AutoSamplingTheory.SALD.cycle102DiscreteForwardKlZeroFluxTraceBoundaryMiddleObligation

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 cycle102DiscreteForwardKlZeroFluxTraceBoundaryMiddleObligation :
    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.discrete_forward_kl.cycle102_zero_flux_trace_boundary_middle

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 102 compiles SALD.generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero and SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundary. Classification: narrows-source-cited-boundary. The cycle-101 raw hzeroBoundary premise in SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientProductRuleBoundary is replaced by two smaller source-facing inputs for appendix.tex:1379-1387: boundaryFlux phi is the boundary integral of the admissible test trace times the normal trace of hatRhoS * barB, and that trace product is zero a.e. on the boundary by compact support, decay, or zero normal trace. The product-rule identity, Mathlib divergence-theorem identity, and weak-test gradient bound hgradNormBound remain explicit.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource — 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.cycle101DiscreteForwardKlNoBoundaryProductRuleLowerObligation
2. SALD.generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero
3. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundary
4. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientProductRuleBoundary
5. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound
6. MeasureTheory.integral_congr_ae
7. MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable
8. Mathlib.MeasureTheory.Integral.DivergenceTheorem
9. Mathlib.MeasureTheory.Integral.Bochner.Basic
10. appendix.tex:1379-1387
11. appendix.tex:1368-1377

note : String (explicit)

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

Proof-producing trace-boundary handoff only. It does not prove the Euclidean product rule, instantiate the divergence theorem, prove the boundary-flux integral representation, prove compact-support/decay/zero-normal-trace hypotheses, prove hgradNormBound, close weak FP/KL/theorem status, import SLT, or change Lake dependencies.

def cycle102DiscreteForwardKlZeroFluxTraceBoundaryMiddleObligation :
    ProofObligation where
  id := "sald.discrete_forward_kl.cycle102_zero_flux_trace_boundary_middle"
  statement := "Cycle 102 compiles SALD.generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero and SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundary. Classification: narrows-source-cited-boundary. The cycle-101 raw hzeroBoundary premise in SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientProductRuleBoundary is replaced by two smaller source-facing inputs for appendix.tex:1379-1387: boundaryFlux phi is the boundary integral of the admissible test trace times the normal trace of hatRhoS * barB, and that trace product is zero a.e. on the boundary by compact support, decay, or zero normal trace. The product-rule identity, Mathlib divergence-theorem identity, and weak-test gradient bound hgradNormBound remain explicit."
  source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle101DiscreteForwardKlNoBoundaryProductRuleLowerObligation",
    "SALD.generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundary",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientProductRuleBoundary",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound",
    "MeasureTheory.integral_congr_ae",
    "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable",
    "Mathlib.MeasureTheory.Integral.DivergenceTheorem",
    "Mathlib.MeasureTheory.Integral.Bochner.Basic",
    "appendix.tex:1379-1387",
    "appendix.tex:1368-1377"
  ]
  note := "Proof-producing trace-boundary handoff only. It does not prove the Euclidean product rule, instantiate the divergence theorem, prove the boundary-flux integral representation, prove compact-support/decay/zero-normal-trace hypotheses, prove hgradNormBound, close weak FP/KL/theorem status, import SLT, or change Lake dependencies."

/-- Cycle-102 lower handoff reducing the trace-product condition to zero test
trace on the boundary.

This removes the supplied `htraceProductZero` premise from the trace-boundary
route when admissible tests have zero boundary trace a.e.  The analytic theorem
that supplies that trace fact remains an obligation.
-/