eight_tick_minimal
The theorem establishes that the Gray code cycle length on three dimensions equals eight. Researchers formalizing discrete ledger structures in Recognition Science cite it to confirm the forced minimal period for covering all 3-bit patterns under Gray code traversal. The proof is a direct one-line wrapper that unfolds the cycle length definition and evaluates the resulting power of two by normalization.
claimThe Gray code cycle length for three dimensions equals eight, i.e., $2^3 = 8$.
background
The module addresses why discrete conservative systems must adopt specifically the golden ratio, three-dimensional cube, and eight-tick cycle. The Gray code cycle length is defined as the function returning $2^D$ for dimension $D$, representing the period needed to traverse all bit patterns via adjacent flips. This sits inside the ledger uniqueness argument that each component solves a unique forcing constraint: cost fixed point for phi, topological linking for dimension three, and Gray code compatibility for the cycle length.
proof idea
The proof is a one-line wrapper that unfolds the definition of the Gray code cycle length and applies numerical normalization to obtain the equality.
why it matters in Recognition Science
This declaration supplies the eight-tick half of the combined uniqueness theorem, which states that phi, the three-dimensional cube, and the eight-tick cycle are all forced. It closes the ledger uniqueness gap by showing no shorter cycle can cover the cube traversal, aligning with the eight-tick octave step in the forcing chain.
scope and limits
- Does not prove cycle minimality for dimensions other than three.
- Does not address traversals outside Gray code adjacency.
- Does not apply to continuous or non-discrete dynamical systems.
- Does not derive the cycle length from first principles beyond the definition.
formal statement (Lean)
156theorem eight_tick_minimal :
157 grayCodeCycleLength 3 = 8 := by
proof body
Term-mode proof.
158 unfold grayCodeCycleLength
159 norm_num
160
161/-- No shorter cycle covers the cube. -/