Remark: Family 216. Critical logarithmic corrections and BKT scaling for the planar XY model

For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations $C_{{\beta }_c}(r)\sim A{r}^{-1/4}(\log r{)}^{1/8}$ and the Berezinskii–Kosterlitz–Thouless essential singularity $\sqrt{{\beta }_c-\beta }\log \xi (\beta )\to B$, with $A,B>0$ after the free-box thermodynamic limit. For finite square-symmetric interactions containing nearest neighbors, discrete Gaussian heights converge to Gaussian fields throughout the rough phase, including its threshold, along geometric torus sizes. Critical center-magnetization and spin-field conclusions retain their stated height, renormalization, and field-input assumptions.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.