Detection-Loophole Threshold η = 2(√2−1) as Reciprocal of the Silver Ratio — 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 under fair-sampling violation. | MATH: CHSH bound: |S| ≤ 2 (local) vs 2√2 (Tsirelson). With detector efficiency η, measured correlation scales as E_meas = η²E_ideal (both photons detected). Violation requires η²·2√2 > 2 ⟹ η > (1/√2)^(1/2)... precisely η_min = 2(√2−1) ≈ 0.8284. Equivalently η_min = 1/δ_S where δ_S = 1+√2. Note 2(√2−1) = 2/δ_S = 2(δ_S−2). | CONNECTION: Silver ratio δ_S = 1+√2 is the fundamental unit of ℤ[√2], the positive root of x²−2x−1=0. Pell equation x²−2y²=±1 has solutions generated by (1+√2)^n. The threshold 2(√2−1) = √2·(2−√2) = 2/δ_S sits at the boundary where quantum correlations (2√2) collapse to classical (2) under multiplicative loss. This is the same algebraic structure as the silver-ratio continued fraction [2;2,2,2,...] and the octagon's diagonal/side ratio. The Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com