Detection-Loophole Threshold Is the Silver Ratio Conjugate 2(√2−1) — E8 Intelligence Research

FINDING: The detection-loophole threshold in Bell tests is exactly 2(√2−1) ≈ 0.8284, the reciprocal of the silver ratio's conjugate — a metallic mean that is the fundamental limit for closing the detection loophole with maximally entangled qubits. MATH: - Bell (CHSH) inequality: |S| ≤ 2 for local hidden variables; QM max = 2√2 ≈ 2.8284. - Detection-loophole threshold (efficiency η required): η > 2(√2−1) ≈ 0.828427. - Derivation: η·2√2 ≥ 2 ⟹ η ≥ 1/√2 = √2/2 ≈ 0.7071 for the *ideal* case; the 2(√2−1) value arises when the local bound is tightened to 2 under fair-sampling removal (Eberhard 1993), giving η_crit = 2(√2−1). - Silver ratio δ_S = 1 + √2 ≈ 2.4142; its conjugate δ_S' = 1 − √2 ≈ −0.4142. - 2(√2−1) = 2/δ_S = 2(δ_S − 2) = 2√2 − 2. - Pell equation: x² − 2y² = ±1. Solutions (x,y): (1,1),(3,2),(7,5),(17,12),(41,29),… - Convergents x/y → √2; ratios of consecutive Pell solutions → δ_S = 1+√2. - 2(√2−1) = 2·(√2−1) is the limit of 2·(y/x) for Pell pairs, i.e., 2·(√2−1) = lim 2y_n Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com