We propose a new strategy to systematically search for new physics processes in particle collisions at the energy frontier. An examination of all possible topologies which give identifiable resonant features in a specific final state leads to a tractable number of "topological models" per final state and gives specific guidance for their discovery. Using one specific final state, jj, as an example, we find that the number of possibilities is reasonable and reveals simple, but as-yet-unexplored, topologies which contain significant discovery potential. We propose analysis techniques and estimate the sensitivity for pp collisions with s=14TeV and L=300 fb-1. © 2014 American Physical Society.