Lie groupoid integration of singular isometries of the Poincaré disk

What the math gives to ML

The paper gives a constructive way to integrate incomplete, singular infinitesimal symmetries into a Lie groupoid rather than forcing them into a globally defined group action. The transferable asset is the combination of source-dependent arrows, path-homotopy equivalence constrained to avoid a singular set, and composition by concatenation. In neural networks this suggests a local-equivariant layer whose transformations are valid only for a given latent source point and whose path class is retained, allowing singular or branched symmetries without introducing invalid global coordinate transformations.