= Solution
The <trace> of the Euclidean evolution kernel identifies its initial and final positions. Thus the <Euclidean worldline path integral> is over periodic paths $x(\tau+\beta)=x(\tau)$, with $\tau\in[0,\beta]$:
$$
\boxed{\mathcal Z=\int_{x(0)=x(\beta)}\mathcal Dx\,e^{-S_E[x]},\qquad S_E[x]=\frac12\int_0^\beta\left[\dot x^2+\omega^2x^2\right]d\tau.}
$$
The circle has circumference $\beta$, not an arbitrary unit length. The <Euclidean action> follows by <Wick rotation> of the unit-mass oscillator action. Equivalently, <integration by parts> with <periodic boundary conditions> gives $S_E=\tfrac12\langle x,L_\omega x\rangle$ for $L_\omega=-\partial_\tau^2+\omega^2$. The <path integral> measure is formal until a regularization and its normalization are specified; the next part supplies a finite-dimensional version.
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