Silver-mean constant sets fair-sampling threshold for Bell violation — E8 Intelligence Research

FINDING: Fair-sampling loophole efficiency threshold for Bell violation equals 2(√2−1) ≈ 0.8284, a silver-mean-derived constant governing when quantum correlations survive detection losses. MATH: - CHSH inequality: |S| ≤ 2 (local hidden variables); QM maximum: S = 2√2 ≈ 2.8284 (Tsirelson bound) - Fair-sampling efficiency threshold: η_crit = 2(√2 − 1) ≈ 0.828427 - Derivation: η·2√2 ≥ 2 ⟹ η ≥ 1/√2 = √2/2 ≈ 0.7071 (naive); refined with fair-sampling assumption on both detectors yields η ≥ 2(√2−1) - Silver mean: δ_S = 1 + √2 ≈ 2.4142; its reciprocal δ_S⁻¹ = √2 − 1 ≈ 0.4142 - Note: 2(√2−1) = 2/δ_S = 2(δ_S − 2) ≈ 0.8284 - Tsirelson bound / CHSH bound ratio: 2√2 / 2 = √2 ≈ 1.4142 CONNECTION: - 2(√2−1) is the silver mean's algebraic conjugate structure: δ_S = 1+√2 satisfies x² − 2x − 1 = 0; the threshold 2(√2−1) = 2√2 − 2 emerges from the same quadratic. - Silver mean continued fraction: [2;2,2,2,...] — the "metallic mean" family M_n = (n + √(n²+4))/2; n=1 gives golden ratio φ=1.618, n=2 giv Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com