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 (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 (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 > 0 there are constants A and m > 0 such that the spin correlation ⟨σ_x·σ_y⟩ is at most A e^{−m‖x − y‖} 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 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^−πβ ≤ gap(β) ≤ C√β e^−πβ 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 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_lat(β) ~ 32 e^{π/4 − 1/2} √β e^−πβ 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 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 The canonical massive continuum limit of the two-dimensional O(3) model and An isolated particle pole for the two-dimensional O(3) spin field.
• Formalization, as described by the repository. The scope note 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 ClassicalON.lean.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 18 October 2026)
2026-10-07	The GitHub tier list 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	MathVet (GitHub; review of openai/math, tables at repository commit c76f1e4, 14 October 2026)	Review (fidelity of the formal statements)	Rates the family "partial": the Lean statement of ClassicalON 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 ClassicalON (8 October 2026).	The review states that its ratings concern whether the Lean states the headline claims.
2026-10-10 to 2026-10-14	con-leche development journal (task #348)	Verification (second kernel)	In its run over all Comparator challenges of the collection at commit fd4aeeb, the journal's table of exceptions does not list ClassicalON, 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 JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) of the four manuscripts without Lean: O(4) sharp bounds, O(4) exact mass, O(3) continuum limit and 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.