72° Rotations in Non-Abelian Anyon Braid Groups Link to Fibonacci Anyons and Golden Ratio — E8 Intelligence Research

Abstract

FINDING: Non-Abelian anyon braid group representations exhibit eigenvalues tied to 72° rotations, linking to Fibonacci anyons and golden ratio phases. | MATH: Braid group \( B_n \) representations via Hecke algebras yield eigenvalues \( e^{\pm 2\pi i/5} \) (72° rotations) for the braid generator \( \sigma_i \), satisfying \( \sigma_i^2 = 1 \) or \( \sigma_i^4 = 1 \) in certain anyon models. The Fibonacci anyon fusion rule \( \tau \times \tau = 1 + \tau \) gives quantum dimension \( d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 \), with braid eigenvalues \( e^{\pm 4\pi i/5} \) (144°) related to \( \phi \). | CONNECTION: 72° = 360°/5, a pentagonal angle; golden ratio \( \phi = 2\cos(36^\circ) = 1.618 \), with \( 2\cos(72^\circ) = 0.618 \) (inverse of \( \phi \)). This ties to \( D_5 \) dihedral symmetry and the root system \( H_2 \) (icosahedral symmetry), a non-crystallographic Coxeter group. | DEPTH: 8 — Directly links topological quantum computing (non-Abelian anyons) to golden ratio g Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com