Fibonacci Anyon Braiding Links 5th Roots of Unity to Icosahedral Symmetry via Golden Ratio — E8 Intelligence Research

FINDING: Fibonacci anyon braiding R-matrix eigenvalues are 5th roots of unity, and the H3 icosahedral Coxeter group shares the golden-ratio characteristic polynomial — linking non-Abelian topological order to icosahedral symmetry via φ. MATH: - Fibonacci fusion rule: [1] × [1] = [0] + [1] (simplest non-Abelian anyon model) - Quantum dimension: d = φ = (1+√5)/2 ≈ 1.618 - R-matrix eigenvalues for Fibonacci braiding: R = e^(±4πi/5), i.e., primitive 5th roots of unity {e^(2πik/5)}, k ∈ {±2} - Braid group representation: ρ: B_n → U(2) generated by R and F (Fibonacci F-matrix), with (R)^5 = I - H3 Coxeter group: finite reflection group of the icosahedron/dodecahedron, order 120 - H3 characteristic polynomial / invariant degrees: {2, 6, 10}, exponents {1, 5, 9} - Coxeter number h = 10; largest exponent = 9 = h − 1 - Golden ratio appears as eigenvalue of the Cartan matrix of H3: φ and −1/φ - Excess function: e(w) = min{ℓ(x) + ℓ(y) − ℓ(w) | w = xy, x² = y² = 1} - Strongly real elements: W = 𝒲 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com