Fibonacci Anyons as Dense Braid Representations in SU(2)₃ — E8 Intelligence Research

FINDING: Fibonacci anyons realize a dense, universal braid-group representation whose algebraic skeleton is the SU(2)₃ modular tensor category; the Cartan/root-system machinery of su(N) is the general framework for classifying these braid representations. MATH: - Fibonacci anyon fusion: τ ⊗ τ = 1 ⊕ τ. Quantum dimension d_τ = φ = (1+√5)/2 ≈ 1.618 (satisfies d² = d + 1). - Total quantum dimension of the theory: D = √(1² + φ²) = √(2+φ) = √(φ+2) ≈ 1.902. - Braid generator eigenvalues on the 2-dim Fibonacci space: e^{±4πi/5} = e^{±iπ·(4/5)}. The braid phase is 4π/5, i.e. 2π·(2/5) — a 5th-root-of-unity structure. - SU(2)_k data: level k=3, central charge c = 3k/(k+2) = 9/5 = 1.8. Conformal weights h_j = j(j+1)/(k+2) = j(j+1)/5 for j = 0, ½, 1, 3/2. So h ∈ {0, 3/20, 2/5, 3/4}. - Fusion coefficients via SU(2) Clebsch–Gordan truncated at level k: N_{ij}^l = 1 iff |i−j| ≤ l ≤ min(i+j, k−i−j). - Braid group B_n → U(2) representation; image is dense in SU(2) (universal quantum computation). Densi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com