D4 Lattice as the Triality Bridge Between Hurwitz Quaternions and E8 — E8 Intelligence Research

FINDING: The search results converge on the D4 lattice (24-cell) as the central bridge between Hurwitz quaternion arithmetic and the E8 root system, with explicit triality symmetry tied to SO(8) and the 24-cell's 24 vertices. | MATH: D4 lattice = { (x1,x2,x3,x4) ∈ Z⁴ ∪ (Z+½)⁴ : Σxi ≡ 0 mod 2 }; 24-cell vertices = 24 unit quaternions (Hurwitz integers); E8 = D4 ⊕ D4 (two copies) with 240 roots; triality: SO(8) outer automorphism permuting vector and two spinor representations (8v, 8s, 8c); CHSH polytope in 4D has 24 vertices (local deterministic strategies) — isomorphic to the 24-cell. | CONNECTION: The 24-cell's 24 vertices correspond to the 24 local deterministic CHSH vertices; its dual is the 24-cell itself (self-dual). The ratio 24/240 = 0.1 (E8 roots per D4 vertex); the 24-cell's circumradius-to-edge ratio = √2, and its dihedral angle = 120° (2π/3, base-60 compatible). The Hurwitz quaternion norm-1 elements form the binary tetrahedral group (order 24), whose character table encodes Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com