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Math-2870-A1 · revision r1 · on Math-2870

OpenAI claim summary for Math-2870

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.

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. 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. Metric–measure universality of subcritical FK and tree-weighted maps
  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

Metric–measure universality of subcritical FK and tree-weighted maps

OpenAI's claim

OpenAI presents a proof in the selected manuscript. 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.

Mathematical statement of the claimed result

Result 1.
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

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