Gwynne and Miller's graph-metric conjecture for critical FK planar maps (0 < q < 4) and spanning-tree-weighted maps, finite spherical case

Random planar maps decorated by statistical-mechanics models are expected to converge, as metric measure spaces, to Liouville quantum gravity (LQG) surfaces with the matching parameter $\gamma$ : uniform maps to the Brownian map ( $\gamma=\sqrt{8/3}$ , Le Gall, Miermont), and other ensembles to the $\gamma$ -LQG sphere equipped with the intrinsic metric constructed by Ding-Dubedat-Dunlap-Falconet, Gwynne-Miller and others. Gwynne and Miller's Conjecture 1.7 states that graph distances, suitably rescaled, converge to the $\gamma$ -LQG metric for several ensembles, explicitly including critical Fortuin-Kasteleyn (FK) maps with $q\in(0,4)$ , $q=2+2\cos(\pi\gamma^2/2)$ , $\gamma\in(\sqrt2,2)$ , and spanning-tree-weighted maps at $\gamma=\sqrt2$ . For these finite spherical ensembles, do rescaled graph distances converge in law to the $\gamma$ -LQG sphere?

Result: Proved(see note)
Status: Partial result
AI contribution: AI-discovered
Method: Argument
Field: Random planar maps; Liouville quantum gravity
Posed by: Ewain Gwynne and Jason Miller (Conjecture 1.7)
Year posed: 2019
Years open: 7y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 48 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Principal theorem: for fixed $q\in(0,4)$ (critical FK maps weighted by $q^{\ell/2}$ , $\ell$ the number of interfaces) and for uniform map-spanning-tree pairs, $n$ -edge spherical maps with degree measure and rescaled graph distance converge in GHP law to the unit-area $\gamma$ -quantum sphere with its LQG metric and area. Companions: joint conformal (flag-triangle uniformization), metric, area and CLE-interface limits for $0<q<4$ ; the critical sphere and CLE $_4$ -decorated limits at $q=4$ ; stationary random walk converging to Liouville Brownian motion with time acceleration exactly the number of edges for FK-Ising and spanning-tree maps; eigenvalue and heat-trace convergence for FK-Ising. Not shown: the other ensembles in Conjecture 1.7, infinite-volume or boundary versions, rates, or uniformity in $q$ .

What the AI did

The release README says the vast majority of results were obtained with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. The family has seven manuscripts (September 24 and October 5, 2026): metric-measure limits for $0<q<4$ and spanning-tree maps (principal here), conformal-embedding limits, the critical $q=4$ case, a continuum-random-tree limit for $q>4$ (a separate entry), and three papers on random walk, spectral and heat-trace convergence built on these geometric results.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the metric-measure manuscript was read against Gwynne-Miller Conjecture 1.7. It gives, for each fixed $q\in(0,4)$ and for spanning-tree maps, deterministic $a_n\to0$ with $(V(M_n),a_nd_n,\mu_n)\to(S,D_h,\mu_h)$ in law in the Gromov-Hausdorff-Prokhorov topology, using the degree measure. The paper itself says it does not address every ensemble or version of the conjecture, asserts no uniformity in $q$ , normalisation power law or rate, and that the uniform vertex measure would need a separate argument. The family's Lean formalization covers only the $q>4$ tree limit, not this theorem. Not refereed. The FK-Ising walk and spectral companions use stated Brownian/LQG inputs.

Gwynne and Miller's graph-metric conjecture for critical FK planar maps (0 < q < 4) and spanning-tree-weighted maps, finite spherical case

What was actually shown

What the AI did

Verification

Sources

Discussion