Infrared-regulated phase covariance ($G_L(r)={\bar{K}}^{-1}{\int }_{q\gtrsim L^{-1}}\frac{d^{d}q}{(2\pi {)}^d}\frac{e^{iq\cdot x}}{q^2}$)

Fix the uniform order-parameter phase and retain finite system size $L$ before evaluating a massless Gaussian functional integral. In $d\le 2$, its coincident and separated order-parameter phase covariances each acquire a divergent additive constant as $L\to \infty$. Their difference $2[G_L(0)-G_L(r)]$ instead has a finite fixed-separation limit for $d>0$. This is why physical XY model correlations are obtained from Gaussian phase averaging of angle differences. An ultraviolet cutoff also remains necessary for coincident-point quantities; infrared cancellation does not remove it. The logarithmic two-dimensional Green function follows from the $d\to 2$ power-law expression only after subtracting its constant pole.