D3 Root System and the Geometric Origin of Tsirelson's Bound — E8 Intelligence Research

FINDING: The CHSH inequality's maximal quantum violation (Tsirelson bound) is geometrically rooted in the qubit manifold, with the octahedral root system D3 encoding the measurement algebra. | MATH: CHSH operator \( B = AB + AB' + A'B - A'B' \). Classical bound: \( |B| ≤ 2 \). Quantum bound (Tsirelson): \( |B|ₘₐₓ = 2√2 ≈ 2.828 \). This equals \( 2 × 1.414 \), where \( √2 \) is the diagonal of the unit square — the same ratio appearing in the D3 root system's long roots. The qubit manifold \( CP^1 S^2 \) carries the octahedral symmetry (24 elements, Weyl group of D3). | CONNECTION: \( 2√2 \) relates to \( 1.618 \) via \( √2 ≈ 1.414 \), and \( 2.828/1.618 ≈ 1.748 \), not a direct golden ratio. However, the D3 root system has 6 roots at 90° intervals (octahedron vertices), and the CHSH measurement directions in the optimal Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com