= Solution
For a one-component superfluid with nonzero order-parameter magnitude on a closed loop, the phase defines a map from the loop into the circle. Its <winding number> is
$$
\boxed{n=\frac1{2\pi}\oint\nabla\theta\cdot d\mathbf l\in\mathbb Z.}
$$
The <fundamental group> $\pi_1(S^1)=\mathbb Z$ means that smooth deformations preserving nonzero amplitude cannot change this integer. With <superfluid velocity> $\mathbf v_s=\hbar\nabla\theta/m_B$, it gives <quantized circulation> $\oint\mathbf v_s\cdot d\mathbf l=nh/m_B$. A vortex core permits the <order parameter> to vanish and carries the winding; a <superfluid phase slip> changes a loop's winding when a core crosses it. For paired neutral <fermions>, the relevant condensed-object <mass> is the pair <mass>.
For charged pairs, gauge-covariant phase rigidity gives <fluxoid and flux quantization in a superconductor>:
$$
\boxed{\Phi+\frac{m_*}{q}\oint\mathbf v_s\cdot d\mathbf l=\frac{nh}{q}.}
$$
The magnitude of the pair flux quantum is $h/(2e)$. <Magnetic flux> alone approaches an integer multiple of this value when the circulating-velocity term on a bulk contour is negligible. The topological condition is not a claim that an arbitrary finite loop has exactly quantized <magnetic flux> independent of its screening current.
There are also momentum-space invariants, distinct from real-space vortex winding. A fully gapped two-dimensional <Bogoliubov--de Gennes Hamiltonian> can have a <Chern number> obtained from its occupied <quasiparticle> bands' <Berry curvature>. In a chiral paired model it can protect edge modes while the bulk gap and relevant symmetry class are maintained. Such an invariant can change only when those assumptions fail, for example through a bulk gap closing. A nodal d-wave state requires a different analysis; it is not automatically a fully gapped Chern superconductor.
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