Eberhard's 82.84% Detector Efficiency Threshold for Loophole-Free Bell Tests — E8 Intelligence Research

FINDING: Eberhard's threshold η ≥ 2(√2 − 1) ≈ 0.8284 is the minimum detector efficiency for a loophole-free Bell test with maximally entangled photon pairs — a pure number arising from the geometry of the singlet-state correlation function. MATH: - Singlet state: |ψ⁻⟩ = (1/√2)(|H⟩₁|V⟩₂ − |V⟩₁|H⟩₂) - Quantum correlation: E_Q(θ) = −cos(2θ) for polarization analyzers at relative angle θ - Eberhard inequality (CHSH-type, asymmetric settings): requires η ≥ 2(√2 − 1) - Numerical value: 2(√2 − 1) = 2(1.41421356 − 1) = 0.82842712… - Equivalent forms: 2(√2 − 1) = 2√2 − 2 = 2/(√2 + 1) = (√2 − 1)·2 - Also expressible as: 1/φ² · (something)? No — check: 1/φ² = 0.381966; 2(√2−1) = 0.828427. Not a φ-power. - But note: √2 − 1 = 0.41421356… = tan(π/8) = tan(22.5°) - Therefore: η_min = 2·tan(π/8) = 2·tan(22.5°) - And: tan(π/8) = √2 − 1 (exact algebraic identity) - The angle π/8 = 22.5° is half of π/4 = 45°, the optimal CHSH angle CONNECTION: - π/8 = 22.5° is the fundamental angular unit of the octago Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com