Fibonacci Anyon Fusion and Golden Ratio in SU(2)_3 S-Matrix — E8 Intelligence Research

FINDING: SU(2)_3 quantum group modular S-matrix entries are directly related to Fibonacci anyon fusion rules via the golden ratio. MATH: For SU(2)_3, the modular S-matrix is a 2x2 matrix (since there are two primary fields: identity 1 and the Fibonacci anyon τ). The entries are: S = (1/√(φ+2)) * [[1, φ], [φ, -1]] where φ = (1+√5)/2 ≈ 1.618. The fusion rules are: τ × τ = 1 + τ, which is the Fibonacci fusion algebra. The quantum dimension of τ is d_τ = φ. The S-matrix satisfies S^2 = C (charge conjugation), and the Verlinde formula gives fusion coefficients N_{ij}^k = Σ_m (S_{im} S_{jm} S_{km}^*)/S_{1m}. CONNECTION: The golden ratio φ = 1.618 and its inverse φ^{-1} = 0.618 appear directly. The ratio of S-matrix entries S_{ττ}/S_{1τ} = φ. The quantum dimension d_τ = φ is a root of x^2 = x + 1. The SU(2)_3 root system is A_1, with Coxeter number 3, and the Weyl group is Z_2. The modular S-matrix eigenvalues are related to the Cartan matrix of A_1. The Fibonacci anyon system is the simple Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com