Compact phase winding term ($i{\rho }_0\int {\partial }_{\tau }\varphi$)

A periodic complex field permits $\varphi (\beta ,\mathbf{r})-\varphi (0,\mathbf{r})=2\pi w(\mathbf{r})$. Its uniform-density first-order term is therefore a winding-dependent imaginary action, not generally a removable global constant. It vanishes in the smooth zero-winding sector used for Gaussian phonons. Density defects and vortices require more than that smooth-sector approximation.