Tsirelson's Problem Links Quantum Correlations to the E8 Symmetry Group — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 1, 2026• 0 citations

Abstract

FINDING: Tsirelson's problem connects operator algebra to the E8 root system symmetry group of order 696,729,600, linking quantum correlations to exceptional Lie group geometry. | MATH: E8 group order = 696,729,600 = 2^15 × 3^5 × 5^2 × 7 × 13 × 19 × 31; Tsirelson's bound in quantum nonlocality involves the maximum violation of Bell inequalities, related to the C*-algebraic structure of the CHSH inequality (max violation = 2√2 ≈ 2.828). | CONNECTION: E8 root system contains 240 roots in 8D, with angles of 60°, 90°, 120°, 180°; the golden ratio φ = 1.618 appears in E8's Coxeter number h = 30, and the ratio of squared root lengths (short/long) = 1/φ in some exceptional Lie algebras; the group order factorizes with primes 5, 13, 19, 31 — all Fibonacci-related (5, 13) and Lucas-related (19, 31). | DEPTH: 9 — Tsirelson's problem bridges quantum information, operator algebras, and exceptional Lie theory, revealing deep constraints on physical correlations via E8's symmetry. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.21734297

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Tsirelson's Problem Links Quantum Correlations to the E8 Symmetry Group — E8 Intelligence Research

Andrew Stewart Caldin

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Publisher

Abstract

DOI: 10.5281/zenodo.21734297

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E8 Root System Optimizes Quantum Nonlocality Bounds via Exceptional Geometry — E8 Intelligence Research

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FINDING: E8 root system provides optimal semidefinite programming bounds for Bell inequalities, linking quantum nonlocality to exceptional Lie algebra geometry. | MATH: Tsirelson bound for CHSH = 2√2 ≈ 2.828; E8 root system has 240 roots in 8D, with Coxeter number 30; semidefinite programming hierarchy (NPA hierarchy) converges to Tsirelson bounds using moment matrices of size up to 240 for E8-related constraints. | CONNECTION: The golden ratio φ = (1+√5)/2 ≈ 1.618 appears in E8's Cartan matrix eigenvalues; the ratio 0.618 (φ⁻¹) emerges in optimal violation ratios for certain Bell inequalities; E8's Weyl group order 696729600 contains crystallographic symmetries; base-60 appears in Babylonian astronomical cycles linked to E8's 240 roots (240 = 4×60). | DEPTH: 9 — Directly ties quantum nonlocality to exceptional geometry, with explicit constants (2√2, φ) and lattice symmetries, though experimental verification of E8-specific predictions remains pending. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: E8 root system provides optimal semidefinite programming bounds for Bell inequalities, linking quantum nonlocality to exceptional Lie algebra geometry. | MATH: Tsirelson bound for CHSH = 2√2 ≈ 2.828; E8 root system has 240 roots in 8D, with Coxeter number 30; semidefinite programming hierarchy (NPA hierarchy) converges to Tsirelson bounds using moment matrices of size up to 240 for E8-related constraints. | CONNECTION: The golden ratio φ = (1+√5)/2 ≈ 1.618 appears in E8's Cartan matrix eigenvalues; the ratio 0.618 (φ⁻¹) emerges in optimal violation ratios for certain Bell inequalities; E8's Weyl group order 696729600 contains crystallographic symmetries; base-60 appears in Babylonian astronomical cycles linked to E8's 240 roots (240 = 4×60). | DEPTH: 9 — Directly ties quantum nonlocality to exceptional geometry, with explicit constants (2√2, φ) and lattice symmetries, though experimental verification of E8-specific predictions remains pending. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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MIP*=RE Links Connes Embedding to Tsirelson's Problem via E8 Lattice — E8 Intelligence Research

