  We investigate the hidden amplitude zeros discovered by Arkani-Hamed et al,
which describe a non-trivial vanishing of scattering amplitudes on special
external kinematics. We prove the conjecture that amplitude zeros uniquely fix
Tr$(\phi^3)$, by first showing that any amplitude can be divided into special
subsets, which independently satisfy the zero condition. We further prove that
this condition is in fact equivalent to a novel enhancement under
Britto-Cachazo-Feng-Witten shifts of the same subsets. More generally, for one
particular zero type, this equivalence can be extended to arbitrary rational
functions built from planar variables (such as the non-linear sigma model). In
the case of Yang-Mills theory, we conjecture the zeros, combined with
color-kinematic duality, uniquely fix the $\lfloor n/2\rfloor$ different tensor
structures of $n$-point gluon amplitudes. Our approach suggests a
straightforward avenue for understanding previous similar uniqueness results,
in which unitarity appears as an emergent property, and naively independent
physical principles contain identical (amplitude) information.
