  On-shell, analytic S-matrix elements in massless theories are constructed
from a finite set of primitive three-point amplitudes, which are fixed by
Poincare invariance up to an overall numerical constant. We classify \emph{all}
such three-point amplitudes in four-dimensions. Imposing the simplest
incarnation of Locality and Unitarity on four-particle amplitudes constructed
from these three-particle amplitudes rules out all but an extremely small
subset of interactions among higher-spin massless states. Notably, the
equivalence principle, and the Weinberg-Witten theorem, are simple corollaries
of this principle. Further, no massless states with helicity larger than two
may consistently interact with massless gravitons. Chromodynamics,
electrodynamics, Yukawa and $\phi^3$-theories are the only marginal and
relevant interactions between massless states. Finally, we show that
supersymmetry naturally emerges as a consistency condition on four-particle
amplitudes involving spin-3/2 states, which must always interact
gravitationally.
