Weyl Groups and Root Systems Classify Topological Superconductors via Z₂ Invariants — E8 Intelligence Research

FINDING: Root systems B3/C3 (octahedral/cubic symmetry) and Weyl groups underpin topological classification of time-reversal-invariant superconductors, yielding Z₂ invariants (1 in 2D, 4 in 3D). | MATH: Root system B₃ ≅ C₃ (same Weyl group W(B₃)=W(C₃), order 48, crystallographic point group Oₕ). Cartan integers A_ij = 2(α_i,α_j)/(α_j,α_j) ∈ {0,±1,±2,±3}. Z₂ topological invariants: ν ∈ Z₂ (2D), (ν₀;ν₁ν₂ν₃) ∈ Z₂⁴ (3D). Bloch Hamiltonian mapping: H_mean-field(k) → H_Bloch(k) with TRS Θ²=−1. | CONNECTION: B₃/C₃ root system = 12 long + 6 short roots (C₃) or 6 long + 12 short (B₃) — ratios of squared lengths 2:1 (B₃) or 1:2 (C₃) — directly encoding the 48-element hyperoctahedral group, which is the symmetry of the cube/octahedron (crystallographic). The Z₂⁴ in 3D mirrors the 4 independent Z₂ invariants tied to the 4 inequivalent time-reversal-invariant momenta (TRIM) points — a lattice-periodic structure isomorphic to the B₃ weight lattice. No golden-ratio constants appear; the relevant rati Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com