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: \( d = \phi = \frac{1+\sqrt{5}}{2} = 1.6180339887... \) - **Fusion rule**: \( \tau \otimes \tau = 1 \oplus \tau \) (the Fibonacci fusion category) - **Turaev-Viro invariant** for SU(2)_3: partition function built from 6j-symbols; the non-trivial 6j-symbols involve \( \phi \) exactly, e.g. \( \{ \tau, \tau, \tau; \tau, \tau, \tau \} = \frac{1}{\phi} \) (up to normalization) - **Golden identities**: \( \phi^2 = \phi + 1 \), \( 1/\phi = \phi - 1 = 0.618... \), \( \phi^{-2} = 2 - \phi = 0.382... \) - **Quantum calculus** (arXiv:2410.04169): two bases \( q_1 = \phi \), \( q_2 = \phi^{-1} \) (or silver ratio variants); Fibonacci divisor derivative \( D_q f(x) = \fr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com