Golden-Ratio Fusion Rules and Quantum 6j-Symbols in Fibonacci Anyon Models — E8 Intelligence Research

FINDING: Fibonacci anyons and SU(2) quantum 6j-symbols at level k=3 yield golden-ratio fusion rules, with recent work extending 6j-symbol calculus to quantum oscillators via golden/silver ratio bases. | MATH: Fibonacci anyons: fusion rule τ⊗τ = 1⊕τ, quantum dimension d_τ = φ = (1+√5)/2 ≈ 1.618; SU(2)_k 6j-symbols at k=3 reduce to Fibonacci category; quantum 6j-symbols satisfy pentagon/hexagon equations; Spiridonov's SL(2,C) 6j-symbols link to Feynman diagrams; arXiv:2410.04169v2 introduces Fibonacci divisor derivative D_q f(x) = [f(qx)-f(q⁻¹x)]/[(q-q⁻¹)x] with q = φ or silver ratio σ = 1+√2, yielding Binet-form number operators N = (φ^n - (-φ)⁻ⁿ)/√5. | CONNECTION: Golden ratio φ = 1.618 appears as quantum dimension — its inverse φ⁻¹ = 0.618 and φ⁻² = 0.382 are the non-trivial fusion probabilities; the Fibonacci anyon braid group representation has Jones polynomial at q = e^{iπ/5}, linking to E₈ root system (crystallographic, rank 8, Weyl group order 696729600) via the 120-cell/600-cell Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com