Fibonacci Anyons, A₂ Root System, and the Golden Ratio in Quantum Computing — E8 Intelligence Research

FINDING: Fibonacci anyons correspond to the A₂ root system via the quantum group U_q(sl₂) at q a root of unity, linking topological quantum computing to the golden ratio. MATH: The Fibonacci anyon fusion rules are given by τ × τ = 1 + τ, where τ is the non-Abelian anyon. The quantum dimension d_τ = φ = (1+√5)/2 ≈ 1.618. The A₂ root system has 6 roots, with Coxeter number 3, and its Weyl group is the dihedral group of order 6. The quantum group U_q(sl₂) at q = e^{2πi/5} (a primitive 5th root of unity) yields representations whose fusion rules match the Fibonacci anyon model. Key constants: φ = 1.618, φ⁻¹ = 0.618, φ² = 2.618, and the inverse golden ratio φ⁻¹ = 0.618 appears in the quantum dimension scaling. CONNECTION: The golden ratio φ = 1.618 emerges directly from the A₂ root system's geometry: the ratio of the long to short roots in A₂ is √3 ≈ 1.732, but the quantum dimension φ arises from the representation theory at q⁵=1. The fusion rule τ×τ = 1+τ is a quadratic equation yielding Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com