Sheffield's finite-volume continuum random tree prediction for critical FK planar maps with q > 4

A critical Fortuin-Kasteleyn planar map with $n$ edges is a rooted planar map $M$ with an edge subset $A$ , sampled with weight $q^{\ell(M,A)/2}$ , where $\ell$ counts primal-dual interfaces. Sheffield's inventory bijection (hamburgers and cheeseburgers) shows that the encoding walk has two-dimensional Brownian fluctuations for $q<4$ and one-dimensional fluctuations for $q>4$ . In the appendix of his paper Sheffield predicted that above $q=4$ the maps lose their surface geometry and become tree-like; Feng (2026) restated this as Conjecture 1.2 and proved an infinite-volume local version. For fixed $q>4$ , do finite FK maps with graph distances rescaled by a constant times $n^{-1/2}$ converge to the Brownian continuum random tree in the Gromov-Hausdorff-Prokhorov topology?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Random planar maps; scaling limits
Posed by: Scott Sheffield (appendix of 'Quantum gravity and inventory accumulation'); restated by Yuyang Feng as Conjecture 1.2
Year posed: 2016
Years open: 10y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 28 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every fixed $q>4$ there is $c_q>0$ such that the $n$ -edge critical FK map, with graph distance scaled by $c_qn^{-1/2}$ and the normalised degree measure, converges in distribution to the Brownian continuum random tree in the GHP topology, through all positive integers $n$ . It is proved directly under the finite law (block decomposition into a simply generated tree with finite offspring variance), not deduced from Feng's infinite-volume local limit. Not shown: the value of $c_q$ , rates, uniformity in $q$ , or anything at $q=4$ (treated in a companion).

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. This entry's principal manuscript (September 24, 2026) is one of seven in the family; it has a Lean formalization of its main theorem. The other manuscripts treat $q\le4$ and are listed under the Gwynne-Miller entry.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Sheffield's prediction as restated by Feng. Lean: lean/ComparatorChallenges/FKCRT.json exists with solution module OAI.Probability.FKMaps.Convergence present at the pinned commit; it is not listed in lean/formalization.yaml. Its statement OAI.FKCRT.finite_fk_maps_converge_to_brownian_crt was read here: for every $q>4$ there is $c>0$ such that, for every bounded GHP-continuous test function, the exact finite FK expectation with distances scaled by $c/\sqrt{n+1}$ and degree measure converges to the expectation under the tree coded by a normalised Brownian excursion. The FK weight in the statement matches $q^{\ell/2}$ . This is the headline claim. Not rebuilt here.

Sheffield's finite-volume continuum random tree prediction for critical FK planar maps with q > 4

What was actually shown

What the AI did

Verification

Sources

Discussion