  The total mass of the Local Group (LG) and the masses of its primary
constituents, the Milky Way and M31, are important anchors for several
cosmological questions. In recent years, independent measurements have
consistently yielded halo masses close to $10^{12} \mathrm{M_\odot}$ for the
MW, and $1-2 \times 10^{12} \mathrm{M_\odot}$ for M31, while estimates derived
from the pair's kinematics via the `timing argument' have yielded a combined
mass of around $5 \times 10^{12} \mathrm{M_\odot}$. Here, we analyse the
extremely large Uchuu simulation to constrain the mass of the Local Group and
its two most massive members. First, we demonstrate the importance of selecting
LG analogues whose kinematics are dominated by mutual interactions to a similar
extent as the LG. Adopting the observed separation and radial velocity, we
obtain a weighted posterior of $75_{-40}^{+65}$ kms$^{-1}$ for the uncertain
transverse velocity. Via Gaussian process regression, we infer a total mass of
$3.2^{+1.2}_{-0.9} \times 10^{12} \mathrm{M_\odot}$, significantly below the
timing argument prediction. Importantly, we show that the remaining uncertainty
is not rooted in the analysis or observational errors, but in the irreducible
scatter in the kinematics-mass relation. We further find a mass for the less
massive halo of $0.9_{-0.3}^{+0.6} \times 10^{12} \mathrm{M_\odot}$ and for the
more massive halo of $2.3_{-0.9}^{+1.0} \times 10^{12} \mathrm{M_\odot}$,
consistent with independent measurements of the masses of MW and M31,
respectively. Incorporating the mass of the MW as an additional prior allows us
to further constrain all measurements and determine that the MW is very likely
to be the lower mass object of the two.
