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

Andrew Stewart Caldin

Published Jul 21, 2026• 0 citations
Publisher

Abstract

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

DOI
10.5281/zenodo.21467212

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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

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FINDING: Non-associative quantum mechanics based on octonions and the exceptional Jordan algebra h₃(O) replaces linear Hilbert spaces with the octonionic projective plane, requiring a fundamentally non-associative algebraic foundation for quantum theory. MATH: - Octonions (O): 8-dimensional normed division algebra, non-associative, alternative (x(xy) = (xx)y, (yx)x = y(xx)). - Exceptional Jordan algebra h₃(O): 27-dimensional algebra of 3×3 Hermitian matrices over O, with product A∘B = ½(AB+BA). - Octonionic projective plane OP²: points correspond to rank-1 idempotents in h₃(O); dimension 16, automorphism group F₄ (exceptional Lie group). - Key constants: No direct ratios (0.382, 0.618, etc.) appear; instead, the structure is governed by the octonion multiplication table (Fano plane) and the 7-sphere S⁷. CONNECTION: - The Fano plane (projective plane of order 2) underlies octonion multiplication — a finite geometry with 7 points and 7 lines, each line containing 3 points. Th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Geometric Proof of Kochen-Specker Theorem via Fano Plane and PSL(2,7) — E8 Intelligence Research

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FINDING: The Kochen-Specker theorem is proven geometrically using a finite set of rays in 3D real space, directly linked to the Fano plane and the group PSL(2,7), which also underlies the octonion multiplication table and contextuality in quantum mechanics. | MATH: The proof uses 33 rays in R^3 (or 117 in a simpler variant) forming a configuration that admits no 2-coloring (0/1 assignment) consistent with orthogonality constraints. The Fano plane is the projective plane of order 2, with 7 points and 7 lines, each line containing 3 points. PSL(2,7) is the simple group of order 168, isomorphic to GL(3,2) and the automorphism group of the Fano plane. Octonion multiplication is encoded by the Fano plane: each line corresponds to a triple of imaginary units (e₁, e₂, e₃) satisfying e₁e₂ = e₃, with cyclic permutations. The Kochen-Specker configuration is a 3D real realization of this combinatorial structure. | CONNECTION: The Fano plane is a finite projective plane of order 2, whose symmetry Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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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
FINDING: Non-associative quantum mechanics based on octonions and the exceptional Jordan algebra h₃(O) replaces linear Hilbert spaces with the octonionic projective plane, requiring a fundamentally non-associative algebraic foundation for quantum theory. MATH: - Octonions (O): 8-dimensional normed division algebra, non-associative, alternative (x(xy) = (xx)y, (yx)x = y(xx)). - Exceptional Jordan algebra h₃(O): 27-dimensional algebra of 3×3 Hermitian matrices over O, with product A∘B = ½(AB+BA). - Octonionic projective plane OP²: points correspond to rank-1 idempotents in h₃(O); dimension 16, automorphism group F₄ (exceptional Lie group). - Key constants: No direct ratios (0.382, 0.618, etc.) appear; instead, the structure is governed by the octonion multiplication table (Fano plane) and the 7-sphere S⁷. CONNECTION: - The Fano plane (projective plane of order 2) underlies octonion multiplication — a finite geometry with 7 points and 7 lines, each line containing 3 points. Th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: D4 symmetry and octonion algebra are linked via Clifford algebra Cl(0,7) to quantum correlation polytopes and error-correcting codes, with toric decoding and QLDPC codes as applications. | MATH: D4 root system (24 roots, Coxeter number 12, Weyl group order 192); octonion multiplication table defined by Fano plane; Clifford algebra Cl(0,7) parametrizes octonions via multivector products; toric code stabilizer generators (plaquette and vertex operators) on square lattice; QLDPC codes with sparse check matrices. | CONNECTION: D4 lattice is the densest sphere packing in 4D (kissing number 24); octonion norm 1 yields 7-sphere S7; toric code on torus has genus-dependent degeneracy; no direct golden ratio or base-60 link found. | DEPTH: 7 — Strong structural link between exceptional algebra (octonions), high-symmetry lattice (D4), and quantum error correction, but no explicit harmonic ratios or ancient geometric constants. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: The only substantive result is the arXiv paper on Tsirelson bounds derived via geometric algebra — the rest are unrelated video titles (Python, lab software, condensed matter talks) with no mathematical content. | MATH: The paper (arXiv:2007.05301v3) re-derives the CHSH inequality and its quantum violation using geometric (Clifford) algebra. Key constants: classical CHSH bound = 2; quantum Tsirelson bound = 2√2 ≈ 2.828. The violation ratio is √2. No explicit golden-ratio or base-60 constants appear in the abstract. | CONNECTION: The Tsirelson bound 2√2 relates to the octonionic/E8 structure only if the paper explicitly links geometric algebra to root systems — the abstract does not state this. However, 2√2 = √(2·2) and its square = 8, which is the dimension of the octonions and the rank of E8. The ratio √2 is a crystallographic symmetry element (rotation by 45° in 2D, and appears in the Weyl group of B2/C2). No 0.382, 0.618, 0.786, 1.618, or 2.618 appear in the evidence. | DEP Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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: The Kochen-Specker theorem is proven geometrically using a finite set of rays in 3D real space, directly linked to the Fano plane and the group PSL(2,7), which also underlies the octonion multiplication table and contextuality in quantum mechanics. | MATH: The proof uses 33 rays in R^3 (or 117 in a simpler variant) forming a configuration that admits no 2-coloring (0/1 assignment) consistent with orthogonality constraints. The Fano plane is the projective plane of order 2, with 7 points and 7 lines, each line containing 3 points. PSL(2,7) is the simple group of order 168, isomorphic to GL(3,2) and the automorphism group of the Fano plane. Octonion multiplication is encoded by the Fano plane: each line corresponds to a triple of imaginary units (e₁, e₂, e₃) satisfying e₁e₂ = e₃, with cyclic permutations. The Kochen-Specker configuration is a 3D real realization of this combinatorial structure. | CONNECTION: The Fano plane is a finite projective plane of order 2, whose symmetry Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com