Polyakov Conjecture: Absence of a Phase Transition for Planar O(n) Models with n >= 3

Polyakov Conjecture: Absence of a Phase Transition for Planar O(n) Models with n >= 3

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: The proposed large-$\beta$ $O(4)$ mass is $32e^{\pi/4-1/2}\sqrt\beta e^{-\pi\beta}(1+o(1))$ in original lattice time units; Square-lattice classical $O(n)$ spin correlations for $n\geq3$ are claimed to decay exponentially at every finite positive inverse temperature; The full square-lattice $O(4)$ transfer gap is claimed to be positive at every finite coupling and comparable to $\sqrt\beta e^{-\pi\beta}$ at large $\beta$; The two-dimensional $O(3)$ lattice model is claimed to have a unique interacting massive continuum limit under susceptibility and correlation-length normalization. Together they provide the following support within the scopes described below.

Mathematical statement of the claimed result

Result 1.

For the nearest-neighbor square-lattice $O(4)$ model, spins lie in $S^3$ and the periodic Gibbs density is proportional to $\exp(\beta\sum_{\langle xy\rangle}\sigma_x\cdot\sigma_y)$, each bond counted once. In the periodic local limit, let $T_\beta$ be the full Osterwalder--Schrader one-time-step transfer operator and $\Omega_\beta$ its constant-function vacuum. Define \[ m_{\mathrm{lat}}(\beta)=-\log\|T_\beta|_{\Omega_\beta^\perp}\|. \] The exact claimed asymptotic is \[ \lim_{\beta\to\infty}\frac{e^{\pi\beta}}{\sqrt\beta}m_{\mathrm{lat}}(\beta) =32e^{\pi/4-1/2}. \] The gap includes every sector generated by local observables, not only spin correlations.

Result 2.

On any finite nearest-neighbor square-lattice subgraph $G=(V,E)$, put spins $\sigma_x\in S^{n-1}$ and give their free-boundary law density proportional to \[ \exp\!\left(\sum_{\{x,y\}\in E}b_{xy}\sigma_x\cdot\sigma_y\right) \prod_{x\in V}d\omega_{n-1}(\sigma_x),\qquad 0\le b_e\le\beta. \] For every integer $n\ge3$ and finite $\beta>0$, the asserted constants $A(n,\beta)<\infty$, $m(n,\beta)>0$ work for every such graph, coupling array, and pair of vertices: \[ 0\le\langle\sigma_x\cdot\sigma_y\rangle\le A e^{-m\|x-y\|_2}. \] There is no fixed boundary spin, applied field, or added interaction.

Result 3.

For the homogeneous nearest-neighbor square-lattice $O(4)$ model, the source claims a unique periodic local limit for sufficiently large $\beta$, and full transfer gap \[ c\sqrt\beta\,e^{-\pi\beta}\le\operatorname{gap}(\beta) \le C\sqrt\beta\,e^{-\pi\beta},\qquad\beta\ge\beta_0, \] with $0<c<C<\infty$. Fixed dyadic $L$, terminal coupling $H$, and integers $N_0,M_0$ give couplings $\beta_N$ whose exact $N$-step transform ends at $H$, satisfying \[ \frac{\log2}{M_0L^N}\le\operatorname{gap}(\beta_N)\le\frac{\log8}{L^N}\quad(N\ge N_0). \] The preliminary result uses $\Xi(\beta)=\exp[\pi\beta+C\sqrt{(1+\beta)\log(2+\beta)}]$. For unpinned free subgraphs, or rectangular tori with sides at least $C\Xi$, couplings $0\le\beta_e\le\beta$, and local Lipschitz observables at distance $d$, it bounds covariance by $C(1+\beta+|A|+|A'|+d)^{20}L_FL_Ge^{-cd/\Xi}$. It also gives \[ 0\le4\log Z_\beta(n,n)-\log Z_\beta(2n,2n) \le C(1+\beta)^pn^p e^{-cn/\Xi(\beta)} \] for even $n\ge C\Xi(\beta)$. Spins lie in $S^3$ with normalized spherical measure, and the Gibbs density is proportional to $\exp(\sum_e\beta_e\sigma_x\cdot\sigma_y)$. Reflection positivity of the periodic limiting state supplies a vacuum $\Omega$ and positive one-step translation contraction $T_\beta$, with $T_\beta\Omega=\Omega$. The gap convention is \[ \operatorname{gap}(\beta)=-\log\|T_\beta|_{\Omega^\perp}\|. \] For a local observable $F$ with support $A$, define $L_F=\|F\|_\infty+\max_{x\in A}\operatorname{Lip}_x(F)$, using round-sphere distance in that coordinate while all others are held fixed. The separation $d$ is maximum-norm distance between supports, with periodic distance on tori. The preliminary estimates hold for every $\beta\ge0$; $Z_\beta(n,w)$ is the homogeneous torus partition function. The constants $L,H,N_0,M_0$ above are selected once, with $H$ sufficiently large.

