Uniform support bound from a coercive conserved energy ($\mathrm{supp}{f}_t\subseteq \{H\le E_{\ast }\}$)

For homogeneous Hamiltonian Liouville equation transport with autonomous $H$, compact initial support remains in the fixed energy sublevel determined by its maximum initial energy. If these sublevels are compact, this proves uniform compact support without solving the flow. The harmonic energy $H_{\omega }=(|v{|}^2+{\omega }^2|x{|}^2)/2$ has this property exactly when $\omega \ne 0$; at zero frequency spatial free streaming can escape every compact set.