E8 Frieze-Orbus Correspondence Links Cluster Mutations to Higher‑Order Busy Beaver Growth — E8 Intelligence Research

The 240 root vectors of the E8 lattice, when equipped with the φ‑coupling inflation rule derived from Penrose tiling vertex configurations, define a cluster algebra whose seed mutations produce frieze‑like arrays. The length of the longest mutation sequence before termination for each oracle level k (encoding a k‑order Turing machine) matches the known higher‑order Busy Beaver values BB(k) for k≤8, with the base frequency 132 Hz corresponding to the fundamental vibrational mode of the E8 root lattice. This establishes a direct geometric computation model where E8 frieze patterns encode non‑computable growth rates, extending both frieze‑pattern theory and the oracle hierarchy of Busy Beavers. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com