Tsirelson Bound Rooted in Singular Values of Correlation Matrix — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 4, 2026• 0 citations

Abstract

FINDING: The CHSH inequality's maximal quantum violation (Tsirelson bound) is exactly $2\sqrt{2}$, a value rooted in the singular-value structure of a 4×4 correlation matrix, not an arbitrary limit. MATH: - CHSH operator: $S = A_1 \otimes B_1 + A_1 \otimes B_2 + A_2 \otimes B_1 - A_2 \otimes B_2$, with $A_i, B_j \in \{\pm1\}$ observables. - Classical bound: $\langle S \rangle \leq 2$. - Quantum bound (Tsirelson): $\langle S \rangle_{\max} = 2\sqrt{2} \approx 2.828$. - Singular value decomposition of the 4×4 correlation matrix $C_{ij} = \langle A_i B_j \rangle$: the maximum of $\sum_{ij} M_{ij} C_{ij}$ over all quantum correlations equals $2\sqrt{2} \cdot \sigma_{\max}(M)$, where $M$ is the CHSH coefficient matrix $\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$. Here $\sigma_{\max}(M) = \sqrt{2}$, so the bound is $2\sqrt{2}$. - Equivalently: $\sqrt{2} = \frac{2}{\sqrt{2}} = \frac{1}{\sin(\pi/4)} = \frac{1}{\cos(\pi/4)}$. - The ratio \(2\sqrt{2}/2 = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23131930

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Tsirelson's Bound: Algebraic Proof of the Quantum CHSH Maximum — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Tsirelson's bound is the algebraic maximum quantum correlation $2\sqrt{2}$ for the CHSH inequality, proven via operator norm constraints on anticommuting observables. | MATH: CHSH inequality classical bound = 2; quantum maximum = $2\sqrt{2} \approx 2.828$; Tsirelson's bound emerges from $\|A_1B_1 + A_1B_2 + A_2B_1 - A_2B_2\| \leq 2\sqrt{2}$ for self-adjoint operators $A_i, B_j$ with $\|A_i\|, \|B_j\| \leq 1$ and $[A_i, B_j] = 0$. The proof uses the algebraic identity $(A_1 + A_2)B_1 + (A_1 - A_2)B_2$ and the C*-algebra norm inequality $\|X + Y\|^2 + \|X - Y\|^2 \leq 2(\|X\|^2 + \|Y\|^2)$. | CONNECTION: $2\sqrt{2}$ is not a golden ratio constant, but the ratio $2\sqrt{2} / 2 = \sqrt{2} \approx 1.414$ appears in crystallographic root systems (e.g., $B_n$ lattice spacing) and in the diagonal of a unit square. The bound's algebraic structure involves anticommutation relations reminiscent of Clifford algebras, which underlie spinor representations and certain Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Jul 20, 20260 citations

Tsirelson Bound and C*-Algebraic Structure of Maximal CHSH Violation — E8 Intelligence Research

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.com

Jul 24, 20260 citations

Tsirelson Bound for CHSH Inequality: 2√2 from Root System A1 and Octagonal Symmetry — E8 Intelligence Research

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Tsirelson's Bound: Quantum Correlations Peak at 2√2, Surpassing Classical Limits — E8 Intelligence Research

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Tsirelson Bound Rooted in Singular Values of Correlation Matrix — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 4, 2026• 0 citations

Publisher

Abstract

FINDING: The CHSH inequality's maximal quantum violation (Tsirelson bound) is exactly (2\sqrt{2}), a value rooted in the singular-value structure of a 4×4 correlation matrix, not an arbitrary limit. MATH: - CHSH operator: (S = A_1 \otimes B_1 + A_1 \otimes B_2 + A_2 \otimes B_1 - A_2 \otimes B_2), with (A_i, B_j \in {\pm1}) observables. - Classical bound: (\langle S \rangle \leq 2). - Quantum bound (Tsirelson): (\langle S \rangle_{\max} = 2\sqrt{2} \approx 2.828). - Singular value decomposition of the 4×4 correlation matrix (C_{ij} = \langle A_i B_j \rangle): the maximum of (\sum_{ij} M_{ij} C_{ij}) over all quantum correlations equals (2\sqrt{2} \cdot \sigma_{\max}(M)), where (M) is the CHSH coefficient matrix (\begin{pmatrix} 1 & 1 \ 1 & -1 \end{pmatrix}). Here (\sigma_{\max}(M) = \sqrt{2}), so the bound is (2\sqrt{2}). - Equivalently: (\sqrt{2} = \frac{2}{\sqrt{2}} = \frac{1}{\sin(\pi/4)} = \frac{1}{\cos(\pi/4)}). - The ratio (2\sqrt{2}/2 = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23131930