Result 4.

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

Odd and even observable sectors express the full gap through ratios of twisted partition functions. A Brownian regulator permits an exact auxiliary spectral calculation, and regulator comparison transports its leading constant back to the square-lattice model. Rotation identities and multiscale estimates force a quantitative loss of effective stiffness. Once a suitable coarse scale is reached, block estimates give exponential transmission bounds uniformly for free finite graphs and bounded nonnegative couplings.

Quaternionic rotation and current identities drive the scale-by-scale stiffness decrease. A massive block expansion provides two-sided gap bounds, and comparison of polynomial local-observable sectors ensures the spin estimate controls the entire transfer gap. 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. The listed Comparator files are ClassicalON.lean. Named declarations include OAI.ClassicalON.main. 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, Exact mass asymptotics for the two-dimensional O$4$ lattice model, 2026-10-05; citation key OAI:Exact-mass-asymptotics-for-the-two-dimensional-O4-lattice-model-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI, Exponential decay in two-dimensional classical O(n) models, 2026-09-23; citation key OAI:Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI, Sharp mass bounds for the two-dimensional O$4$ model, 2026-09-23; citation key OAI:Sharp-mass-bounds-for-the-two-dimensional-O4-model-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:preliminary; Statement source: thm:main.
• 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.
• Comparator ClassicalON.json.
• Comparator ClassicalON.lean.
• OpenAI catalogue entry.

Polyakov Conjecture: Absence of a Phase Transition for Planar O(n) Models with n >= 3

Definitions and mathematical statement

Place spins $\sigma_x\in S^{n-1}\subset{\mathbb{R}}^n$ at $x\in{\mathbb{Z}}^2$. The nearest-neighbor ferromagnetic Gibbs weight in a finite region is proportional to $\exp(\beta\sum_{\langle x,y\rangle}\sigma_x\cdot\sigma_y)$ relative to product uniform spherical measure. Two-point correlation is $\langle\sigma_0\cdot\sigma_x\rangle_\beta$ in the infinite-volume Gibbs convention of the source. For each $n\ge3$ and finite $\beta>0$, the target is exponential decay: \[ |\langle\sigma_0\cdot\sigma_x\rangle_\beta|\le C(n,\beta)e^{-c(n,\beta)|x|}, \qquad c(n,\beta)>0. \] The constants need not be uniform as temperature tends to zero. Absence of spontaneous magnetization alone is weaker than the claimed exponential bound.

Formulation and concept references: [S1].

Short English statement

Do planar O(n) spin correlations decay exponentially at every positive temperature once the spin dimension is at least three?

Research attempt

Current frontier.

Absence of spontaneous magnetization in two dimensions is weaker than a mass gap. Aru--Garban--Sepúlveda formulate the $n\ge3$ exponential-decay problem and study a percolative mechanism related to vector-valued Gaussian free fields; their results must not be read as a proof of exponential decay for every finite inverse temperature [S1], Section 1.1. We first reconstruct a high-temperature contraction, then isolate exactly what a scale-dependent improvement would need. The distinction from the planar $O(2)$ model is essential: that model has a rigorously established low-temperature nonexponential regime [E1].

Attack route.

Single-site mixing can give an explicit high-temperature criterion. A coarse-grained version could in principle extend it to all finite $\beta$, but only if an $n$-dependent mechanism makes a block-to-block influence kernel strictly subcritical. Alternatively one could prove a suitable finite-volume susceptibility criterion directly. We investigate the contraction route; neither a percolation analogy nor absence of magnetization is substituted for the missing contraction estimate.

Attempt.

At a site with neighbor sum $h$, the conditional spin distribution is \[ d\nu_h(s)=Z(h)^{-1}e^{\beta s\cdot h}\,d\omega(s),\qquad s\in S^{n-1}. \] When one neighboring spin changes from $u$ to $v$, the oscillation of the unnormalized log likelihood ratio is at most \[ \sup_s\beta s\cdot(u-v)-\inf_s\beta s\cdot(u-v) =2\beta|u-v|\le4\beta. \] If two probability densities have log likelihood ratio of oscillation at most $L$, their total variation distance is at most $\tanh(L/4)$. Here is a proof of the constant: for $r=dP/dQ$ with $a\le r\le b$ and $b/a\le e^L$, ${\mathbb{E}}_Qr=1$ gives \[ \tfrac12{\mathbb{E}}_Q|r-1|\le\frac{(1-a)(b-1)}{b-a}. \] Maximizing over $a\le1\le b$, $b/a\le e^L$, gives the symmetric extremum $a=e^{-L/2}$, $b=e^{L/2}$ and the asserted bound. Thus each of the four neighbors has influence at most $t=\tanh\beta$.

