In a gravitational theory with a massless photon the maximum charge-to-mass ratio of black holes approaches the prediction of the Einstein-Maxwell theory as black hole mass increases: $Q_{rm ext}/M =1+ alpha/M^2$ for some constant $alpha$. We will show that $alpha>0$ if below the quantum gravity scale $Lambda$ there are many degrees of freedom with a hierarchically small mass gap $log(Lambda/m_{rm gap})gg 1$. In this regime one can treat gravity as a non-dynamical background field and derive field-theoretic sum-rules for the coefficients of the leading corrections to the Einstein-Maxwell theory. The positivity of $alpha$ follows from the sum-rules. As a consequence, gravitational attraction gets weaker than the electric force among maximally charged black holes as they become lighter, and large extremal black holes can decay into smaller black holes.
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