D4 Root System's Coxeter Number Links Anyons to Topological Quantum Computing — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
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
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BibTeX
@misc{caldin2026root,
title = {{D4 Root System's Coxeter Number Links Anyons to Topological Quantum Computing — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23431400},
url = {https://zenodo.org/records/23431400},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - D4 Root System's Coxeter Number Links Anyons to Topological Quantum Computing — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/17
PB - Zenodo
DO - 10.5281/zenodo.23431400
UR - https://zenodo.org/records/23431400
AB - 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
ER -