  The most important aspects of scattering amplitudes have long been thought to
be associated with their poles. But recently a very different sort of "split"
factorizations for a wide range of particle and string tree amplitudes have
been discovered away from poles. In this paper, we give a simple, conceptual
origin for these splits arising from natural properties of the binary geometry
of the curve integral formulation for scattering amplitudes for Tr$(\Phi^3)$
theory. The most natural way of "joining" smaller surfaces to build larger ones
directly produces a choice of kinematics for which higher amplitudes factor
into lower ones. This gives a generalization of splits to all orders in the
topological expansion. These splits allow us to access and compute
loop-integrated multi-soft limits for particle and string amplitudes, at all
loop orders. This includes split factorizations and multi-soft limits for pion
and gluon amplitudes, that are related to Tr$(\Phi^3)$ theory by a simple
kinematical shift.
