Isomorphism of tournaments with bounded VC dimension

What the math gives to ML

The useful transferable object is the Sauer–Shelah restriction bound: a set system with VC dimension d induces at most 1+|Y|^d distinct restrictions on any sampled subset Y, even when the ambient universe is large. Thresholded attention neighborhoods form such a set system, so queries with identical sampled-key incidence patterns can share routing or key-value aggregation work. The dual-family inequality permits estimating complexity from either query-side or key-side patterns, selecting the cheaper grouping direction. This is most promising as adaptive attention and KV-cache compression, although experiments must quantify the approximation error introduced by thresholding real-valued attention.