Foundation T7 Cycle Realization Gray Cycle3 Closed Walk Image Is Circle

The Gray cycle as a circle

A Gray code is a sequence of binary strings in which each consecutive pair differs in exactly one bit. The classic 3-bit Gray code runs 000, 001, 011, 010, 110, 111, 101, 100, and then wraps back to 000, which also differs by one bit. This wrap-around makes the sequence a closed walk on the vertices of a cube, where each vertex is a 3-bit string and each edge connects strings that differ in one bit. The walk visits all eight vertices exactly once, a property called Hamiltonian, and every step, including the final wrap, changes exactly one bit, a property called edge-distinct.

The declaration grayCycle3ClosedWalk_image_is_circle is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It states that this specific Gray cycle, read as a closed walk on the 3-cube, has a combinatorial circle image: it is both Hamiltonian and edge-distinct. The proof is direct: it combines two earlier theorems, one showing the walk visits every vertex exactly once and another showing each consecutive pair differs by one bit. A companion theorem, grayCycle3_no_higher_sphere, proves that no closed walk on a cube graph realizes a sphere of dimension two or higher. Together they establish that the T7 recognition cycle, a structural feature of the framework's cost-forcing chain, has the combinatorial shape of a circle and not a higher-dimensional sphere.

In Recognition Science, the framework models a ledger, a discrete record of recognition events, and derives an eight-tick recognition cycle as part of its forcing chain. The T7 cycle realization module adds a theorem surface connecting that cycle to graph theory. The term RealizedDefect, the framework's label for the shape a closed walk produces, is defined to be the circle at this layer. The framework proves the combinatorial fact: the canonical 3-bit Gray cycle is a circle. What remains a MODEL, not a theorem, is the full CW/covering realization of the walk as a topological circle S¹. The predicate layer records the conclusion, but the continuous geometric realization is not yet formalized.

The consequence for a reader is concrete: within the framework, the eight-tick cycle has the combinatorial signature of a circle, not a higher sphere. This supports the framework's route toward three spatial dimensions, where the cycle's circular shape is one step. The framework does not claim that every Hamiltonian closed walk on a cube is edge-distinct; that statement is false, since a path function that is bijective does not force one-bit steps. The Gray cycle and its explicit 3-bit instance are the real content, and they are what the theorem establishes.

/-- The explicit Gray walk has combinatorial circle image. -/
theorem grayCycle3ClosedWalk_image_is_circle :
    ImageIsCircle grayCycle3ClosedWalk :=
  closed_walk_image_is_circle grayCycle3ClosedWalk
    grayCycle3ClosedWalk_hamiltonian grayCycle3ClosedWalk_edge_distinct

/-- The explicit Gray walk has combinatorial circle image. -/
theorem grayCycle3ClosedWalk_image_is_circle :
    ImageIsCircle grayCycle3ClosedWalk :=
  closed_walk_image_is_circle grayCycle3ClosedWalk
    grayCycle3ClosedWalk_hamiltonian grayCycle3ClosedWalk_edge_distinct

/-- The canonical 3-bit Gray cycle does not realize as `S^p` for any `p ≥ 2`. -/
theorem grayCycle3_no_higher_sphere (p : ℕ) (hp : 2 ≤ p) :
    ¬ ImageIsSpherePofDim grayCycle3ClosedWalk p :=
  no_higher_sphere_from_closed_walk grayCycle3ClosedWalk p hp

/-- A Gray cycle's path, read as a closed walk, is edge-distinct by the
Gray-cycle adjacency field. -/
theorem edge_distinct_of_grayCycle (D : ℕ) (W : Patterns.GrayCycle D) :
    EdgeDistinct ⟨W.path⟩ :=
  W.oneBit_step

What this page does not claim

The full CW/covering realization of the walk as a topological circle S¹ is proved; it remains a MODEL. Every Hamiltonian closed walk on a cube is edge-distinct; the framework explicitly deletes that false claim. The T7 cycle realization theorem alone derives three spatial dimensions; it is one step in a larger chain.

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)

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: