Upper-shadow comparisons on the slice and the Frankl--Tokushige product conjectures

What the math gives to ML

The paper gives a sharp, dimension-free expansion profile for lifting a family of fixed-cardinality subsets to larger-cardinality supersets. The transferable asset is not the extremal set-theory conclusion itself, but an explicit coverage guarantee that quantifies how many distinct larger masks must be reached from a prescribed collection of smaller masks. This can support combinatorial data augmentation or sparse routing: instead of sampling larger feature masks blindly, construct them as upper-shadow expansions of useful smaller masks and use the profile to choose expansion levels and estimate coverage. The resulting method is most plausible for set-valued inputs, sparse subnetworks, and mixture-of-experts routers whose decisions can be represented by binary feature or expert subsets.