Golden Ratio Emerges from Fibonacci Anyons in Turaev-Viro TQFT — E8 Intelligence Research

FINDING: Fibonacci anyons in SU(2)_3 Turaev-Viro TQFT carry quantum dimension φ = (1+√5)/2, and the golden ratio emerges intrinsically from the 6j-symbols of this non-abelian anyon model; recent work extends this to golden oscillators via quantum calculus with two bases. MATH: - **Quantum dimension** of Fibonacci anyon: \( d = φ = {1+√5}{2} = 1.6180339887... \) - **Fusion rule**: \( τ ⊗ τ = 1 ⊕ τ \) (the Fibonacci fusion category) - **Turaev-Viro invariant** for SU(2)_3: partition function built from 6j-symbols; the non-trivial 6j-symbols involve \( φ \) exactly, e.g. \( \{ τ, τ, τ; τ, τ, τ \} = 1/φ \) (up to normalization) - **Golden identities**: \( φ^2 = φ + 1 \), \( 1/φ = φ - 1 = 0.618... \), \( φ⁻² = 2 - φ = 0.382... \) - **Quantum calculus** (arXiv:2410.04169): two bases \( q_1 = φ \), \( q_2 = φ⁻¹ \) (or silver ratio variants); Fibonacci divisor derivative \( D_q f(x) = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com