Fox $p$-Colorings as Fixed Points of Braid Representations

What the math gives to ML

The paper constructs an explicit finite-dimensional representation of the braid group using local, invertible 2-by-2 crossing operators, and proves that these operators satisfy braid and far-commutativity relations exactly. The transferable asset is not Fox coloring itself, but a principled library of local reversible channel-mixing transformations whose global behavior is invariant under braid-word rewrites. These operators can replace or augment token-mixing, channel-mixing, or reversible residual blocks, providing structured mixing with linear cost in the number of local crossings rather than quadratic cost for a dense matrix. The most direct experiment is a braid-word mixer inserted into a small Transformer or MLP and compared with permutations, butterfly mixing, and dense linear mixing at equal parameter count and FLOPs.