  Neutrinos may acquire small Dirac or Majorana masses by new low-energy
physics in terms of the chiral gravitational anomaly, as proposed by Dvali and
Funcke (2016). This model predicts fast neutrino decays, $\nu_i\to\nu_j+\phi$
and $\nu_i\to\bar{\nu}_j+\phi$, where the gravi-majorons $\phi$ are
pseudoscalar Nambu-Goldstone bosons. The final-state neutrino and antineutrino
distributions differ depending on the Dirac or Majorana mass of the initial
state. This opens a channel for distinguishing these cases, for example in the
spectrum of high-energy astrophysical neutrinos. In particular, we put bounds
on the neutrino lifetimes in the Majorana case, ${\tau_2}/{m_2}> 1.1\times
10^{-3}(6.7\times 10^{-4})~{\rm s/eV}$ and ${\tau_3}/{m_3}> 2.2\times
10^{-5}(1.3\times 10^{-4})~{\rm s/eV}$ at 90% CL for hierarchical (degenerate)
masses, using data from experiments searching for antineutrino appearance from
the Sun.
