SpecSphere: Dual-Pass Spectral-Spatial Graph Neural Networks with Certified Robustness

We introduce SpecSphere, the first dual-pass spectral-spatial GNN that certifies every prediction against both $\ell\_{0}$ edge flips and $\ell\_{\infty}$ feature perturbations, adapts to the full homophily-heterophily spectrum, and surpasses the expressive power of 1-Weisfeiler-Lehman while retaining linear-time complexity. Our model couples a Chebyshev-polynomial spectral branch with an attention-gated spatial branch and fuses their representations through a lightweight MLP trained in a cooperative-adversarial min-max game. We further establish (i) a uniform Chebyshev approximation theorem, (ii) minimax-optimal risk across the homophily-heterophily spectrum, (iii) closed-form robustness certificates, and (iv) universal approximation strictly beyond 1-WL. SpecSphere achieves state-of-the-art node-classification accuracy and delivers tighter certified robustness guarantees on real-world benchmarks. These results demonstrate that high expressivity, heterophily adaptation, and provable robustness can coexist within a single, scalable architecture.