Bloch Sphere Geometry Yields Tsirelson's Bound — E8 Intelligence Research

FINDING: Tsirelson's bound (2√2 ≈ 2.828) emerges from the geometry of the Bloch sphere and SO(3) rotations, not from ad hoc quantum postulates. | MATH: CHSH inequality classical bound = 2; quantum maximum = 2√2. Derivation via Bloch vectors: for observables A(a), B(b) with outcomes ±1, correlation E(a,b) = a·b (for spin-1/2). CHSH expression S = E(a,b) + E(a,b') + E(a',b) − E(a',b'). Maximizing over unit vectors a,a',b,b' ∈ S² yields S_max = 2√2. This is equivalent to the maximum of |a·b + a·b' + a'·b − a'·b'| = 2√2, attained when the four vectors form a regular tetrahedron (tetrahedral angle 109.47°, cos⁻¹(−1/3)). | CONNECTION: The tetrahedron is a root system of A₃ (SU(4) Weyl group), and its vertices correspond to the 4 vertices of the 3-simplex. The ratio 2√2/2 = √2 ≈ 1.414 relates to the diagonal of a square — a crystallographic ratio. The Bloch sphere's SO(3) symmetry group has the same Lie algebra as SU(2), whose root system is A₁. The tetrahedral configuration is the maximal sy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com