= Solution
The propagator is the quantum amplitude for evolving from one angular position to another. It evolves a wavefunction by $\Psi(\phi_F,t)=\int_0^{2\pi}d\phi_I\,K(\phi_F,\phi_I;t)\Psi(\phi_I,0)$. The <path integral> weights each trajectory by $e^{iS/\hbar}$, with $S=\int dt\,I(\partial_t\phi)^2/2$. Since the configuration space is a circle, paths must include all possible lifts of the final angle to the real line.
Take $t=-i\hbar\beta$ by <Wick rotation>. Let $\tau=t_E/\hbar$, which runs from zero to $\beta$, and let the dot denote $d/d\tau$. The dimensionless Euclidean exponent is then $I\int_0^\beta\dot\phi^2d\tau/(2\hbar^2)$. Tracing the kernel identifies final and initial angles modulo $2\pi$, while the lift can change by $2\pi m$. Thus the <rotor winding-sector path integral> is
$$
\boxed{\mathcal Z=\int_0^{2\pi}d\phi_0\sum_{m\in\mathbb Z}
\int_{\phi(0)=\phi_0}^{\phi(\beta)=\phi_0+2\pi m}\mathcal D\phi\,
\exp\left[-\frac{I}{2\hbar^2}\int_0^\beta d\tau\,\dot\phi^2\right].}
$$
The integration measure is fixed by the short-time free-rotor kernel. The dot on $\phi$ is essential; it is present in the PDF but lost in the TeX aid.
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