Cyclic Shuffle Groups: Universal Two-Transitivity and Complete Classification

What the math gives to ML

The paper studies two explicit permutations on N=kn positions: a perfect shuffle and a cyclic pile rotation. Its transferable asset is the constructive group action: when n is not a power of k, these two O(N)-cost reindexings generate a 2-transitive group, so every ordered pair of positions can in principle be transported to every other ordered pair. This suggests sparse token-mixing architectures that replace dense N-by-N attention with repeated fixed permutations plus learned channel transformations. The theorem guarantees pair reachability, while word length, optimization behavior, and practical information mixing must be tested empirically.