Information Causality as the Sole Origin of the Quantum Correlation Law

This work presents a complete, non-circular, and principle-based derivation of the quantum correlation law E () = - () from three operational postulates: no-signaling with fair marginals, rotational covariance, and Information Causality (IC). The central contribution is the proof that IC alone enforces the kernel positivity (positive semidefiniteness) of the correlation matrix K (, ) = -E (-). This closes a long-standing axiomatic gap, as prior derivations often required kernel positivity as an independent mathematical assumption. Combined with the Herglotz theorem and Tsirelson saturation, this yields the unique harmonic form of the cosine law without additional ad-hoc postulates. Furthermore, numerical analysis of the hyperbolic extension (SO (2, 1) ) reveals that while IC still implies kernel positivity in non-Euclidean geometries, the Tsirelson bound can be exceeded. This suggests a deep interplay between spacetime geometry and the observed limits of quantum correlations.