The Alcalde Lopez-Heeney-Lis conjecture: joint scaling limit of critical Ashkin-Teller heights and current clusters
On the critical line of the Ashkin-Teller model (, , ), Lis's representation gives a height function and primal and dual current-cluster collections. At the Ising point (double random currents) the height converges to a Gaussian free field and the clusters to two-valued local sets of that field. Alcalde Lopez, Heeney and Lis proved the joint limit of the height and both complete cluster collections in Jordan domains at that point and conjectured (Conjecture 1.2) the critical-line extension with replaced by , . Does this joint scaling limit hold at every point of the critical line, including the four-state Potts endpoint?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Statistical mechanics; conformally invariant scaling limits
- Posed by
- Tomas Alcalde Lopez, Lorca Heeney and Marcin Lis (arXiv 2602.05886, 2026, Conjecture 1.2)
- Year posed
- 2026
- Years open
- 0y
- Solved
- 2026-09-25
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 16 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
For every fixed point of the critical Ashkin-Teller line, including the four-state Potts endpoint, every bounded Jordan domain and every admissible polygonal approximation, the height, the wired and free current-cluster collections and the distinguished boundary cluster converge jointly: the height to a GFF with the predicted coupling constant , and the clusters to canonical recursive two-valued local sets of the same field, at all nesting depths, in positive-diameter Hausdorff matching topology. Not claimed: cluster area measures, magnetization fields, all spin-domain-wall curves, or metric exponents.
What the AI did
The release README says all results were produced by an unreleased internal OpenAI model using one fixed procedure, about three hours of ChatGPT Pro thinking compute per result on average. 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.
Verification
No independent mathematician has checked this yet. Checked here: the introduction and the main theorem ('Joint critical-line limit', Section 2) were read against Conjecture 1.2 as the paper describes it; the conjecture's source was not opened. The family has no Lean formalization (lean/docs/233.md does not exist at the pinned commit). The result is for bounded Jordan domains with admissible polygonal approximations and wired primal / free dual boundary conditions. It uses six-vertex inputs from Duminil-Copin, Kozlowski, Lammers and Manolescu (2026) and cites another release preprint on the balanced six-vertex GFF limit only as a comparison.