  In this paper, we study a novel behavior developed by certain tree-level
scalar scattering amplitudes, including the biadjoint, NLSM, and special
Galileon, when a subset of kinematic invariants vanishes without producing a
singularity. This behavior exhibits properties which we call $\textit{smooth
splitting}$ and $\textit{semi-locality}$. The former means that an amplitude
becomes the product of exactly three amputated Berends-Giele currents, while
the latter means that any two currents share one external particle. We call
these smooth splittings 3-splits. In fact, there are exactly $\binom{n}{3}-n$
such 3-splits, one for each generic, interior triangle in a polygon; as they
cannot be obtained from standard factorization, they are a new phenomenon in
Quantum Field Theory. In fact, the resulting splitting is analogous to the one
first seen in Cachazo-Early-Guevara-Mizera (CEGM) amplitudes which generalize
standard cubic scalar amplitudes from their ${\rm Tr}\, G(2,n)$ formulation to
${\rm Tr}\, G(k,n)$, where ${\rm Tr}\, G(k,n)$ is the tropical Grassmannian.
Along the way, we show how smooth splittings naturally lead to the discovery of
mixed amplitudes in the NLSM and special Galileon theories and to novel
BCFW-like recursion relations for NLSM amplitudes.
