  We theoretically investigate the low-temperature phase of the recently
synthesized Lu$_2$Mo$_2$O$_5$N$_2$ material, an extraordinarily rare
realization of a $S=1/2$ three-dimensional pyrochlore Heisenberg
antiferromagnet in which Mo$^{5+}$ are the $S=1/2$ magnetic species. Despite a
Curie-Weiss temperature ($\Theta_{\rm CW}$) of $-121(1)$ K, experiments have
found no signature of magnetic ordering $or$ spin freezing down to
$T^*\approx0.5$ K. Using density functional theory, we find that the compound
is well described by a Heisenberg model with exchange parameters up to third
nearest neighbors. The analysis of this model via the pseudofermion functional
renormalization group method reveals paramagnetic behavior down to a
temperature of at least $T=|\Theta_{\rm CW}|/100$, in agreement with the
experimental findings hinting at a possible three-dimensional quantum spin
liquid. The spin susceptibility profile in reciprocal space shows
momentum-dependent features forming a "gearwheel" pattern, characterizing what
may be viewed as a molten version of a chiral noncoplanar incommensurate spiral
order under the action of quantum fluctuations. Our calculated reciprocal space
susceptibility maps provide benchmarks for future neutron scattering
experiments on single crystals of Lu$_2$Mo$_2$O$_5$N$_2$.
