Fixed-endpoint rotor fluctuation normalization ($Z_0=\sqrt{I/(2\pi \beta {\hslash }^2)}$)

In a temporal winding sector of a quantum rotor, decompose the lifted angle into a straight winding path and a fluctuation vanishing at both endpoints. The cross term in the kinetic action vanishes, so all sectors have the same Gaussian path integral. With inverse-energy imaginary time its value is the displayed diagonal free-particle kernel on the line. Integrating the initial angle separately supplies $2\pi$. Thus this fixed-endpoint factor is an amplitude density, not an additional rotor partition function or an uncounted angular zero mode.