= Solution
Let $\mathcal N\in\mathbb Z$ be the vortex <winding number>, distinct from the particle number $N$. Outside its core, the <superfluid velocity> is $u_\theta=\hbar\mathcal N/(mr)$. The outer flow energy is therefore multiplied by $\mathcal N^2$, while the resolved core contribution contains the appropriate constant $L_{0|\mathcal N|}$. The same matching integration gives
$$
\boxed{\Delta E_{\mathcal N}\simeq
\frac{\pi n_c\hbar^2}{m}\mathcal N^2
\left[\log\frac R\xi+L_{0|\mathcal N|}-\frac12\right]}.
$$
The core equation depends on $\mathcal N^2$, so reversing circulation leaves this energy unchanged. On the other hand, $-i\hbar\partial_\theta[f(r)e^{i\mathcal N\theta}]=\hbar\mathcal N f(r)e^{i\mathcal N\theta}$, giving the exact result
$$
\boxed{L_z=N\hbar\mathcal N}.
$$
Negative winding reverses <angular momentum>; zero winding has zero excess vortex energy and zero angular momentum. The energy approximation applies to each fixed nonzero winding in the TF limit. More generally its core radius, of order $|\mathcal N|\xi$, must be much smaller than $R$ so that a matching interval exists. It is not valid uniformly for arbitrary growing $|\mathcal N|$ at fixed cloud size. The leading quadratic energy cost also explains why multiply quantized vortices can lower their energy by separating into singly quantized vortices; this energy comparison does not by itself establish a particular dynamical splitting rate.
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