Spectral Graph Neural Networks offer a principled approach to graph filtering but face a fundamental "Stability-vs-Adaptivity" trade-off. This trade-off is dictated by the choice of spectral domain. Filters in the finite [-1, 1] domain (e.g., ChebyNet) are numerically stable at high polynomial degrees (K) but are static and low-pass, causing them to fail on heterophilic graphs. Conversely, filters in the semi-infinite [0, infty) domain (e.g., KrawtchoukNet) are highly adaptive and achieve SOTA results on heterophily by learning non-low-pass responses. However, as we demonstrate, these adaptive filters can also suffer from numerical instability, leading to catastrophic performance collapse at high K. In this paper, we propose to resolve this trade-off by designing a hybrid-domain GNN, HybSpecNet, which combines a stable `ChebyNet` branch with an adaptive `KrawtchoukNet` branch. We first demonstrate that a "naive" hybrid architecture, which fuses the branches via concatenation, successfully unifies performance at low K, achieving strong results on both homophilic and heterophilic benchmarks. However, we then prove that this naive architecture fails the stability test. Our K-ablation experiments show that this architecture catastrophically collapses at K=25, exactly mirroring the collapse of its unstable `KrawtchoukNet` branch. We identify this critical finding as "Instability Poisoning," where `NaN`/`Inf` gradients from the adaptive branch destroy the training of the model. Finally, we propose and validate an advanced architecture that uses "Late Fusion" to completely isolate the gradient pathways. We demonstrate that this successfully solves the instability problem, remaining perfectly stable up to K=30 while retaining its SOTA performance across all graph types. This work identifies a critical architectural pitfall in hybrid GNN design and provides the robust architectural solution.

Spectral Graph Neural Networks (GNNs) based on polynomial filters, such as ChebyNet, suffer from two critical limitations: 1) performance collapse on "heterophilic" graphs and 2) performance collapse at high polynomial degrees (K), known as over-smoothing. Both issues stem from the static, low-pass nature of standard filters. In this work, we propose `KrawtchoukNet`, a GNN filter based on the discrete Krawtchouk polynomials. We demonstrate that `KrawtchoukNet` provides a unified solution to both problems through two key design choices. First, by fixing the polynomial's domain N to a small constant (e.g., N=20), we create the first GNN filter whose recurrence coefficients are \textit{inherently bounded}, making it exceptionally robust to over-smoothing (achieving SOTA results at K=10). Second, by making the filter's shape parameter p learnable, the filter adapts its spectral response to the graph data. We show this adaptive nature allows `KrawtchoukNet` to achieve SOTA performance on challenging heterophilic benchmarks (Texas, Cornell), decisively outperforming standard GNNs like GAT and APPNP.