= Solution
The internal orientation $\alpha_0$ is a periodic <collective coordinate> with period $2\pi$. Its conjugate <momentum> is the integrated canonical charge. The <Bohr-Sommerfeld quantization> for a positively oriented circuit of this angle is, in units $\hbar=1$,
$$
\oint p_\alpha\,d\alpha=2\pi Q_\theta=2\pi\ell,\qquad\ell\in\mathbb Z.
$$
There is no oscillator turning point and hence no half-integer Maslov correction for this <cyclic coordinate>. Equivalently, a wavefunction $e^{iQ_\theta\alpha_0}$ must be single valued. At $\theta=0$, the allowed signed charges and leading semiclassical <masses> in the $T=1$ family are therefore
$$
\boxed{Q=\ell\in\mathbb Z,\qquad M_\ell=m\sqrt{r^2+\ell^2}.}
$$
The signed integers include both rotation directions and the nonrotating $\ell=0$ state. Every finite integer gives $|\omega|<m$.
For nonzero theta it is still the canonical charge that is integral, while the mechanical charge is shifted:
$$
\boxed{Q_\theta=\ell,\qquad
Q_0=\ell+\frac{\theta}{2\pi}T,\qquad
M_{T,\ell}=m\sqrt{r^2+\left(\ell+\frac{\theta}{2\pi}T\right)^2}.}
$$
For the stated family set $T=1$; the reversed <kink> gives the corresponding $T=-1$ family. This is the <semiclassical charged-kink spectrum with a theta angle>. One may equivalently describe the spectrum by fractional mechanical charges instead of integral canonical charges, but the two conventions must not be mixed. The theta periodicity follows by relabeling the integer when $\theta\mapsto\theta+2\pi$. These are leading semiclassical energies; the quantization argument does not assert that quantum fluctuation corrections vanish.
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