Information Causality Implies Kernel Positivity and Geometric Robustness of Quantum Correlations (Version 3)

Version 3 of this paper provides the final numerical verification for a research series investigating the axiomatic foundations of quantum mechanics. Building upon previous work on Entanglement, No-Signaling, and Symmetry, this paper establishes Information Causality (IC) as the sole origin of the positive semi-definite (PSD) structure of bipartite correlation kernels. Key Scientific Updates in v3: •Numerical Proof (Turbo-Scan v6): Rigorous eigenvalue-based PSD testing of 500 random kernels. •Resolution of the "Gray Zone": Demonstration that higher-order IC protocols ($N=3$) achieve a 100% detection rate for non-positivity. •Geometric Universality: Verification of the Tsirelson bound (2√2) in the hyperbolic plane (H²). •Reproducibility: Inclusion of the full Python verification suite and raw simulation data.This work demonstrates that the mathematical requirement of kernel positivity is a direct physical consequence of the causal structure of information exchange.