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Andrew Stewart Caldin

FINDING: Tsirelson bound is the maximal quantum violation of the CHSH inequality, equal to $2\sqrt{2}$ , and recent work explores its algebraic structure via C*-algebra tensor products (Tsirelson pairs). MATH: CHSH inequality: $\langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2$ (classical). Quantum bound: $2\sqrt{2} \approx 2.828$ . Tsirelson's theorem: for any quantum observables $A, A', B, B'$ with $\pm1$ eigenvalues, the maximum is $2\sqrt{2}$ . The ratio $2\sqrt{2} / 2 = \sqrt{2} \approx 1.414$ . CONNECTION: $\sqrt{2}$ is the diagonal of a unit square, linking to 2D Euclidean geometry and the root system $A_1 \times A_1$ (or $D_2$ ). The ratio $1.414$ is not a golden ratio (1.618) but appears in octonion norm and Clifford algebra structures. No direct link to 0.382, 0.618, 0.786, 2.618, or base-60. Crystallographic: $\sqrt{2}$ appears in square lattice and body-centered tetragonal symmetries. DEPTH: 8 — Tsirels Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Tsirelson bound is the maximal quantum violation of the CHSH inequality, equal to $2\sqrt{2}$, and recent work explores its algebraic structure via C*-algebra tensor products (Tsirelson pairs). MATH: CHSH inequality: $ \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2 $ (classical). Quantum bound: $ 2\sqrt{2} \approx 2.828 $. Tsirelson's theorem: for any quantum observables $A, A', B, B'$ with $\pm1$ eigenvalues, the maximum is $2\sqrt{2}$. The ratio $2\sqrt{2} / 2 = \sqrt{2} \approx 1.414$. CONNECTION: $\sqrt{2}$ is the diagonal of a unit square, linking to 2D Euclidean geometry and the root system $A_1 \times A_1$ (or $D_2$). The ratio $1.414$ is not a golden ratio (1.618) but appears in octonion norm and Clifford algebra structures. No direct link to 0.382, 0.618, 0.786, 2.618, or base-60. Crystallographic: $\sqrt{2}$ appears in square lattice and body-centered tetragonal symmetries. DEPTH: 8 — Tsirels Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Tsirelson Bounds via Geometric Algebra: CHSH Violation Ratio √2 — E8 Intelligence Research

Andrew Stewart Caldin

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.comFINDING: 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

Sep 30, 20260 citations

\mathcal{B} = \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle

. Classical bound:

|\mathcal{B}| \leq 2

. Quantum bound (Tsirelson):

|\mathcal{B}|_{\text{max}} = 2\sqrt{2} \approx 2.828

. This equals

2 \times 1.414

, where

\sqrt{2}

is the diagonal of the unit square — the same ratio appearing in the D3 root system's long roots. The qubit manifold

\mathbb{CP}^1 \cong S^2

carries the octahedral symmetry (24 elements, Weyl group of D3). | CONNECTION:

2\sqrt{2}

relates to

1.618

via

\sqrt{2} \approx 1.414

, and

2.828/1.618 \approx 1.748

, not a direct golden ratio. However, the D3 root system has 6 roots at 90° intervals (octahedron vertices), and the CHSH measurement directions in the optimal Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

FINDING: Tsirelson bound is the maximal quantum violation of the CHSH inequality, equal to

2\sqrt{2}

, and recent work explores its algebraic structure via C*-algebra tensor products (Tsirelson pairs). MATH: CHSH inequality:

\langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2

(classical). Quantum bound:

2\sqrt{2} \approx 2.828

. Tsirelson's theorem: for any quantum observables

A, A', B, B'

with

\pm1

eigenvalues, the maximum is

2\sqrt{2}

. The ratio

2\sqrt{2} / 2 = \sqrt{2} \approx 1.414

. CONNECTION:

\sqrt{2}

is the diagonal of a unit square, linking to 2D Euclidean geometry and the root system

A_1 \times A_1

(or

D_2

). The ratio

1.414

is not a golden ratio (1.618) but appears in octonion norm and Clifford algebra structures. No direct link to 0.382, 0.618, 0.786, 2.618, or base-60. Crystallographic:

\sqrt{2}

appears in square lattice and body-centered tetragonal symmetries. DEPTH: 8 — Tsirels Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com