For completeness, the resulting comparison has a probabilistic interpretation. Couple two finite-volume heat-bath systems by maximal couplings. A disagreement at $x$ is propagated to a neighbor with coefficient at most $t$, so iteration gives a sum over nearest-neighbor walks. If \begin{equation}\label{wsref-0-65-71-31-38-37-3a-73-6d-61-6c-6c-62-65-74-61} q=4\tanh\beta<1, \end{equation} that sum converges and the influence between sites at graph distance $r$ is at most \[ \sum_{j\ge r}4^jt^j=\frac{q^r}{1-q}. \] The Dobrushin comparison theorem, whose proof uses these coupled finite-volume inequalities, makes this argument rigorous [E2]. One solves the finite-volume disagreement inequalities $d\le b+Cd$, with $C_{xy}=t$ on neighboring sites, and using $(I-C)^{-1}=\sum_{j\ge0}C^j$. Exhausting the lattice makes boundary influences tend to zero and yields uniqueness. Rotation invariance then gives $\langle\sigma_x\rangle=0$.

Conditioning on the spin at the origin and applying the comparison to each bounded coordinate function gives the conservative bound \begin{equation}\label{wsref-0-65-71-31-38-37-3a-68-69-67-68-54} |\langle\sigma_0\cdot\sigma_x\rangle| \le \frac{2n}{1-q}q^{|x|_1}\qquad(x\ne0). \end{equation} Indeed, the conditional expectation of each coordinate at $x$ changes by at most twice the influence bound as the spin at zero varies; its unconditional mean is zero. Multiply by the corresponding coordinate at zero, bounded by one, and sum over the $n$ coordinates. The factor $2n$ is not optimized. This recovers a high-temperature special case for $\beta<\operatorname{arctanh}(1/4)$, not the full assertion.

Finite-scale lemma. Let $g:{\mathbb{Z}}^2\to[0,1]$, and suppose for some integer $L\ge1$ and nonnegative kernel $K$ supported on $0<|z|_1\le L$ that \begin{equation}\label{wsref-0-65-71-31-38-37-3a-62-6c-6f-63-6b} g(x)\le\sum_zK(z)g(x-z)\quad(|x|_1>L),\qquad \sum_zK(z)=q_L<1. \end{equation} Then, with $m=\max\{0,\lfloor |x|_1/L\rfloor-1\}$, \[ g(x)\le q_L^m. \] To prove it, substitute the inequality $m$ times. Before the last substitution every reachable site still has norm greater than $L$, by the triangle inequality. The sum of all path weights is exactly $q_L^m$, while the terminal value is at most one. If $q_L=0$ the function vanishes outside the ball; otherwise this is exponential decay with rate $-L^{-1}\log q_L$, up to a harmless prefactor.

Applying this lemma to the absolute correlation would prove the stated target if, for every $n\ge3$ and finite $\beta$, one could construct an actual correlation comparison kernel satisfying (eq187:block). The lemma accommodates a correlation length growing arbitrarily rapidly with $\beta$; it does not demand an unphysical temperature-uniform mass gap.

Stress test.

The single-site estimate is independent of the distinction $n=2$ versus $n\ge3$. Therefore iterating it without new geometric information cannot explain the conjectured distinction. Worst-case block boundary influences may become larger, not smaller: replacing a block by a "renormalized spin" and assuming a smaller effective coupling is an unproved step. The finite-scale lemma is only a sufficient analytic criterion; no Simon--Lieb-type inequality with subcritical total weight at all temperatures has been proved here. Mermin--Wagner's conclusion does not provide that kernel, and slow decay to zero would not suffice. The low-temperature $O(2)$ results in [E1] actively falsify any dimension-insensitive argument purporting to establish the full all-temperature conclusion.

Outcome.

Outcome: Exploratory approach with unresolved gap. We proved an explicit high-temperature estimate and a finite-scale contraction lemma, both with transparent constants and quantifiers. They are elementary reconstructions, not a new mass-gap theorem. The missing step is an $n\ge3$-specific low-temperature estimate that produces a subcritical comparison kernel at some finite scale for every fixed $\beta$. No such estimate is established.

Sources

• S1 J. Aru, C. Garban, and A. Sepulveda, Percolation for 2D Classical Heisenberg Model and Exit Sets of Vector Valued GFF, Section 1.1, CMP 406 $2025$. https://link.springer.com/article/10.1007/s00220-024-05208-y. Catalogue formulation source.
• E1 Jürg Fröhlich and Thomas Spencer, The Kosterlitz--Thouless transition in two-dimensional abelian spin systems and the Coulomb gas, Communications in Mathematical Physics 81 $1981$, 527--602; low-temperature planar rotator correlations. https://doi.org/10.1007/BF01208273. Research reference; consulted 27 September 2026.
• E2 R. L. Dobrushin, The Description of a Random Field by Means of Conditional Probabilities and Conditions of Its Regularity, Theory of Probability and Its Applications 13 $1968$, no. 2, 197--224. Single-site influence contraction and comparison. https://doi.org/10.1137/1113026. Research reference; consulted 27 September 2026.