OpenAI claim summary for Math-2871

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

• OpenAI, The critical Liouville quantum sphere and geometric limits of FK maps at q=4, 2026-09-24; citation key OAI:The-critical-Liouville-quantum-sphere-and-geometric-limits-of-FK-maps-at-q-equals-4-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 211: Lean scope.
• OpenAI catalogue entry.