Tsirelson Saturation as a Uniqueness Principle: The Quantum Correlation is the Only Extremal PSD Kernel on S²

We prove that among all positive semi-definite (PSD) correlation functions E(d) = −∑ₗ≥₁ aₗPₗ(cos d) with aₗ ≥ 0 on the Bloch sphere S², the quantum correlation E(d) = −cos d is the unique function that achieves the Tsirelson bound S_CHSH = 2√2. For even l ≥ 2 the result follows analytically from the non-zero constant Chebyshev term of Pₗ. For odd l ≥ 3 it follows from an algebraic incompatibility: the Chebyshev modes hₖ(u) = 3cos(ku) − cos(3ku) for k = 1 and k = 3 cannot simultaneously attain their individual maxima 2√2, because the triple-angle identity forces cos(3u) = −1/√2 whenever cos(u) = 1/√2. Combined with the companion result that Information Causality enforces aₗ ≥ 0, this yields the complete axiomatic derivation: Information Causality together with Tsirelson saturation uniquely selects E(d) = −cos d on S².