The scattering predictions of a web of theories including Yang-Mills (YM), gravity, bi-adjoint scalar, the non-linear sigma model (NLSM), Dirac-Born-Infeld-Volkov-Akulov (DBI-VA) and the special Galileon (sGal) form a class of special objects with two fascinating properties: they are related by the double-copy procedure, and they can be defined purely by on-shell constraints. We expand on both of these properties. First we show that NLSM tree-level amplitudes are fully determined by imposing color-dual structure together with cyclic invariance and locality. We then consider how hard-scaling can be used to constrain the predictions of these theories, as opposed to the usual soft-scaling. We probe the UV by generalizing the familiar BCFW shift off-shell to a novel single hard limit. We show that UV scalings are sufficient to fully constrain: 1. Bi-adjoint doubly-ordered amplitudes, assuming locality; 2. NLSM and BI, assuming locality and unitarity; 3. Special Galileon, assuming locality, unitarity, and a UV bound for the general Galileon vertex. We see how potentially distinct aspects of this UV behavior can be understood and unified via double-copy relations. Surprisingly, we find evidence that assuming unitarity for these theories may not be necessary, and can emerge via UV considerations and locality alone. These results complete the observations that, like IR considerations, UV scaling is sufficient to fully constrain a wide range of tree-level amplitudes, for both gauge, gravity, and effective field theories.
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