Closing the gap and settling the problem of queens on an $n\times n$ board, each attacking at most one other

What the math gives to ML

The paper develops a scalable modular construction for placing many objects on a grid while limiting collisions along rows, columns, and both diagonal directions. Its transferable asset is a permutation whose projections i, i-p(i), and i+p(i) are all injective modulo B, followed by block lifting that preserves these collision guarantees across a larger grid. This can be adapted into deterministic sparse attention or routing masks with balanced query and key degrees and low overlap under several geometric projections. The most promising target is a 2D-token transformer, where the structure can reduce attention cost while avoiding purely random sparsity.