OpenAI · 216-03 · 2026-10-05 · Probability and statistical mechanics

The Critical Spin Field of the Planar XY Model

preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026

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

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations $C_{β_{c}} (r) \sim A r^{- 1/4} (lo g r)^{1/8}$ and the Berezinskii–Kosterlitz–Thouless essential singularity $β_{c} - β lo g ξ (β) \to B$ , 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.

Using the companion critical Bessel-height and pin-limit theorems, we prove that at the mass-defined critical inverse temperature of the nearest-neighbor XY model on the square lattice, the spin field on a square with boundary angles fixed to zero converges along the full sequence to the full-variance imaginary exponential of a zero-Dirichlet Gaussian free field with stiffness $2/ π$ . The normalization uses the exact center magnetization and the lattice Green-function factor. Convergence holds in law in $H^{- 3}_{loc} ((- 1, 1)^{2})$ ; all mixed moments and joint laws of fields smeared against smooth compactly supported test functions in $(- 1, 1)^{2}$ converge as well.

Preprint source and Lean scope

The abstract above is the catalogue abstract. The sections below are the full preprint source, 0 sections. This page is not only the abstract. The manuscript file is loading.

No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.