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

Andrew Stewart Caldin

Published Jul 20, 2026• 0 citations

Abstract

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

DOI: 10.5281/zenodo.21449532

Related papers

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 Rooted in Singular Values of Correlation Matrix — E8 Intelligence Research

Andrew Stewart Caldin

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

Oct 4, 20260 citations

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

Andrew Stewart Caldin

Oct 13, 20260 citations

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

Andrew Stewart Caldin

Jul 31, 20260 citations

Bloch Sphere Geometry Yields Tsirelson's Bound — E8 Intelligence Research

Andrew Stewart Caldin

Sep 21, 20260 citations

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

Andrew Stewart Caldin

Sep 30, 20260 citations

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

Andrew Stewart Caldin

Published Jul 20, 2026• 0 citations

Publisher

Abstract

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

DOI: 10.5281/zenodo.21449532

Related papers

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 Rooted in Singular Values of Correlation Matrix — E8 Intelligence Research

Andrew Stewart Caldin

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.comFINDING: 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

Oct 4, 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 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

Bloch Sphere Geometry Yields Tsirelson's Bound — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Tsirelson's bound (2√2 ≈ 2.828) emerges from the geometry of the Bloch sphere and SO(3) rotations, not from ad hoc quantum postulates. | MATH: CHSH inequality classical bound = 2; quantum maximum = 2√2. Derivation via Bloch vectors: for observables A(a), B(b) with outcomes ±1, correlation E(a,b) = a·b (for spin-1/2). CHSH expression S = E(a,b) + E(a,b') + E(a',b) − E(a',b'). Maximizing over unit vectors a,a',b,b' ∈ S² yields S_max = 2√2. This is equivalent to the maximum of |a·b + a·b' + a'·b − a'·b'| = 2√2, attained when the four vectors form a regular tetrahedron (tetrahedral angle 109.47°, cos⁻¹(−1/3)). | CONNECTION: The tetrahedron is a root system of A₃ (SU(4) Weyl group), and its vertices correspond to the 4 vertices of the 3-simplex. The ratio 2√2/2 = √2 ≈ 1.414 relates to the diagonal of a square — a crystallographic ratio. The Bloch sphere's SO(3) symmetry group has the same Lie algebra as SU(2), whose root system is A₁. The tetrahedral configuration is the maximal sy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Tsirelson's bound (2√2 ≈ 2.828) emerges from the geometry of the Bloch sphere and SO(3) rotations, not from ad hoc quantum postulates. | MATH: CHSH inequality classical bound = 2; quantum maximum = 2√2. Derivation via Bloch vectors: for observables A(a), B(b) with outcomes ±1, correlation E(a,b) = a·b (for spin-1/2). CHSH expression S = E(a,b) + E(a,b') + E(a',b) − E(a',b'). Maximizing over unit vectors a,a',b,b' ∈ S² yields S_max = 2√2. This is equivalent to the maximum of |a·b + a·b' + a'·b − a'·b'| = 2√2, attained when the four vectors form a regular tetrahedron (tetrahedral angle 109.47°, cos⁻¹(−1/3)). | CONNECTION: The tetrahedron is a root system of A₃ (SU(4) Weyl group), and its vertices correspond to the 4 vertices of the 3-simplex. The ratio 2√2/2 = √2 ≈ 1.414 relates to the diagonal of a square — a crystallographic ratio. The Bloch sphere's SO(3) symmetry group has the same Lie algebra as SU(2), whose root system is A₁. The tetrahedral configuration is the maximal sy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 21, 20260 citations

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

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

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