Discrete Unique Continuation on Simplex

What the math gives to ML

The paper gives a discrete uncertainty principle for functions on a composition simplex: local oriented-simplicial cancellation constraints force any function that is nonzero at the balanced index to have polynomially large support. This is transferable as an anti-collapse mechanism for simplex-indexed neural parameters, such as structured expert routers, polynomial feature coefficients, or multi-index convolution kernels. The practical route is to enforce the linear relations exactly by projection onto their nullspace, retain a nonzero balanced coefficient, and use the theorem's support exponent as a falsifiable target rather than merely adding an unstructured sparsity penalty. The main limitation is that the guarantee applies to exact constraints and signed coefficients, so the first experiments should use a small signed feature or routing module rather than ordinary probability-valued attention.