  The Weak Gravity Conjecture (WGC) demands the existence of superextremal
particles in any consistent quantum theory of gravity. The standard lore is
that these particles are introduced to ensure that extremal black holes are
either unstable or marginally stable, but it is not clear what is wrong if this
doesn't happen. This note shows that, for a generic Einstein quantum theory of
gravity in AdS, exactly stability of extremal black branes is in tension with
rigorously proven quantum information theorems about entanglement entropy.
Avoiding the contradiction leads to a nonperturbative version of the WGC, which
reduces to the usual statement at weak coupling. The argument is general, and
it does not rely on either supersymmetry or a particular UV completion,
assuming only the validity of Einsteinian gravity, effective field theory, and
holography. The pathology is related to the development of an infinite throat
in the near-horizon region of the extremal solutions, which suggests a
connection to the ER=EPR proposal.
