AutoSamplingTheory.SALD.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation

Data definition / provenance and workflow record

Meaning and type

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

def cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation :
    ProofObligation

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.

id : String (explicit)

Stable obligation identifier.

sald.general_moving_target_discrete.cycle136_em_weak_fp_std_basis_source_density_lower

statement : String (explicit)

Desired mathematical or workflow content as String; it is not a proposition in Prop and the def does not prove it.

Cycle 136 compiles SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula, SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula, and SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField for appendix.tex:1379-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining hweakFpStdBasisDef source formula is no longer the primitive weak-FP input; it follows from a smaller weak-FP-to-density-Laplacian action equality plus a source density-Laplacian standard-basis formula for the selected weak test. The density-Laplacian standard-basis formula is further narrowed to a source action definition plus a pointwise standard-basis source field equality, and lower_2 narrows that pointwise standard-basis field equality to hsourceDensityLaplacianEqLaplacian, the source-field identification with Mathlib's Laplacian of the selected weak test. The theorem does not reintroduce htestLaplacianOperator, htestLaplacianActionDef, or hweakFpLaplacianDef, and all second-Green, box-divergence, trace, diffusion, and first-Green leaves remain explicit.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource — audited data reference, not expanded and not a compiled dependency edge

status : AutoSamplingTheory.ProofStatus (explicit)

Stored ProofStatus, default obligation; even an explicitly stored formalized does not independently certify a Lean theorem.

AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification

dependsOn : List String (explicit)

List of declared dependency names as strings; may mix theorem names, obligations, source labels, or descriptions. Not the compiled dependency DAG.

Ordered data items

1. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula
2. SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula
3. SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField
4. SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula
5. SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula
6. SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv
7. Mathlib.Analysis.InnerProductSpace.Laplacian
8. appendix.tex:1379-1387
9. appendix.tex:1392-1427
10. sald.general_moving_target_discrete.em_interpolation_fp

note : String (explicit)

Recorded evidence/caveats; may distinguish a compiled scalar helper from still-open source analysis.

Dynamic-leaf worker packet. It narrows only hweakFpStdBasisDef to the source density-Laplacian standard-basis formula and the equality selecting that density-Laplacian action as the weak-FP abstract Laplacian; lower_1 further narrows hdensityLaplacianStdBasisDef to a source action definition plus a pointwise standard-basis source formula; lower_2 narrows that pointwise formula to hsourceDensityLaplacianEqLaplacian, the source-field equality to Mathlib's Laplacian of the selected weak test. It does not claim hweakFpDensityLaplacianAction, hdensityLaplacianActionDef, htestLaplacianStdBasisDef, second-Green, box-divergence, trace, diffusion, first-Green, KL differentiation, LSI, DV, Gronwall, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content.

def cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle136_em_weak_fp_std_basis_source_density_lower"
  statement := "Cycle 136 compiles SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula, SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula, and SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField for appendix.tex:1379-1427. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining hweakFpStdBasisDef source formula is no longer the primitive weak-FP input; it follows from a smaller weak-FP-to-density-Laplacian action equality plus a source density-Laplacian standard-basis formula for the selected weak test. The density-Laplacian standard-basis formula is further narrowed to a source action definition plus a pointwise standard-basis source field equality, and lower_2 narrows that pointwise standard-basis field equality to hsourceDensityLaplacianEqLaplacian, the source-field identification with Mathlib's Laplacian of the selected weak test. The theorem does not reintroduce htestLaplacianOperator, htestLaplacianActionDef, or hweakFpLaplacianDef, and all second-Green, box-divergence, trace, diffusion, and first-Green leaves remain explicit."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula",
    "SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula",
    "SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula",
    "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula",
    "SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv",
    "Mathlib.Analysis.InnerProductSpace.Laplacian",
    "appendix.tex:1379-1387",
    "appendix.tex:1392-1427",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "Dynamic-leaf worker packet. It narrows only hweakFpStdBasisDef to the source density-Laplacian standard-basis formula and the equality selecting that density-Laplacian action as the weak-FP abstract Laplacian; lower_1 further narrows hdensityLaplacianStdBasisDef to a source action definition plus a pointwise standard-basis source formula; lower_2 narrows that pointwise formula to hsourceDensityLaplacianEqLaplacian, the source-field equality to Mathlib's Laplacian of the selected weak test. It does not claim hweakFpDensityLaplacianAction, hdensityLaplacianActionDef, htestLaplacianStdBasisDef, second-Green, box-divergence, trace, diffusion, first-Green, KL differentiation, LSI, DV, Gronwall, theorem closure, SLT import, Lake/toolchain changes, theorem-status promotion, fake closure, broad audit, wrapper churn, non-EM fallback, or sald_version_2 content."

/-- Cycle-136 proof-DAG pane for the weak-FP standard-basis source-density
boundary. -/