= Single-vortex energy-entropy balance
{title2=$\Delta F/(k_BT)=E_c/(k_BT)+(\pi K-2)\log(L/a)$}
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 gives
$$
E_{\rm v}=E_c+\frac{\rho_s}{2}\int_a^L\frac{2\pi r\,dr}{r^2}
=E_c+\pi\rho_s\log(L/a).
$$
Boundary shape changes finite terms. There are order $(L/a)^2$ core positions, so the positional <entropy> is $S_{\rm v}=2k_B\log(L/a)+O(k_B)$, where $k_B$ is the <Boltzmann constant>. With $K=\rho_s/(k_BT)$, subtracting $TS_{\rm v}$ gives the displayed <free energy>.
For $L/a=1000$ and $E_c/(k_BT)=2$, 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.
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