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

FINDING: The CHSH inequality's maximal quantum violation (Tsirelson bound) is exactly \(2√2\), a value rooted in the singular-value structure of a 4×4 correlation matrix, not an arbitrary limit. MATH: - CHSH operator: \(S = A_1 ⊗ B_1 + A_1 ⊗ B_2 + A_2 ⊗ B_1 - A_2 ⊗ B_2\), with \(A_i, B_j ∈ \{±1\}\) observables. - Classical bound: \( S ≤ 2\). - Quantum bound (Tsirelson): \( S ₘₐₓ = 2√2 ≈ 2.828\). - Singular value decomposition of the 4×4 correlation matrix \(Cᵢⱼ = A_i B_j \): the maximum of \(∑ᵢⱼ Mᵢⱼ Cᵢⱼ\) over all quantum correlations equals \(2√2 · σₘₐₓ(M)\), where \(M\) is the CHSH coefficient matrix \({pmatrix} 1 & 1 \\ 1 & -1 {pmatrix}\). Here \(σₘₐₓ(M) = √2\), so the bound is \(2√2\). - Equivalently: \(√2 = {2}{√2} = 1/sin(π/4) = 1/cos(π/4)\). - The ratio \(2√2/2 = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com