Bell Test Detection Loophole Threshold as Reciprocal Silver Ratio from Pell Equation — E8 Intelligence Research

FINDING: The detection-loophole efficiency threshold for Bell tests is η = 2(√2−1) ≈ 0.8284, the reciprocal of the silver ratio δ_S = 1+√2 ≈ 2.4142, arising from the Pell equation x²−2y²=±1. MATH: - Silver ratio: δ_S = 1+√2 = 2.41421356…, satisfies x²−2x−1=0 - Reciprocal: 1/δ_S = √2−1 = 0.41421356… (the "silver mean" conjugate) - Detection threshold: η_min = 2(√2−1) = 2√2−2 ≈ 0.8284271 - Pell equation: x²−2y²=±1; fundamental solution (x,y)=(1,1)→(3,2)→(7,5)→(17,12)… - Convergents of δ_S: 1/1, 3/2, 7/5, 17/12, 41/29… (ratios of Pell numbers) - CHSH inequality: |S| ≤ 2 (classical), |S| ≤ 2√2 (Tsirelson bound) - Tsirelson bound: 2√2 = 2δ_S − 2 = 2(√2−1)·δ_S = η_min · δ_S - Key identity: η_min · δ_S = 2√2 − 2 = 2(√2−1) · (1+√2) = 2(2−1) = 2 - Therefore: η_min = 2/δ_S = 2(√2−1) CONNECTION: - The threshold 2(√2−1) is the inverse silver ratio scaled by 2 — a direct geometric-harmonic constant - Silver ratio δ_S = 1+√2 is the second metallic mean (after golden ratio φ = (1+√5)/2) - Metallic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com