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

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 $q$ .

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

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

Confidence: Not scored

Registry verification: unreviewed · announcement · partial

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

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

specialist registry · passed

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.

Source ↗

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

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Companion: Canonical conformal limits of subcritical FK planar maps

paper · pending