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The Hasenfratz-Maggiore-Niedermayer exact mass gap for the two-dimensional O(4) lattice model

For the nearest-neighbour O(4)O(4) model on Z2\mathbb Z^2 (spins in S3S^3, weight exp⁡(β∑x∼yσx⋅σy)\exp(\beta\sum_{x\sim y}\sigma_x\cdot\sigma_y)), asymptotic freedom predicts a mass mlat(β)m_{\mathrm{lat}}(\beta) of order β e−πβ\sqrt\beta\,e^{-\pi\beta} as β→∞\beta\to\infty. Hasenfratz, Maggiore and Niedermayer (1990) computed the exact ratio m/ΛMS‾=32/(πe)m/\Lambda_{\overline{\mathrm{MS}}}=\sqrt{32/(\pi e)} for O(3)O(3) and O(4)O(4) by matching the Bethe ansatz with perturbation theory, which fixes the numerical prefactor of the lattice mass. Does the gap of the full lattice transfer operator satisfy the predicted exact asymptotic, with the predicted constant?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Statistical mechanics; asymptotically free sigma models, mass gap
Posed by
Peter Hasenfratz, Michele Maggiore and Ferenc Niedermayer (Phys. Lett. B, 1990); extended to all O(n), n >= 3, by Hasenfratz and Niedermayer (1990)
Year posed
1990
Years open
36y
Solved
2026-10-05
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
28 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for the periodic infinite-volume state, the gap of the full Osterwalder-Schrader transfer operator, including rotation-invariant sectors and measured in original lattice time steps, satisfies lim⁡β→∞eπββ−1/2mlat(β)=32exp⁡(π/4−1/2)\lim_{\beta\to\infty}e^{\pi\beta}\beta^{-1/2}m_{\mathrm{lat}}(\beta)=32\exp(\pi/4-1/2). The input companion gives cβe−πβ≤mlat≤Cβe−πβc\sqrt\beta e^{-\pi\beta}\le m_{\mathrm{lat}}\le C\sqrt\beta e^{-\pi\beta} and a positive full gap at every β>0\beta>0. Not shown: the corresponding exact constant for O(3)O(3) or general O(n)O(n), a continuum O(4)O(4) theory, or any statement about nonperiodic Gibbs states.

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 (October 5, 2026) takes the family's 'Sharp mass bounds for the two-dimensional O(4) model' (September 23) as its constructive starting point and adapts arguments from the two O(3) companions of October 4.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and the historical section of 'Exact mass asymptotics for the two-dimensional O(4) lattice model' were read against the Hasenfratz-Maggiore-Niedermayer prediction as the paper cites it. The constant 32 exp(pi/4 - 1/2) was not independently re-derived from the MS-bar ratio here. The proof (sector comparison, a Brownian U(2) regulator solved by Bethe-type integral equations, and regulator matching with a finite coupling shift) was not refereed. No Lean formalization: the family's Lean scope note excludes the O(4) gap claims. The result depends on the unreviewed companion 'Sharp mass bounds' for the two-sided bounds, the periodic state and the renormalization trajectories, and on two O(3) companions for adapted arguments.

Sources

Changelog1 change

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