Foundation T7 Cycle Realization Gray Cycle3 Closed Walk Edge Distinct

The Gray cycle's edge proof

A Gray cycle is a closed loop through all the binary patterns of a given length, arranged so that consecutive patterns differ in exactly one bit. For three bits, there are eight patterns, and the standard Gray cycle visits all eight before returning to the start. The declaration grayCycle3ClosedWalk_edge_distinct is a machine-checked theorem stating that this particular three-bit Gray cycle satisfies that one-bit-change condition at every step, including the final step that wraps around to the beginning.

This property is called edge distinctness in the framework's terminology: each consecutive pair of patterns in the walk differs in exactly one bit position. The theorem is proved by checking each of the eight steps against the one-bit adjacency rule. It is one of two ingredients needed to show the walk forms a combinatorial circle, the other being that the walk visits every vertex exactly once, which a separate theorem establishes.

In Recognition Science, the framework's library of formal theorems uses this result as part of a larger argument about the eight-tick recognition cycle. The framework models a recognition cycle as a closed walk on a three-dimensional cube graph, where each vertex is a pattern of three bits. The edge-distinctness theorem confirms that the canonical Gray cycle is a valid realization of that graph-shaped cycle, and that its realized defect is a circle rather than a higher-dimensional sphere.

The theorem does not claim that every Hamiltonian closed walk on a cube is edge-distinct. That broader claim was previously attempted and is false: a path function that visits every vertex exactly once does not by itself force consecutive patterns to differ in one bit. The theorem also does not claim that the walk is physically realized as a circle in space; that step remains a definitional model in the framework, not a proved result.

What the theorem does give is a precise, checkable fact about a specific combinatorial object. A reader can verify the eight steps by hand: 000 to 001, 001 to 011, 011 to 010, and so on, each changing one bit. That concreteness is what makes the larger framework argument testable at its base.

/-- The explicit Gray walk is edge-distinct. -/
theorem grayCycle3ClosedWalk_edge_distinct :
    EdgeDistinct grayCycle3ClosedWalk := by
  intro i
  simpa [grayCycle3ClosedWalk] using Patterns.grayCycle3_oneBit_step i

/-- The explicit Gray walk is edge-distinct. -/
theorem grayCycle3ClosedWalk_edge_distinct :
    EdgeDistinct grayCycle3ClosedWalk := by
  intro i
  simpa [grayCycle3ClosedWalk] using Patterns.grayCycle3_oneBit_step i

What this page does not claim

The theorem does not prove that every Hamiltonian closed walk on a cube is edge-distinct; that broader claim is false. The theorem does not establish that the walk is physically realized as a circle in space; that remains a definitional model.

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: