  Recently announced results from the KATRIN collaboration imply an upper bound
on the effective electron anti-neutrino mass $m_{\nu_{e}}$, $m_{\nu_{e}}<
0.8~{\rm eV}/c^{2}$. Here we explore the implications of combining the KATRIN
upper bound using a previously inferred lower bound on the smallest neutrino
mass state, $m_{i,{\rm min}}\gtrsim 0.4~{\rm eV}/c^{2}$ implied by the
stability of white dwarfs and neutron stars in the presence of long-range
many-body neutrino-exchange forces. By combining a revised lower bound estimate
with the expected final upper bound from KATRIN, we find that the available
parameter space for $m_{\nu_{e}}$ may be closed completely within the next few
years. We then extend the argument when a single light sterile neutrino flavor
is present to set a lower mass limit on sterile neutrinos.
