Weyl Groups as Discrete Symmetry Engines Linking Icosahedral H₃ to E₈ and Langlands — E8 Intelligence Research

FINDING: Weyl groups of root systems (including icosahedral H₃/H₄) are the discrete symmetry engines of Lie theory, with super-Weyl generalizations extending to Lie superalgebras; the icosahedral group A₅ ≅ H₃ Weyl group is the exceptional finite reflection group linked to E₈ via folding and to geometric Langlands through root data. | MATH: Weyl group W = ⟨s_α : α ∈ Δ⟩, s_α(β) = β − 2⟨β,α⟩/⟨α,α⟩·α; Cartan matrix A_ij = 2⟨α_i,α_j⟩/⟨α_j,α_j⟩; H₃ has Coxeter matrix m_ij = 5 (icosahedral), order |W(H₃)| = 120; H₄ order = 14400; E₈ root system has 240 roots, Weyl group order = 696,729,600; super Weyl groups (arXiv:2401.11068) are quotients of classical Weyl groups by parity/odd root actions. | CONNECTION: Icosahedral symmetry is the rank-3 Weyl group H₃ — its golden ratio φ = (1+√5)/2 = 1.618 appears in the Cartan integers: for adjacent simple roots, 4cos²(π/5) = φ+2 ≈ 3.618, and the squared length ratio of long/short roots in non-crystallographic H₃ is φ² = 2.618. The H₃ root system coordi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com