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 \( \mathcal{B} = \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \). Classical bound: \( |\mathcal{B}| \leq 2 \). Quantum bound (Tsirelson): \( |\mathcal{B}|_{\text{max}} = 2\sqrt{2} \approx 2.828 \). This equals \( 2 \times 1.414 \), where \( \sqrt{2} \) is the diagonal of the unit square — the same ratio appearing in the D3 root system's long roots. The qubit manifold \( \mathbb{CP}^1 \cong S^2 \) carries the octahedral symmetry (24 elements, Weyl group of D3). | CONNECTION: \( 2\sqrt{2} \) relates to \( 1.618 \) via \( \sqrt{2} \approx 1.414 \), and \( 2.828/1.618 \approx 1.748 \), 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