Golden Ratio Eigenvalues Link Fibonacci Anyon Braiding to Modular Group SL(2,Z) — E8 Intelligence Research

FINDING: Fibonacci anyon braiding matrices yield eigenvalues equal to the golden ratio, linking topological quantum computing to modular group SL(2,Z) via the Fibonacci sequence's matrix representation. MATH: - Fibonacci recurrence: \( Fₙ₊₁ = F_n + Fₙ₋₁ \) → matrix \( M = {pmatrix} 1 & 1 \\ 1 & 0 {pmatrix} \) with eigenvalues \( φ = (1+√5)/2 ≈ 1.618 \) and \( ψ = (1-√5)/2 ≈ -0.618 \). - Binet's formula: \( F_n = (φ^n - ψ^n)/√5 \). - Fibonacci anyon braiding matrices (e.g., \( R \)-matrix and \( F \)-matrix) have eigenvalues \( φ \) and \( -1/φ \), satisfying the Yang-Baxter equation. - Modular group SL(2,Z) generated by \( S = {pmatrix} 0 & -1 \\ 1 & 0 {pmatrix} \) and \( T = {pmatrix} 1 & 1 \\ 0 & 1 {pmatrix} \); Fibonacci anyon fusion rules (1 × 1 = 1, 1 × τ = τ, τ × τ = 1 + τ) correspond to the golden ratio's algebraic properties. CONNECTION: - Golden ratio \( φ = 1.618 \) and its re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com