<p>We study the recently introduced curve integral formalism that defines a new family of formulas for the scattering amplitudes of the colored scalar tr<i>ϕ</i>³ theory. We find that the curve integral manifests a very surprising fact about these amplitudes: the dependence on the number of particles, <i>n</i>, and the loop order, <i>L</i>, is effectively decoupled in these formulas. We derive the curve integrals at tree-level for all <i>n</i>. We then show that, for higher loop-order, it suffices to study the curve integrals for <i>L</i>-loop tadpole-like amplitudes, which have just one particle per color trace-factor. By combining these tadpole-like formulas with the tree-level results, we find formulas for the all <i>n</i> amplitudes at <i>L</i> 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 <i>n</i>.</p>