  We present how a neutrino condensate and small neutrino masses emerge from a
topological formulation of gravitational anomaly. We first recapitulate how a
gravitational $\theta$-term leads to the emergence of a new bound neutrino
state analogous to the $\eta'$ meson of QCD. Then we show the consequent
formation of a neutrino vacuum condensate, which effectively generates small
neutrino masses. Afterwards we outline several phenomenological consequences of
our neutrino mass generation model. The cosmological neutrino mass bound
vanishes since we predict the neutrinos to be massless until the phase
transition in the late Universe, $T\sim {\rm meV}$. Deviations from an equal
flavor rate due to enhanced neutrino decays in extraterrestrial neutrino fluxes
can be observed in future IceCube data. The current cosmological neutrino
background only consists of the lightest neutrinos, which, due to enhanced
neutrino-neutrino interactions, either bind up, form a superfluid, or
completely annihilate into massless bosons. Strongly coupled relic neutrinos
could provide a contribution to cold dark matter in the late Universe, together
with the new proposed particles and topological defects, which may have formed
during neutrino condensation. These enhanced interactions could also be a
source of relic neutrino clustering in our Galaxy, which possibly makes the
overdense cosmic neutrino background detectable in the KATRIN experiment. The
neutrino condensate provides a mass for the hypothetical $B-L$ gauge boson,
leading to a gravity-competing force detectable in short-distance measurements.
Gravitational waves detections have the potential to probe our neutrino mass
generation mechanism.

High Energy Physics - Phenomenology;Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Theory