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: \( F_{n+1} = F_n + F_{n-1} \) → matrix \( M = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \) with eigenvalues \( \phi = (1+\sqrt{5})/2 \approx 1.618 \) and \( \psi = (1-\sqrt{5})/2 \approx -0.618 \). - Binet's formula: \( F_n = (\phi^n - \psi^n)/\sqrt{5} \). - Fibonacci anyon braiding matrices (e.g., \( R \)-matrix and \( F \)-matrix) have eigenvalues \( \phi \) and \( -1/\phi \), satisfying the Yang-Baxter equation. - Modular group SL(2,Z) generated by \( S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \) and \( T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \); 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 \( \phi = 1.618 \) and its re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com