  This is part of a series of papers describing the new curve integral
formalism for scattering amplitudes of the colored scalar tr$\phi^3$ theory. We
show that the curve integral manifests a very surprising fact about these
amplitudes: the dependence on the number of particles, $n$, and the loop order,
$L$, is effectively decoupled. We derive the curve integrals at tree-level for
all $n$. We then show that, for higher loop-order, it suffices to study the
curve integrals for $L$-loop tadpole-like amplitudes, which have just one
particle per color trace-factor. By combining these tadpole-like formulas with
the the tree-level result, we find formulas for the all $n$ amplitudes at $L$
loops. We illustrate this result by giving explicit curve integrals for all the
amplitudes in the theory, including the non-planar amplitudes, through to two
loops, for all $n$.
