No source addresses detection-loophole threshold 2(√2−1) or Pell-equation structure — E8 Intelligence Research

FINDING: Search returned only pedagogical/experimental Bell-test material — no source in the set actually addresses the detection-loophole threshold 2(√2−1) or its Pell-equation / metallic-mean structure. The mathematical content below is reconstructed from established theory, not from the listed URLs. MATH: - CHSH inequality: |S| ≤ 2 (local hidden variables); Tsirelson bound: |S| ≤ 2√2 (quantum). - Detection-loophole (fair-sampling) threshold: to close it with the standard Eberhard/Mermin form, required detection efficiency η > 2(√2 − 1) ≈ 0.8284. - 2(√2 − 1) = 2/φ² · ... — evaluate directly: √2 − 1 ≈ 0.41421, so 2(√2−1) ≈ 0.82843. - Relation to silver ratio δ_S = 1 + √2 ≈ 2.41421 (the metallic mean of order 2). Note 2(√2−1) = 2/(1+√2) = 2/δ_S. So the threshold is 2 divided by the silver mean. - Pell equation: x² − 2y² = ±1. Fundamental solution (1,1) → 1+√2 = δ_S. Convergents of √2 = [1;2,2,2,…] give δ_S powers; δ_S^n = x_n + y_n√2. - General metallic means: M_n = (n + √(n²+4))/2; n Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com