test_laplacian_std_basis_definition
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled dynamic-leaf worker theorem: SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula narrows htestLaplacianActionDef to htestLaplacianStdBasisDef, a source formula saying the test-calculus abstract laplacian action is the standard-basis second-derivative action on the selected weak test. The theorem uses the cycle-134 Mathlib Laplacian formula and leaves hweakFpStdBasisDef plus all second-Green, box-divergence, trace, diffusion, and first-Green leaves 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. SALD.cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisLowerObligation
    2. SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula
    3. SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv
    4. InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis
    5. Mathlib.Analysis.InnerProductSpace.Laplacian
    6. appendix.tex:1379-1427
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback
    2. sald.general_moving_target_discrete.em_interpolation_fp
    3. 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
  2. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle135.lower_1_packet.operator_std_basis_composition
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled lower_1 scout theorem: SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula derives the old htestLaplacianOperator boundary directly from the two smaller source-facing leaves htestLaplacianStdBasisDef and hweakFpStdBasisDef. It composes the cycle-134 weak-FP standard-basis bridge, the cycle-135 test-action standard-basis bridge, and the cycle-134 source-pullback operator theorem; it does not discharge either source-facing standard-basis leaf.
    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.cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisLowerObligation
    2. SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula
    3. SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula
    4. SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula
    5. SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback
    6. InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis
    7. Mathlib.Analysis.InnerProductSpace.Laplacian
    8. appendix.tex:1392-1427
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization
    2. SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfSourceLaplacianPullback
    status : AutoSamplingTheory.ProofStatus (explicit)

    Recorded workflow status, default planned.

    AutoSamplingTheory.ProofStatus.formalized — stored label only; no proof certification
  3. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle135.lower_2_packet.second_green_std_basis_consumer
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled lower_2 continuation: SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula derives the downstream second-Green diffusion-source conclusion from htestLaplacianStdBasisDef and hweakFpStdBasisDef, so the consumer no longer takes htestLaplacianActionDef or hweakFpLaplacianDef. The theorem keeps the two source-facing standard-basis formulas and all second-Green, box-divergence, trace, diffusion, and first-Green leaves explicit.
    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.cycle135GeneralMovingTargetDiscreteEmSecondGreenStdBasisConsumerLowerObligation
    2. SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula
    3. SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula
    4. SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula
    5. SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula
    6. SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization
    7. appendix.tex:1379-1387
    8. 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
    status : AutoSamplingTheory.ProofStatus (explicit)

    Recorded workflow status, default planned.

    AutoSamplingTheory.ProofStatus.formalized — stored label only; no proof certification
  4. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle135.remaining_source_laplacian_definitions
    interface : String (explicit)

    Text describing intended mathematical interface.

    Remaining exact theorem boundary after cycle 135 lower_2: prove the source-facing htestLaplacianStdBasisDef for the local test-calculus action and hweakFpStdBasisDef for the weak-FP action's standard-basis source formula. Once both are supplied, SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula feeds the downstream second-Green consumer; 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.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula
    4. SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula
    5. SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula
    6. SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula
    7. SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback
    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
  5. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle135.reviewer_test_laplacian_std_basis_check
    interface : String (explicit)

    Text describing intended mathematical interface.

    Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, the downstream consumer removes htestLaplacianActionDef and hweakFpLaplacianDef in favor of htestLaplacianStdBasisDef plus hweakFpStdBasisDef through SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula, hsecondGreenTestTracePointwiseEq and htestTracePointwiseZero remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, 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.cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisLowerObligation
    2. SALD.cycle135GeneralMovingTargetDiscreteEmSecondGreenStdBasisConsumerLowerObligation
    3. SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula
    4. SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula
    5. SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula
    6. appendix.tex:1379-1427
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. cycle 135 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 cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle135.lower_packet.test_laplacian_std_basis_definition"
      interface := "Compiled dynamic-leaf worker theorem: SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula narrows htestLaplacianActionDef to htestLaplacianStdBasisDef, a source formula saying the test-calculus abstract laplacian action is the standard-basis second-derivative action on the selected weak test. The theorem uses the cycle-134 Mathlib Laplacian formula and leaves hweakFpStdBasisDef plus all second-Green, box-divergence, trace, diffusion, and first-Green leaves separate."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisLowerObligation",
        "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula",
        "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv",
        "InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis",
        "Mathlib.Analysis.InnerProductSpace.Laplacian",
        "appendix.tex:1379-1427"
      ]
      reusedBy := [
        "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback",
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle135.lower_1_packet.operator_std_basis_composition"
      interface := "Compiled lower_1 scout theorem: SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula derives the old htestLaplacianOperator boundary directly from the two smaller source-facing leaves htestLaplacianStdBasisDef and hweakFpStdBasisDef. It composes the cycle-134 weak-FP standard-basis bridge, the cycle-135 test-action standard-basis bridge, and the cycle-134 source-pullback operator theorem; it does not discharge either source-facing standard-basis leaf."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisLowerObligation",
        "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula",
        "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula",
        "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula",
        "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback",
        "InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis",
        "Mathlib.Analysis.InnerProductSpace.Laplacian",
        "appendix.tex:1392-1427"
      ]
      reusedBy := [
        "SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfSourceLaplacianPullback"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle135.lower_2_packet.second_green_std_basis_consumer"
      interface := "Compiled lower_2 continuation: SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula derives the downstream second-Green diffusion-source conclusion from htestLaplacianStdBasisDef and hweakFpStdBasisDef, so the consumer no longer takes htestLaplacianActionDef or hweakFpLaplacianDef. The theorem keeps the two source-facing standard-basis formulas and all second-Green, box-divergence, trace, diffusion, and first-Green leaves explicit."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle135GeneralMovingTargetDiscreteEmSecondGreenStdBasisConsumerLowerObligation",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula",
        "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula",
        "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula",
        "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula",
        "SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization",
        "appendix.tex:1379-1387",
        "appendix.tex:1392-1427"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle135.remaining_source_laplacian_definitions"
      interface := "Remaining exact theorem boundary after cycle 135 lower_2: prove the source-facing htestLaplacianStdBasisDef for the local test-calculus action and hweakFpStdBasisDef for the weak-FP action's standard-basis source formula. Once both are supplied, SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula feeds the downstream second-Green consumer; 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.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula",
        "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula",
        "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula",
        "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula",
        "SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle135.reviewer_test_laplacian_std_basis_check"
      interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, the downstream consumer removes htestLaplacianActionDef and hweakFpLaplacianDef in favor of htestLaplacianStdBasisDef plus hweakFpStdBasisDef through SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula, hsecondGreenTestTracePointwiseEq and htestTracePointwiseZero remain explicit, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, or sald_version_2 use."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisLowerObligation",
        "SALD.cycle135GeneralMovingTargetDiscreteEmSecondGreenStdBasisConsumerLowerObligation",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula",
        "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula",
        "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula",
        "appendix.tex:1379-1427"
      ]
      reusedBy := ["cycle 135 reviewer"]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 136: EM weak-FP standard-basis source-density formula -/

/-- Cycle-136 lower obligation narrowing the weak-FP standard-basis source
formula to a density-Laplacian action formula. -/

Existing module entry · Audited data-reader index · All teaching coverage