Silver Ratio and Pell Equation in the Bell Test Detection-Loophole Threshold — E8 Intelligence Research

FINDING: The detection-loophole efficiency threshold in Bell tests is η ≥ 2(√2−1) ≈ 0.8284, the reciprocal of the silver ratio δ_S = 1+√2 ≈ 2.4142, arising from the geometry of the CHSH inequality and connected to the Pell equation x²−2y²=±1. MATH: - Silver ratio: δ_S = 1 + √2 = 2.41421356… ; 1/δ_S = √2 − 1 = 0.41421356… - Threshold: η_crit = 2(√2 − 1) = 2√2 − 2 = 0.82842712… - Equivalent form: η_crit = 2/δ_S = 2/(1+√2) - CHSH bound: |S| ≤ 2√2 (Tsirelson); classical |S| ≤ 2 - Pell equation: x² − 2y² = ±1, solutions (x,y) = (1,0),(1,1),(3,2),(7,5),(17,12),(41,29)… - Convergents of √2: 1/1, 3/2, 7/5, 17/12, 41/29 → ratios approach δ_S - Continued fraction: √2 = [1; 2,2,2,…] (all-2 periodicity = "silver" metallic mean) - Metallic means: δ_n = (n + √(n²+4))/2 ; n=1 → φ=1.618, n=2 → δ_S=2.414 CONNECTION: - 0.8284 = 2(√2−1) is the silver-ratio conjugate doubled. Silver ratio δ_S = 1+√2 is the n=2 metallic mean, sibling of golden ratio φ = (1+√5)/2 (n=1). - Pell equation x²−2y²=±1 is the in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com