  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,
$\ell\ell jj$, 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 $\sqrt{s}=14$ TeV and
$\mathcal{L}=300$ fb$^{-1}$.