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Tsirelson's Bound: Algebraic Proof of the Quantum CHSH Maximum — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Tsirelson's bound is the algebraic maximum quantum correlation $2\sqrt{2}$ for the CHSH inequality, proven via operator norm constraints on anticommuting observables. | MATH: CHSH inequality classical bound = 2; quantum maximum = $2\sqrt{2} \approx 2.828$ ; Tsirelson's bound emerges from $\|A_1B_1 + A_1B_2 + A_2B_1 - A_2B_2\| \leq 2\sqrt{2}$ for self-adjoint operators $A_i, B_j$ with $\|A_i\|, \|B_j\| \leq 1$ and $[A_i, B_j] = 0$ . The proof uses the algebraic identity $(A_1 + A_2)B_1 + (A_1 - A_2)B_2$ and the C*-algebra norm inequality $\|X + Y\|^2 + \|X - Y\|^2 \leq 2(\|X\|^2 + \|Y\|^2)$ . | CONNECTION: $2\sqrt{2}$ is not a golden ratio constant, but the ratio $2\sqrt{2} / 2 = \sqrt{2} \approx 1.414$ appears in crystallographic root systems (e.g., $B_n$ lattice spacing) and in the diagonal of a unit square. The bound's algebraic structure involves anticommutation relations reminiscent of Clifford algebras, which underlie spinor representations and certain Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Tsirelson's bound is the algebraic maximum quantum correlation $2\sqrt{2}$ for the CHSH inequality, proven via operator norm constraints on anticommuting observables. | MATH: CHSH inequality classical bound = 2; quantum maximum = $2\sqrt{2} \approx 2.828$; Tsirelson's bound emerges from $\|A_1B_1 + A_1B_2 + A_2B_1 - A_2B_2\| \leq 2\sqrt{2}$ for self-adjoint operators $A_i, B_j$ with $\|A_i\|, \|B_j\| \leq 1$ and $[A_i, B_j] = 0$. The proof uses the algebraic identity $(A_1 + A_2)B_1 + (A_1 - A_2)B_2$ and the C*-algebra norm inequality $\|X + Y\|^2 + \|X - Y\|^2 \leq 2(\|X\|^2 + \|Y\|^2)$. | CONNECTION: $2\sqrt{2}$ is not a golden ratio constant, but the ratio $2\sqrt{2} / 2 = \sqrt{2} \approx 1.414$ appears in crystallographic root systems (e.g., $B_n$ lattice spacing) and in the diagonal of a unit square. The bound's algebraic structure involves anticommutation relations reminiscent of Clifford algebras, which underlie spinor representations and certain Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Jul 20, 20260 citations

Tsirelson Bound and C*-Algebraic Structure of Maximal CHSH Violation — E8 Intelligence Research

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

Jul 24, 20260 citations

Tsirelson Bound for CHSH Inequality: 2√2 from Root System A1 and Octagonal Symmetry — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Tsirelson bound for CHSH inequality is 2√2 ≈ 2.828, linking quantum correlations to root system A1 and octagonal symmetry. MATH: CHSH inequality: ⟨AB⟩ + ⟨AB'⟩ + ⟨A'B⟩ - ⟨A'B'⟩ ≤ 2 (classical). Tsirelson bound: ≤ 2√2 (quantum). Optimal violation achieved by observables A = σ_z, A' = σ_x, B = (σ_z + σ_x)/√2, B' = (σ_z - σ_x)/√2. CONNECTION: 2√2 = √(4+4) = 2 * 1.414; ratio 1.414 approximates √2, not directly 0.618/1.618. However, the angle between optimal measurement directions is 45°, which is π/4 — octagonal symmetry (8-fold rotation). The root system A1 (SU(2)) generates the Pauli matrices; the square lattice emerges from the two-dimensional Bloch sphere projection. The octagonal symmetry appears in the 45° rotation between Alice's and Bob's bases. DEPTH: 8 — This bound is fundamental to quantum nonlocality, revealing that quantum correlations are constrained by the geometry of SU(2) and the square lattice of measurement directions. The 45° angle is a crystallographic ro Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Tsirelson bound for CHSH inequality is 2√2 ≈ 2.828, linking quantum correlations to root system A1 and octagonal symmetry. MATH: CHSH inequality: ⟨AB⟩ + ⟨AB'⟩ + ⟨A'B⟩ - ⟨A'B'⟩ ≤ 2 (classical). Tsirelson bound: ≤ 2√2 (quantum). Optimal violation achieved by observables A = σ_z, A' = σ_x, B = (σ_z + σ_x)/√2, B' = (σ_z - σ_x)/√2. CONNECTION: 2√2 = √(4+4) = 2 * 1.414; ratio 1.414 approximates √2, not directly 0.618/1.618. However, the angle between optimal measurement directions is 45°, which is π/4 — octagonal symmetry (8-fold rotation). The root system A1 (SU(2)) generates the Pauli matrices; the square lattice emerges from the two-dimensional Bloch sphere projection. The octagonal symmetry appears in the 45° rotation between Alice's and Bob's bases. DEPTH: 8 — This bound is fundamental to quantum nonlocality, revealing that quantum correlations are constrained by the geometry of SU(2) and the square lattice of measurement directions. The 45° angle is a crystallographic ro Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Jul 31, 20260 citations

