The Bell Inequality as a Stabiliser-Range Theorem: Tsirelson's Bound from Symmetry Breaking

The violation of Bell inequalities by quantum systems is among the most precisely tested predictions in physics, yet the origin of the Tsirelson bound CHSH ≤ 2√2 is rarely stated in fully algebraic terms. We show that this bound is an instance of the stabiliser-range theorem first established for check-digit codes over finite alphabets. The three-element framework (G, F, E) of that work translates to the quantum bipartite setting as follows: the state group G is the unitary group U(HA ⊗ HB), the fold F is the singlet state |ψ−⟩, and the perturbation family E is the set of local unitary transformations. The product stabiliser Stab× (the classical, local-hidden-variable description) is a strict subset of the tensor stabiliser Stab⊗ (the quantum description); the difference is exactly the diagonal SU(2) subgroup {U ⊗ U : U ∈ SU(2)}. We prove a Lie-algebra stretch factor theorem: every generator X ∈ su(2) satisfies ‖dφdiag(X)‖HS = √2 · ‖dφprod(X)‖HS, where φdiag and φprod are the diagonal and product embeddings of SU(2) into U(4). This stretch factor is the algebraic source of the √2 in the Tsirelson bound. We further establish a Stab-Tsirelson equivalence chain: the diagonal stabiliser condition (U ⊗ U)|ψ−⟩ = |ψ−⟩ implies, through a single algebraic sequence, that the CHSH operator has eigenvalue −2√2 on |ψ−⟩. For N-particle systems we prove inductively that the MABK (Mermin-Ardehali-Belinskii-Klyshko) ratio satisfies M(N) = 2(N−1)/2, confirmed numerically for N = 1, …, 5. No-signaling emerges as a corollary: the diagonal stabiliser is inaccessible from either subsystem alone. The framework places the Tsirelson bound, the multiparticle scaling, and the no-signaling principle within a single algebraic structure, providing a unified account of quantum nonlocality that is traceable to the check-digit symmetry-matching condition Stab(G, F) ∩ E = {e}. Complete proofs are provided in the supplementary document.