  We report the discovery of KMT-2020-BLG-0414Lb, with a planet-to-host mass
ratio $q_2 = 0.9$--$1.2 \times 10^{-5} = 3$--$4~q_{\oplus}$ at $1\sigma$, which
is the lowest mass-ratio microlensing planet to date. Together with two other
recent discoveries ($4 \lesssim q/q_\oplus \lesssim 6$), it fills out the
previous empty sector at the bottom of the triangular $(\log s, \log q)$
diagram, where $s$ is the planet-host separation in units of the angular
Einstein radius $\theta_{\rm E}$. Hence, these discoveries call into question
the existence, or at least the strength, of the break in the mass-ratio
function that was previously suggested to account for the paucity of very
low-$q$ planets. Due to the extreme magnification of the event, $A_{\rm
max}\sim 1450$ for the underlying single-lens event, its light curve revealed a
second companion with $q_3 \sim 0.05$ and $|\log s_3| \sim 1$, i.e., a factor
$\sim 10$ closer to or farther from the host in projection. The measurements of
the microlens parallax $\pi_{\rm E}$ and the angular Einstein radius
$\theta_{\rm E}$ allow estimates of the host, planet, and second companion
masses, $(M_1, M_2, M_3) \sim (0.3M_{\odot}, 1.0M_{\oplus}, 17M_{J})$, the
planet and second companion projected separations, $(a_{\perp,2}, a_{\perp,3})
\sim (1.5, 0.15~{\rm or}~15)$~au, and system distance $D_{\rm L} \sim 1$ kpc.
The lens could account for most or all of the blended light ($I \sim 19.3$) and
so can be studied immediately with high-resolution photometric and
spectroscopic observations that can further clarify the nature of the system.
The planet was found as part of a new program of high-cadence follow-up
observations of high-magnification events. The detection of this planet,
despite the considerable difficulties imposed by Covid-19 (two KMT sites and
OGLE were shut down), illustrates the potential utility of this program.
