  In this work, we explore the proposed mechanism in which the gravitational
$\theta$ anomaly generates neutrino masses. We highlight that the leading
renormalizable interactions of the neutrino condensate forbid the possibility
of generating hierarchical masses consistent with observation. This conclusion
still holds when Standard Model loop corrections are accounted for. We show
that higher-dimensional operators can alleviate this problem. The
higher-dimensional operators could be generated from the gravitational anomaly
itself, but there is no clear way to know without a deeper understanding of the
low-energy description of this mechanism. Because of that, we explore the
possibility of new particles generating neutrino mass splittings. We show that
both new particles that alter the scalar potential of the condensate or new
particles in loops for the neutrino self-energy can solve this problem.
