View source for Mass generation in two-dimensional O(n) models

'''Mass generation in two-dimensional O(''n'') models''' refers to Polyakov's prediction that the two-dimensional classical O(''n'') spin model with ''n'' ≥ 3 has exponentially decaying correlations at every positive temperature, with a correlation length that grows exponentially at low temperature (asymptotic freedom), and to the corresponding massive continuum quantum field theories. Five manuscripts dated 23 September to 5 October 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]) claim exponential decay for every ''n'' ≥ 3 at every positive temperature, a massive interacting continuum limit of the O(3) model, and the sharp order and exact leading asymptotic of the lattice mass of the O(4) model. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== Background ==
* As recounted in the introductions of the manuscripts:
** Mermin and Wagner, and Mermin for classical spins, proved that continuous symmetry prevents spontaneous magnetization in two dimensions; McBryan and Spencer obtained algebraic upper bounds on correlations. For two-component spins, Fröhlich and Spencer proved a polynomial lower bound at low temperature (the Berezinskii–Kosterlitz–Thouless phase), so exponential decay at every temperature fails for ''n'' = 2.
** For ''n'' ≥ 3, Polyakov's non-Abelian renormalization argument and the work of Brézin and Zinn-Justin predict mass generation at every positive temperature. Aizenman and Simon proved exponential decay at high temperature, and Kupiainen established a 1/''n'' expansion and a mass gap for a large number of components. Aru, Garban and Sepúlveda record the all-temperature lattice assertion as an open conjecture in their 2025 paper ([https://arxiv.org/abs/2212.06767 arXiv:2212.06767]).
** Hasenfratz, Maggiore and Niedermayer, and Hasenfratz and Niedermayer for general ''n'', obtained exact predictions for the ratio of the physical mass to the renormalization scale by matching Bethe-ansatz calculations with perturbation theory.

== The original results ==
* '''Exponential decay (23 September 2026).''' Theorem 1.1 of ''Exponential decay in two-dimensional classical O(n) models'' states that for every ''n'' ≥ 3 and every finite β &gt; 0 there are constants ''A'' and ''m'' &gt; 0 such that the spin correlation ⟨σ<sub>''x''</sub>·σ<sub>''y''</sub>⟩ is at most ''A'' e<sup>−''m''‖''x'' − ''y''‖</sup> on every finite free-boundary subgraph of the square lattice with edge strengths in [0, β]. The manuscript states that it does not identify the low-temperature correlation length or prove uniqueness of infinite-volume Gibbs states. Source: the manuscript [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026 Exponential decay in two-dimensional classical O(n) models].
* '''Sharp mass bounds for O(4) (23 September 2026).''' The manuscript states two-sided bounds ''c''√β e<sup>−πβ</sup> ≤ gap(β) ≤ ''C''√β e<sup>−πβ</sup> for the full transfer gap of the nearest-neighbour O(4) model at large β, and a positive gap at every finite β. It states that these are order bounds, not an exact prefactor. Source: the manuscript [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Sharp-mass-bounds-for-the-two-dimensional-O4-model-September-23-2026 Sharp mass bounds for the two-dimensional O(4) model].
* '''Exact O(4) mass asymptotics (5 October 2026).''' Theorem 1.1 of the manuscript states that the full transfer gap of the square-lattice O(4) model satisfies ''m''<sub>lat</sub>(β) ~ 32 e<sup>π/4 − 1/2</sup> √β e<sup>−πβ</sup> as β → ∞. The proof is described as comparing the lattice model with an auxiliary Brownian regulator whose spins are loops in U(2) and whose finite-circle spectrum is computed. Source: the manuscript [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Exact-mass-asymptotics-for-the-two-dimensional-O4-lattice-model-October-5-2026 Exact mass asymptotics for the two-dimensional O(4) lattice model].
* '''The O(3) continuum limit (4 October 2026).''' The manuscript states that the nearest-neighbour two-dimensional O(3) model, normalized by its susceptibility and second-moment correlation length, has a continuum limit as the bare coupling tends to infinity. The limiting fields satisfy the Osterwalder–Schrader axioms and reconstruct a local relativistic theory with a unique vacuum, a positive Hamiltonian gap and a nonzero connected four-point function. A companion manuscript states that the two-point spectral measure of the spin field has an isolated atom at the lowest mass. Sources: the manuscripts [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026 The canonical massive continuum limit of the two-dimensional O(3) model] and [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/An-Isolated-Particle-Pole-for-the-Two-Dimensional-O3-Spin-Field-October-4-2026 An isolated particle pole for the two-dimensional O(3) spin field].
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/215.md lean/docs/215.md] states that the formalization proves the exponential decay theorem above, uniformly over finite free-boundary subgraphs and edge strengths at most β, and that the infinite-volume limit and the O(4) spectral-gap claim are not included. Its Comparator challenge is [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/ComparatorChallenges/ClassicalON.lean ClassicalON.lean].

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 18 October 2026)
|-
| 2026-10-07 || The GitHub tier list [https://github.com/evand/open-math-problems/blob/032af2d6856c4da8598467936812b90d6b1f6b64/README.md evand/open-math-problems] || Catalogue || Under the Yang–Mills existence and mass gap problem, the list cites the O(3) continuum manuscript as a two-dimensional toy-model result, noting that the lattice decay is in Lean while the continuum limit is paper-only. || Catalogue commentary; it does not review the proofs.
|-
| 2026-10-08 || [https://github.com/mathvet/mathvet MathVet] (GitHub; review of openai/math, tables at repository commit <code>c76f1e4</code>, 14 October 2026) || Review (fidelity of the formal statements) || Rates the family "partial": the Lean statement of <code>ClassicalON</code> states the exponential decay theorem, while the O(3) continuum limit and the O(4) mass results are not formalized. It records a sandboxed Comparator pass for <code>ClassicalON</code> (8 October 2026). || The review states that its ratings concern whether the Lean states the headline claims.
|-
| 2026-10-10 to 2026-10-14 || [https://github.com/leanprover/con-leche/blob/90577d9e49ee3aed45da3d421bdf5dc780192304/DESIGN.md con-leche development journal] (task #348) || Verification (second kernel) || In its run over all Comparator challenges of the collection at commit <code>fd4aeeb</code>, the journal's table of exceptions does not list <code>ClassicalON</code>, so this challenge was accepted by con-leche and by Lean's official kernel. || Development journal.
|-
| 2026-10-16 || J. Jardine, Lattice24, referee reviews in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995]) of the four manuscripts without Lean: [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_348.json O(4) sharp bounds], [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_043.json O(4) exact mass], [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_071.json O(3) continuum limit] and [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_303.json O(3) particle pole] || Review (AI-assisted audit) || All four reviews report no gap. The O(4) sharp-bounds review says that the renormalization closure and comparison contraction, which carry most of the sharp lower bound, are written as dense sketches it could not verify line by line, and that the O(3) companions import them. The O(4) exact-mass review calls the helicity matching step a sketch of the companion's method. The O(3) continuum review describes its main dependence on the O(4) machinery as "a dependency risk", not a demonstrated gap. || Part of a completed audit of the collection's manuscripts without Lean; the reviews are published under CC BY 4.0.
|}

== Notes ==
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]