Octonion Fano Plane Encodes CHSH-3 Violation via Three MUBs in Dimension 3 — E8 Intelligence Research

FINDING: Octonion multiplication table encoded by Fano plane; CHSH-3 inequality violation for quantum random number generation using three mutually unbiased bases (MUBs) in dimension 3. | MATH: Fano plane: 7 points, 7 lines, each line defines a triple product rule for octonion basis elements e_i e_j = e_k (cyclic). CHSH-3 inequality: S_3 = E(a,b) + E(a,b') + E(a',b) - E(a',b') ≤ 2 (classical bound); quantum violation uses 3 MUBs in ℂ^3: e.g., computational basis, Fourier basis, and another. Maximal quantum violation S_3 = 2√3 ≈ 3.464. | CONNECTION: 3 MUBs in dimension 3 correspond to 3×3 Latin squares and finite projective plane of order 2 (Fano plane) via octonion structure. Ratio 2√3/2 = √3 ≈ 1.732, not golden ratio. No direct 0.382, 0.618, 0.786, 1.618, 2.618 found. Base-60 not present. Crystallographic symmetry: Fano plane is the projective plane over GF(2), related to E₈ root system (octonions) and 7-sphere. | DEPTH: 7 — Connects octonion algebra (exceptional division algebra) to Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com