Proof of the weak gravity conjecture from black hole entropy

We prove that higher-dimension operators contribute positively to the entropy of a thermodynamically stable black hole at fi xed mass and charge. Our results apply whenever the dominant corrections originate at tree level from quantum fi eld theoretic dynamics. More generally, positivity of the entropy shift is equivalent to a certain inequality relating the free energies of black holes. These entropy inequalities mandate new positivity bounds on the coe ffi cients of higher-dimension operators. One of these conditions implies that the charge-to-mass ratio of an extremal black hole asymptotes to unity from above for increasing mass. Consequently, large extremal black holes are unstable to decay to smaller extremal black holes and the weak gravity conjecture is automatically satis fi ed. Our fi ndings generalize to arbitrary spacetime dimension and to the case of multiple gauge fi elds. The assumptions of this proof are valid across a range of scenarios, including string theory constructions with a dilaton stabilized below the string scale.