OpenAI reports Gaussian free-field limits across the balanced six-vertex range

For the balanced-torus plane state of the square-lattice six-vertex model with a=b=1 and 0<c≤2, its unit-jump height field converges modulo constants to a Gaussian free field whose squared multiplier is 1/arcsin(c/2) under the Green-kernel normalization −(2π)^(-1)log|x−y|.

Sources for this record

What was already known

Coulomb-gas calculations predicted the Gaussian coupling, and dimer methods treated the free-fermion point. Prior rigorous work established delocalization, logarithmic fluctuations, or near-free-fermion covariance without a full GFF law throughout this range.

AI involvement AI-led

OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.

OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.

Importance Major

Proves the field scaling limit and its coupling constant across the full stated critical six-vertex range.

Limitations

The plane state is the prescribed iterated balanced-torus limit; the field is defined modulo additive constants, and the result is not claimed for arbitrary six-vertex weights or boundary states. No completed proof check was documented in the available sources.

Technical verification Reported

Reported is the default when evidence for a higher level is insufficient. The OpenAI-authored manuscript reports and argues the result, but the reviewed sources do not document a completed check of the full assessed claim.

Human understanding Few

Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.

Correction and publication history

2026-10-18 · publication

Primary manuscript, attribution, and prior-work review for non-notifying local inclusion; technical check remains undocumented.

Read this published revision