Encyclopedia Foundation Foundation Pair Kernel Canonical Source Green Export S28 Canonical Source Green

ARTICLE 3 claims 3 theorems

Foundation Pair Kernel Canonical Source Green Export S28 Canonical Source Green

A single machine-checked interface bundles several theorem-backed layers of the framework's ledger, but it introduces no new physical premise.

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The assembled source interface

In the Recognition Science framework, a ledger is a discrete record of events that the framework treats as the substrate of physical structure. The declaration canonicalSourceGreenExport3 is a machine-checked bundle that assembles several previously proved layers of that ledger behind one interface. The bundle combines a witnessed event-posting constructor, a signed multiplicity graph with a period-two collision repair, an exact nonlinear first variation with a conserved current and tangent Hessian, and a five-class executable-effect carrier with a scale-covariant readout. The declaration proves that this combined interface is inhabited, meaning that all the pieces can coexist consistently; the proof uses only the framework's existing theorems and adds no new axiom or unfinished proof.

What the declaration does not do is introduce any new physical premise. The exported native-curvature branch is explicitly a tangent model, not a claim about the global field background. The interface deliberately leaves outside its scope the identification of a global field background with the native edge drop, the selection of an exact-J field unit, absolute duration or energy units, and identification with an external raw ontology. These are boundaries of the export, not missing numerical tests. The declaration also records that absolute event action is not invariant under the full unit quotient, and that period two requires the signed multiplicity graph because the simpler relation set alone records multiplicity one where the six-slot operator records two.

In plain terms, this declaration is a consolidation step. It takes theorem-backed components from earlier stages and proves they fit together into one input surface for an executable shadow. The scale-covariant readout survives all duration and energy boundary-unit rescalings, and the forced D+2 exponent survives those rescalings as well. What a reader can take away is that the framework's internal ledger, at this stage, has a single coherent interface whose pieces are each backed by a machine-checked proof, and that the interface's limits are explicitly marked rather than silently assumed.

THEOREM canonicalSourceGreenExport3 · IndisputableMonolith/Foundation/PairKernelCanonicalSourceGreenExportS28.lean
canonicalSourceGreenExport3 · IndisputableMonolith/Foundation/PairKernelCanonicalSourceGreenExportS28.lean:225 
/-- The complete export is inhabited entirely by S8-S27 theorems. -/
theorem canonicalSourceGreenExport3 :
    Nonempty CanonicalSourceGreenExport3 := by
  obtain
    ⟨event, hevent, hledgerGreen, hunitTangent,
      hnativeTangent, hnativeScale, hnativeSource, hnotReciprocal⟩ :=
    canonicalExactJTangentConsumer_exists
  refine ⟨{
    event := event
    event_realized := hevent
    event_has_ledger_witness :=
      realizedPrimitivePostingPair_has_ledger_event event hevent
    every_effect_has_event_witness :=
      everyProductionEffectClass_has_realizedEventWitness
    effects_have_physical_instances :=
      executableEffects_have_physicalInstances
    effect_carrier_complete :=
      executableEffectPhysicalCarrier_complete
    effect_carrier_dimension :=
      executableEffectPhysicalCarrier_dimension_eq_five
    no_silent_effect := no_silentProductionEffect
    distinct_effects_are_observable :=
      distinctProductionEffectClasses_have_distinguishingProbe
    exact_action_is_event_fold :=
      recognitionProductionGraph3_exactJCostAction_eq_eventBatch
    signed_support_is_witnessed :=
      signedGraph_support_eq_recognitionGraph_support
    signed_generator_is_framed :=
      pulledBackSignedPostingLaplacian3_eq_framed
    exact_first_variation := by
      intro σ field variation
      exact
        exactJCostAction_hasDerivAt_line_gauss
          (signedRecognitionProductionGraph3 3 σ)
          field variation
    exact_source_is_conserved := by
      intro σ field
      exact
        exactJNonlinearLaplacian_sum_zero
          (signedRecognitionProductionGraph3 3 σ) field
    exact_action_is_gauge_invariant := by
      intro σ field c
      exact
        exactJCostAction_add_const
          (signedRecognitionProductionGraph3 3 σ) field c
    tangent_hessian_is_symmetric := by
      intro σ background left right
      exact
        exactJHessianPairing_comm
          (signedRecognitionProductionGraph3 3 σ)
          background left right
    tangent_hessian_is_nonnegative := by
      intro σ background perturbation
      exact
        exactJHessianAction_nonneg
          (signedRecognitionProductionGraph3 3 σ)
          background perturbation
    tangent_has_gauge_kernel := by
      intro σ background c i
      exact
        exactJTangentLaplacian_const_zero
          (signedRecognitionProductionGraph3 3 σ)
          background c i
    nonlinear_generator_is_frame_invariant :=
      exactJNonlinearLaplacian_signed_frame_independent
    tangent_generator_is_frame_invariant :=
      exactJTangentLaplacian_signed_frame_independent
    ledger_unit_green_attachment := hledgerGreen
    unit_tangent_green_attachment := hunitTangent
    native_curvature_tangent_model := hnativeTangent
    native_tangent_green_scale := hnativeScale
    native_ordered_source_formula := hnativeSource
    native_ordered_source_is_not_reciprocal := hnotReciprocal
    exact_green_ratio_is_scale_covariant :=
      tangentGreenScale_eq_exactJGreenRatio
    raw_external_ontology_remains_independent :=
      externalRawOntology_realization_remains_independent
  }⟩
THEOREM CanonicalSourceGreenExport3 · IndisputableMonolith/Foundation/PairKernelCanonicalSourceGreenExportS28.lean
CanonicalSourceGreenExport3 · IndisputableMonolith/Foundation/PairKernelCanonicalSourceGreenExportS28.lean:54 
/-- One theorem-backed input surface for the FD1 executable shadow.

