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

From tree- to loop-simplicity in affine Toda theories II: higher-order poles and cut decompositions