Polyakov's mass-generation conjecture: exponential decay at every temperature in the two-dimensional O(n) model, n at least 3

In the classical nearest-neighbour $O(n)$ model on $\mathbb Z^2$ , spins $\sigma_x\in S^{n-1}$ carry the Gibbs weight $\exp(\beta\sum_{x\sim y}\sigma_x\cdot\sigma_y)$ . Mermin and Wagner showed there is no spontaneous magnetization at any $\beta<\infty$ , and McBryan-Spencer gave algebraic upper bounds on correlations, but these allow a zero exponential rate. For $n=2$ the Berezinskii-Kosterlitz-Thouless phase (Frohlich-Spencer) has only power-law decay at low temperature. For $n\ge3$ Polyakov's renormalization argument (1975) predicts mass generation at every positive temperature; rigorous proofs covered only high temperature (Aizenman-Simon) and large $n$ (Kupiainen). Do two-point correlations $\langle\sigma_x\cdot\sigma_y\rangle$ decay exponentially in $|x-y|$ for every $n\ge3$ and every finite $\beta>0$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Statistical mechanics; classical spin systems, nonlinear sigma models
Posed by: Alexander M. Polyakov (Phys. Lett. B 59, 1975), predicting mass generation for n >= 3; recorded as an open conjecture by Aru, Garban and Sepulveda (Comm. Math. Phys., 2025)
Year posed: 1975
Years open: 51y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 62 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every integer $n\ge3$ and finite $\beta>0$ there are $A<\infty$ , $m>0$ with $0\le\langle\sigma_x\cdot\sigma_y\rangle_{G,b}\le Ae^{-m\|x-y\|_2}$ for every finite subgraph $G$ of $\mathbb Z^2$ with free boundary and every coupling array $b\in[0,\beta]^E$ ; Corollary 1.2 passes this to free-boundary box limits, giving finite susceptibility. The companion 'Sharp mass bounds' proves, for $O(4)$ , a positive gap of the full transfer operator (all local observables) in the unique periodic state at every $\beta>0$ , and $m\asymp\sqrt\beta e^{-\pi\beta}$ for large $\beta$ . Not shown: uniqueness of infinite-volume Gibbs states for general $n$ , a full transfer gap for $n\ne4$ in this paper, or the low-temperature correlation length for general $n$ .

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript (September 23, 2026) is self-contained; a companion of the same date proves the stronger full transfer gap for O(4), and later companions in the family build the O(3) continuum limit and the exact O(4) mass asymptotic on these estimates.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of 'Exponential decay in two-dimensional classical O(n) models' were read against Polyakov's prediction as the paper and Aru-Garban-Sepulveda state it. The proof (a fourth-variation rotation inequality on annuli, a sign-cluster crossing bound and a reduction from n to 3 components) was not refereed. Lean: lean/formalization.yaml lists comparator config ComparatorChallenges/ClassicalON.json, declaration OAI.ClassicalON.main in OAI/Probability/ClassicalON/Main.lean, permitted axioms propext, Quot.sound and Classical.choice. The statement file ComparatorChallenges/ClassicalON.lean was read: for every n >= 3 and beta > 0 it gives A and m > 0 such that, on every finite nearest-neighbour subgraph of Z^2 with edge strengths in [0, beta], the normalized product-sphere Gibbs correlation of two spins lies between 0 and A exp(-m |x-y|). That is exactly Theorem 1.1, the headline claim; it does not cover infinite-volume limits or any transfer gap. Not rebuilt here. Scope the paper itself states: spin two-point function only, free boundary conditions and their subsequential limits, constants that may degrade as beta grows, no uniqueness of Gibbs states and no correlation-length asymptotic.

Polyakov's mass-generation conjecture: exponential decay at every temperature in the two-dimensional O(n) model, n at least 3

What was actually shown

What the AI did

Verification

Sources

Discussion