The nested CLE limit of half-plane double-dimer loops
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Statement recorded in the supplied source.
For the half-plane Temperleyan double-dimer ensemble, do all macroscopic loops converge as curves along the full mesh limit to nested $CLE_4$ in the upper half-plane, in the specified loop-collection matching topology?Short English statement
For the half-plane Temperleyan double-dimer ensemble, do all macroscopic loops converge as curves along the full mesh limit to nested $CLE_4$ in the upper half-plane, in the specified loop-collection matching topology?Sources
- S1 Original question source or mathematical context. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026/build/sections/10-introduction.tex.
- S2 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
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OpenAI claim summary for Math-2868
All macroscopic half-plane Temperleyan double-dimer loops are claimed to converge to the nested $\mathrm{CLE}_4$ ensemble along the full mesh limit. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.
The nested CLE limit of half-plane double-dimer loops
OpenAI's claim
OpenAI presents a proof in the selected manuscript. All macroscopic half-plane Temperleyan double-dimer loops are claimed to converge to the nested $\mathrm{CLE}_4$ ensemble along the full mesh limit.Mathematical statement of the claimed result
Result 1.
On the upper-half-plane square lattice, use the Temperleyan matching law obtained by exhausting with coarse-grid simple polygons, deleting a boundary root escaping to infinity, and taking local limits of uniform perfect matchings. Superpose two independent samples and remove doubled edges, leaving loops $\mathcal L_\delta$. For $F(z)=(z-i)/(z+i)$, the source claims tightness and \[ F(\mathcal L_\delta)\Longrightarrow F(\mathcal L), \] where $\mathcal L$ is nested $\operatorname{CLE}_4$ in the upper half-plane, normalized by Brownian-loop-soup central charge one. Convergence matches all loops above every positive diameter threshold in uniform curve distance modulo circle reparametrizations, retaining multiplicity and nesting, along the full mesh limit.Outline of the argument
Rectangle transfer is expressed in particle sectors and controlled by singular modes with marked boundary caps. Traversal and short-segment bounds provide curve tightness; the resulting geometric control upgrades the dimer field information to matching of every macroscopic loop.Formal verification
Family 226 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.Status caveat
OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.Sources
- OpenAI, The Curve Scaling Limit of Half-Plane Double Dimers, 2026-09-23; citation key
OAI:The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - OpenAI catalogue entry.