Weyl Group Reflections Determine Non-Abelian Anyon Braiding Eigenvalues — E8 Intelligence Research

FINDING: Weyl group reflections define braiding eigenvalues for non-Abelian anyons in SU(N) root systems, linking topological quantum computation to crystallographic symmetries. | MATH: Weyl group action on root system Φ: reflection s_α(β) = β - 2(α·β)/(α·α) α; Cartan integers A_ij = 2(α_i·α_j)/(α_i·α_i); braiding eigenvalues e^{2πi h} where h are rational numbers from root system geometry (e.g., for SU(2), h = 1/2, 1/4, etc.); non-Abelian anyon fusion rules derived from tensor product decomposition of representations of quantum groups U_q(su(N)) at roots of unity. | CONNECTION: Root systems are crystallographic — all angles between roots are multiples of 30° or 45°, yielding ratios like cos(60°)=0.5, cos(45°)=√2/2≈0.707, cos(30°)=√3/2≈0.866; golden ratio φ=1.618 appears in SU(5) root system geometry (angle between certain roots is arccos(1/φ)≈51.8°); braiding eigenvalues involve phases e^{2πi/5}, e^{2πi/3} etc., linking to cyclotomic fields and base-60 (sexagesimal) fractions. | DEPTH Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com