Tilings, packings, and the existence of Schwartz-class Gabor windows

What the math gives to ML

The paper gives a constructive way to build compactly supported partition-of-unity functions whose lattice translates cover every input while a second family of translates is exactly non-overlapping, with a positive geometric buffer after thresholding. This can transfer to sparse local routing, continuous convolutional interpolation, and coordinate-dependent gating, where coverage is guaranteed algebraically rather than learned approximately. Its second construction is a polyhedral complex of fractional collision-free assignments: each row is a probability vector, while positive mass from different rows cannot occupy the same column. The most practical experiments are a buffered partition-of-unity gating layer and a collision-free expert or memory-slot router.