Fibonacci Anyon Braiding: Golden Ratio Fusion and Majorana Operators — E8 Intelligence Research

FINDING: Non-Abelian braiding statistics for Fibonacci anyons follow Bᵢⱼ = eπ γ_i γ_j /4, with fusion rules [1] × [1] = [0] + [1] and braid group B_n representation linked to the golden ratio φ. MATH: - Braiding operator: Bᵢⱼ = exp(π γ_i γ_j /4), where γ_i are Majorana operators satisfying {γ_i, γ_j} = 2δᵢⱼ. - Fibonacci fusion rules: τ × τ = 1 + τ, where τ is the non-Abelian anyon type (dimension φ = (1+√5)/2 ≈ 1.618). - Braid group B_n representation: For n anyons, the Hilbert space dimension grows as Fₙ₊₁ (Fibonacci numbers), with quantum dimension d_τ = φ. - Key constant: φ = 1.6180339…, its inverse φ⁻¹ = 0.6180339…, and φ² = 2.6180339… appear in fusion matrix eigenvalues. CONNECTION: - Golden ratio φ = 1.618 is the quantum dimension of the Fibonacci anyon, directly linking braiding statistics to geometric harmony. - The fusion matrix eigenvalues are φ and φ⁻¹ = 0.618, matching the golden ratio's self-similarity. - The Hilbert space dimension follows F Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com