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

I prove that among all positive semi-definite (PSD) correlation functions E(d) = −Σ a_l P_l(cos d) with a_l ≥ 0 on the Bloch sphere S², the quantum correlation E(d) = −cos d is the unique function that saturates the Tsirelson bound 2√2. The proof proceeds through the equality conditions of the Hilbert-space Tsirelson inequality: saturation forces a rigid parallelogram structure in the feature space, which projects onto every active Legendre component and is shown to be harmonically impossible for all l ≥ 2 via a spherical-harmonic obstruction. A supporting appendix gives the independent Chebyshev-based bound for the symmetric one-parameter configuration. Combined with the companion result that Information Causality enforces a_l ≥ 0, this yields a complete information-theoretic characterization of the quantum correlation law on S².