The $\mathcal{L}$-Calculus for Causal Variational Principles: An Exterior Differential Calculus on Non-Smooth Spaces

What the math gives to ML

The paper develops an exterior differential calculus on supports of Radon measures that may be discrete, singular, or otherwise non-manifold, while retaining restriction, extension, pullback, cohomology, and Stokes-type constructions. The most transferable asset is its precise local-to-global gluing criterion: local objects defined on overlapping pieces can be assembled into a global object when the complements are separating and the overlap restrictions agree. This suggests a seam-free architecture for neural representations on irregular point clouds, graphs, meshes, or dynamically selected patches, in which local experts are constrained or projected to form one globally consistent feature field. The construction is more principled than ordinary overlap averaging because it exposes a falsifiable compatibility condition and can be used as a regularizer or differentiable projection.