Strongly-coupled black holes (are not) at weak coupling

We prove sharp bounds on the moduli space excursion that is possible outside the event horizon of a spherically symmetric charged black hole. In regions where the black hole charge $Q (\phi)$ is no less than its asymptotic value, we prove that the total moduli space excursion is less than $\frac{\pi}{2\sqrt{k_N}}$ where $k_N$ is the rationalized Newton constant. We also show that the moduli space excursion in a specified direction in which $Q (\phi)$ is non-decreasing satisfies the same bound. Applying our bound to electrically charged black holes, we rule out a potential loophole in a recent proof [arXiv:2401.14449] of the Weak Gravity Conjecture in perturbative bosonic string theory.