Space-bound discovery by exit reachability ($m\leftarrow 2m$)

For a bounded-space machine, start with a logarithmic work budget and test accepting reachability and reachability of a transition leaving the budget. Accept if acceptance is found, double the budget if an exit is reachable, and otherwise reject. If all paths use at most $Cs(n)$ space, doubling stops at $O(s(n))$. Each budget test uses the Savitch theorem recursion and the storage is reused. The method discovers a sufficient bound without computing $s(n)$.