  Adopting the notation of Teukolsky and Press, we derive the connection
relation for asymptotic solutions of the massless Dirac equation on a Kerr
background. We show that, unlike bosonic fields, the connection relation for
massless Dirac fields (neutrino) provides a rigorous proof of mode stability.
The same relation also implies that every incoming mode can be absorbed by the
black hole or there is no superradiance. Recent works on overspinning black
holes have shown that this can lead to the formation of naked singularities. We
argue that the fact that both the mode stability of the black hole under
neutrino perturbations and the instability of the event horizon (therefore the
instability of the black hole) can be derived from the same connection relation
leads to a paradox. In other words mode, stability implies event horizon
instability as far as neutrino perturbations are concerned.
