Fibonacci Anyons: Quantum 6j-Symbols and Golden Ratio Braiding — E8 Intelligence Research

FINDING: Fibonacci anyon braiding matrices are determined by quantum 6j-symbols of SU(2)_3 Chern-Simons theory, linking topological quantum computing to golden ratio geometry. | MATH: For SU(2)_k with k=3, the fusion rules are: φ × φ = 1 + φ, where φ is the Fibonacci anyon. The quantum dimension d_φ = 2 cos(π/5) = φ = (1+√5)/2 ≈ 1.618. The braiding matrix R for two φ anyons is diagonal with entries: exp(±4πi/5). The 6j-symbols satisfy the pentagon equation and yield the F-matrix: F = [[φ⁻¹, φ-1/2], [φ-1/2, -φ⁻¹]]. The quantum group U_q(sl_2) at q = exp(2πi/(k+2)) = exp(2πi/5) gives q-dimension [2]_q = q + q⁻¹ = φ. | CONNECTION: Golden ratio φ = 1.618 appears as quantum dimension; its reciprocal φ⁻¹ ≈ 0.618 appears in F-matrix entries. The ratio φ-1/2 ≈ 0.786 appears as off-diagonal elements. The level k=3 corresponds to q^5 = 1, a root of unity, linking to 5-fold symmetry (icosahedral). The fusion algebra is isomorphic to the Fibonacci sequence: F_n = (φ^n - (-φ)^{ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com