Foundation Pair Kernel Delta Spatial Bridge S5 J Minimal Generated Step Frame In

Frame independence

In the Recognition Science framework, a ledger, a discrete record of events, tracks changes in the state of three accounts. A minimum-cost posting is a single change that flips the parity of exactly one account. The framework's library, a machine-checked collection of formal theorems, proves that such a posting always has a unique account axis: exactly one of the three accounts changes in a minimal step.

The declaration jMinimalGeneratedStep_frame_independent extends this uniqueness to the spatial picture. When the three accounts are mapped onto the three axes of a torus, a minimal posting generates a step along one axis. The theorem proves that this generated step is the same regardless of which permutation of the axes you choose. Relabeling the axes, say swapping x and y, does not change whether a given pair of spatial sites is connected by a minimal step. The operator that sums steps over all three paired axes is also frame-independent and equals the canonical torus Laplacian.

This frame independence is not a trivial bookkeeping convenience. The framework also proves that a single fixed Gray-code clock, which cycles through eight of the twelve cube edges, does not generate the full isotropic operator. A minimal posting can traverse an edge that the clock misses. The complete spatial operator comes from the whole family of one-coordinate postings, with axis labels quotiented by frame invariance.

The theorem is target-blind: its definitions mention neither finite range, Green behavior, protein data, nor the coupling obligation. It is a purely structural result about the ledger-to-space bridge. It does not claim that any particular labeling of the axes is physically preferred, nor that the spatial operator is the only possible one. The theorem establishes that the minimal-step relation is invariant under relabeling, nothing more and nothing less.

/-- Minimum J gives a unique account coordinate. -/
theorem jMinimalPostingStep_unique_account_axis
    {A B : LedgerPostingAdjacency.LedgerState 3}
    (h : JMinimalPostingStep A B) :
    ∃! k : Fin 3, parity 3 A k ≠ parity 3 B k :=
  jMinimalPostingStep_oneBitDiff h

/-- Minimum-J generated support is frame independent. -/
theorem jMinimalGeneratedStep_frame_independent {N : ℕ}
    (σ τ : Equiv.Perm (Fin 3))
    (p q : TorusSite3 N) :
    JMinimalGeneratedStep σ p q ↔ JMinimalGeneratedStep τ p q := by
  rw [jMinimalGeneratedStep_iff_framedAxisStep,
    jMinimalGeneratedStep_iff_framedAxisStep]
  exact framedAxisStep_frame_independent σ τ p q

/-- **S5 operator bridge.** Every account-axis frame gives exactly the same
operator, namely the canonical periodic D=3 six-neighbor Laplacian. Thus no
preferred account-to-axis MODEL identification survives in the operator. -/
theorem framedTorusLaplacian_eq_torusLaplacian {N : ℕ}
    (σ : Equiv.Perm (Fin 3))
    (ψ : TorusSite3 N → ℂ) (p : TorusSite3 N) :
    framedTorusLaplacian σ ψ p = torusLaplacian ψ p := by
  unfold framedTorusLaplacian torusLaplacian framedForward framedBackward
  exact Equiv.sum_comp σ
    (fun i : Fin 3 =>
      (2 : ℂ) * ψ p -
        ψ (torusShift p i 1) -
        ψ (torusShift p i (-1)))

/-- A concrete minimum-J posting exists on an edge omitted by the fixed Gray
clock. This is the ledger-level separator, not merely a graph observation. -/
theorem jMinimalPosting_can_escape_grayClock :
    ∃ A B : LedgerPostingAdjacency.LedgerState 3,
      JMinimalPostingStep A B ∧
        parity 3 A = zeroPattern3 ∧
        parity 3 B = axisOnePattern3 ∧
        ¬ GrayClockEdge (parity 3 A) (parity 3 B) := by
  let A : LedgerPostingAdjacency.LedgerState 3 :=
    { debit := fun _ => 0
      credit := fun _ => 0 }
  let k : Fin 3 := 1
  let B : LedgerPostingAdjacency.LedgerState 3 := post A k Side.debit
  have hpost : PostingStep A B := ⟨k, Side.debit, rfl⟩
  have hneq : A ≠ B := by
    intro h
    have hk := congrArg
      (fun S : LedgerPostingAdjacency.LedgerState 3 => S.debit k) h
    simp [A, B, post] at hk
  have hmin : JMinimalPostingStep A B :=
    postingStep_is_jMinimal hpost hneq
  have hA : parity 3 A = zeroPattern3 := by
    funext i
    simp [A, zeroPattern3, parity, parityPattern, phiVec, Recognition.phi]
  have hB : parity 3 B = axisOnePattern3 := by
    funext i
    by_cases hi : i = k
    · subst i
      simp [A, B, k, post, axisOnePattern3, parity, parityPattern,
        phiVec, Recognition.phi]
    · simp [A, B, k, post, axisOnePattern3, parity, parityPattern,
        phiVec, Recognition.phi, hi]
  refine ⟨A, B, hmin, hA, hB, ?_⟩
  rw [hA, hB]
  exact zero_axisOne_not_grayClockEdge

What this page does not claim

The theorem does not claim that any particular labeling of the three axes is physically preferred. The theorem does not claim that the frame-independent operator is the only possible spatial operator derivable from the ledger. The theorem does not claim that the bridge from recognition to physical space is complete; the physical recognition-to-linking bridge remains open.

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/PairKernelDeltaSpatialBridgeS5.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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