  The largest tau neutrino dataset to date is IceCube's atmospheric tau
neutrino appearance dataset containing $>1,000$ tau neutrino and antineutrino
events as determined by a fit to a standard three-flavor oscillation framework.
On an event-by-event basis, however, it is impossible to know that any given
event is a tau neutrino as they are identical to either an electron neutrino
charged-current event or a neutral-current interaction of any active flavor.
Nonetheless, we conclusively show that, using only the cascade sample even
without knowledge of the oscillation parameters and without assuming that the
lepton mixing matrix is unitary, tau neutrino identification is still possible
and there is no viable scenario in which all of the tau neutrino candidates are
actually electron neutrinos. This is primarily due to the matter effect and the
tau lepton production threshold, as well as the fact that tau neutrinos are
systematically reconstructed at a lower energy than electron neutrinos due to
one or more outgoing neutrinos. This conclusively shows that it is possible for
an atmospheric neutrino oscillation experiment to confirm that $U_{\tau1}$,
$U_{\tau2}$, and $U_{\tau3}$ are not all zero even with limited particle
identification.
