= Solution
For a neutral condensate with <order parameter> $\psi=\sqrt{n_s}e^{i\phi}$, the <superfluid velocity> away from singularities is $\mathbf v_s=\hbar\nabla\phi/m$. Single-valuedness around a closed contour gives $\oint\nabla\phi\cdot d\mathbf l=2\pi n$, hence <quantization of circulation>:
$$
\boxed{\oint\mathbf v_s\cdot d\mathbf l=n\frac hm.}
$$
The <mass> is that of the condensed object: an atom for a bosonic atomic superfluid, or a pair for a paired-fermion condensate. A nonzero winding cannot be continuously removed while the <order parameter> remains nonzero throughout a spanning region. A vortex allows its phase to wind while its amplitude vanishes in a core.
For a charged condensate, minimal coupling gives $m\mathbf v_s+q\mathbf A=\hbar\nabla\phi$. Thus
$$
\boxed{\Phi+\frac mq\oint\mathbf v_s\cdot d\mathbf l=n\frac hq.}
$$
Using $\mathbf j_s=n_sq\mathbf v_s$ and $\lambda_L^2=m/(\mu_0n_sq^2)$ gives the <fluxoid and flux quantization in a superconductor> relation $\Phi+\mu_0\lambda_L^2\oint\mathbf j_s\cdot d\mathbf l=nh/q$. The signs of oriented flux and winding depend on the signed pair charge; the magnitude of the elementary superconducting flux quantum is \b[$\Phi_0=h/(2e)$]. Flux alone is quantized when the current term is negligible on a contour deep in the bulk. A thin ring or a contour near a core can instead require the full fluxoid.
In a neutral superfluid, vortex flow carries <kinetic energy> and <angular momentum>. Rotation can favor vortices because equilibrium minimizes the rotating-frame quantity $E-\boldsymbol\Omega\cdot\mathbf L$. The core size is set by a coherence or healing scale. In a type-II superconductor, an <Abrikosov vortex> carries quantized <magnetic flux> and circulating supercurrent; its core scale is the <coherence length> and its magnetic screening scale is the penetration depth. Between the lower and upper critical fields, a vortex lattice can be favored. <Flux pinning> affects how that structure responds to current. Type-I bulk superconductors instead favor macroscopic normal/superconducting separation rather than the usual stable dilute Abrikosov-vortex regime.
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