= Constant far-field phase excludes net vortex winding
If a continuous nonzero complex field tends uniformly to one on large circles, its values on a sufficiently large circle lie in a disk about one that excludes zero. That disk is contractible, so the field's <winding number> on that circle is zero. A lone unit-charge <quantum vortex> therefore cannot satisfy the literal boundary $\psi\to1$. For a vortex it is the <wave amplitude>, rather than the entire complex field, that can approach one; $f(r)e^{i\theta}$ gives a simple illustration of direction-dependent far-field <complex argument>.
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