  KMT-2016-BLG-2605, with planet-host mass ratio $q=0.012\pm 0.001$, has the
shortest Einstein timescale, $t_\e = 3.41\pm 0.13\,$days, of any planetary
microlensing event to date. This prompts us to examine the full sample of 7
short ($t_\e<7\,$day) planetary events with good $q$ measurements. We find that
six have clustered Einstein radii $\theta_\e = 115\pm 20\,\muas$ and
lens-source relative proper motions $\mu_\rel\simeq 9.5\pm 2.5\,\masyr$. For
the seventh, these two quantities could not be measured. These distributions
are consistent with a Galactic-bulge population of very low-mass (VLM) hosts
near the hydrogen-burning limit. This conjecture could be verified by imaging
at first adaptive-optics light on next-generation (30m) telescopes. Based on a
preliminary assessment of the sample, "planetary" companions (i.e., below the
deuterium-burning limit) are divided into "genuine planets", formed in their
disks by core accretion, and very low-mass brown dwarfs, which form like stars.
We discuss techniques for expanding the sample, which include taking account of
the peculiar "anomaly dominated" morphology of the KMT-2016-BLG-2605 light
curve.
