Golden Ratio Braids: Topological Quantum Computation Meets Root-System Geometry — E8 Intelligence Research

FINDING: Fibonacci anyons realize braid group representations through the Temperley-Lieb algebra, with the golden ratio as the quantum dimension, linking topological quantum computation to root-system geometry. | MATH: Fibonacci anyons have quantum dimension \( d = \varphi = \frac{1+\sqrt{5}}{2} \approx 1.618 \). The braid group \( B_n \) acts via the Temperley-Lieb algebra \( TL_n(\delta) \) with loop value \( \delta = \varphi + \varphi^{-1} = \sqrt{5} \approx 2.236 \). The Jones polynomial at \( q = e^{i\pi/5} \) (or \( q = e^{2\pi i/5} \)) yields Fibonacci anyon statistics; the fusion rule is \( \tau \otimes \tau = 1 \oplus \tau \), giving the golden ratio as the largest eigenvalue of the fusion matrix \( N_\tau = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} \), whose eigenvalues are \( \varphi \) and \( -\varphi^{-1} \). The two-boundary Temperley-Lieb algebra (Wilbert) extends this with boundary parameters \( \alpha, \beta \) satisfying \( \alpha\beta = \delta^2 - 4 = 1 \) (for \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com