Which Holonomy Signatures Are Realizable? A Complete Answer for Closed Surfaces

What the math gives to ML

The paper gives an explicit local surgery on quad meshes that splits one cone vertex into two while preserving genus, all other vertices, and the subgroup of rotational holonomy. This is useful as a mathematically controlled data-augmentation operator for graph or mesh neural networks: it changes local geometry and combinatorics without changing a chosen global topology label. The strongest transfer is not the classification theorem itself, but the constructive dart-pairing surgery together with the exact Gauss–Bonnet square-count lower bound, which enables invariant-preserving augmentation and minimal mesh benchmarks.