All-order prescription for facet regions in massless wide-angle scattering

We take a step toward answering a long-standing question in the asymptotic expansion of Feynman integrals: how to systematically determine and interpret the regions in the Expansion-by-Regions technique for multiscale processes? Focusing on generic massless wide-angle scattering, we provide an all-order momentum-space prescription for facet regions, which generally dominate -- and in most cases exhaust -- the contributions in a given asymptotic expansion. This extends the Euclidean-space picture, where regions correspond to specific subgraphs, to the complexities of Minkowski space. Our results are derived from a novel analytical approach combining graph theory and convex geometry, and they yield several key byproducts. Theoretically, we uncover for the first time the algebraic structure underlying momentum modes (collinear, soft, and their hierarchies): they form a non-distributive lattice in general. Phenomenologically, we show that ``cascading modes'' are always absent in our scope. Algorithmically, we implement this prescription in "Facet Region Interpreter", a computer program to systematically derive regions in momentum space, by which we validate our prescription through 18,000+ examples.