Stable colored black holes with quartic self-interactions

We analytically prove the linear radial stability of non-Abelian black holes with quartic self-interactions. The background, constructed from the Wu--Yang magnetic monopole ansatz, is an exact black-hole solution carrying a non-Abelian magnetic charge $Q_{\rm NA}^2$ controlled by a single coupling parameter $\chi$, and admits two distinct branches. The odd sector is always stable, while in the even sector the effective potential is positive for branch~I and negative for branch~II, establishing stability and potential instability, respectively. The potential instability of branch~II is consistent with its connection to the perturbatively unstable Einstein--Yang--Mills Reissner--Nordstr\"{o}m solution. Branch~I remains linearly stable throughout the physical domain of $\chi$ where the solutions are regular and free of naked singularities. Our results prove the existence of the first linearly stable asymptotically flat hairy black holes in four dimensions with a minimally coupled non-Abelian Proca self-interaction.