Theorem 7: Minimal Recognition Period T = 2³ = 8
Hypercube visit + atomic tick ⇒ minimal period; generalized T = 2^D
Statement
On the 3‑cube, any ledger‑compatible recognition has minimal period T = 8; more generally T = 2^D on QD.
In plain English
Want the shortest loop that touches every corner of a cube without cheating and with one move per click? It’s eight clicks. That’s an 8‑beat rhythm baked into 3D recognition. In D dimensions, the beat is 2^D.
- Why inevitable: you must visit all vertices (count = 8), and atomicity allows one per tick; Gray code hits the bound.
- What it buys: a natural “bar length” for cycles that larger structures can sync to.
Sketch
- Spatial completeness requires visiting |V| distinct vertices per period.
- Atomicity gives one hop per tick, so T ≥ |V|.
- A Gray code cycle provides existence and shows minimality.