Conformal limits of all mixed Ising spin and energy correlations

Conformal limits of all mixed Ising spin and energy correlations

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Weak square-symmetric finite-range even Ising interactions are claimed to preserve the conformal limits of mixed bulk spin and energy correlations after two field normalizations.

Mathematical statement of the claimed result

Result 1.

Fix $J>0$ and a real potential $V(A)$ on finite nonempty subsets of $\mathbb Z^2$, vanishing for odd $|A|$ or diameter greater than a fixed finite range, and invariant under translations, reflections and quarter-turns. For spins in $D_a=D\cap a\mathbb Z^2$, use \[ H=-J\sum_{|u-v|=a}\sigma_u\sigma_v-\lambda\sum_AV(A)\prod_{x\in A}\sigma_{ax}, \] counting unordered terms once. Free boundary retains interactions contained in $D_a$; constant $+$ or $-$ boundary retains every interaction meeting $D_a$ with exterior spins fixed. Let $D$ be bounded and simply connected with $C^2$ Jordan boundary. There exist $\lambda_0>0$ and factors $\beta_c,Z_\sigma,Z_\epsilon$, continuous at zero, with $\beta_c(0)=\log(1+\sqrt2)/(2J)$ and $Z_\sigma(0)=Z_\epsilon(0)=1$, all factors nonzero for $|\lambda|<\lambda_0$. Define, at $\beta_c(\lambda)$, \[ S_a(x)=a^{-1/8}\sigma_{[x]_a},\qquad E_{a,j}(y)=a^{-1}[\sigma_{[y]_a}\sigma_{[y]_a+ae_j} -\langle\sigma_{[y]_a}\sigma_{[y]_a+ae_j}\rangle], \] where $[x]_a=a\lfloor x/a\rfloor$. For every $b\in\{\mathrm{free},+,-\}$, all $n,m\ge0$ with $n+m\ge1$, distinct interior insertions and $j_k\in\{1,2\}$, \[ \left\langle\prod_{i=1}^nS_a(x_i)\prod_{k=1}^mE_{a,j_k}(y_k)\right\rangle \longrightarrow Z_\sigma(\lambda)^nZ_\epsilon(\lambda)^m \mathcal C_D^b(\boldsymbol x;\boldsymbol y), \] where $\mathcal C_D^b$ is the corresponding pure nearest-neighbour critical limit with identical centering. The same limit holds in the plane after first taking periodic-box thermodynamic limits. For a conformal bijection $\varphi:D\to D'$, the covariance convention is \[ \mathcal C_D^b(\boldsymbol x;\boldsymbol y)= \left(\prod_i|\varphi'(x_i)|^{1/8}\prod_k|\varphi'(y_k)|\right) \mathcal C_{D'}^b(\varphi\boldsymbol x;\varphi\boldsymbol y). \] In particular the plane two-spin limit is a nonzero multiple of $|x-y|^{-1/4}$.

Outline of the argument

Buffered comparisons and ordered pinning isolate targets from boundary modifications. The perturbed critical branch is compared to the nearest-neighbor reference while actual-state energy centering removes additive shifts; multiscale transmission extends the comparison to different domain and boundary states.

Formal verification

For family 218, the formal scope is a finite-graph likelihood comparison through the zero-field FK connection probability after common pins are deleted. Stopping-band resolution and the universality companions are excluded. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. 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, Conformal universality of bulk Ising correlations under weak interactions, 2026-09-23; citation key OAI:Conformal-universality-of-bulk-Ising-correlations-under-weak-interactions-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: model:main.
• Family 218: Lean scope.
• OpenAI catalogue entry.