  We present a semiclassical proof of the weak gravity conjecture in $D = 4$
spacetime dimensions for scalar matter gauged under a $U(1)^N$ gauge group. We
compute the non-perturbative macroscopic entropy of a scalar field in an
extremal black hole background at the level of linearized backreaction on the
metric. The scalar field is assumed to violate or saturate the weak gravity
conjecture. The scalar contributes a logarithmic correction to the entropy in
the black hole geometry that outgrows the classical contribution. We
demonstrate that the entropy of the gauged scalar violates the generalized
second law in the limit of large black hole charge. Our result suggests that
entropy inequalities may directly discriminate between effective field theories
that live in the landscape versus the swampland.
