Foundation Cycle Operator Cycle Perm Injective

The cycle permutation

The recognition cycle, a discrete record of events in the framework's account, walks through eight states. The walk follows a Gray code path, a standard sequence where each step changes exactly one binary digit. The path visits the vertices in the order 0, 1, 3, 2, 6, 7, 5, 4, then returns to 0. The declaration cyclePerm_injective proves that this walk never revisits a vertex before the cycle completes. In mathematical terms, the map from each vertex to its successor is injective, meaning distinct starting points always lead to distinct next states.

This injectivity is one of several properties that the framework's machine-checked library of formal theorems verifies for the cycle. The same library also proves that the cycle has period eight, that no shorter number of steps returns a vertex to itself, and that each step is a single bit flip. Together these results certify that the cycle is a genuine permutation of the eight vertices, a rearrangement with no collisions and no skipped states.

The proof itself is a direct computation over the eight possible vertices. The machine-checked library, a collection of formal theorems verified by a computer, checks each case and confirms the map is injective. This is a finite fact, not a general theorem about all cycles or all Gray codes. It holds for this specific eight-step sequence on the three-bit cube.

What the declaration does not claim is broader. It does not assert that the cycle is the only possible recognition cycle, nor that the Gray code path is unique. It does not claim that the cycle's structure forces any physical property by itself. The injectivity is a building block, one of several properties that the framework's library assembles into a certificate for the cycle operator. The physical significance, such as the emergence of mixing angles, comes from the eigenvalues and eigenstates of the full operator, not from injectivity alone.

/-- The cycle permutation is injective (hence bijective on Fin 8). -/
theorem cyclePerm_injective : Function.Injective cyclePerm := by
  intro a b h
  fin_cases a <;> fin_cases b <;> simp_all [cyclePerm, grayOrderInv, grayOrder, gray8At]

/-- The cycle permutation has period exactly 8. -/
theorem cyclePerm_period : ∀ v, (cyclePerm^[8]) v = v := by
  intro v; fin_cases v <;> native_decide

/-- After fewer than 8 iterations, the permutation is NOT the identity. -/
theorem cyclePerm_not_identity_before_8 :
    ∀ k, 0 < k → k < 8 → ∃ v, (cyclePerm^[k]) v ≠ v := by
  intro k hk hk8
  interval_cases k <;> exact ⟨0, by native_decide⟩

/-- Each step of the cycle equals a single bit flip (the one identified by flippedBit). -/
theorem cycle_step_is_bitflip (t : Fin 8) :
    cyclePerm (grayOrder t) = bitFlipOp (flippedBit t) (grayOrder t) := by
  fin_cases t <;> native_decide

What this page does not claim

The declaration does not claim the Gray code cycle is the only possible recognition cycle. The declaration does not claim injectivity alone forces any physical property. The declaration does not claim the cycle operator's eigenvalues or eigenstates are derived from injectivity.

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

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