The universally quantified fields are the exact formulas the shadow mirrors.
The existential event is shared by the ledger source and tangent Green
branches. -/
structure CanonicalSourceGreenExport3 where
  event : PostingPair3 3
  event_realized :
    event ∈ realizedPrimitivePostingPairs3 3
  event_has_ledger_witness :
    ∃ A B : LedgerPostingAdjacency.LedgerState 3,
      PostingStep A B ∧
        PairKernelOperationalLocalityS4.JMinimalPostingStep A B ∧
        (∑ k : Fin (TorusCard3 3),
          divF (elementaryPosting event.1 event.2) k) = 0
  every_effect_has_event_witness :
    ∀ effect : ProductionOperationEffectClass3,
      ∃ operation : PairKernelProductionOperationChannelSelectionS25.ProductionSourceOperation3,
        (PairKernelProductionOperationChannelSelectionS25.productionSourceOperation_event
          operation).1 ∈ realizedPrimitivePostingPairs3 3 ∧
          productionOperationEffectClass3 operation = effect
  effects_have_physical_instances :
    ProductionEffectsRealizePhysicalChannels3
      executableEffectPhysicalResponseSystem3
  effect_carrier_complete :
    PostingCarrierCoherenceComplete3
      executableEffectPhysicalChannelCarrier3
  effect_carrier_dimension :
    ∀ target : RealizedPostingEvent3 3,
      physicalPostingCarrierDimension3
          executableEffectPhysicalChannelCarrier3 target = 5
  no_silent_effect :
    ∀ effect : ProductionOperationEffectClass3,
      ¬ SilentProductionEffect3 effect
  distinct_effects_are_observable :
    ∀ left right : ProductionOperationEffectClass3,
      left ≠ right →
        ∃ probe : ProductionOperationEffectProbe3,
          productionOperationEffectClassObservation3 probe left ≠
            productionOperationEffectClassObservation3 probe right
  exact_action_is_event_fold :
    ∀ field : Fin (TorusCard3 3) → ℝ,
      exactJCostAction (recognitionProductionGraph3 3) field =
        postingBatchExactJCostAction3
          (realizedPrimitivePostingPairs3 3) field
  signed_support_is_witnessed :
    ∀ (σ : Equiv.Perm (Fin 3))
      (i j : Fin (TorusCard3 3)),
      (signedRecognitionProductionGraph3 3 σ).weight i j ≠ 0 ↔
        (recognitionProductionGraph3 3).weight i j ≠ 0
  signed_generator_is_framed :
    ∀ (σ : Equiv.Perm (Fin 3))
      (field : PairKernelPeriodic3.TorusSite3 3 → ℝ)
      (p : PairKernelPeriodic3.TorusSite3 3),
      (pulledBackSignedPostingLaplacian3 σ field p : ℂ) =
        PairKernelDeltaSpatialBridgeS5.framedTorusLaplacian σ
          (fun x => (field x : ℂ)) p
  exact_first_variation :
    ∀ (σ : Equiv.Perm (Fin 3))
      (field variation : Fin (TorusCard3 3) → ℝ),
      HasDerivAt
        (fun t : ℝ =>
          exactJCostAction
            (signedRecognitionProductionGraph3 3 σ)
            (fun i => field i + t * variation i))
        (2 * ∑ i,
          variation i *
            exactJNonlinearLaplacian
              (signedRecognitionProductionGraph3 3 σ)
              field i)
        0
  exact_source_is_conserved :
    ∀ (σ : Equiv.Perm (Fin 3))
      (field : Fin (TorusCard3 3) → ℝ),
      ∑ i,
        exactJNonlinearLaplacian
          (signedRecognitionProductionGraph3 3 σ)
          field i = 0
  exact_action_is_gauge_invariant :
    ∀ (σ : Equiv.Perm (Fin 3))
      (field : Fin (TorusCard3 3) → ℝ)
      (c : ℝ),
      exactJCostAction
          (signedRecognitionProductionGraph3 3 σ)
          (fun i => field i + c) =
        exactJCostAction
          (signedRecognitionProductionGraph3 3 σ) field
  tangent_hessian_is_symmetric :
    ∀ (σ : Equiv.Perm (Fin 3))
      (background left right : Fin (TorusCard3 3) → ℝ),
      exactJHessianPairing
          (signedRecognitionProductionGraph3 3 σ)
          background left right =
        exactJHessianPairing
          (signedRecognitionProductionGraph3 3 σ)
          background right left
