Sharp criterion for the coarse-grained arrow of time in Hamiltonian dynamics

The Gibbs entropy of a Hamiltonian ensemble is constant, and the growth of entropy is commonly attributed to coarse-graining. We ask when Gaussian coarse-graining actually guarantees that the coarse-grained entropy does not decrease, for every initial ensemble. For linear Hamiltonian flows we prove a necessary and sufficient condition: the coarse-graining window must widen at least as fast as the flow stretches phase space. At fixed resolution no linear flow produces an arrow of time for all states. For nonlinear flows we prove a lower bound on the entropy production that establishes the criterion for every state no broader than the window, show that the threshold cannot be lowered, and prove it to leading order in the limit of fine resolution. The general case is stated as a conjecture and supported numerically for the pendulum. For the quartic oscillator no finite coarse-graining rate yields an arrow of time for all states. Verification code: https://doi.org/10.5281/zenodo.23135448