The full critical XY spin-field limit with fixed boundary

The full critical XY spin-field limit with fixed boundary

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. With zero boundary angles, the normalized critical XY spin field is claimed to approach the imaginary exponential of a zero-Dirichlet Gaussian field of stiffness $2/\pi$.

Mathematical statement of the claimed result

Result 1.

For angles $\theta_x\in\mathbb R/(2\pi\mathbb Z)$ on $\Lambda_R=[-R,R]^2\cap\mathbb Z^2$, use the free-boundary cosine law proportional to $\exp[b\sum_{\{x,y\}\subset\Lambda_R,\,|x-y|=1}\cos(\theta_x-\theta_y)]$, relative to product uniform angles. Count each edge once and omit exterior edges. Define \[ C_b(r)=\lim_{R\to\infty}\mathbb E_{R,b}^{\mathrm{free}}\cos(\theta_0-\theta_{re_1}),\quad m(b)=\lim_{r\to\infty}-r^{-1}\log C_b(r),\quad b_c=\inf\{b>0:m(b)=0\}. \] The thermodynamic limit precedes the separation limit. Set $D=(-1,1)^2$, $D_n=[-1,1]^2\cap n^{-1}\mathbb Z^2$ and fix boundary angles to zero at $b_c$, retaining every nearest-neighbour bond in $D_n$. Let $a_n=\mathbb E_n^0e^{i\theta_0}=\mathbb E_n^0\cos\theta_0>0$. Let $G_n$ be the inverse of the unscaled graph Laplacian $L_nu(x)=\sum_{|x-y|=1/n}(u(x)-u(y))$ with zero boundary. With $K_*=2/\pi$, define \[ S_n=\frac1{n^2}\sum_{x\in D_n\cap D}a_n^{-1} \exp\left(\frac{G_n(x,x)-G_n(0,0)}{2K_*}\right)e^{i\theta_x}\delta_x. \] For the zero-Dirichlet GFF $\Phi_D$ with covariance $(-\Delta_D)^{-1}$, its circle averages define the full-variance imaginary chaos \[ V_*=\lim_{\epsilon\downarrow0}\exp\left(\frac{i\Phi_{D,\epsilon}}{\sqrt{K_*}} +\frac{\mathbb E\Phi_{D,\epsilon}^2}{2K_*}\right). \] Then $S_n\Longrightarrow V_*$ along all integers in $H^{-3}_{\mathrm{loc}}(D)$. For every finite set $f_j\in C_c^\infty(D;\mathbb C)$ and integers $p_j,q_j\ge0$, \[ \mathbb E_n^0\prod_jS_n(f_j)^{p_j}\overline{S_n(f_j)}^{q_j} \longrightarrow\mathbb E\prod_jV_*(f_j)^{p_j}\overline{V_*(f_j)}^{q_j}. \] The joint laws also converge. The square geometry, aligned fixed boundary, centre normalization and diagonal Green correction are all part of this claim.

Outline of the argument

Critical Bessel-height and pin-limit inputs give the joint mixed-moment asymptotics. Exact center-magnetization and lattice Green normalizations identify the field amplitude; distributional tightness upgrades moment comparisons to joint smeared-field convergence.

Formal verification

Family 216 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

• OpenAI, The Critical Spin Field of the Planar XY Model, 2026-10-05; citation key OAI:The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI catalogue entry.