= Solution
For a periodic two-dimensional lattice the <vortex charge neutrality> constraint is $\boxed{\sum_xq_x=0}$. In the winding form of the <Villain model>, $q$ is the oriented plaquette curl of the integer bond variable $p$, up to an orientation sign. Summing all plaquette curls cancels each bond against its neighboring plaquette. This proves neutrality directly. Equivalently, restoring the common dual-height mode modulo its integer shift gives $\int_0^1d\phi_0\,e^{2\pi i\phi_0\sum q}$, which vanishes unless the total charge is zero; only then is the inverse lattice <Laplacian> well-defined on the charge source.
On an infinite plane, the same condition is required for a localized finite-energy configuration with constant phase at infinity. Net charge $Q$ produces $\theta\sim Q\arg x$ far away, so its exterior action is
$$
\frac\beta2\int_a^L2\pi r\,dr\,\frac{Q^2}{r^2}=\pi\beta Q^2\log(L/a),
$$
which diverges unless $Q=0$. With an open boundary the total charge instead equals the boundary winding, and the finite planar duality in part (b) can sum nonneutral configurations. Thus neutrality follows from periodicity or the stated finite-energy boundary condition, not from arbitrary boundaries. In three dimensions the analogous local constraint is that <vortex> lines cannot end in the bulk.
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