Extension Problems for the Vladimirov--Taibleson Operator and the Hierarchical Laplacian

What the math gives to ML

The paper provides a non-Archimedean analogue of the Caffarelli–Silvestre construction: a nonlocal boundary operator is represented as a local weighted-harmonic computation on a tree. The transferable asset is the explicit scale-dependent edge conductance, which turns long-range hierarchical interactions into sparse multilevel message passing, together with an energy identity that can be used as a principled regularizer. A practical neural adaptation is a differentiable tree-extension layer over a hierarchy of tokens, patches, or graph clusters, with the induced Dirichlet-to-Neumann signal fed back as a learned multiscale interaction operator.