= Finite-scale vortex renormalization
{title2=$\ell_{\max}=\log(L/a)$}
An experiment of size $L$ stops the <renormalization-group flow> at a finite $\ell_{\max}$. In the leading scaled model take the original initial conditions $Y_0=0.06$ and $X_0=-0.10,-0.06,-0.03$. The invariant is respectively $0.0064,0,-0.0027$. The first approaches $X=-0.08$, hence $K=(2.08)/\pi$; the second follows the critical <separatrix>; the third eventually leaves the small-<vortex fugacity> regime. These are chosen model trajectories, with no fitted experimental parameters.
On the critical <separatrix>, $X=-Y$ implies $Y'=-Y^2$, hence
$$
Y(\ell)=\frac{Y_0}{1+Y_0\ell},\qquad
K(\ell)=\frac2\pi+\frac{Y_0}{\pi(1+Y_0\ell)}.
$$
Even a critical finite sample therefore has a <phase stiffness> above its asymptotic jump value, with a slow logarithmic correction in $L$. Slightly above the transition, a sample with $L\ll\xi$ can resemble the algebraic phase before the flow reaches unbound <vortices>. A trap additionally varies the local density and <phase stiffness>; neither its whole-cloud coherence nor a finite observation window automatically measures the homogeneous <thermodynamic limit>.
\Image[/bkt-scaled-vortex-renormalization-flow.png]
{title=Original scaled vortex flow and finite-scale stiffness}
{description=The arrows increase observation distance in the leading near-critical model $X'=Y^2$, $Y'=XY$. All three colored trajectories start at $Y_0=0.06$. The right panel uses $K=(2-X)/\pi$ along the same trajectories. The plot is a mathematical illustration, not historical data; the dilute model must be abandoned when the vortex fugacity grows large.}
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