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Border Gateway Protocol (BGP) allows providers to express complex routing policies preserving high degrees of autonomy. However, unrestricted routing policies can adversely impact routing stability. A key concept to understand the interplay between autonomy and expressiveness on one side, and stability on the other side, is safety under filtering, i.e., guaranteed stability under autonomous usage of route filters. BGP route filters are used to selectively advertise specific routes to specific neighbors. In this paper, we provide a characterization of safety under filtering, filling the large gap between previously known necessary and sufficient conditions. Our characterization is based on the absence of a particular kind of dispute wheel, a structure involving circular dependencies among routing preferences. We exploit our result to show that networks admitting multiple stable states are provably unsafe under filtering, and the troublesome portion of the configuration can be pinpointed starting from the stable states alone. This is especially interesting from an operational point of view since networks with multiple stable states actually happen in practice (BGP wedgies). Finally, we show that adding filters to an existing configuration may lead to oscillations even if the configuration is safe under any link failure. Unexpectedly, we find policy configurations where misconfigured filters can do more harm than network faults.