We theoretically investigate the low-temperature phase of the recently synthesized Lu2Mo2O5N2 material, an extraordinarily rare realization of a S=1/2 three-dimensional pyrochlore Heisenberg antiferromagnet in which Mo5+ are the S=1/2 magnetic species. Despite a Curie-Weiss temperature (ΘCW) of −121(1) K, experiments have found no signature of magnetic ordering or spin freezing down to T∗≈0.5 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=|ΘCW|/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 Lu2Mo2O5N2.