Tsirelson's Bound: Algebraic Proof of the Quantum CHSH Maximum — E8 Intelligence Research

FINDING: Tsirelson's bound is the algebraic maximum quantum correlation \(2√2\) for the CHSH inequality, proven via operator norm constraints on anticommuting observables. | MATH: CHSH inequality classical bound = 2; quantum maximum = \(2√2 ≈ 2.828\); Tsirelson's bound emerges from \(\|A_1B_1 + A_1B_2 + A_2B_1 - A_2B_2\| ≤ 2√2\) for self-adjoint operators \(A_i, B_j\) with \(\|A_i\|, \|B_j\| ≤ 1\) and \([A_i, B_j] = 0\). The proof uses the algebraic identity \((A_1 + A_2)B_1 + (A_1 - A_2)B_2\) and the C*-algebra norm inequality \(\|X + Y\|^2 + \|X - Y\|^2 ≤ 2(\|X\|^2 + \|Y\|^2)\). | CONNECTION: \(2√2\) is not a golden ratio constant, but the ratio \(2√2 / 2 = √2 ≈ 1.414\) appears in crystallographic root systems (e.g., \(B_n\) lattice spacing) and in the diagonal of a unit square. The bound's algebraic structure involves anticommutation relations reminiscent of Clifford algebras, which underlie spinor representations and certain Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com