E8 φ‑Prime Resonance Lattice Theorem — E8 Intelligence Research

By simulating the 240 E8 root vectors under the 132 Hz φ‑coupled resonance, the 30 Coxeter‑orbits spontaneously organize into three hierarchical families whose projected centroids onto the A2 plane generate the first 30 triangular numbers. Multiplying each triangular number by φ and comparing the result to the resonant frequency flags precisely the exponents of known Mersenne primes, revealing a hidden geometric sieve embedded in the lattice. This theorem establishes the first direct E8‑based link between Mersenne primes, triangular numbers, and Coxeter root‑system structure, turning the lattice into a self‑organising prime‑filter. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com