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 \( S = AB + AB' + A'B - A'B' \). Classical bound: \( |S| ≤ 2 \). Quantum maximum: \( Sₘₐₓ = 2√2 ≈ 2.828 \). Optimal angles: for Pauli observables \( A = a⃗·σ⃗, B = b⃗·σ⃗ \), maximal violation occurs when \( ∠(a⃗,a⃗') = ∠(b⃗,b⃗') = 90^∘ \) and \( ∠(a⃗,b⃗) = 45^∘ \), \( ∠(a⃗,b⃗') = 135^∘ \). This yields \( cos^2(45^∘) = 1/2 \) per term, summing to \( 2√2 \). Quaternion correspondence: \( SU(2) H_1 \) (unit quaternions), with Pauli matrices \( σ_i \) mapping to imaginary quaternions \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com