  The global Schwinger formula, introduced by Cachazo and Early as a single
integral over the positive tropical Grassmannian, provides a way to uncover
properties of scattering amplitudes which are hard to see in their standard
Feynman diagram formulation. In a recent work, Cachazo and one of the authors
extended the global Schwinger formula to general $\phi^p$ theories. When $p=4$,
it was conjectured that the integral decomposes as a sum over cones which are
in bijection with non-crossing chord diagrams, and further that these can be
obtained by finding the zeroes of a piece-wise linear function, $H(x)$. In this
note we give a proof of this conjecture. We also present a purely combinatorial
way of computing $\phi^p$ amplitudes by triangulating a trivial extended
version of non-crossing $(p-2)$-chord diagrams, called extended diagrams, and
present a proof of the bijection between triangulated extended diagrams and
Feynman diagrams when $p=4$. This is reminiscent of recent constructions using
Stokes polytopes and accordiohedra. However, the $\phi^p$ amplitude is now
partitioned by a new collection of objects, each of which characterizes a
polyhedral cone in the positive tropical Grassmannian in the form of an
associahedron or of an intersection of two associahedra. Moreover, we comment
on the bijection between extended diagrams and double-ordered biadjoint scalar
amplitudes. We also conjecture the form of the general piece-wise linear
function, $H^{\phi^p}(x)$, whose zeroes generate the regions in which the
$\phi^p$ global Schwinger formula decomposes into.
