PR-Box Extremality and the Limits of Quantum Uniqueness in Bell Polytopes — E8 Intelligence Research

FINDING: The Popescu-Rohrlich (PR) box defines the maximal nonlocal vertex of the Bell polytope; its extremal structure is characterized by convex geometry, and recent work shows PR-box fraction cannot uniquely single out quantum theory among generalized nonsignaling theories. | MATH: PR box correlations: \(P(a,b|x,y) = 1/2\) if \(a ⊕ b = xy\), else 0. Bell polytope \(B\): convex hull of local deterministic strategies; its extremal points in (2,2,2) scenario are the 8 local vertices plus 4 PR-box vertices (the "nonlocal" extreme points). The PR box saturates the CHSH inequality: \( A_0B_0 + A_0B_1 + A_1B_0 - A_1B_1 = 4\) (Tsirelson bound for quantum: \(2√2 ≈ 2.828\)). | CONNECTION: The ratio \(2√2/4 = 1/√2 ≈ 0.7071\) — not directly in the golden set, but note \(0.7071^2 = 0.5\), and the quantum-to-PR gap \(4 - 2√2 ≈ 1.1716\) relates to \(2(√2-1) ≈ 0.82 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com