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Math-2872 · mathematics · Probability

Conformal metric and loop convergence of FK maps below four

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each fixed FK weight $0<q<4$, do canonical flag-triangulation conformal embeddings of spherical n-edge FK maps admit deterministic distance rescalings $a_n\to 0$ so that the area measure, the entire distance graph and every macroscopic nested loop converge jointly, through all n, to the corresponding Liouville quantum sphere and CLE decoration? Require vanishing spherical flag mesh and the stated conditional independence of the loop decoration and embedded marked quantum surface.

Short English statement

For each fixed FK weight $0<q<4$, do canonical flag-triangulation conformal embeddings of spherical n-edge FK maps admit deterministic distance rescalings $a_n\to 0$ so that the area measure, the entire distance graph and every macroscopic nested loop converge jointly, through all n, to the corresponding Liouville quantum sphere and CLE decoration? Require vanishing spherical flag mesh and the stated conditional independence of the loop decoration and embedded marked quantum surface.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2872-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2872

The conformal-metric manuscript claims the full package asked for here: the flag embedding, vanishing mesh, embedded metric and area limit, the nested loop decoration and its specified conditional independence, for every fixed $0<q<4$. The metric--measure companion supplies the undecorated Gromov--Hausdorff--Prokhorov limit. That companion alone would not establish the conformal and decorated clauses, but the conformal-metric theorem does. The aggregate match is therefore full.

Conformal metric and loop convergence of FK maps below four

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: For each $0<q<4$, flag-triangle embeddings of finite FK maps are claimed to converge with area, graph metric, and all nested interfaces to a Liouville sphere with an independent conformal loop ensemble; Subcritical FK-decorated spherical maps and tree-weighted maps are claimed to have their corresponding ordinary Liouville quantum spheres as metric-measure limits through all edge counts. Together they give a claimed affirmative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
Fix $0<q<4$ and set $q=2+2\cos(\pi\gamma^2/2)$, $\kappa=16/\gamma^2$, $\sqrt2<\gamma<2$. Rooted $n$-edge FK map/subset pairs have weight $q^{\ell/2}$. Their $4n$ equilateral flag triangles carry normalized area $m_n$, and three independent area marks fix the uniformization $\phi_n$ to $0,1,\infty$. With all-edge distance $d_n$, define \[ K_n(a)=\{(\phi_n(u),\phi_n(v),a d_n(u,v)):u,v\in V(M_n)\}. \] The source claims a deterministic $a_n\to0$ giving \[ ((\phi_n)_*m_n,K_n(a_n),\Gamma_n) \Longrightarrow(\mu_h,\{(z,w,D_h(z,w))\},\Gamma), \] for the similarly marked ordinary unit-area $\gamma$-LQG sphere and a nested $\operatorname{CLE}_\kappa$ independent of the embedded surface. Area, distance graphs, and curves use weak, Hausdorff, and uniform loop-matching topologies, respectively. Maximum embedded flag-triangle diameter vanishes in probability.
Result 2.
For each fixed $q\in(0,4)$, sample rooted $n$-edge planar map/subset pairs $(M,A)$ with weight $q^{\ell(M,A)/2}$, $\ell=2k(A)+|A|-|V(M)|$. Separately, sample rooted map/spanning-tree pairs uniformly. Distances $d_n$ use every primal edge and $\mu_n(v)=\deg(v)/(2n)$. The source claims deterministic $a_n>0$, $a_n\to0$, such that \[ (V(M_n),a_nd_n,\mu_n)\Longrightarrow(S,D_h,\mu_h) \] in Gromov--Hausdorff--Prokhorov law, through all integer sizes. The limiting sphere has the ordinary LQG law normalized to area one: $q=2+2\cos(\pi\gamma^2/2)$, $\sqrt2<\gamma<2$, for FK, and $\gamma=\sqrt2$ for spanning trees. Root and decoration are forgotten.

Outline of the argument

Contour sewing puts the discrete flags on a common topological sphere. Protected finite visit records retain exact primal and dual incidences, resampling identifies their conditional laws, and local extremal-length comparisons upgrade the topological coupling to canonical conformal and metric convergence.

Finite contour records retain every visit and edge incidence needed to read graph distances. Conditional resampling identifies local limiting laws, and metric tightness plus Liouville uniqueness identifies the globally coupled degree measure and intrinsic distance after deterministic normalization.

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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