Foundation T7 Cycle Realization

The circle from a cycle

A Gray cycle is a sequence of binary strings, each differing from the next by exactly one bit, that returns to its start after visiting every possible string once. For three bits, the cycle runs 000, 001, 011, 010, 110, 111, 101, 100, then back to 000. This object has been known since 1953, when Frank Gray patented it for pulse-code encoding, and it appears in telegraphy, rotary encoders, and puzzle design. The cycle's defining property is that consecutive entries flip a single digit, a constraint that makes it a Hamiltonian cycle on the three-dimensional cube graph.

The framework Recognition Science studies how a discrete record of events, its ledger, can force geometric structure. Within that framework, the T7 cycle is the eight-tick recognition cycle that emerges from the golden ratio and the cost function J(x) = (x + 1/x)/2 - 1. The module T7CycleRealization takes this cycle and asks what shape its closed walk realizes. The answer, proved in the machine-checked library of formal theorems, is a circle.

The proof works in two steps. First, a theorem shows that any closed walk on a cube graph that is Hamiltonian, meaning it visits every vertex exactly once, and edge-distinct, meaning consecutive steps flip one bit, has a combinatorial circle image. Second, a specialization shows that the canonical three-bit Gray cycle satisfies both conditions. The realized defect, the shape the cycle traces in a cellular completion, is definitionally set to Circle. A further theorem rules out higher spheres: no closed walk on a cube graph realizes S^p for any p ≥ 2.

What this establishes in plain language is that the eight-step recognition cycle, when embedded as a graph walk, has the topology of a circle and nothing higher. The framework's library proves this for the explicit Gray cycle on three bits, and the result extends to any dimension D ≥ 2 where a Hamiltonian, edge-distinct closed walk exists. The consequence is that the T7 cycle cannot produce a two-dimensional sphere or any higher-dimensional analogue; it is confined to one-dimensional circular structure.

This matters because it constrains what the recognition cycle can generate. A circle is the only sphere dimension available to a graph-shaped closed walk, so any attempt to derive higher-dimensional defects from the T7 cycle alone fails. The module thus closes a route toward three-dimensional space, leaving that derivation to the separate T8 dimension-forcing results. The circle result is a clean topological fact about Gray cycles, independent of the framework's broader claims.

/-- Combinatorial circle from Hamiltonian plus edge-distinct hypotheses. -/
theorem closed_walk_image_is_circle
    {D : ℕ} (W : ClosedWalkOnCube D)
    (hHam : Hamiltonian W) (hEdge : EdgeDistinct W) :
    ImageIsCircle W :=
  ⟨hHam, hEdge⟩

/-- Part (iv): no closed walk in a graph-shaped cube realizes a higher sphere. -/
theorem no_higher_sphere_from_closed_walk
    {D : ℕ} (W : ClosedWalkOnCube D) (p : ℕ) (hp : 2 ≤ p) :
    ¬ ImageIsSpherePofDim W p := by
  intro hp1
  dsimp [ImageIsSpherePofDim] at hp1
  subst p
  omega

/-- Specialization of the realization theorem to the canonical 3-bit Gray cycle. -/
theorem grayCycle3_realizes_circle
    (cell : SubstrateAxioms.CellularCompletion 3) :
    RealizedDefect cell grayCycle3ClosedWalk = Circle := by
  exact t7_cycle_realizes_circle 3 (by decide) cell
    grayCycle3ClosedWalk grayCycle3ClosedWalk_hamiltonian

What this page does not claim

The module does not prove that the T7 cycle produces three-dimensional space; that is the separate T8 result. The realized defect is definitionally set to Circle at the predicate layer, not constructed as a full CW complex. The framework does not claim that every Hamiltonian closed walk is edge-distinct; the former theorem claiming this was deleted as false.

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/T7CycleRealization.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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