Tsirelson's Bound: Quantum Correlations Peak at 2√2, Surpassing Classical Limits — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: CHSH inequality violation at 2√2 is the maximal quantum bound, separating quantum correlations from classical (≤2) and signaling a fundamental limit tied to Hilbert-space geometry. | MATH: Classical bound: ⟨CHSH⟩ ≤ 2. Quantum bound (Tsirelson): ⟨CHSH⟩ ≤ 2√2 ≈ 2.828. Violation ratio: 2√2 / 2 = √2 ≈ 1.414. The gap (2√2 − 2) ≈ 0.828. In base-60: 2√2 ≈ 2°49'41" (2.8284 × 60 = 169.704′ → 2′49.7″). | CONNECTION: 2√2 = √(8) = 2 × 1.414. Note 1.414² = 2. The ratio 2/2√2 = 1/√2 ≈ 0.7071, which is sin(45°) — the diagonal of a unit square. The golden ratio φ = 1.618 appears in the *ratio* (2√2)/φ ≈ 1.748, not directly. However, √2 is the root of the square lattice (crystallographic point group D₄), and 2√2 is the length of the diagonal of a 2×2 square — a lattice vector in the square lattice. The CHSH-3 inequality (arXiv:2003.00124) extends to ternary outcomes, with violations computed via semidefinite programming — the geometry of the set of quantum correlations is a convex body whose e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: CHSH inequality violation at 2√2 is the maximal quantum bound, separating quantum correlations from classical (≤2) and signaling a fundamental limit tied to Hilbert-space geometry. | MATH: Classical bound: ⟨CHSH⟩ ≤ 2. Quantum bound (Tsirelson): ⟨CHSH⟩ ≤ 2√2 ≈ 2.828. Violation ratio: 2√2 / 2 = √2 ≈ 1.414. The gap (2√2 − 2) ≈ 0.828. In base-60: 2√2 ≈ 2°49'41" (2.8284 × 60 = 169.704′ → 2′49.7″). | CONNECTION: 2√2 = √(8) = 2 × 1.414. Note 1.414² = 2. The ratio 2/2√2 = 1/√2 ≈ 0.7071, which is sin(45°) — the diagonal of a unit square. The golden ratio φ = 1.618 appears in the *ratio* (2√2)/φ ≈ 1.748, not directly. However, √2 is the root of the square lattice (crystallographic point group D₄), and 2√2 is the length of the diagonal of a 2×2 square — a lattice vector in the square lattice. The CHSH-3 inequality (arXiv:2003.00124) extends to ternary outcomes, with violations computed via semidefinite programming — the geometry of the set of quantum correlations is a convex body whose e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 13, 20260 citations

Tsirelson bound as geometric maximum of a correlation polytope under Pauli group symmetry — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: CHSH violation is bounded by the Tsirelson limit 2√2, which is the geometric maximum of a correlation polytope inscribed in the Bloch sphere's state space; the octahedral symmetry of the Pauli group constrains the measurement directions that achieve it. MATH: - CHSH operator: S = A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁, with |⟨S⟩| ≤ 2 (classical) and |⟨S⟩| ≤ 2√2 ≈ 2.828 (quantum/Tsirelson). - Optimal quantum value 2√2 = 2·√2 arises from measurement directions at 45° in the equatorial plane of the Bloch sphere (A₀,A₁ and B₀,B₁ each separated by π/2, with relative offset π/4). - Qubit state: |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ}sin(θ/2)|1⟩; Bloch vector r = (sinθcosφ, sinθsinφ, cosθ), |r| ≤ 1. - Pauli group {I, X, Y, Z} generates the octahedral rotation group O ⊂ SO(3); the six ±X, ±Y, ±Z eigenstates sit at the six vertices of a regular octahedron inscribed in the Bloch sphere. - Tsirelson bound: 2√2 = 4·cos(π/4) = 4/√2. Equivalently, the maximum is 4·(1/√2), where 1/√2 = 0.7071 is the projection facto Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: CHSH violation is bounded by the Tsirelson limit 2√2, which is the geometric maximum of a correlation polytope inscribed in the Bloch sphere's state space; the octahedral symmetry of the Pauli group constrains the measurement directions that achieve it. MATH: - CHSH operator: S = A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁, with |⟨S⟩| ≤ 2 (classical) and |⟨S⟩| ≤ 2√2 ≈ 2.828 (quantum/Tsirelson). - Optimal quantum value 2√2 = 2·√2 arises from measurement directions at 45° in the equatorial plane of the Bloch sphere (A₀,A₁ and B₀,B₁ each separated by π/2, with relative offset π/4). - Qubit state: |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ}sin(θ/2)|1⟩; Bloch vector r = (sinθcosφ, sinθsinφ, cosθ), |r| ≤ 1. - Pauli group {I, X, Y, Z} generates the octahedral rotation group O ⊂ SO(3); the six ±X, ±Y, ±Z eigenstates sit at the six vertices of a regular octahedron inscribed in the Bloch sphere. - Tsirelson bound: 2√2 = 4·cos(π/4) = 4/√2. Equivalently, the maximum is 4·(1/√2), where 1/√2 = 0.7071 is the projection facto Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 27, 20260 citations

