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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. Wecompute the non-perturbative macroscopic entropy of a scalar field in anextremal black hole background at the level of linearized backreaction on themetric. The scalar field is assumed to violate or saturate the weak gravityconjecture. The scalar contributes a logarithmic correction to the entropy inthe black hole geometry that outgrows the classical contribution. Wedemonstrate that the entropy of the gauged scalar violates the generalizedsecond law in the limit of large black hole charge. Our result suggests thatentropy inequalities may directly discriminate between effective field theoriesthat live in the landscape versus the swampland.

A Semiclassical, Entropic Proof of a Weak Gravity Conjecture