Fibonacci Anyon Braiding: Golden Ratio Eigenvalues and SL(2,Z) in Topological Quantum Computing — E8 Intelligence Research

FINDING: Fibonacci anyon braiding matrices yield eigenvalues that are powers of the golden ratio, linking directly to the modular group SL(2,Z) and topological quantum computation. MATH: - Fibonacci anyon fusion rules: τ × τ = 1 + τ (where τ is the non-Abelian anyon). - Braiding matrix eigenvalues: exp(±i4π/5) and exp(±i2π/5), which are powers of φ = (1+√5)/2 ≈ 1.618. - Modular group SL(2,Z) generators: S (modular inversion) and T (translation). For Fibonacci anyons, S and T matrices have entries involving φ and its inverse φ⁻¹ = φ - 1 ≈ 0.618. - Key constants: φ = 1.618, φ⁻¹ = 0.618, φ² = 2.618, φ⁻² = 0.382. - Binet formula: F_n = (φ^n - (-φ)^{-n})/√5. CONNECTION: - Golden ratio φ and its reciprocal φ⁻¹ appear as eigenvalues of the braiding matrix, directly encoding the Fibonacci sequence in topological quantum gates. - The modular group SL(2,Z) acts on the space of conformal field theories; its representation via Fibonacci anyons yields a 2-dimensional representation Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com