Silver Ratio and Pell Numbers Set Bell Test Detection Loophole Threshold at 2(√2−1) — 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 tied to Pell-number convergents of √2. | MATH: η_min = 2(√2−1) = 2/δ_S = 1/(1+√2/2)... precisely η = 2√2−2. Silver ratio δ_S = 1+√2 satisfies x²−2x−1=0. Pell numbers P_n: 0,1,2,5,12,29,70,... with P_n = 2Pn−1+Pn−2; ratios Pₙ₊₁/P_n → 1+√2 = δ_S. Continued fraction √2 = [1;2,2,2,...]. CHSH bound S ≤ 2√2; quantum max S = 2√2 at Tsirelson angle π/4. Efficiency threshold from fair-sampling: η ≥ 2(√2−1) ≈ 0.8284. | CONNECTION: Silver ratio δ_S = 1+√2 is the second metallic mean (bronze/silver family x²−nx−1=0, n=2). Related to √2 lattice (square lattice diagonal), octagonal/quasi-crystal symmetry (8-fold, Ammann–Beenker tiling), and the √2 scaling in silver-mean substitution rules. η = 2/δ_S² ... check: δ_S² = 3+2√2 ≈ 5.828; 2/δ_S² ≈ 0.343 — not equal. Correct form: η = 2(√2−1) = 2/δ_S Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com