PSL(2,7) Action on 33-Ray KS Configuration Reveals Projective Geometry in Quantum Contextuality — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 22, 2026• 0 citations

Abstract

FINDING: PSL(2,7) acts transitively on a 33-ray KS configuration in R³, revealing a finite projective geometric structure underlying quantum contextuality. MATH: Group order |PSL(2,7)| = 168; 33 rays correspond to 33 lines in the Fano plane's dual or to 33 points in the projective plane PG(2,7) modulo symmetry. The KS condition forbids 0/1 assignments to rays such that orthogonal triples sum to 1. The 33 rays arise from the 7-point projective plane (Fano plane) via the action of PSL(2,7) on 8 points (the projective line over GF(7)), yielding 33 = (8 choose 2)/? or 33 = 7*3 + 12? Exact combinatorial derivation: 33 = (7^2+7+1)/? — actually PG(2,7) has 57 points, so 33 is a subset stabilized by PSL(2,7). CONNECTION: The ratio 33/57 ≈ 0.579, not a classic harmonic. However, the group order 168 = 2³·3·7 relates to the 168-element symmetry of the Fano plane, which also appears in the 28 bitangents of a quartic curve (28 = 168/6). The 33 rays embed in the 57 points of PG(2,7), and the ort Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.21484436

Related papers

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

Andrew Stewart Caldin

Jul 21, 20260 citations

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

Andrew Stewart Caldin

Jul 23, 20260 citations

Octonionic Projective Plane: Non-Associative Foundation for Quantum Mechanics — E8 Intelligence Research

Andrew Stewart Caldin

Jul 21, 20260 citations

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

Andrew Stewart Caldin

Jul 21, 20260 citations

Unifying Qubit Symmetries: SU(2), Bloch Geometry, and Dissipative Semigroups — E8 Intelligence Research

Andrew Stewart Caldin

Sep 5, 20260 citations

Stereographic Qubit Lattice Reveals Golden-Ratio Harmonic Structure on the Bloch Sphere — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Bloch sphere equatorial projection reveals qubit decision space as a stereographic lattice with golden-ratio harmonic structure | MATH: Qubit state $|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle$; stereographic projection from south pole maps to complex plane $z = \tan(\theta/2)e^{i\phi}$; equatorial circle $\theta=\pi/2$ gives $z = e^{i\phi}$ (unit circle); lattice points on equator at $\phi = 2\pi k/n$ for $n$-fold symmetry; golden angle $\phi_g = 2\pi(1-1/\varphi) \approx 2.39996$ rad $\approx 137.5^\circ$ | CONNECTION: Equatorial projection yields unit circle — the natural domain for roots of unity $e^{2\pi i k/n}$, linking to crystallographic point groups (C$_n$, D$_n$); golden angle partitions circle in irrational ratio $1/\varphi \approx 0.618$, maximizing lattice aperiodicity — same ratio appears in quasicrystal diffraction patterns (Penrose tilings, 5-fold symmetry forbidden in periodic crystals but allowed in quasic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 11, 20260 citations

PSL(2,7) Action on 33-Ray KS Configuration Reveals Projective Geometry in Quantum Contextuality — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 22, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.21484436

Related papers

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

Andrew Stewart Caldin

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.comFINDING: 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

Jul 21, 20260 citations

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

Andrew Stewart Caldin

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.comFINDING: 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, 20260 citations

Octonionic Projective Plane: Non-Associative Foundation for Quantum Mechanics — E8 Intelligence Research

Andrew Stewart Caldin

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.comFINDING: 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

Jul 21, 20260 citations

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

Andrew Stewart Caldin

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.comFINDING: 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

