Tsirelson Bound from Bloch Sphere Geometry and SU(2) Rotations — E8 Intelligence Research

FINDING: The CHSH inequality's quantum bound (Tsirelson bound = 2√2) emerges from the geometry of the qubit manifold (Bloch sphere), where optimal measurement angles are fixed by SU(2) rotations and Pauli observable non-commutativity — not by arbitrary choice. | MATH: CHSH operator \( S = \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \). Classical bound: \( |S| \leq 2 \). Quantum maximum: \( S_{\text{max}} = 2\sqrt{2} \approx 2.828 \). Optimal angles: for Pauli observables \( A = \vec{a}\cdot\vec{\sigma}, B = \vec{b}\cdot\vec{\sigma} \), maximal violation occurs when \( \angle(\vec{a},\vec{a}') = \angle(\vec{b},\vec{b}') = 90^\circ \) and \( \angle(\vec{a},\vec{b}) = 45^\circ \), \( \angle(\vec{a},\vec{b}') = 135^\circ \). This yields \( \cos^2(45^\circ) = 1/2 \) per term, summing to \( 2\sqrt{2} \). Quaternion correspondence: \( SU(2) \cong \mathbb{H}_1 \) (unit quaternions), with Pauli matrices \( \sigma_i \) mapping to imaginary quaternions \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com