Critical Center Magnetization in the Planar XY Model

Assuming the stated critical-height, local-renormalization, and spin-field inputs from the companion papers, we determine the center magnetization of the planar XY model in a square with aligned boundary spins at its mass-defined critical threshold. As $n\to \infty$, the magnetization is $A_{\mathrm{XY}}{n}^{-1/8}(\log n{)}^{1/16}(1+o(1))$, where $A_{\mathrm{XY}}$ is a finite, strictly positive model-specific constant.