D4 and B2: Square Symmetry in Quantum Contextuality Bounds — E8 Intelligence Research
Andrew Stewart Caldin
FINDING: Dihedral group D4 and root system B2 are structurally identical, forming the symmetry group of a square (order 8), and this crystallographic symmetry appears in quantum contextuality bounds. MATH: - D4 = ⟨r, s | r⁴ = s² = e, srs = r⁻¹⟩, order 8. - Root system B2: two orthogonal simple roots of equal length, angle 90°, Weyl group = D4 (order 8). - Quantum contextuality: semidefinite relaxations bound maximal violation of noncontextuality inequalities; B2/D4 symmetry constrains correlation polytopes. CONNECTION: - B2 root lattice is the square lattice (Z²). - Ratio of long to short roots = √2 ≈ 1.414 (not a golden ratio, but a crystallographic constant). - No direct 0.382, 0.618, 1.618, 2.618, or base-60 appears. - However, D4 is the symmetry of the square, whose diagonal/half-diagonal ratio = √2, and the angle 90° yields cos(90°)=0, sin(90°)=1 — no harmonic ratio. DEPTH: 6 - Solid connection between finite group theory, root systems, and quantum contextuality Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
•Jul 23, 2026•0 citations