Foundation Pair Kernel Signed Posting Transport S11 Signed Recognition Productio

Frame independence of the signed graph

In the Recognition Science framework, a ledger is a discrete record of events, and a recognition event is a posting between two accounts. The declaration signedRecognitionProductionGraph3_weight_frame_independent is a theorem in the framework's machine-checked library of formal theorems. It states that a certain weighted graph, built from signed posting events on a three-axis torus, is unchanged when the three axes are relabeled. In plainer terms: the graph's edge weights do not depend on which axis you call first, second, or third.

The graph in question is signedRecognitionProductionGraph3. It assigns a weight to each ordered pair of sites on a finite torus. The weight is the number of forward and backward posting events along each of the three axes, counted with sign. When periodicity makes a forward and a backward event land on the same ordered pair, the weight is 2, not 1. The theorem signedPostingWeight3_frame_independent proves this count is invariant under any permutation of the three axes. The graph-level theorem then extends that invariance to every edge weight of the full graph.

This matters because the framework's earlier construction, called S8, stored realized ordered pairs as a relation set. At period two, that set forgot one occurrence when forward and backward events coincided. The signed graph repairs that collision by retaining both occurrences. The frame independence theorem shows the repair is stable: relabeling axes does not change any weight. A separate theorem, pulledBackSignedPostingLaplacian3_eq_framed, proves the graph's Laplacian equals the framed six-neighbor operator for every nonempty period, including period two, with no collision-free hypothesis needed.

The theorem does not claim that the graph's support is frame independent in a trivial way, nor that the weights themselves are derived from physical measurement. It claims only a formal invariance property of a defined object. The framework's library proves the statement with no sorry and no new axiom; the audit rests on the ambient type theory's standard postulates, not on a premise-free foundation. The physical interpretation of the graph as a model of recognition production is a definitional choice, not a proved physical law.

/-- Account-axis relabeling preserves every constructor coefficient. -/
theorem signedRecognitionProductionGraph3_weight_frame_independent
    {N : ℕ} [NeZero N]
    (σ τ : Equiv.Perm (Fin 3))
    (i j : Fin (TorusCard3 N)) :
    (signedRecognitionProductionGraph3 N σ).weight i j =
      (signedRecognitionProductionGraph3 N τ).weight i j :=
  signedPostingWeight3_frame_independent σ τ _ _

/-- Signed generator-event multiplicity between two torus sites. Each axis has
one forward and one backward occurrence, even when periodicity makes their
targets coincide. -/
def signedPostingWeight3
    {N : ℕ}
    (σ : Equiv.Perm (Fin 3))
    (p q : TorusSite3 N) : ℝ :=
  ∑ k : Fin 3,
    ((if q = framedForward σ p k then 1 else 0) +
      (if q = framedBackward σ p k then 1 else 0))

/-- The multiplicity-preserving graph fold is exactly the framed six-neighbor
operator. No collision-free hypothesis is needed. -/
theorem pulledBackSignedPostingLaplacian3_eq_framed
    {N : ℕ} [NeZero N]
    (σ : Equiv.Perm (Fin 3))
    (field : TorusSite3 N → ℝ)
    (p : TorusSite3 N) :
    (pulledBackSignedPostingLaplacian3 σ field p : ℂ) =
      framedTorusLaplacian σ
        (fun x => (field x : ℂ)) p := by
  have hreal :
      pulledBackSignedPostingLaplacian3 σ field p =
        ∑ k : Fin 3,
          ((2 : ℝ) * field p -
            field (framedForward σ p k) -
            field (framedBackward σ p k)) := by
    unfold pulledBackSignedPostingLaplacian3 laplacian
      signedRecognitionProductionGraph3
    rw [← Equiv.sum_comp (torusSiteEquivFin N)]
    simp only [Equiv.symm_apply_apply, Equiv.apply_symm_apply]
    unfold signedPostingWeight3
    calc
      ∑ q : TorusSite3 N,
          (∑ k : Fin 3,
            ((if q = framedForward σ p k then (1 : ℝ) else 0) +
              (if q = framedBackward σ p k then 1 else 0))) *
            (field p - field q) =
          ∑ q : TorusSite3 N, ∑ k : Fin 3,
            (((if q = framedForward σ p k then (1 : ℝ) else 0) +
              (if q = framedBackward σ p k then 1 else 0)) *
                (field p - field q)) := by
                  apply Finset.sum_congr rfl
                  intro q _
                  rw [Finset.sum_mul]
      _ = ∑ k : Fin 3, ∑ q : TorusSite3 N,
            (((if q = framedForward σ p k then (1 : ℝ) else 0) +
              (if q = framedBackward σ p k then 1 else 0)) *
                (field p - field q)) := Finset.sum_comm
      _ = ∑ k : Fin 3,
            ((field p - field (framedForward σ p k)) +
              (field p - field (framedBackward σ p k))) := by
                apply Finset.sum_congr rfl
                intro k _
                simp only [add_mul, Finset.sum_add_distrib]
                simp
      _ = ∑ k : Fin 3,
            ((2 : ℝ) * field p -
              field (framedForward σ p k) -
              field (framedBackward σ p k)) := by
                apply Finset.sum_congr rfl
                intro k _
                ring
  rw [hreal]
  unfold framedTorusLaplacian
  push_cast
  rfl

What this page does not claim

The theorem does not claim the graph's weights are physically measured values. The theorem does not claim the graph is the only possible repair for the S8 collision. The theorem does not claim the framework's physical interpretation of the graph is a proved law.

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/PairKernelSignedPostingTransportS11.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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