Encyclopedia Foundation Foundation Pair Kernel Action Extensionality S7 Global Torus Graph3 Violates Pos

ARTICLE 4 claims 4 theorems

Foundation Pair Kernel Action Extensionality S7 Global Torus Graph3 Violates Pos

A specific graph on a three-dimensional torus shows that the framework's earlier assumptions do not force a key identity, and the proof is a machine-checked theorem.

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A counterexample's reach

In the Recognition Science framework, a ledger (a discrete record of events) can be represented as a weighted graph. The framework's cost function, J(x) = (x + 1/x)/2 - 1, assigns a real number to each graph based on the differences between its vertices' potentials. A central question is whether this cost function alone can identify the graph's structure. The declaration globalTorusGraph3_violates_positiveRealizedProductionActionIdentity answers a specific version of this question in the negative.

The declaration is a theorem: it states that a particular graph, called globalTorusGraph3 on a three-dimensional torus, does not satisfy the ProductionActionEqualsPositiveRealizedPostingAction3 identity. This identity would mean the graph's cost function equals a sum over edge-wise positive posting weights. The theorem shows this is false for this specific graph. More importantly, it is used to prove a broader result: the framework's existing premises do not force this identity to hold for all graphs. The graph is a counterexample, a concrete case where the premises hold but the identity fails.

This counterexample is not a failure of the framework; it is a precise boundary. It shows that the cost function, while powerful, cannot by itself determine every graph's structure. The cost function is blind to a graph's diagonal entries (its self-loops), as shown by the theorem exactJCostAction_eq_of_offDiagonal. This means the cost function cannot distinguish between a graph and another graph with different self-loop weights. The counterexample graph exploits this blindness to violate the identity.

The proof is a machine-checked theorem in the framework's library of formal theorems. It contains no sorry and no new axioms, meaning it is derived from the framework's existing definitions and theorems. This is not a conjecture or a heuristic; it is a verified mathematical fact within the system. The declaration therefore establishes a definitive limit on what the cost function can achieve, and it does so with the full weight of a formal proof.

What the declaration does not claim is that the identity is false for all graphs. The canonical posting graph, for instance, does satisfy it. The declaration only shows that the identity is not a consequence of the existing premises. It leaves open the possibility that additional conditions could force the identity, or that other graphs might satisfy it. The counterexample is a specific, narrow result, not a general statement about the framework's power.

THEOREM globalTorusGraph3_violates_positiveRealizedProductionActionIdentity · IndisputableMonolith/Foundation/PairKernelActionExtensionalityS7.lean
globalTorusGraph3_violates_positiveRealizedProductionActionIdentity · IndisputableMonolith/Foundation/PairKernelActionExtensionalityS7.lean:1008 
/-- 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
THEOREM existing_premises_do_not_force_positiveRealizedProductionActionIdentity · IndisputableMonolith/Foundation/PairKernelActionExtensionalityS7.lean
existing_premises_do_not_force_positiveRealizedProductionActionIdentity · IndisputableMonolith/Foundation/PairKernelActionExtensionalityS7.lean:1126 
/-- The graph-independent exact-action, Gauss, source, and Pi-free premises
isolated in S6 do not force the weakest edge-wise positive realized action
identity. -/
theorem existing_premises_do_not_force_positiveRealizedProductionActionIdentity :
    ¬ (∀ G : WeightedLedgerGraph (TorusCard3 3),
      ExistingActionLedgerPremises3 G →
        ProductionActionEqualsPositiveRealizedPostingAction3 G) := by
  intro hforce
  exact globalTorusGraph3_violates_positiveRealizedProductionActionIdentity
    (hforce (globalTorusGraph3 3)
      globalTorusGraph3_satisfies_existing_premises)
THEOREM exactJCostAction_eq_of_offDiagonal · IndisputableMonolith/Foundation/PairKernelActionExtensionalityS7.lean
exactJCostAction_eq_of_offDiagonal · IndisputableMonolith/Foundation/PairKernelActionExtensionalityS7.lean:226 
/-- Off-diagonal coefficient agreement is also sufficient for exact-action
agreement. Diagonal coefficients never contribute. -/
theorem exactJCostAction_eq_of_offDiagonal
    {n : ℕ} (G H : WeightedLedgerGraph n)
    (hweight :
      ∀ i j : Fin n, i ≠ j → G.weight i j = H.weight i j)
    (ε : Fin n → ℝ) :
    exactJCostAction G ε = exactJCostAction H ε := by
  unfold exactJCostAction
  apply Finset.sum_congr rfl
  intro i _
  apply Finset.sum_congr rfl
  intro j _
  by_cases hij : i = j
  · subst j
    simp
  · rw [hweight i j hij]
THEOREM globalTorusGraph3_violates_positiveRealizedProductionActionIdentity · IndisputableMonolith/Foundation/PairKernelActionExtensionalityS7.lean
globalTorusGraph3_violates_positiveRealizedProductionActionIdentity · IndisputableMonolith/Foundation/PairKernelActionExtensionalityS7.lean:1008 
/-- 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 is false for all graphs; the canonical posting graph satisfies it. The counterexample graph is the only one that violates the identity. The framework's cost function is unable to determine any graph structure; it can determine off-diagonal entries.

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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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