AutoSamplingTheory.SALD.cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundDag

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 cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundDag :
    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.cycle100.lower_packet.barB_inner_gradient_contraction

   interface : String (explicit)

   Text describing intended mathematical interface.

   Compiled local handoff: Cauchy--Schwarz proves the hpairNormBound contraction for fieldPairing phi barB x = inner (testGrad phi x) (barB x), then routes through the cycle-100 weakGrad law-integral definition alignment. The old hpairNormBound supplied premise is replaced by the smaller hgradNormBound weak-test gradient estimate.

   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.cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundLowerObligation
   2. SALD.generalMovingTargetDiscreteBarBPairNormBoundOfInnerGradientBound
   3. SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryInnerGradientBound
   4. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound
   5. SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef
   6. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction
   7. Mathlib.Analysis.InnerProductSpace.Basic

   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
   3. thm:forward-KL-discrete
   4. thm:general-moving-target-SALD-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.cycle100.remaining_no_boundary_after_inner_gradient

   interface : String (explicit)

   Text describing intended mathematical interface.

   Remaining exact theorem after the inner-gradient contraction: prove the weak-test gradient norm bound ||testGrad phi x|| <= pairBound phi and prove driftDiv phi = -int x, inner (testGrad phi x) (barB x) d hatRhoS by the no-boundary divergence theorem for hatRhoS * barB.

   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. Mathlib.MeasureTheory.Integral.DivergenceTheorem
   2. Mathlib.Analysis.InnerProductSpace.Basic
   3. Mathlib.MeasureTheory.Integral.Bochner.Basic
   4. appendix.tex:1379-1387
   5. 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 cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle100.lower_packet.barB_inner_gradient_contraction"
      interface := "Compiled local handoff: Cauchy--Schwarz proves the hpairNormBound contraction for fieldPairing phi barB x = inner (testGrad phi x) (barB x), then routes through the cycle-100 weakGrad law-integral definition alignment. The old hpairNormBound supplied premise is replaced by the smaller hgradNormBound weak-test gradient estimate."
      source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundLowerObligation",
        "SALD.generalMovingTargetDiscreteBarBPairNormBoundOfInnerGradientBound",
        "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryInnerGradientBound",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound",
        "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction",
        "Mathlib.Analysis.InnerProductSpace.Basic"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp",
        "thm:forward-KL-discrete",
        "thm:general-moving-target-SALD-discrete"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle100.remaining_no_boundary_after_inner_gradient"
      interface := "Remaining exact theorem after the inner-gradient contraction: prove the weak-test gradient norm bound ||testGrad phi x|| <= pairBound phi and prove driftDiv phi = -int x, inner (testGrad phi x) (barB x) d hatRhoS by the no-boundary divergence theorem for hatRhoS * barB."
      source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "Mathlib.MeasureTheory.Integral.DivergenceTheorem",
        "Mathlib.Analysis.InnerProductSpace.Basic",
        "Mathlib.MeasureTheory.Integral.Bochner.Basic",
        "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 101: post-cycle-100 discrete pressure-test synchronization -/

/-- Cycle-101 middle synchronization for the discrete forward-KL closure pressure
test.

This is intentionally not another broad theorem-route wrapper.  Cycles 89 and
95 already record the route through the discrete theorem wrappers.  After the
cycle-100 weak-pairing and inner-gradient handoffs, the pressure test stops at
the exact no-boundary/weak-test-gradient boundary consumed by the shared EM
weak-FP backend.
-/