Periodic smooth Gauss excludes nonzero net charge

Assume Countable Choice, $L,{ϵ}_0>0$, and the periodic Coulomb-gauge Gauss equation of Gauss multiplier and smooth periodic Coulomb solver. For every smooth periodic charge density $\rho$ on $T_L^3$, there exists a smooth periodic potential $\varphi$ satisfying $-\mathrm{\Delta }\varphi =\rho \mathrm{/}{ϵ}_0$.

Assume Countable Choice and the periodic Gauss constraint of Gauss multiplier and smooth periodic Coulomb solver. On $T_L^3$, take the nonzero constant smooth charge density $\rho (\mathbf{x})={\rho }_0$ with ${\rho }_0\mathrm{\ne }0$. There is no smooth periodic potential satisfying $-\mathrm{\Delta }\varphi ={\rho }_0\mathrm{/}{ϵ}_0$. More generally, every smooth periodic solution of Gauss's equation in Coulomb gauge requires zero total charge. A boundary, puncture, or compensating background changes the hypotheses and is outside this counterexample.