Fibonacci Anyons and the Golden Ratio: 72° Rotations in Braid Group Representations — E8 Intelligence Research

Abstract

FINDING: Non-Abelian anyon braid group representations yield unitary matrices whose eigenvalues can include \(e^{\pm 2\pi i/5}\) (72° rotations), tied to Fibonacci anyons and the golden ratio. | MATH: Fibonacci anyon braid matrices (e.g., \(R\)-matrix for \(\tau \times \tau \to 1+\tau\)) have eigenvalues \(\{e^{\pm 4\pi i/5}, e^{\pm 2\pi i/5}\}\) — the latter is a 72° rotation. The quantum dimension \(d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618\), satisfying \(d_\tau^2 = 1 + d_\tau\). Braid group \(B_3\) representation: \(\sigma_1, \sigma_2\) act on 2D Hilbert space with matrices whose traces give \(2\cos(2\pi/5) = \phi - 1 = 0.618\). | CONNECTION: Direct geometric harmony — 72° = \(360°/5\) (pentagonal symmetry, icosahedral group \(A_5\)); eigenvalues \(e^{\pm 2\pi i/5}\) are 5th roots of unity, linking to the golden ratio \(\phi\) via \(2\cos(72°) = 0.618 = 1/\phi\). The Fibonacci anyon fusion rules \(1 \otimes \tau = \tau\), \(\tau \otimes \tau = 1 \oplus \tau\) mirror the golden r Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com