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Research Army Evolution — Gen 596 — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 24, 2026• 0 citations
Publisher

Abstract

Domain: Quantum Physics Finding: Octonion MUBs and Fano Plane Yield Maximal CHSH-3 Violation Depth: 8/10 Domain weight evolved to: 5.00 Total domain hits: 297 Active search terms: 322 Timestamp: 15/07/2026 10:46 BST The research army self-evolved — domains producing deeper findings get more attention. New search queries auto-generated from discoveries. Generation 596. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.21524870

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

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The Fano Plane, Octonion Non-Associativity, and the Peres-Mermin Proof of Contextuality — E8 Intelligence Research

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FINDING: The Fano plane encodes the non-associative multiplication table of octonions, and this algebraic structure underlies the Peres-Mermin square proof of quantum contextuality. MATH: Octonion multiplication is defined by 7 imaginary units \( e_1, \dots, e_7 \) with \( e_i^2 = -1 \) and \( e_i e_j = \epsilon_{ijk} e_k \) where \( \epsilon_{ijk} \) is given by the 7 lines (each a triple) of the Fano plane. The Peres-Mermin square is a 3×3 array of observables whose row and column products yield ±1, demonstrating contextuality via the Kochen-Specker theorem. No new constants or ratios appear. CONNECTION: The Fano plane is the projective plane of order 2, with 7 points and 7 lines, each line containing 3 points. This is a finite geometry with symmetry group PSL(2,7) of order 168. No direct link to golden ratio, base-60, or crystallographic root systems. However, the octonion algebra is intimately tied to the exceptional Lie group \( G_2 \) (automorphism group of octonions) and the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Jul 23, 2026•0 citations
Domain: Quantum Physics Finding: Octonion MUBs and Fano Plane Yield Maximal CHSH-3 Violation Depth: 8/10 Domain weight evolved to: 5.00 Total domain hits: 297 Active search terms: 322 Timestamp: 15/07/2026 10:46 BST The research army self-evolved — domains producing deeper findings get more attention. New search queries auto-generated from discoveries. Generation 596. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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
FINDING: The Fano plane encodes the non-associative multiplication table of octonions, and this algebraic structure underlies the Peres-Mermin square proof of quantum contextuality. MATH: Octonion multiplication is defined by 7 imaginary units e1,…,e7 e_1, \dots, e_7 e1​,…,e7​ with ei2=−1 e_i^2 = -1 ei2​=−1 and eiej=ϵijkek e_i e_j = \epsilon_{ijk} e_k ei​ej​=ϵijk​ek​ where ϵijk \epsilon_{ijk} ϵijk​ is given by the 7 lines (each a triple) of the Fano plane. The Peres-Mermin square is a 3×3 array of observables whose row and column products yield ±1, demonstrating contextuality via the Kochen-Specker theorem. No new constants or ratios appear. CONNECTION: The Fano plane is the projective plane of order 2, with 7 points and 7 lines, each line containing 3 points. This is a finite geometry with symmetry group PSL(2,7) of order 168. No direct link to golden ratio, base-60, or crystallographic root systems. However, the octonion algebra is intimately tied to the exceptional Lie group G2 G_2 G2​ (automorphism group of octonions) and the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com