The Learned Snake

Gauthier, Goupil, and Toure (arXiv:2603.12400) train a diffusion model to generate maximal snake polyominoes — self-avoiding paths on a grid that can't be extended without creating adjacency violations. The model learns maximality and adjacency constraints implicitly, without explicit encoding.

The through-claim: neural networks can learn combinatorial maximality from examples. Snake polyominoes are defined by global constraints — no two cells share an edge unless they're consecutive in the path, and no cell can be added without violating this rule. These constraints are inherently non-local: whether a cell can be added depends on the entire existing path. But the diffusion model, trained on examples up to moderate grid sizes, generates valid maximal snakes on grids up to 28×28, approaching the limits of brute-force computation.

The surprise is the cross-scale generalization. Training on small grids produces correct behavior on large grids, even though the constraint structure changes qualitatively with scale (more potential violations, longer-range dependencies). The model appears to learn the principle of maximality, not just the examples of maximality. It makes errors — branching, cycles, disconnected components — but these errors are structurally informative: they're the ways maximality can fail, not random noise.

This connects to a broader question about learning mathematical structure: can a neural network learn a proof strategy from examples of theorems? The snake polyomino result suggests a qualified yes — the model captures the constraint pattern well enough to generalize, but not well enough to guarantee correctness. It learns the shape of the proof without its rigor.