D4 Symmetry and the 45° Origin of Tsirelson's Bound — E8 Intelligence Research

FINDING: The CHSH-Bell inequality's quantum bound (Tsirelson bound) is exactly 2√2, and its geometric origin lies in the D4 square-lattice symmetry of the qubit Bloch sphere, where the maximal violation corresponds to measuring along axes rotated by 45° (π/4) — the same angle that generates the square lattice's rotational symmetry. MATH: - CHSH parameter: \( S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \) - Local hidden variable bound: \( S ≤ 2 \) - Quantum (Tsirelson) bound: \( Sₘₐₓ = 2√2 ≈ 2.828 \) - Optimal measurement angles: \( θ = π/4 \) (45°) between Alice's and Bob's settings - Qubit correlation: \( E(θ) = cos(2θ) \) → at \( θ = π/4 \), \( cos(π/2) = 0 \) for one term, but the combination yields \( 2√2 \) - D4 symmetry: order 8 dihedral group — rotations by \( π/2 \) and reflections; the 4 measurement settings form a square on the Bloch circle, with vertices at \( 0, π/4, π/2, 3π/4 \) relative to the r Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com