A local integer-variance lower bound for indirect Coulomb energy

The Lieb–Oxford program asks for universal lower bounds on the indirect Coulomb energy of a many-body state in terms of its one-particle density alone. We give such a bound that uses only one nonperturbative fact: the number of particles in a ball is an integer-valued random variable. Starting from the Fefferman–de la Llave representation of the Coulomb kernel, the indirect Coulomb energy is rewritten as an exact integral of variances of local ball counts minus their means. The integer-valued constraint then gives the sharp single-variable variance bound $\Var X\ge \theta(m)(1-\theta(m))$, which produces a closed-form local density functional lower bound involving only the local ball mass. The resulting universal constant is $C_{\rm IV}=1.569\ldots$, lying below the Lieb–Oxford constant $1.68$ and above the Dirac constant. The bound is exact for one particle and isolates a clean number-quantization contribution to the Lieb–Oxford problem.