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The Kosterlitz essential singularity of the XY correlation length

For the nearest-neighbor XY model on Z2\mathbb Z^2, let m(b)m(b) be the exponential decay rate of the spin correlation along an axis and ξ(b)=1/m(b)\xi(b)=1/m(b) the correlation length, finite for b<bcb<b_c. Unlike ordinary critical points, where ξ\xi diverges as a power of bc−bb_c-b, Kosterlitz's renormalization-group analysis predicts an essential singularity: log⁡ξ(b)∼A/bc−b\log\xi(b)\sim A/\sqrt{b_c-b} as b↑bcb\uparrow b_c. Rigorous work established the transition and the exponential-versus-polynomial dichotomy (Frohlich-Spencer, van Engelenburg-Lis, Aizenman-Harel-Peled-Shapiro, Lammers) but not the rate of divergence. Is there a constant A∈(0,∞)A\in(0,\infty) with bc−b log⁡ξ(b)→A\sqrt{b_c-b}\,\log\xi(b)\to A as b↑bcb\uparrow b_c?

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
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
48 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there is AXY∈(0,∞)A_{XY}\in(0,\infty) with lim⁡b↑bcbc−b log⁡ξ(b)=AXY\lim_{b\uparrow b_c}\sqrt{b_c-b}\,\log\xi(b)=A_{XY}, for the square-lattice nearest-neighbor cosine model. The constant is expressed through the block construction and is not evaluated. It does not cover other lattices, the Villain model, the behaviour of the susceptibility, or corrections beyond the leading order.

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. It builds on two companion manuscripts of the same family (the critical exponent and logarithmic-correction papers).

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Kosterlitz's prediction. It is for the ordinary cosine model on the square lattice, with ξ\xi defined from the free-box limit and then the axis separation limit, and bb the coupling in the Hamiltonian; the constant AA is identified only implicitly. The proof depends on the two companion papers for the critical inputs and on a renormalization map whose leading recursion is the Kosterlitz flow. The proof was not refereed. No Lean formalization exists for this family.

Sources

Changelog1 change

Discussion