The double-dimer loop ensemble converges to CLE4

What OpenAI claims

Resolves the half-plane Temperleyan form of the double-dimer scaling-limit conjecture: the complete loop ensemble formed by two independent dimer coverings of the Temperleyan square lattice converges to nested CLE4. Convergence matches every macroscopic loop as an unparametrized curve, upgrading convergence of loop observables to convergence of the loops themselves.

Paper

1. The curve scaling limit of half-plane double dimers

   23 September 2026 · Preprint, not peer reviewed · PDF

   We prove that the complete double-dimer loop ensemble for the Temperleyan square lattice in the upper half-plane converges to nested CLE4. The convergence holds along the full mesh limit and matches every macroscopic loop as an unparametrized curve. This resolves the half-plane Temperleyan form of the double-dimer CLE4 scaling-limit conjecture.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.