A non-Gaussian massive continuum theory from the planar O(3) model

A non-Gaussian massive continuum theory from the planar O$3$ model

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The two-dimensional $O(3)$ lattice model is claimed to have a unique interacting massive continuum limit under susceptibility and correlation-length normalization.

Mathematical statement of the claimed result

Result 1.

For square-lattice $O(3)$ spins, the source specifies dyadic $L>1$, terminal coupling $H>0$, field normalizations $B_N>0$, and \[ a_N=L^{-N},\qquad\beta_N=H+\frac{\log L}{2\pi}N+O(\log(N+1)). \] The smeared fields $\phi_N(f)=B_Na_N^2\sum_x f(a_Nx)\sigma_x$ are claimed to converge jointly in law and all moments, with Schwinger distributions converging as tempered distributions at every order. The limiting distributions are Euclidean and internally $O(3)$ invariant, reflection positive, symmetric, clustering, and satisfy $|S_n(F)|\le C^nn!\sup_x(1+|x|^2)^{2n}|F(x)|$. The reconstructed $1+1$-dimensional local unitary theory has a unique vacuum, nonzero orthogonal complement, and \[ H_{\mathrm{phys}}|_{\Omega^\perp}\ge mI\quad(m>0). \] The connected distribution of order four does not vanish on smooth tests with separated times, excluding Wick factorization. For some $b_0>0$ and every $b\ge b_0$, the preliminary lattice mixing bound is exponential on scale $X(b)=Ce^{(4\pi-\eta)b}$, $\eta>0$, uniformly for $0\le\beta_e\le b$, including free subgraphs and sufficiently large unpinned tori; the homogeneous periodic local limit is unique. The spin law uses normalized spherical measure on $S^2$ and density proportional to $\exp(\sum_e\beta_e\sigma_x\cdot\sigma_y)$. For supports $A,A'$ at maximum-norm separation $d$, taking periodic distance on tori, define \[ L_F=\|F\|_\infty+\max_{x\in A}\operatorname{Lip}_x(F),\qquad |\operatorname{Cov}(F,G)|\le C(1+b+|A|+|A'|+d)^p L_FL_G e^{-cd/X(b)}. \] Each coordinate Lipschitz constant uses round distance on $S^2$, keeping other spins fixed. Constants $C,c,p,\eta$ are positive. The estimate applies to finite free subgraphs and tori with each side at least $CX(b)$. Write $Z_b(n,w)$ for the homogeneous torus partition function. Given an aspect bound $\kappa\ge1$, for even side lengths obeying $\kappa^{-1}\le n/w\le\kappa$ and $\min(n,w)\ge CX(b)$, the constants can be chosen to give \[ 0\le4\log Z_b(n,w)-\log Z_b(2n,2w)\le C(1+b+n+w)^p e^{-c\min(n,w)/X(b)}. \]

Outline of the argument

A prescribed physical block scale first produces limiting Schwinger fields with uniform finite-volume and moment estimates. Renormalization comparisons remove subsequence dependence and convert the block prescription to canonical normalization; reflection positivity and clustering reconstruct a gapped theory with a nonzero connected four-point function.

Formal verification

For family 215, the selected theorem gives exponential finite-free-graph $O(n)$ spin decay for $n\geq3$ and bounded ferromagnetic edge strengths. Plane limits, full $O(4)$ spectral gaps, and continuum $O(3)$ results are beyond that entry. 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, The canonical massive continuum limit of the two-dimensional O$3$ model, 2026-10-04; citation key OAI:The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: prop:preliminary; Statement source: thm:main.
• Family 215: Lean scope.
• OpenAI catalogue entry.