aspg-40-4460-1792533445 20261020215725 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Pure Mathematics for Theoretical Computer Science PMTCS 2995-3162 2025 5 2 Dwi Retnowardani Universitas PGRI Argopuro Jember, Indonesia Polynomial graph filters Ha(S) = ΣKk =0 akSk are local, scalable, and central to spectral graph neural computation, but their output may vary sharply when the graph shift S is perturbed. This paper develops a finite-radius perturbation calculus that requires neither commutativity nor eigengaps. For ∥S∥2 ≤ ρ and ∥E∥2 ≤ ε, we prove the certificate ∥Ha(S+E)−Ha(S)∥2 ≤ ΣKk =1 |ak|(ρ +ε)k−ρk, show equality for aligned positive polynomials, derive an explicit second-order remainder, and propagate the certificate through multilayer graph neural networks. The bound induces a sensitivity-weighted quadratic design whose unique minimizer is available in closed form. Controlled analysis uses three spectral responses, degree K = 8, and 2,304 filter evaluations on Erd˝os–Rényi, stochastic-block, Barabási– Albert, and random-geometric graphs. Relative to unconstrained least squares, the proposed design reduces the mean finite-radius certificate by 74.9%, the first-order coefficient sensitivity by 72.9%, and observed operator sensitivity by 6.9%. It matches accuracy-controlled ridge filtering within 0.3% in observed sensitivity while yielding a 7.3% smaller certificate. The result is a compact, auditable robustness layer for graph filters and polynomial graph-convolution banks. 2025 07 11 4460 https://creativecommons.org/licenses/by/4.0/ 10.54216/PMTCS.050202 https://americaspg.com/journal/40/article/4460 https://americaspg.com/storage/articles/manuscripts/01KY60E5AQ9JPAENKKXFJ3DF87.pdf