= Solution
First lift the angle to the real line. The <time-sliced configuration-space path integral> for a free coordinate with inertia I has kinetic action $\int dt\,I\dot\phi^2/2$. The rotor kernel sums lifted endpoints separated by $\phi_F-\phi_I+2\pi m$, because all such endpoints represent the same final angle on the circle.
Continue real time as $t=-i\hbar\tau$. Then $e^{-iHt/\hbar}=e^{-H\tau}$ and the end of the thermal interval is $\tau=\beta$, in inverse-energy units. Thus
$$
\frac{S_E}{\hbar}=\frac{I}{2\hbar^2}\int_0^\beta d\tau\,\dot\phi^2.
$$
The <trace> integrates the initial angle over one copy of the circle, not the whole lifted line. It yields the <rotor winding-sector path integral>
$$
\boxed{Z=\int_0^{2\pi}d\phi\sum_{m\in\mathbb Z}
\int_{\phi(0)=\phi}^{\phi(\beta)=\phi+2\pi m}\mathcal D\phi\,
\exp\left[-\frac{I}{2\hbar^2}\int_0^\beta d\tau\,\dot\phi^2\right].}
$$
The integer m is temporal <winding number>, not the angular-momentum label n. Locally the measure has the free-kernel normalization $[I/(2\pi\hbar^2\Delta\tau)]^{1/2}$ per time slice. Summing topological sectors and retaining this normalization are necessary for the correct thermal trace.
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