  We systematically explore the space of scalar effective field theories (EFTs)
consistent with a Lorentz invariant and local S-matrix. To do so we define an
EFT classification based on four parameters characterizing 1) the number of
derivatives per interaction, 2) the soft properties of amplitudes, 3) the
leading valency of the interactions, and 4) the spacetime dimension. Carving
out the allowed space of EFTs, we prove that exceptional EFTs like the
non-linear sigma model, Dirac-Born-Infeld theory, and the special Galileon lie
precisely on the boundary of allowed theory space. Using on-shell momentum
shifts and recursion relations, we prove that EFTs with arbitrarily soft
behavior are forbidden and EFTs with leading valency much greater than the
spacetime dimension cannot have enhanced soft behavior. We then enumerate all
single scalar EFTs in d<6 and verify that they correspond to known theories in
the literature. Our results suggest that the exceptional theories are the
natural EFT analogs of gauge theory and gravity because they are one-parameter
theories whose interactions are strictly dictated by properties of the
S-matrix.
