The double-dimer CLE4 scaling-limit conjecture in the half-plane Temperleyan setting
Superposing two independent uniform dimer coverings of a planar lattice graph produces doubled edges and a collection of loops. Kenyon proved that dimer height fluctuations converge to the Gaussian free field, Kenyon and Wilson matched boundary pairing probabilities with GFF contour lines, and Kenyon, Dubédat, Basok-Chelkak and Bai-Wan identified topological loop observables with those of nested . The double-dimer conjecture asks that the loops themselves, as curves, converge to nested ; topological observables alone do not exclude thin excursions or retraced arcs. For the standard Temperleyan dimer law on the square lattice in the upper half-plane, does the full double-dimer loop ensemble converge to nested in the topology of loop collections, matching every macroscopic loop as a curve?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Dimer model; conformal loop ensembles
- Posed by
- Kenyon and collaborators (conjecture motivated by Kenyon's GFF limit and Kenyon-Wilson); the manuscript cites no single posing source
- Year posed
- —
- Years open
- —
- 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
- 28 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for two independent dimer coverings of the Temperleyan square lattice in the upper half-plane with the standard infinite-volume law, the loop collection , mapped to the disk, is tight and converges in law as to with nested , matching every macroscopic loop as an unparametrized curve with all nesting generations. New inputs: a bound on disjoint traversals of a slab and control of signed crossing counts, from a column transfer matrix. Not shown: bounded or general simply connected domains, non-Temperleyan boundary conditions, or other lattices.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, 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 manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript, 'The curve scaling limit of half-plane double dimers' (September 23, 2026).
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture: for the half-plane Temperleyan law the compactified loop ensembles are tight and converge to standard nested in a loop-collection topology defined up to reparametrization, along the full mesh limit. This is one domain and one boundary condition, which the paper itself calls the half-plane Temperleyan form, so the entry is partial. There is no Lean formalization. The proof imports topological convergence from Basok and Izyurov's 2025 preprint (Theorem 4.2(2)) and adds transfer-matrix estimates for slab traversals and signed crossings; none of this was refereed here.