AutoSamplingTheory.SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianLawIntegralLowerObligation

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 cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianLawIntegralLowerObligation :
    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.cycle142_em_generator_trace_laplacian_law_integral_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 142 lower_2 compiles SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula for appendix.tex:984-995 and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining hemGeneratorLaplacianStateIntegral premise under SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula is no longer primitive; it follows from the law-space selected-test Laplacian generator fact hemGeneratorLaplacianLawIntegral, the source-law identity hatRhoS = Measure.map hatXAtS P, AEMeasurable hatXAtS, and source-Laplacian field measurability by SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral. The theorem keeps hlaplacianEqEmGenerator, htraceFieldStdBasis, hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, htestLaplacianStdBasisDef, density-Laplacian source facts, Green/trace/box-divergence leaves, and diffusion leaves 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.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula
2. SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula
3. SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral
4. MeasureTheory.integral_map
5. Mathlib.Analysis.InnerProductSpace.Laplacian
6. appendix.tex:984-995
7. appendix.tex:1379-1387
8. 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 lower_2 packet inside the cycle-142 EM generator trace-state illness area. It narrows only hemGeneratorLaplacianStateIntegral to the smaller law-space selected-test Laplacian generator integral hemGeneratorLaplacianLawIntegral plus already explicit map-law and measurability transport facts. It does not prove that law integral, htraceFieldStdBasis, hlaplacianEqEmGenerator, source-field equality, source-field measurability, source-functional construction, density-Laplacian source facts, Green/trace/box-divergence leaves, diffusion construction, KL differentiation, 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 cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianLawIntegralLowerObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle142_em_generator_trace_laplacian_law_integral_lower"
  statement := "Cycle 142 lower_2 compiles SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula for appendix.tex:984-995 and appendix.tex:1379-1387. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the remaining hemGeneratorLaplacianStateIntegral premise under SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula is no longer primitive; it follows from the law-space selected-test Laplacian generator fact hemGeneratorLaplacianLawIntegral, the source-law identity hatRhoS = Measure.map hatXAtS P, AEMeasurable hatXAtS, and source-Laplacian field measurability by SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral. The theorem keeps hlaplacianEqEmGenerator, htraceFieldStdBasis, hweakFpSourceFieldEqLaplacian, hsourceLaplacianFieldMeas, hsourceLaplacianFunctional, htestLaplacianStdBasisDef, density-Laplacian source facts, Green/trace/box-divergence leaves, and diffusion leaves explicit."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula",
    "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula",
    "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral",
    "MeasureTheory.integral_map",
    "Mathlib.Analysis.InnerProductSpace.Laplacian",
    "appendix.tex:984-995",
    "appendix.tex:1379-1387",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "Dynamic-leaf lower_2 packet inside the cycle-142 EM generator trace-state illness area. It narrows only hemGeneratorLaplacianStateIntegral to the smaller law-space selected-test Laplacian generator integral hemGeneratorLaplacianLawIntegral plus already explicit map-law and measurability transport facts. It does not prove that law integral, htraceFieldStdBasis, hlaplacianEqEmGenerator, source-field equality, source-field measurability, source-functional construction, density-Laplacian source facts, Green/trace/box-divergence leaves, diffusion construction, KL differentiation, 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-141 proof-DAG pane for the EM generator source-action split. -/