AutoSamplingTheory.SALD.cycle108DiscreteForwardKlHatRhoBarBFluxContinuityLowerObligation

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 cycle108DiscreteForwardKlHatRhoBarBFluxContinuityLowerObligation :
    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.cycle108_hatRho_barB_flux_continuity_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 108 lower compiles SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldContinuousOnBox and SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox. Classification: narrows-source-cited-boundary. The remaining cycle-107 box-trace instantiation for weightedField=hatRhoS*barB is narrowed by discharging the generic hcontinuous input from separate continuity of the density representative hatRhoDensity and the conditional drift barB on the source box. The Mathlib box divergence theorem is still consumed through SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox, while off-countable Frechet differentiability, divergence integrability, boundaryFlux equals the interior divergence integral, signed-face-to-testTrace identification, hproductRule, hdivergenceTheorem, hgradNormBound, and htestTraceZero 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.generalMovingTargetDiscreteHatRhoBarBWeightedFieldContinuousOnBox
2. SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox
3. SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox
4. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero
5. MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable
6. Mathlib.MeasureTheory.Integral.DivergenceTheorem
7. Mathlib.MeasureTheory.Integral.Bochner.Basic
8. appendix.tex:1379-1387
9. appendix.tex:1368-1377
10. sald.general_moving_target_discrete.em_interpolation_fp
11. sald.discrete_forward_kl.em_interpolation_fp

note : String (explicit)

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

Proof-producing continuity/product-flux handoff only. It does not prove the Frechet derivative of hatRhoS*barB, divergence integrability, boundaryFlux/divergence identification, signed-face trace parametrization, product rule, hgradNormBound, zero trace, weak FP/KL closure, theorem-status promotion, SLT import, Lake dependency change, or sald_version_2 use.

def cycle108DiscreteForwardKlHatRhoBarBFluxContinuityLowerObligation :
    ProofObligation where
  id := "sald.discrete_forward_kl.cycle108_hatRho_barB_flux_continuity_lower"
  statement := "Cycle 108 lower compiles SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldContinuousOnBox and SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox. Classification: narrows-source-cited-boundary. The remaining cycle-107 box-trace instantiation for weightedField=hatRhoS*barB is narrowed by discharging the generic hcontinuous input from separate continuity of the density representative hatRhoDensity and the conditional drift barB on the source box. The Mathlib box divergence theorem is still consumed through SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox, while off-countable Frechet differentiability, divergence integrability, boundaryFlux equals the interior divergence integral, signed-face-to-testTrace identification, hproductRule, hdivergenceTheorem, hgradNormBound, and htestTraceZero remain explicit."
  source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldContinuousOnBox",
    "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox",
    "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero",
    "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable",
    "Mathlib.MeasureTheory.Integral.DivergenceTheorem",
    "Mathlib.MeasureTheory.Integral.Bochner.Basic",
    "appendix.tex:1379-1387",
    "appendix.tex:1368-1377",
    "sald.general_moving_target_discrete.em_interpolation_fp",
    "sald.discrete_forward_kl.em_interpolation_fp"
  ]
  note := "Proof-producing continuity/product-flux handoff only. It does not prove the Frechet derivative of hatRhoS*barB, divergence integrability, boundaryFlux/divergence identification, signed-face trace parametrization, product rule, hgradNormBound, zero trace, weak FP/KL closure, theorem-status promotion, SLT import, Lake dependency change, or sald_version_2 use."

/-- Cycle-108 proof-DAG pane for the concrete `hatRhoS * barB` continuity
piece of the box-trace boundary. -/