 Linear-time multiplication in an automatic structure

Fix the first tape of a group multiplier automaton and use dynamic programming over automaton states and output-padding status, retaining predecessor pointers. The pumping of a group multiplier automaton bounds the search by $|u|+N$ layers, each of fixed size. This constructs a representative of $\overline{u}x$ in time $O(|u|+1)$ without assuming uniqueness of representatives.