Hidden zeros = secret BCFW scaling, and a new path to uniqueness

We explore several features related to the newly discovered hidden amplitude zeros arXiv:2312.16282, which describes the vanishing of scattering amplitudes on special external kinematics. We prove the conjecture that amplitude zeros uniquely fix Tr φ³, by showing that any amplitude can be divided into special subsets, which independently satisfy the zero condition. We further prove that the same subsets satisfy a previously unknown secret enhancement under Britto-Cachazo-Feng-Witten shifts, which turns out to be equivalent to the amplitude zero condition. Similar but weaker statements also hold for non-linear sigma model, while for Yang-Mills, when imposed together with color-kinematic duality, amplitude zeros uniquely fix the different tensor components of gluon amplitudes. This direction suggests a straightforward avenue for understanding previous uniqueness results, as well as the connections between naively independent properties of scattering amplitudes.