Bang--bang representation of $3\times 3$ embeddable stochastic matrices

What the math gives to ML

The paper gives a constructive sparse factorization of embeddable 3-state Markov transition matrices into at most seven elementary row-mixing matrices. Its transferable asset is exact stochasticity combined with a bounded sequence of cheap two-row updates: each factor changes one row by interpolating it with another. This suggests replacing a dense learned 3-state transition or routing matrix with a product of at most seven interpretable factors, reducing parameters and imposing stable probabilistic structure. The guarantee is specialized to 3x3 embeddable matrices, so the first experiment should test whether this restriction improves long-horizon behavior or regularization rather than assuming a universal benefit.