CHSH Inequalities as Unified Rules Linking Entanglement to Device-Independent Tests — E8 Intelligence Research

FINDING: Bell/CHSH inequalities define the boundary between local-hidden-variable (classical) and quantum correlations; recent work (Buscemi) recasts them as a unified family of "rules" linking entanglement to device-independent tests. | MATH: CHSH form S = E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′) ≤ 2 (LHV bound); quantum (Tsirelson) bound S_max = 2√2 ≈ 2.828. Quantum correlation set is strictly larger than the classical polytope but strictly smaller than the no-signaling polytope (PR-box value 4). | CONNECTION: 2√2 = 2·√2 is the Tsirelson constant; note √2 is the diagonal of the unit square and the ratio of the octahedron/cube dual pair. The CHSH polytope is a hypercube (8 vertices, 2⁴ sign choices) — a 4-dimensional measure polytope, one of the crystallographic/root-system family (B₄/C₄ symmetry). The classical bound 2 and quantum bound 2√2 differ by factor √2 ≈ 1.414, not a Fibonacci/φ ratio. No 0.618/1.618 structure appears in the inequality itself. | DEPTH: 7 Notes on the other item Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com