  We conjecture that the leading two-derivative tree-level amplitudes for
gluons and gravitons can be derived from gauge invariance together with mild
assumptions on their singularity structure. Assuming locality (that the
singularities are associated with the poles of cubic graphs), we prove that
gauge-invariance in just (n-1) particles together with minimal power-counting
uniquely fixes the amplitude. Unitarity in the form of factorization then
follows from locality and gauge invariance. We also give evidence for a
stronger conjecture: assuming only that singularities occur when the sum of a
subset of external momenta go on-shell, we show in non-trivial examples that
gauge-invariance and power-counting demand a graph structure for singularities.
Thus both locality and unitarity emerge from singularities and gauge
invariance. Similar statements hold for theories of Goldstone bosons like the
non-linear sigma model and Dirac-Born-Infeld, by replacing the condition of
gauge invariance with an appropriate degree of vanishing in soft limits.