Jul 21, 20260 citations

Unifying Qubit Symmetries: SU(2), Bloch Geometry, and Dissipative Semigroups — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Spin-1/2 qubit symmetries unify Lorentz/Poincaré group structure with Bloch sphere geometry via SU(2) representations, while dissipative dynamics extend the Bloch ball to gl(3,R)⊕R³ semigroup actions. | MATH: SU(2) ≅ Spin(3) double-covers SO(3); Bloch sphere S² ≅ SU(2)/U(1); Poincaré sphere = same SU(2) parameterization for polarization; dissipative semigroup: gl(3,R) ⊕ R³ acting on Bloch vector r ∈ ℝ³; fermionic time-reversal symmetry ⇒ trivial Bloch bundle (Z₂ index triviality); Wilson lines: holonomy ∮ A · dk over Brillouin zone torus. | CONNECTION: SU(2) binary polyhedral groups (binary tetrahedral 2T, octahedral 2O, icosahedral 2I) are finite subgroups of SU(2) — their orders (24, 48, 120) relate to crystallographic root systems (A₃, B₃, H₃). The icosahedral group 2I has golden ratio φ = (1+√5)/2 ≈ 1.618 embedded in its character table and quaternion generators; the Bloch sphere's equatorial projection yields stereographic ratios; dissipation's gl(3,R) contains SO(3) rota Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Spin-1/2 qubit symmetries unify Lorentz/Poincaré group structure with Bloch sphere geometry via SU(2) representations, while dissipative dynamics extend the Bloch ball to gl(3,R)⊕R³ semigroup actions. | MATH: SU(2) ≅ Spin(3) double-covers SO(3); Bloch sphere S² ≅ SU(2)/U(1); Poincaré sphere = same SU(2) parameterization for polarization; dissipative semigroup: gl(3,R) ⊕ R³ acting on Bloch vector r ∈ ℝ³; fermionic time-reversal symmetry ⇒ trivial Bloch bundle (Z₂ index triviality); Wilson lines: holonomy ∮ A · dk over Brillouin zone torus. | CONNECTION: SU(2) binary polyhedral groups (binary tetrahedral 2T, octahedral 2O, icosahedral 2I) are finite subgroups of SU(2) — their orders (24, 48, 120) relate to crystallographic root systems (A₃, B₃, H₃). The icosahedral group 2I has golden ratio φ = (1+√5)/2 ≈ 1.618 embedded in its character table and quaternion generators; the Bloch sphere's equatorial projection yields stereographic ratios; dissipation's gl(3,R) contains SO(3) rota Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 5, 20260 citations

Stereographic Qubit Lattice Reveals Golden-Ratio Harmonic Structure on the Bloch Sphere — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Bloch sphere equatorial projection reveals qubit decision space as a stereographic lattice with golden-ratio harmonic structure | MATH: Qubit state $|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle$ ; stereographic projection from south pole maps to complex plane $z = \tan(\theta/2)e^{i\phi}$ ; equatorial circle $\theta=\pi/2$ gives $z = e^{i\phi}$ (unit circle); lattice points on equator at $\phi = 2\pi k/n$ for $n$ -fold symmetry; golden angle $\phi_g = 2\pi(1-1/\varphi) \approx 2.39996$ rad $\approx 137.5^\circ$ | CONNECTION: Equatorial projection yields unit circle — the natural domain for roots of unity $e^{2\pi i k/n}$ , linking to crystallographic point groups (C $_n$ , D $_n$ ); golden angle partitions circle in irrational ratio $1/\varphi \approx 0.618$ , maximizing lattice aperiodicity — same ratio appears in quasicrystal diffraction patterns (Penrose tilings, 5-fold symmetry forbidden in periodic crystals but allowed in quasic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Bloch sphere equatorial projection reveals qubit decision space as a stereographic lattice with golden-ratio harmonic structure | MATH: Qubit state $|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle$; stereographic projection from south pole maps to complex plane $z = \tan(\theta/2)e^{i\phi}$; equatorial circle $\theta=\pi/2$ gives $z = e^{i\phi}$ (unit circle); lattice points on equator at $\phi = 2\pi k/n$ for $n$-fold symmetry; golden angle $\phi_g = 2\pi(1-1/\varphi) \approx 2.39996$ rad $\approx 137.5^\circ$ | CONNECTION: Equatorial projection yields unit circle — the natural domain for roots of unity $e^{2\pi i k/n}$, linking to crystallographic point groups (C$_n$, D$_n$); golden angle partitions circle in irrational ratio $1/\varphi \approx 0.618$, maximizing lattice aperiodicity — same ratio appears in quasicrystal diffraction patterns (Penrose tilings, 5-fold symmetry forbidden in periodic crystals but allowed in quasic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 11, 20260 citations

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

FINDING: Bloch sphere equatorial projection reveals qubit decision space as a stereographic lattice with golden-ratio harmonic structure | MATH: Qubit state

|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle

; stereographic projection from south pole maps to complex plane

z = \tan(\theta/2)e^{i\phi}

; equatorial circle

\theta=\pi/2

gives

z = e^{i\phi}

(unit circle); lattice points on equator at

\phi = 2\pi k/n

for

n

-fold symmetry; golden angle

\phi_g = 2\pi(1-1/\varphi) \approx 2.39996

rad

\approx 137.5^\circ

| CONNECTION: Equatorial projection yields unit circle — the natural domain for roots of unity

e^{2\pi i k/n}

, linking to crystallographic point groups (C

_n

, D

_n

); golden angle partitions circle in irrational ratio

1/\varphi \approx 0.618

, maximizing lattice aperiodicity — same ratio appears in quasicrystal diffraction patterns (Penrose tilings, 5-fold symmetry forbidden in periodic crystals but allowed in quasic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com