Two spatial regions $B$ and $R$ are hyperentangled if the generalized entropy
satisfies $S_{\text{gen}}^{B\cup R}<S_{\text{gen}}^R$. If in addition all
future (or all past) directed inward null shape deformations of $B$ decrease
$S_{\text{gen}}^{B\cup R}$, then we show that the causal development of $B$,
with $R$ held fixed, must be incomplete. This result eliminates the Null Energy
Condition from the assumptions of a recently proven singularity theorem.
Instead, we assume a quantum version of the Bousso bound. Taking $R$ to contain
the Hawking radiation after the Page time, our theorem predicts a singularity
in the past causal development of the black hole interior. This is surprising
because the classical spacetime is nonsingular in the past. However, one finds
that Cauchy slices that are required to contain $R$ do not remain in the
semiclassical regime. The quantum singularities predicted by our theorem are an
obstruction to further semiclassical evolution, generalizing the singularities
of classical general relativity.
