Result 215, Probability and statistical mechanics

Canonical O(3)O(3) continuum limit and exact O(4)O(4) mass asymptotics

Constructs the canonical continuum limit of the two-dimensional nearest-neighbor O(3)O(3) model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice O(4)O(4) model, determines the exact leading asymptotic of the full transfer gap, mlat(β)∼32eπ/4−1/2β e−πβm_{\mathrm{lat}}(\beta)\sim32e^{\pi/4-1/2}\sqrt\beta\,e^{-\pi\beta}. The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor O(n)O(n) models with n ≥ 3 at every positive temperature.

Lean formalization Proof

The bigger picture

Why it matters

The manuscripts report how a two-dimensional lattice of interacting spins can produce a continuum theory with genuine interactions and a positive energy threshold for excitations. They also quantify that threshold precisely in a related lattice model.

What changes?

The O(3) model has three-component unit spins interacting only with nearest neighbors. With no external field or topological term, the manuscript reports a continuum limit as the bare coupling grows through all positive real values, without selecting subsequences. Susceptibility and second-moment correlation length fix field and distance scales. The reported limit is non-Gaussian, local and relativistic, with a unique vacuum and a positive mass gap: every non-vacuum excitation has energy bounded away from zero.

What does that help mathematicians do?

For square-lattice O(4), the reported full transfer gap is asymptotic to 32 times exp(pi/4 - 1/2) times sqrt(beta) times exp(-pi beta), where beta is inverse temperature and tends to infinity. This fixes the leading coefficient, not just the decay rate. Because the gap includes rotation-invariant local observables, it also rules out a slower excitation scale hidden in those sectors. This lattice conclusion does not itself establish an O(4) continuum theory.

Are there practical applications?

The immediate value is foundational: these claims clarify how short-range spin interactions generate finite correlation scales. The family also reports exponential spin-correlation decay for square-lattice O(n) models with n at least 3 at every positive temperature, uniformly over finite free-boundary subgraphs and bounded nonnegative edge strengths. This constrains how strongly distant spins remain correlated, rather than demonstrating a technological application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

5 manuscripts

The canonical massive continuum limit of the two-dimensional O(3) model

October 4, 2026 122 pages

We construct a canonical interacting massive continuum limit of the two-dimensional nearest-neighbor O(3)O(3) model with unit-length spins, no external field, and no topological term. Normalized by susceptibility and second-moment correlation length, the limit exists as the bare coupling tends to infinity through all positive real values, without selecting subsequences. The limiting fields satisfy the Osterwalder–Schrader axioms and have a nonzero connected four-point correlation on separated time supports. Their reconstructed theory has a unique vacuum, a nonzero vacuum complement, and a positive Hamiltonian gap on that entire complement.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026,
  author = {{OpenAI}},
  title = {{The canonical massive continuum limit of the two-dimensional O(3) model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/massive-continuum-o3.pdf}{OAI:The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026}},
  year = {2026}
}

An Isolated Particle Pole for the Two-Dimensional O(3) Spin Field

October 4, 2026 26 pages

We consider the continuum spin field constructed from the nearest-neighbor two-dimensional O(3)O(3) model by the fixed prescription of the companion paper. We prove that its vector two-point spectral measure has a positive atom corresponding to the lowest mass, separated by a positive gap from all remaining mass support.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:An-Isolated-Particle-Pole-for-the-Two-Dimensional-O3-Spin-Field-October-4-2026,
  author = {{OpenAI}},
  title = {{An Isolated Particle Pole for the Two-Dimensional O(3) Spin Field}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-Isolated-Particle-Pole-for-the-Two-Dimensional-O3-Spin-Field-October-4-2026/o3-particle-pole.pdf}{OAI:An-Isolated-Particle-Pole-for-the-Two-Dimensional-O3-Spin-Field-October-4-2026}},
  year = {2026}
}

Exact mass asymptotics for the two-dimensional O(4) lattice model

October 5, 2026 109 pages

For the nearest-neighbor O(4)O(4) model on the square lattice at inverse temperature β, we prove the exact low-temperature asymptotic

mlat(β)∼32exp⁡(π/4−1/2)β exp⁡(−πβ)(β→∞).\displaystyle m_{\mathrm{lat}}(\beta)\sim 32\exp(\pi/4-1/2)\sqrt\beta\,\exp(-\pi\beta) \qquad (\beta\to\infty).

Here the mass is the gap of the full Osterwalder–Schrader transfer operator, including rotation-invariant local-observable sectors, in units of one original lattice time step.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Exact-mass-asymptotics-for-the-two-dimensional-O4-lattice-model-October-5-2026,
  author = {{OpenAI}},
  title = {{Exact mass asymptotics for the two-dimensional O(4) lattice model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Exact-mass-asymptotics-for-the-two-dimensional-O4-lattice-model-October-5-2026/exact-mass-o4.pdf}{OAI:Exact-mass-asymptotics-for-the-two-dimensional-O4-lattice-model-October-5-2026}},
  year = {2026}
}

Sharp mass bounds for the two-dimensional O(4) model

September 23, 2026 102 pages

We prove sharp mass bounds for the two-dimensional nearest-neighbor O(4)O(4) model. For all sufficiently large inverse couplings β, the full transfer gap, including rotation-invariant local-observable sectors, is bounded above and below by positive multiples of βe−πβ\sqrt\beta e^{-\pi\beta}. We also prove that the model has a unique periodic local limit and a positive full gap at every finite β > 0, resolving the all-temperature lattice mass-generation conjecture for this periodic state. Under the stated cutoff scaling, the mass bounds are uniform in independently fixed physical units. A continuum spectral interpretation requires separate convergence hypotheses for the transfer semigroup and the block observables.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Sharp-mass-bounds-for-the-two-dimensional-O4-model-September-23-2026,
  author = {{OpenAI}},
  title = {{Sharp mass bounds for the two-dimensional O(4) model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Sharp-mass-bounds-for-the-two-dimensional-O4-model-September-23-2026/paper.pdf}{OAI:Sharp-mass-bounds-for-the-two-dimensional-O4-model-September-23-2026}},
  year = {2026}
}

Exponential decay in two-dimensional classical O(n) models

September 23, 2026 20 pages Main result formalized in Lean

We prove exponential decay of two-point correlations for the classical nearest-neighbor O(n)O(n) model on the square lattice, for every n ≥ 3 and every finite positive inverse temperature. The estimate is uniform over finite free-boundary subgraphs and bounded nonnegative edge strengths. This resolves positively the all-temperature exponential spin-decay conjecture for these models.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026,
  author = {{OpenAI}},
  title = {{Exponential decay in two-dimensional classical $O(n)$ models}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026/paper.pdf}{OAI:Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/215.md.

Canonical ‘O(3)‘`O(3)` continuum limit and exact ‘O(4)‘`O(4)` mass asymptotics

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized result proves exponential decay of spin correlations for the two-dimensional classical O(n)O(n) model at every temperature when n≥3n\ge3. For each interaction bound β>0\beta>0, constants AA and m>0m>0 work for every finite square-lattice subgraph with free boundary and nonnegative edge strengths at most β\beta: the correlation between sites x,yx,y is at most Ae−m∣x−y∣Ae^{-m|x-y|}. The infinite-volume limit and the separate O(4)O(4) spectral-gap claim are not included.

Comparator links

Result Comparator statement
Exponential correlation decay for classical O(n)O(n) models ClassicalON.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.