= Solution
A straight <quantum vortex> with integer <winding number> $\mathcal N$ has $\widetilde\psi=R(r)e^{i\mathcal N\theta}$, independent of $z$. Here $r$ is measured in the unit of distance from the preceding part. The polar <Laplacian> supplies the centrifugal term, giving
$$
\boxed{R''+\frac{R'}r-\frac{\mathcal N^2}{r^2}R
+(1-\alpha R^2-\beta R^4)R=0}.
$$
For $\mathcal N\ne0$, regularity requires $R\sim c r^{|\mathcal N|}$ near the axis; the bulk condition is $R\to1$. The full <wavefunction> tends to $e^{i\mathcal N\theta}$ rather than one. This is the <constant far-field phase excludes net vortex winding> distinction: the equation remains applicable to a vortex, but the nonwinding boundary condition must be interpreted as an amplitude condition.
Let $F(R)=R-\alpha R^3-\beta R^5$. Since $\alpha+\beta=1$, one has $F(1)=0$ and $F'(1)=1-3\alpha-5\beta=-2(1+\beta)$. Insert $R=1-p/r^2+\cdots$. Its derivative terms start at order $r^{-4}$, while
$$
-\frac{\mathcal N^2R}{r^2}=-\frac{\mathcal N^2}{r^2}+O(r^{-4}),\qquad
F(R)=\frac{2(1+\beta)p}{r^2}+O(r^{-4}).
$$
Cancellation of the order-$r^{-2}$ terms proves the <cubic-quintic quantum vortex tail>
$$
\boxed{R(r)=1-\frac{\mathcal N^2}{2(1+\beta)r^2}+O(r^{-3}),\qquad
p=\frac{\mathcal N^2}{2(1+\beta)}}.
$$
For a stable nondegenerate bulk state in this scaling, $1+\beta=n_0(V_0+2W_0n_0)/\mu>0$. At $\beta=-1$ the amplitude restoring derivative vanishes, and an inverse-square perturbation cannot cancel the centrifugal term, so the stated expansion fails. For zero winding the uniform solution has $p=0$.
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