AutoSamplingTheory.SALD.cycle108DiscreteForwardKlHatRhoBarBFluxContinuityDag

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 cycle108DiscreteForwardKlHatRhoBarBFluxContinuityDag :
    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.cycle108.lower_packet.hatRho_barB_flux_continuity

   interface : String (explicit)

   Text describing intended mathematical interface.

   Compiled lower theorem: SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldContinuousOnBox proves continuity of x |-> hatRhoDensity x • barB x on the Mathlib source box from separate density and barB continuity, and SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox feeds that concrete product flux into the cycle-107 box theorem.

   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.cycle108DiscreteForwardKlHatRhoBarBFluxContinuityLowerObligation
   2. SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldContinuousOnBox
   3. SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox
   4. SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox
   5. Mathlib.Topology.Algebra.MulAction
   6. Mathlib.MeasureTheory.Integral.DivergenceTheorem
   7. appendix.tex:1379-1387
   8. 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.cycle108.remaining_hatRho_barB_box_trace_boundary

   interface : String (explicit)

   Text describing intended mathematical interface.

   Remaining exact boundary after the compiled product-flux continuity handoff: prove the Frechet derivative of x |-> hatRhoDensity x • barB x off a countable interior set, 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.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox
   2. SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox
   3. SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq
   4. ASTIS.SALD.forward_KL_discrete.cycle107.remaining_box_trace_instantiation_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 cycle108DiscreteForwardKlHatRhoBarBFluxContinuityDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.forward_KL_discrete.cycle108.lower_packet.hatRho_barB_flux_continuity"
      interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldContinuousOnBox proves continuity of x |-> hatRhoDensity x • barB x on the Mathlib source box from separate density and barB continuity, and SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox feeds that concrete product flux into the cycle-107 box theorem."
      source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle108DiscreteForwardKlHatRhoBarBFluxContinuityLowerObligation",
        "SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldContinuousOnBox",
        "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox",
        "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox",
        "Mathlib.Topology.Algebra.MulAction",
        "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.cycle108.remaining_hatRho_barB_box_trace_boundary"
      interface := "Remaining exact boundary after the compiled product-flux continuity handoff: prove the Frechet derivative of x |-> hatRhoDensity x • barB x off a countable interior set, 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.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox",
        "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox",
        "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq",
        "ASTIS.SALD.forward_KL_discrete.cycle107.remaining_box_trace_instantiation_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 derivative -/

/-- Cycle-108 lower obligation for the concrete product-flux Frechet
derivative sub-boundary.

This packet stays below the active EM no-boundary backend.  It acts only on
the Frechet differentiability piece of
`ASTIS.SALD.forward_KL_discrete.cycle108.remaining_hatRho_barB_box_trace_boundary`:
the derivative of `hatRhoDensity x • barB x` is obtained from separate density
and `barB` derivatives, off the union of their countable exception sets.
-/