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

Abstract

We prove that every embeddable $3\times 3$ row-stochastic matrix is a product of at most seven elementary row-stochastic matrices. Frydman's example shows that this bound is sharp. The proof combines Frydman's six-factor criterion and spiral theorem with a reduction to a three-parameter critical family and an analysis of the membership certificates that persist along it.