A deterministic linear walk clock on tree-weighted planar maps

A deterministic linear walk clock on tree-weighted planar maps

OpenAI's claim

OpenAI presents a proof in the selected manuscript. On tree-weighted planar maps, the stationary walk is claimed to converge to Brownian motion of a $\sqrt2$-Liouville sphere with exactly the edge count as its time factor.

Mathematical statement of the claimed result

Result 1.

Choose $(M_n,T_n)$ uniformly from rooted planar maps with $n$ edges and a distinguished spanning tree. Loops and multiple edges are permitted. On its vertex set $V_n$, let $d_n$ be graph distance and $\mu_n=\sum_v\deg(v)/(2n)\,\delta_v$, counting a loop twice in the degree. Start a rate-one walk $X^n$ from $\mu_n$ and choose each incident half-edge uniformly at a jump. Its generator is \[ L_nf(v)=\deg(v)^{-1}\sum_{e\text{ incident half-edge}}[f(\operatorname{other}(e))-f(v)]. \] Let $(S,D_h,\mu_h)$ have the ordinary Liouville quantum sphere law at parameter $\sqrt2$, normalized to area one. Its stationary Liouville Brownian motion $B^h$ has closed form $\mathcal E_h(f,g)=\frac12\int_{S^2}\langle\nabla f,\nabla g\rangle\,d\operatorname{vol}$ on $L^2(\mu_h)$. For a random real $Y$, define $m(Y)$ by $\mathbb E\arctan(Y-m(Y))=0$, and set \[ a_n=\exp\{m(\log\operatorname{diam}(S,D_h))-m(\log\max(1,\operatorname{diam}(V_n,d_n)))\}. \] With $Q_n^T=\operatorname{Law}((X^n_{nt})_{0\le t\le T}\mid M_n,T_n)$ and $Q_h^T=\operatorname{Law}((B_t^h)_{0\le t\le T}\mid h)$, \[ (V_n,a_nd_n,\mu_n,Q_n^T)\Longrightarrow(S,D_h,\mu_h,Q_h^T)\quad(T>0). \] The topology embeds both spaces into a common metric space and combines Hausdorff distance, Prokhorov distance of the area measures, and Prokhorov distance of the path laws in Skorokhod $J_1$ topology. The geometric metric-measure and contour reconstruction results are inputs; the walk clock is exactly $nt$, with multiplier one.

Outline of the argument

Contour reconstruction is extended to constrained energy minima and extremal lengths. A primal--dual obstruction prevents annular energy degeneracy even without face-shape control; energy convergence and cutoff estimates then retain the conditional path law and determine the clock constant.

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, A Linear Clock for Random Walk on Tree-Weighted Planar Maps, 2026-10-05; citation key OAI:A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 211: Lean scope.
• OpenAI catalogue entry.