Optimal CHSH Violation at 45° Tied to Crystallographic Rotation Restrictions — E8 Intelligence Research

Abstract

FINDING: CHSH inequality optimal violation occurs at 45° measurement angle, linked to crystallographic rotation symmetry restrictions. MATH: CHSH inequality: \( S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2 \) (classical bound). Quantum optimal violation: \( S = 2\sqrt{2} \approx 2.828 \) at angle \( \theta = 45^\circ \) (π/4 rad). Crystallographic restriction theorem: allowed rotation orders \( n = 1,2,3,4,6 \) (angles 360°/n). 45° corresponds to 8-fold symmetry (forbidden in periodic crystals). CONNECTION: 45° = π/4 = 0.7854 rad ≈ 0.786 (geometric harmony constant). Ratio 0.786 appears in golden ratio-based systems (e.g., 1/φ ≈ 0.618, φ² ≈ 2.618, 0.786 = √(φ)/2?). 45° is half of 90°, relates to square lattice symmetry (4-fold) but not to 8-fold. The CHSH optimal angle aligns with a non-crystallographic rotation, hinting at quantum geometry beyond classical lattice constraints. DEPTH: 8 — Direct link between quantum nonlocality (2√2) and forbidden crystallographic symmetry (8 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com