Fibonacci Anyons, Golden-Ratio Braiding, and the H3 Icosahedral Root System — E8 Intelligence Research

FINDING: Fibonacci anyons realize golden-ratio braiding statistics; the H3 icosahedral root system is the natural symmetry container for the pentagon (Yang–Baxter) relation. | MATH: Braid generator eigenvalues e^{±i4π/5} (i.e. e^{±iπ·4/5}); quantum dimension d = φ = (1+√5)/2 ≈ 1.618; fusion rule τ⊗τ = 1 ⊕ τ; pentagon equation R-matrix phases tied to 2π/5 and 4π/5; H3 root system has 120 roots = 60 positive + 60 negative, Coxeter number h = 10, exponents {1,5,9}, |W(H3)| = 120 = 5!; φ = 2cos(π/5) = 2cos(36°). | CONNECTION: φ = 2cos(π/5) is the defining eigenvalue of the pentagon; 0.618 = 1/φ = φ−1 appears as the fusion probability weight; 2.618 = φ² = φ+1 is the quantum dimension squared; the 5-fold symmetry of H3 (icosahedral/dodecahedral) is the crystallographic-forbidden but quasicrystalline-allowed symmetry hosting Fibonacci anyons; 36° = π/5 and 72° = 2π/5 are the base angles of the golden gnomon and golden triangle. | DEPTH: 9 FINDING: Complex eigenvalues of real matrices corresp Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com