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Critical two-point decay r−1/4(log⁡r)1/8r^{-1/4}(\log r)^{1/8} in the planar XY model

The nearest-neighbor XY model on Z2\mathbb Z^2 at inverse temperature bb gives angles θx\theta_x the weight exp⁡(b∑x∼ycos⁡(θx−θy))\exp(b\sum_{x\sim y}\cos(\theta_x-\theta_y)). Berezinskii, Kosterlitz and Thouless predicted a transition at bcb_c driven by vortex unbinding, with algebraic decay of the spin correlation Cb(r)=⟨cos⁡(θ0−θre1)⟩C_b(r)=\langle\cos(\theta_0-\theta_{re_1})\rangle for b≥bcb\ge b_c. Kosterlitz's 1974 renormalization-group analysis predicts at criticality the universal exponent η=1/4\eta=1/4 with a multiplicative logarithmic correction, Cbc(r)≍r−1/4(log⁡r)1/8C_{b_c}(r)\asymp r^{-1/4}(\log r)^{1/8} (as recorded, e.g., by Janke for the Villain model). Rigorous work gave power-law bounds (McBryan-Spencer, Frohlich-Spencer, van Engelenburg-Lis) but not the critical exponent. Does Cbc(r)=B r−1/4(log⁡r)1/8(1+o(1))C_{b_c}(r)=B\,r^{-1/4}(\log r)^{1/8}(1+o(1)) hold with 0<B<∞0<B<\infty?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Statistical mechanics; Berezinskii-Kosterlitz-Thouless transition
Posed by
J. M. Kosterlitz (renormalization-group prediction), building on Berezinskii and Kosterlitz-Thouless
Year posed
1974
Years open
52y
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
50 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Log-correction paper, Theorem 1.1: Cbc(r)=BXYr−1/4(log⁡r)1/8(1+o(1))C_{b_c}(r)=B_{XY}r^{-1/4}(\log r)^{1/8}(1+o(1)) as r→∞r\to\infty along the axis, with BXY∈(0,∞)B_{XY}\in(0,\infty), for the free-box limit. Exponent paper (September 24): Cbc(r)=r−1/4+o(1)C_{b_c}(r)=r^{-1/4+o(1)}. The constant is not computed; off-axis directions, other lattices and the Villain model at criticality are not covered here, and the next-order log⁡log⁡r/log⁡r\log\log r/\log r correction is not claimed. Companions prove a Gaussian free field limit for discrete Gaussian heights through the roughening threshold with universal value 8π8\pi, and (conditional on stated inputs) the critical center magnetization n−1/8(log⁡n)1/16n^{-1/8}(\log n)^{1/16} and spin-field scaling limit.

What the AI did

The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, 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, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The family has six manuscripts; the September 24 paper proves the exponent r−1/4+o(1)r^{-1/4+o(1)} and the October 5 paper the logarithmic correction, building on it.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the logarithmic-correction paper and the main theorem of the September 24 exponent paper were read against the prediction. Both concern the ordinary cosine model on the square lattice, with bcb_c defined by the vanishing of the mass and the free-box thermodynamic limit taken before r→∞r\to\infty, along a coordinate axis. The log paper relies on the exponent paper (critical height coefficient 8π8\pi) and two other cited OpenAI companions for analytic maps and annular estimates. The proofs were not refereed. No Lean formalization exists for this family.

Sources

Changelog1 change

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