AutoSamplingTheory.SALD.cycle107DiscreteForwardKlBoundaryFluxIntegralDag

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.

def cycle107DiscreteForwardKlBoundaryFluxIntegralDag :
    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.forward_KL_discrete.cycle107.lower_packet.boundary_flux_integral_box

   interface : String (explicit)

   Text describing intended mathematical interface.

   Compiled lower theorem: SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox converts a box divergence integral for the weighted field hatRhoS * barB to the signed face sum by MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable, then uses the supplied face-to-trace identification to produce the exact hboundaryFluxIntegral premise.

   source : AutoSamplingTheory.SourceAnchor (explicit)

   Source pointer.

   AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource — 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.cycle107DiscreteForwardKlBoundaryFluxIntegralLowerObligation
   2. SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox
   3. MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable
   4. Mathlib.MeasureTheory.Integral.DivergenceTheorem
   5. appendix.tex:1379-1387
   6. appendix.tex:1368-1377

   reusedBy : List String (explicit)

   Declared consumers as strings, default empty.

   Ordered data items

   1. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero
   2. sald.general_moving_target_discrete.em_interpolation_fp
   3. sald.discrete_forward_kl.em_interpolation_fp
   4. thm:forward-KL-discrete

   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.forward_KL_discrete.cycle107.remaining_box_trace_instantiation_boundary

   interface : String (explicit)

   Text describing intended mathematical interface.

   Remaining exact boundary after the compiled box theorem: instantiate weightedField with the concrete hatRhoS * barB flux, prove continuity on the source box, off-countable Frechet differentiability, divergence integrability, boundaryFlux equals the interior divergence integral, and the signed Mathlib face sum equals the existing testTrace/normalFluxTrace boundary integral. Keep hproductRule, hdivergenceTheorem, hgradNormBound, and htestTraceZero separate.

   source : AutoSamplingTheory.SourceAnchor (explicit)

   Source pointer.

   AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource — 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.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox
   2. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero
   3. SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq
   4. ASTIS.SALD.cycle106.remaining_named_barB_version_boundary
   5. Mathlib.MeasureTheory.Integral.DivergenceTheorem
   6. appendix.tex:1379-1387
   7. appendix.tex:1368-1377

   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

def cycle107DiscreteForwardKlBoundaryFluxIntegralDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.forward_KL_discrete.cycle107.lower_packet.boundary_flux_integral_box"
      interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox converts a box divergence integral for the weighted field hatRhoS * barB to the signed face sum by MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable, then uses the supplied face-to-trace identification to produce the exact hboundaryFluxIntegral premise."
      source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle107DiscreteForwardKlBoundaryFluxIntegralLowerObligation",
        "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox",
        "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable",
        "Mathlib.MeasureTheory.Integral.DivergenceTheorem",
        "appendix.tex:1379-1387",
        "appendix.tex:1368-1377"
      ]
      reusedBy := [
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero",
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp",
        "thm:forward-KL-discrete"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.forward_KL_discrete.cycle107.remaining_box_trace_instantiation_boundary"
      interface := "Remaining exact boundary after the compiled box theorem: instantiate weightedField with the concrete hatRhoS * barB flux, prove continuity on the source box, off-countable Frechet differentiability, divergence integrability, boundaryFlux equals the interior divergence integral, and the signed Mathlib face sum equals the existing testTrace/normalFluxTrace boundary integral. Keep hproductRule, hdivergenceTheorem, hgradNormBound, and htestTraceZero separate."
      source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero",
        "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq",
        "ASTIS.SALD.cycle106.remaining_named_barB_version_boundary",
        "Mathlib.MeasureTheory.Integral.DivergenceTheorem",
        "appendix.tex:1379-1387",
        "appendix.tex:1368-1377"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp"
      ]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 108: concrete `hatRhoS * barB` product-flux continuity -/

/-- Cycle-108 lower obligation for the concrete product-flux instantiation of
the cycle-107 Mathlib box theorem.

This packet stays on the active EM backend.  It acts only on the continuity
piece of
`ASTIS.SALD.forward_KL_discrete.cycle107.remaining_box_trace_instantiation_boundary`
for the weighted field `hatRhoS * barB`; the derivative, divergence
integrability, boundaryFlux/interior-divergence, and signed-face/trace
identifications remain explicit.
-/