  tangent_hessian_is_nonnegative :
    ∀ (σ : Equiv.Perm (Fin 3))
      (background perturbation : Fin (TorusCard3 3) → ℝ),
      0 ≤ exactJHessianAction
        (signedRecognitionProductionGraph3 3 σ)
        background perturbation
  tangent_has_gauge_kernel :
    ∀ (σ : Equiv.Perm (Fin 3))
      (background : Fin (TorusCard3 3) → ℝ)
      (c : ℝ) (i : Fin (TorusCard3 3)),
      exactJTangentLaplacian
          (signedRecognitionProductionGraph3 3 σ)
          background (fun _ => c) i = 0
  nonlinear_generator_is_frame_invariant :
    ∀ (σ τ : Equiv.Perm (Fin 3))
      (field : Fin (TorusCard3 3) → ℝ)
      (i : Fin (TorusCard3 3)),
      exactJNonlinearLaplacian
          (signedRecognitionProductionGraph3 3 σ)
          field i =
        exactJNonlinearLaplacian
          (signedRecognitionProductionGraph3 3 τ)
          field i
  tangent_generator_is_frame_invariant :
    ∀ (σ τ : Equiv.Perm (Fin 3))
      (background perturbation : Fin (TorusCard3 3) → ℝ)
      (i : Fin (TorusCard3 3)),
      exactJTangentLaplacian
          (signedRecognitionProductionGraph3 3 σ)
          background perturbation i =
        exactJTangentLaplacian
          (signedRecognitionProductionGraph3 3 τ)
          background perturbation i
  ledger_unit_green_attachment :
    SignedPostingSourceAttachment3 1 event
      (Equiv.refl (Fin 3))
      (signedRealGreenField3 1 event)
  unit_tangent_green_attachment :
    ConstantCurvatureSignedPostingAttachment3
      1 (by norm_num) 1 event
      (Equiv.refl (Fin 3))
      (constantCurvatureSignedGreenField3 1 1 event)
  native_curvature_tangent_model :
    ConstantCurvatureSignedPostingAttachment3
      (1 + Constants.hbar) nativeCurvature_pos.le
      nativeOrderedExactJSource event
      (Equiv.refl (Fin 3))
      (constantCurvatureSignedGreenField3
        (1 + Constants.hbar)
        nativeOrderedExactJSource event)
  native_tangent_green_scale :
    realGreenScaleFromPostingMagnitude
        (nativeOrderedExactJSource /
          (1 + Constants.hbar)) =
      nativeExactJConjugateSource /
        (1 + Constants.hbar)
  native_ordered_source_formula :
    nativeOrderedExactJSource =
      2 * Real.sqrt
        (Constants.hbar * (Constants.hbar + 2))
  native_ordered_source_is_not_reciprocal :
    nativeOrderedExactJSource ≠
      PairKernelNativeActionQuantum.nativeActionQuantumInv
  exact_green_ratio_is_scale_covariant :
    ∀ drop : ℝ,
      realGreenScaleFromPostingMagnitude
          ((2 * exactJOneEdgeSourceAtDrop drop) /
            exactJCurvatureAtDrop drop) =
        exactJGreenRatioAtDrop drop
  raw_external_ontology_remains_independent :
    CommittedProductionOperations3 ∧
      ∃ left right : PostingPhysicalResponseSystem3.{0} 3,
        ProductionEffectsRealizePhysicalChannels3 left ∧
          ¬ ProductionEffectsRealizePhysicalChannels3 right
THEOREM canonicalSourceGreen_scaleCovariantReadout · IndisputableMonolith/Foundation/PairKernelCanonicalSourceGreenExportS28.lean
canonicalSourceGreen_scaleCovariantReadout · IndisputableMonolith/Foundation/PairKernelCanonicalSourceGreenExportS28.lean:306 
/-- The canonical effect system's scale-covariant physical readout remains
available as part of the S28 input surface. -/
def canonicalSourceGreen_scaleCovariantReadout :=
  executableEffectPhysical_scaleCovariant_consumer_exists

What this page does not claim

The declaration does not prove that the global field background is identical to the native edge drop. It does not select an exact-J field unit or absolute duration or energy units. It does not identify the framework's objects with any external raw ontology.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/PairKernelCanonicalSourceGreenExportS28.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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