D4 Root System's Coxeter Number Links Anyons to Topological Quantum Computing — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: The D4 root system's Coxeter number (h=6) and its braid group action directly yield the quantum dimension formula for anyons, linking crystallographic symmetry to topological quantum computation. | MATH: For D4 (simply-laced, rank 4), Coxeter number h=6. The braid group B_n acts on anyonic Hilbert space via R-matrix and B-matrix satisfying Yang-Baxter. Quantum dimension d_a = [2]_{q} = q + q^{-1} with q = e^{iπ/h} = e^{iπ/6}. Thus d_a = 2cos(π/6) = √3 ≈ 1.732. For the defining representation of D4, the twist factor θ = e^{2πi h_a / h} where h_a are conformal weights. The fusion rules follow the D4 Dynkin diagram adjacency matrix (Cartan matrix A_D4), with Perron-Frobenius eigenvalue = 2cos(π/6) = √3. | CONNECTION: √3 = 1.732 is NOT in the golden ratio family (0.382, 0.618, 0.786, 1.618, 2.618). However, √3 relates to hexagonal close-packing and the 60° angle of the D4 root lattice (base-60 geometry). The ratio 2cos(π/6) = √3 emerges from the Coxeter number 6 (h=6 → π/h = 30°). Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222807626
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