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

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.com