The Weak Gravity Conjecture as an Entropy Theorem

We establish the Weak Gravity Conjecture as a theorem derived from quantum information principles. Using only the Quantum Null Energy Condition (QNEC) and generalized entropy, we show that any consistent semiclassical theory of gravity must contain a charged state with charge-to-mass ratio $q/m\ge1$. The argument proceeds by contradiction: if all charged states are strictly subextremal, extremal charged horizons form a one-sided boundary of the semiclassical state space. On such horizons, the affine-parameter half-sided modular inclusion structure required for null entropy variations becomes algebraically obstructed, rendering the horizon algebra modularly terminal. Terminality forces an upper divergence of the second null derivative of the outside entropy along complete null generators, which is incompatible with QNEC when the local stress tensor remains finite. Our result reframes a central swampland conjecture as a consequence of semiclassical entropy consistency, without invoking string theory, holography, or dynamical decay assumptions.