= Solution
Continue the <free-particle propagator> to $t=-i\mathcal T$ and replace its coordinate mass by the angular inertia $I$. The <Euclidean free-particle bridge amplitude> at coincident endpoints is
$$
\boxed{Z_0=\sqrt{\frac I{2\pi\hbar\mathcal T}}=\sqrt{\frac I{2\pi\hbar^2\beta}}.}
$$
This normalization includes the time-sliced <Gaussian measure>, not just a formal <determinant> with an unspecified multiplicative constant. Set $A=2\pi^2I/(\hbar^2\beta)>0$. The correct <Poisson summation formula> is $\sum_mF(m)=\sum_n\int_{-\infty}^{\infty}dx\,F(x)e^{-2\pi inx}$; the source PDF omits the integral sign before its $d\theta$. For $F(x)=e^{-Ax^2+i2\pi fx}$ the Gaussian Fourier integral gives
$$
\int_{-\infty}^{\infty}dx\,e^{-Ax^2+i2\pi(f-n)x}=\sqrt{\frac\pi A}\,e^{-\pi^2(n-f)^2/A}.
$$
The overall prefactor cancels exactly: $2\pi Z_0\sqrt{\pi/A}=1$, and $\pi^2/A=\beta\hbar^2/(2I)$. Hence
$$
\boxed{Z=\sum_n e^{-\beta\hbar^2(n-f)^2/(2I)},}
$$
in agreement with the angular-momentum <trace>. Pairing $m$ and $-m$ makes the winding expression real, and the <momentum> expression shows positivity. Both are periodic in flux and invariant under $f\mapsto-f$. At high <temperature> the zero-winding term gives $Z\simeq\sqrt{2\pi I/(\beta\hbar^2)}$, with exponentially small flux dependence; at low <temperature> the nearest integer angular-momentum state dominates, with two equal lowest terms at half-integer flux. These independent limits also check the normalization and the flux phase.
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