Quantum Violations of CHSH-3 Inequality via Tensor Product Hilbert Spaces — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: The CHSH-3 inequality extends the standard CHSH framework to three measurement settings per party, with quantum violations computable via semidefinite programming; the Hilbert space structure underpinning these violations is the tensor product of local Hilbert spaces, with singular values of the correlation matrix determining maximal violation bounds. | MATH: Standard CHSH: $S = \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2$ (local), $S_{\text{quantum}} \leq 2\sqrt{2} \approx 2.828$ . CHSH-3 generalizes to three settings per party, with the quantum bound given by the maximal singular value of the correlation tensor $T_{ij} = \langle A_i \otimes B_j \rangle$ , i.e., $\|T\|_\infty \leq \sqrt{3}$ for the specific CHSH-3 form (Tsirelson-type bound). The Hilbert space is $\mathcal{H}_A \otimes \mathcal{H}_B$ , and the singular value decomposition of the $3 \times 3$ correlation matrix yields singular values $\sigma_1 \geq \si Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: The CHSH-3 inequality extends the standard CHSH framework to three measurement settings per party, with quantum violations computable via semidefinite programming; the Hilbert space structure underpinning these violations is the tensor product of local Hilbert spaces, with singular values of the correlation matrix determining maximal violation bounds. | MATH: Standard CHSH: \(S = \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2$ (local), $S_{\text{quantum}} \leq 2\sqrt{2} \approx 2.828$. CHSH-3 generalizes to three settings per party, with the quantum bound given by the maximal singular value of the correlation tensor $T_{ij} = \langle A_i \otimes B_j \rangle$, i.e., $\|T\|_\infty \leq \sqrt{3}$ for the specific CHSH-3 form (Tsirelson-type bound). The Hilbert space is $\mathcal{H}_A \otimes \mathcal{H}_B$, and the singular value decomposition of the $3 \times 3$ correlation matrix yields singular values \(\sigma_1 \geq \si Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 17, 20260 citations

FINDING: Tsirelson's bound is the algebraic maximum quantum correlation

2\sqrt{2}

for the CHSH inequality, proven via operator norm constraints on anticommuting observables. | MATH: CHSH inequality classical bound = 2; quantum maximum =

2\sqrt{2} \approx 2.828

; Tsirelson's bound emerges from

\|A_1B_1 + A_1B_2 + A_2B_1 - A_2B_2\| \leq 2\sqrt{2}

for self-adjoint operators

A_i, B_j

with

\|A_i\|, \|B_j\| \leq 1

and

[A_i, B_j] = 0

. The proof uses the algebraic identity

(A_1 + A_2)B_1 + (A_1 - A_2)B_2

and the C*-algebra norm inequality

\|X + Y\|^2 + \|X - Y\|^2 \leq 2(\|X\|^2 + \|Y\|^2)

. | CONNECTION:

2\sqrt{2}

is not a golden ratio constant, but the ratio

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

appears in crystallographic root systems (e.g.,

B_n

lattice spacing) and in the diagonal of a unit square. The bound's algebraic structure involves anticommutation relations reminiscent of Clifford algebras, which underlie spinor representations and certain 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

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

(local),

S_{\text{quantum}} \leq 2\sqrt{2} \approx 2.828

. CHSH-3 generalizes to three settings per party, with the quantum bound given by the maximal singular value of the correlation tensor

T_{ij} = \langle A_i \otimes B_j \rangle

, i.e.,

\|T\|_\infty \leq \sqrt{3}

for the specific CHSH-3 form (Tsirelson-type bound). The Hilbert space is

\mathcal{H}_A \otimes \mathcal{H}_B

, and the singular value decomposition of the

3 \times 3

correlation matrix yields singular values \(\sigma_1 \geq \si Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com