Congruence classes of monodromies of even triangulations

What the math gives to ML

The paper exposes a discrete holonomy construction: transporting a 3-coloring across adjacent triangular faces produces permutations in the finite group \(\mathfrak{S}_3\), and the product around a closed dual walk depends only on its homotopy class. This is transferable as a discrete gauge structure for graph neural networks on triangulated or mesh-like data, where messages are transported by permutation representations instead of being compared in arbitrary local coordinate frames. The most promising implementation is a holonomy-aware face GNN with \(\mathfrak{S}_3\)-equivariant message passing and cycle-consistency losses, especially on data with nontrivial topology or multiple local labeling conventions.