Golden Ratio Eigenvalues in SU(2)₃ Chern-Simons Braiding — E8 Intelligence Research

FINDING: SU(2)₃ Chern-Simons theory braiding matrix eigenvalues are powers of the golden ratio, linking quantum topology to quadratic irrationals. | MATH: For SU(2)₃, the braiding matrix eigenvalues are exp(±2πi·3/8) and exp(±2πi·1/8) in the spin-1/2 and spin-1 channels; the quantum dimension d = [2]ₙ = sin(π·3/5)/sin(π/5) = φ = (1+√5)/2 ≈ 1.618. The S-matrix entry S_{0,1/2} = √(2/5) sin(π/5) = √(φ⁻¹/√5) ≈ 0.5878. The fusion rules: 1/2 ⊗ 1/2 = 0 ⊕ 1, 1/2 ⊗ 1 = 1/2, 1 ⊗ 1 = 0. | CONNECTION: Golden ratio φ = 1.618 appears as quantum dimension of fundamental representation; φ⁻¹ = 0.618; φ² = 2.618; 1/φ = 0.618. The level k=3 gives q = exp(2πi/5), linking to pentagonal symmetry (crystallographic point group 5m). The braiding eigenvalues are roots of unity of order 8 and 5, connecting to base-60 sexagesimal fractions (1/8=0.125, 1/5=0.2) and the 5-fold symmetry of icosahedral quasicrystals. | DEPTH: 9 — Direct embedding of golden ratio into topological quantum field theory; braiding matrice Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com