Foundation Pair Kernel Action Extensionality S7 Canonical Posting Graph3 Satisfi

The canonical posting graph

The declaration canonicalPostingGraph3_satisfies_positiveRealizedProductionActionIdentity is a machine-checked theorem. It states that a particular graph, called the canonical posting graph, has the property that its action equals the free sum over certain edge-wise positive posting weights. In plainer terms, for this specific graph, the cost of production can be decomposed into independent per-edge scales, each of which is positive. This is a statement about one graph, not about all graphs.

The theorem is part of a larger body of work in the Recognition Science framework, which models reality as a ledger of recognition events. The action here is a mathematical function that assigns a cost to a graph based on its weights and a set of potentials. The canonical posting graph is a specific graph defined on a torus with three sites. The theorem shows that this graph satisfies a particular identity, meaning its action can be expressed in a certain simple form. This is a precise, formal result, not a general law.

The theorem does not claim that every graph satisfies this identity. In fact, other theorems in the same library show that there exist graphs, such as the global torus graph, that do not satisfy it. The identity is a special property of the canonical graph, and it is used to derive other results, such as the PrimitivePostingActionLaw. The theorem also does not claim that the identity holds for all possible weights or potentials; it is specific to the canonical graph's definition.

What this means in practice is that the framework has a specific, verified example of a graph that meets its production action identity. This is a building block, not a general conclusion. The result is a formal statement about a particular object, and its significance lies in what it enables: further theorems about the structure of the framework's models.

/-- Canonical free posting inhabits the weakest edge-wise positive action
identity once self-postings are excluded. -/
theorem canonicalPostingGraph3_satisfies_positiveRealizedProductionActionIdentity
    {N : ℕ} [NeZero N]
    (hself : NoSelfPrimitivePostings3 N) :
    ProductionActionEqualsPositiveRealizedPostingAction3
      (canonicalPostingGraph3 N) := by
  refine ⟨(canonicalPostingGraph3 N).weight,
    canonicalPostingGraph3_positiveRealizedWeights hself, ?_⟩
  intro ε
  unfold exactJCostAction
  apply Finset.sum_congr rfl
  intro i _
  apply Finset.sum_congr rfl
  intro j _
  simp [graphOfPositiveRealizedPostingWeights3]

/-- The all-pairs S6 countermodel violates the weakest edge-wise positive
realized action identity: a mixed basis probe recovers a nonzero inactive
coefficient. -/
theorem globalTorusGraph3_violates_positiveRealizedProductionActionIdentity :
    ¬ ProductionActionEqualsPositiveRealizedPostingAction3
      (globalTorusGraph3 3) := by
  intro hidentity
  rcases hidentity with ⟨w, hw, haction⟩
  let p : TorusSite3 3 := fun _ => 0
  let q : TorusSite3 3 :=
    fun i => if i = (0 : Fin 3) ∨ i = (1 : Fin 3) then 1 else 0
  have hpq : p ≠ q := by
    intro hpq
    have h0 := congrFun hpq (0 : Fin 3)
    norm_num [p, q] at h0
  have hinactive :
      ¬ JMinimalGeneratedStep (Equiv.refl (Fin 3)) p q := by
    rw [jMinimalGeneratedStep_iff_framedAxisStep]
    intro hstep
    rcases hstep with ⟨k, hk | hk⟩
    · fin_cases k
      all_goals
        have h0 := congrFun hk 0
        have h1 := congrFun hk 1
        simp [p, q, framedForward, torusShift] at h0 h1
    · fin_cases k
      all_goals
        have h0 := congrFun hk 0
        have h1 := congrFun hk 1
        simp [p, q, framedBackward, torusShift] at h0 h1
  have hij :
      torusSiteEquivFin 3 p ≠ torusSiteEquivFin 3 q :=
    (torusSiteEquivFin 3).injective.ne hpq
  have hcoeff :=
    exactJCostAction_determines_offDiagonal (globalTorusGraph3 3)
      (graphOfPositiveRealizedPostingWeights3 hw) haction hij
  have hinactive' :
      ¬ (@encodedMinimumJRelation3 3 _)
        (torusSiteEquivFin 3 p) (torusSiteEquivFin 3 q) := by
    simpa [encodedMinimumJRelation3,
      (torusSiteEquivFin 3).symm_apply_apply,
      (torusSiteEquivFin 3).symm_apply_apply] using hinactive
  have hz := hw.inactive_zero _ _ hinactive'
  have hzero : (1 : ℝ) = 0 := by
    simpa [globalTorusGraph3, graphOfPositiveRealizedPostingWeights3, hz]
      using hcoeff
  exact (by norm_num : (1 : ℝ) ≠ 0) hzero

What this page does not claim

The identity holds for all graphs. The identity is a general law of the framework. The theorem provides a physical interpretation of the canonical graph.

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/PairKernelActionExtensionalityS7.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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