= Solution
In the usual smooth localized finite-energy class, the positive potential requires $|\phi|\to1$ at infinity and the <gauge covariant derivative> tends to zero there. On a sufficiently large circle write $\phi\simeq e^{i\chi}$ locally. Then the tangential <gauge potential> approaches $d\chi$. The normalized Higgs phase defines a continuous map $S^1_\infty\to U(1)$ with integer <winding number>
$$
\boxed{N=\frac1{2\pi}\oint d\chi=\frac1{2\pi}\int_{\mathbb R^2}B\,d^2x\in\mathbb Z.}
$$
The second equality is <Stokes theorem> with the chosen orientation and $D_j=\partial_j-ia_j$. Suitable decay makes the integrated covariant-derivative error vanish. Thus <magnetic flux quantization of an Abelian Higgs vortex> labels disconnected boundary sectors. This uses the standard continuous vacuum asymptotics, not a claim that bare integral finiteness alone supplies every pointwise boundary limit.
A nonzero winding requires zeros of the complex <Higgs field> somewhere inside: a nowhere-zero field on the whole disk would extend its normalized boundary map and make its winding zero. For isolated zeros the signed local indices sum to $N$. For arbitrary configurations, vortex-antivortex pairs may add zeros without changing $N$, so $|N|$ need not count all unsigned zeros.
The <critical coupling> relates the charge to a minimum energy. Set $j_i=\operatorname{Im}(\bar\phi D_i\phi)$. The covariant derivative commutator $[D_1,D_2]\phi=-iB\phi$ gives
$$
|D_1\phi|^2+|D_2\phi|^2=|(D_1+iD_2)\phi|^2+B|\phi|^2+\partial_1j_2-\partial_2j_1.
$$
Discarding its vanishing boundary integral and completing the magnetic square yields
$$
E=\int\left\{\frac12|(D_1+iD_2)\phi|^2+\frac12\left[B-\frac{1-|\phi|^2}{2}\right]^2\right\}d^2x+\frac12\int B\,d^2x.
$$
For positive $N$, the <Bogomolny vortex equations> are $(D_1+iD_2)\phi=0$ and $B=(1-|\phi|^2)/2$. Reverse both signs for negative charge. Hence
$$
\boxed{E\ge\pi|N|.}
$$
Critical-coupling vortex solutions attain this bound; their energy depends only on $|N|$. Positive-charge solutions have $N$ zeros counted with multiplicity, and their positions provide $2N$ real coordinates on the <Abelian Higgs vortex moduli space>. Coincident zeros are allowed. Position variations leave the static minimum energy unchanged, expressing the absence of static forces among same-sign Bogomolny vortices. The sign of $N$ reverses flux and orientation. Topology alone does not determine an arbitrary field's profile or full energy: the zero-charge vacuum has energy zero, while nonvacuum configurations in the same sector can have positive energy.
For the given radial ansatz, $a_\theta$ is the coordinate one-form coefficient: the <gauge potential> is $a(r)d\theta$, and the orthonormal azimuthal component is $a(r)/r$. Thus $B=a'(r)/r$, consistently with the printed energy. Since $h(\infty)=1$, the phase $e^{i\theta}$ has winding one; equivalently $a(0)=0$, $a(\infty)=1$ gives
$$
\boxed{N=1,\qquad\int B\,d^2x=2\pi[a(\infty)-a(0)]=2\pi.}
$$
For the trial functions all energy is inside $0<r<r_0$. Their four contributions to $E/\pi$ are
$$
\begin{aligned}
\text{magnetic: }&\int_0^{r_0}\frac{(2r/r_0^2)^2}{r^2}r\,dr=\frac2{r_0^2},\\
\text{radial gradient: }&\int_0^{r_0}\frac1{r_0^2}r\,dr=\frac12,\\
\text{angular gradient: }&\int_0^{r_0}\frac{(1-r^2/r_0^2)^2}{r_0^2}r\,dr=\frac16,\\
\text{potential: }&\frac14\int_0^{r_0}(1-r^2/r_0^2)^2r\,dr=\frac{r_0^2}{24}.
\end{aligned}
$$
Therefore the <compact-core variational estimate for an Abelian Higgs vortex> is
$$
\boxed{E(r_0)=\pi\left(\frac2{r_0^2}+\frac23+\frac{r_0^2}{24}\right).}
$$
Writing $s=r_0^2>0$, the derivative is $-2/s^2+1/24$, and the energy diverges at both ends of the $s$ range. The unique global minimum is
$$
\boxed{r_0^4=48,\quad r_0^2=4\sqrt3,\quad E_{\rm trial,min}=\pi\left(\frac23+\frac1{\sqrt3}\right)\simeq1.24402\pi.}
$$
The magnetic and potential energies agree at this optimum, as required by radial scale balance. The piecewise fields are continuous with square-integrable first derivatives; their derivative jumps do not create an additional surface energy. They are trial fields, not exact solutions. Their optimized energy is an upper-bound estimate, about 24.4% above the exact critical one-vortex energy $\pi$.
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