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For a periodic two-dimensional lattice the vortex charge neutrality constraint is $\boxed{\sum _x{q}_x=0}$. 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 ${\int }_0^{1}d{\varphi }_0{e}^{2\pi i{\varphi }_0\sum q}$, 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 iswhich 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.

Consider an original counting example: a square two-dimensional XY model of side $L$, a microscopic core radius $a$, and a fixed phase stiffness ${\rho }_s$. This deliberately neglects screening by other vortices. Away from a unit phase vortex, $\theta =\arg (x+iy)$ and $|\nabla \theta |=1/r$. Integrating the Goldstone-mode effective free energy over an annulus givesBoundary shape changes finite terms. There are order $(L/a{)}^2$ core positions, so the positional entropy is $S_{\mathrm{v}}=2k_B\log (L/a)+O(k_B)$, where $k_B$ is the Boltzmann constant. With $K={\rho }_s/(k_{B}T)$, subtracting $T{S}_{\mathrm{v}}$ gives the displayed free energy.

For $L/a=1000$ and $E_c/(k_{B}T)=2$, the dimensionless free energy cost is about $5.546$ at $K=0.8$ and $-0.965$ at $K=0.5$. The same logarithm appears in the cost and the positional reward: increasing the system size strengthens whichever has the larger coefficient. The corresponding one-defect Boltzmann factors, about $0.00390$ and $2.62$, are relative statistical weights, not normalized probabilities. The latter signals failure of a dilute single-defect estimate.

A single net phase vortex requires compatible boundaries: on a periodic sample use neutral pairs, as required by vortex charge neutrality. This argument identifies a possible threshold, but predicting the universal stiffness jump requires the renormalized phase stiffness, after smaller pairs have responded. It does not justify setting the transition temperature equal to a bare lattice coupling divided by a universal constant.