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Conformal universality of bulk spin and energy correlations for weakly interacting square-lattice Ising models

The critical nearest-neighbor Ising model has rigorous conformally covariant scaling limits of its spin and energy correlations (Chelkak-Hongler-Izyurov, Hongler-Smirnov). Universality predicts the same limits, up to two field normalizations, for models with weak finite-range even multispin perturbations at their own critical temperature. Rigorous results covered energy correlations in the plane and on cylinders (Giuliani-Greenblatt-Mastropietro, Antinucci-Giuliani-Greenblatt) and boundary spins in the half-plane, but not bulk spins. Giuliani's ICM 2022 survey (Section 2, Open Problems 1-2) asks for mixed spin and energy limits. For weak even finite-range perturbations, do the critical mixed bulk spin and energy correlations converge, after two normalizations, to the nearest-neighbor conformal limits?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Two-dimensional statistical mechanics; Ising model; conformal invariance
Posed by
Alessandro Giuliani, Scaling limits and universality of Ising and dimer models (ICM 2022), Section 2 and Open Problems 1-2
Year posed
2022
Years open
4y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
20 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For sufficiently small square-symmetric finite-range even multispin perturbations of either sign, with one critical-temperature branch and two field normalizations, mixed bulk spin and energy correlations converge to the nearest-neighbor conformal limits in every bounded simply connected C2C^2 Jordan domain with free, plus or minus boundary conditions, and in the periodic plane state; energies are centered in their actual states. Companion: weakly perturbed contour energies have a domain-independent temperature at which the Dobrushin interface converges to chordal SLE3\mathrm{SLE}_3. Not shown: perturbations without square symmetry, multiply connected domains, boundary spin observables, or strong perturbations.

What the AI did

The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family 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 manuscripts are authored 'OpenAI' and name no human author. This entry draws on three manuscripts of September 23, 2026: the bulk correlation paper, a companion proving chordal SLE3 limits for weakly perturbed contour energies, and a technical companion on buffered comparison and stopping bands. The family's three October 5 random-bond manuscripts are a separate entry.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and main theorem statement of 'Conformal universality of bulk Ising correlations under weak interactions' were read against Giuliani's program as the manuscript summarizes it; the paper itself says it covers the square-symmetric model in smooth simply connected domains and the plane, while the broader program includes other topologies and boundary observables. Giuliani's survey was not read here. Lean challenge BufferedIsing (lean/docs/218.md) covers only a finite-graph likelihood comparison lemma from the companion 'Buffered comparison and stopping-band resolution in critical Ising'; its scope note says the stopping-band theorem is outside it. It does not state this entry's headline, so the entry is unreviewed, not Lean-checked.

Sources

Changelog1 change

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