  Recently we showed how, in two-dimensional scalar theories, one-loop
threshold diagrams can be cut into the product of one or more tree-level
diagrams arXiv:2206.09368. Using this method on the ADE series of Toda models,
we computed the double- and single-pole coefficients of the Laurent expansion
of the S-matrix around a pole of arbitrary even order, finding agreement with
the bootstrapped results. Here we generalise the cut method explained in
arXiv:2206.09368 to multiple loops and use it to simplify large networks of
singular diagrams. We observe that only a small number of cut diagrams survive
and contribute to the expected bootstrapped result, while most of them cancel
each other out through a mechanism inherited from the tree-level integrability
of these models. The simplification mechanism between cut diagrams inside
networks is reminiscent of Gauss's theorem in the space of Feynman diagrams.
