Hitting Moving Targets on Eulerian Digraphs in O(mn) Expected Time

Let G be a connected loopless Eulerian directed multigraph with n vertices and m arcs, and let X be its half-lazy simple random walk. For every deterministic target sequence (u_t), we prove that the expected hitting time is at most t_unif(1/4)+10m(n-1)=O(mn), uniformly in the starting vertex. This removes the logarithmic factor in the general moving-target bound of Boczkowski, Peres and Sousi and answers their explicit question following Corollary 2.3. The proof combines a point Dirichlet inequality, a killed-operator contraction and burn-in of the unconditioned law. The same bound holds for independent random targets. An explicit simple nonreversible 18-vertex, 51-arc graph has fixed-target expectation 936>918=mn, refuting coefficient one for arbitrary deterministic targets but not settling that coefficient for two independent walks. The ambiguous full AIM Problem 3.1 is not claimed completely resolved. This is an unrefereed preprint with reproducible exact-rational checks; no independent review, formal verification or absolute priority is claimed.