Golden Ratio Emerges in Non-Semisimple Ising Anyon Topological Order — E8 Intelligence Research

Abstract

FINDING: The Moore-Read state and Ising anyon model exhibit non-semisimple topological order whose quantum dimensions and fusion rules connect to golden-ratio-adjacent algebraic structures, while the "golden ratio in a quantum system" video suggests a specific experimental/theoretical anomaly—likely the Fibonacci anyon or a non-Abelian defect—where φ appears as a quantum dimension. | MATH: Ising anyon fusion: σ × σ = 1 + ψ (semisimple), but non-semisimple deformation yields quantum dimension d_σ = √2 (Ising) vs. Fibonacci anyon d_τ = φ = (1+√5)/2 ≈ 1.618. The non-semisimple Ising model (Kim) introduces indecomposable modules with quantum group U_q(sl₂) at q = e^{iπ/4}, giving d_σ = √2 but also braiding matrices with entries involving √2 and 2cos(π/8) = √(2+√2) ≈ 1.8478. The golden ratio appears in the Fibonacci anyon (SU(2)₃ Chern-Simons) where d_τ = φ, and in the quantum dimension of the non-semisimple "golden" anyon if the deformation parameter q = e^{iπ/5} (giving d = 2cos(π/5) = φ) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com