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Math-2871 · mathematics · Probability

Conformal geometry of spherical FK maps at weight four

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For FK weight $q=4$, do canonical conformal embeddings of finite spherical n-edge maps converge jointly in area measure, the full rescaled distance graph and the complete nested loop collection to the critical $\gamma =2$ quantum sphere with its critical metric and loop decoration, for a deterministic $a_n\to 0$ and every $n\to \infty $? Include vanishing spherical flag-triangle mesh.

Short English statement

For FK weight $q=4$, do canonical conformal embeddings of finite spherical n-edge maps converge jointly in area measure, the full rescaled distance graph and the complete nested loop collection to the critical $\gamma =2$ quantum sphere with its critical metric and loop decoration, for a deterministic $a_n\to 0$ and every $n\to \infty $? Include vanishing spherical flag-triangle mesh.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2871-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2871

At $q=4$, the proposed FK-map limit is a critical unit-area quantum sphere, jointly with its intrinsic metric and nested $\mathrm{CLE}_4$ interfaces. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Conformal geometry of spherical FK maps at weight four

OpenAI's claim

OpenAI presents a proof in the selected manuscript. At $q=4$, the proposed FK-map limit is a critical unit-area quantum sphere, jointly with its intrinsic metric and nested $\mathrm{CLE}_4$ interfaces.

Mathematical statement of the claimed result

Result 1.
For rooted $n$-edge spherical FK maps at $q=4$, weight map/subset pairs by $4^{\ell/2}$, where $\ell=2k(A)+|A|-|V|$. Uniformize the equilateral flag-triangle sphere by three independent area marks sent to $0,1,\infty$; denote the embedding $\phi_n$, normalized flag area $m_n$, full embedded distance graph $K_n(a_n)$, and nested interfaces $\Gamma_n$. The source claims deterministic $a_n\to0$ with \[ ((\phi_n)_*m_n,K_n(a_n),\Gamma_n) \Longrightarrow(\mu_h^{\mathrm{crit}},\{(z,w,D_h(z,w))\},\Gamma). \] The target is its area-one critical $\gamma=2$ quantum-sphere law and nested $\operatorname{CLE}_4$. Topologies are weak area, Hausdorff distance graphs, and uniform loop matching with nesting/multiplicities. In the same joint limit, the maximum spherical diameter of any embedded flag triangle tends to zero in probability. The limiting nested whole-plane $\operatorname{CLE}_4$ is independent of the entire marked quantum surface. The distance graph is \[ K_n(a)=\{(\phi_n(u),\phi_n(v),a d_n(u,v)):u,v\in V(M_n)\}, \] where $d_n$ counts all primal map edges.

Outline of the argument

The field and area law are first obtained by taking the Liouville parameter to its critical value, with area-mark normalization and norm convergence. Critical contour sewing and local reconstruction then couple the embedded map metric, measure, and all macroscopic loop interfaces to this surface.

Formal verification

For family 211, the selected Lean theorem is the $q>4$ Brownian tree metric-measure limit, with degree measure and the $n^{-1/2}$ distance scale. The subcritical, critical-Liouville, walk, and spectral companions are not included. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. 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

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