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Math-2868-A1 · revision r1 · on Math-2868

OpenAI claim summary for Math-2868

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. The nested CLE limit of half-plane double-dimer loops
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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