  We investigate the physics of black hole formation from the head-on
collisions of boosted equal mass Oscillatons (OS) in full numerical relativity,
for both the cases where the OS have equal phases or are maximally off-phase
(anti-phase). While unboosted OS collisions will form a BH as long as their
initial compactness $\mathcal{C}\equiv GM/R$ is above a numerically determined
critical value $\mathcal{C}>0.035$, we find that imparting a small initial
boost counter-intuitively \emph{prevents} the formation of black holes even if
$\mathcal{C}> 0.035$. If the boost is further increased, at very high boosts
$\gamma>1/12\mathcal{C}$, BH formation occurs as predicted by the hoop
conjecture. These two limits combine to form a "stability band" where
collisions result in either the OS "passing through" (equal phase) or "bouncing
back" (anti-phase), with a critical point occurring around ${\cal C}\approx
0.07$. We argue that the existence of this stability band can be explained by
the competition between the free fall and the interaction timescales of the
collision.
