GOE bulk universality for random regular graphs of fixed degree
Let be the adjacency matrix of a uniform random -regular graph on vertices with fixed. Its eigenvalue density tends to the Kesten-McKay law, and Jakobson, Miller, Rivin and Rudnick predicted, with numerical evidence, that the local eigenvalue statistics in the bulk are those of the Gaussian Orthogonal Ensemble. Bulk universality was proved for growing degrees (Bauerschmidt-Huang-Knowles-Yau; Bourgade-Huang for ), and edge universality at fixed degree (Huang-McKenzie-Yau). Bourgade and Huang state the fixed-degree case as Conjecture 2.11. For fixed and fixed bulk energy, does the unfolded eigenvalue point process converge to the GOE bulk process?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Random matrix theory; spectra of sparse random graphs
- Posed by
- Dmitry Jakobson, Stephen D. Miller, Igor Rivin and Zeev Rudnick, Eigenvalue spacings for regular graphs, IMA Vol. 109 (1999); stated as Conjecture 2.11 by Bourgade and Huang (arXiv:2607.07617, 2026)
- Year posed
- 1999
- Years open
- 27y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 40 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims fixed-energy GOE bulk universality of the full microscopic eigenvalue point process for uniform random -regular graphs, every fixed including cubic graphs, at every energy in the open Kesten-McKay bulk. The companion extends this to sufficiently weak fixed iid uniform diagonal disorder on compact subintervals of the clean band. It does NOT address the spectral edges (already known), energy-averaged spacing statistics in the original JMRR form beyond what the point-process limit implies, or strong disorder.
What the AI did
The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. The principal manuscript (23 September) proves the clean fixed-degree case; a companion (5 October) adapts its entropy and Gaussian insertion arguments to weak fixed Anderson disorder. No Lean formalization.
Verification
No independent mathematician has checked this yet. Theorem 1.1 was read against the conjecture: for every fixed and fixed in the open Kesten-McKay bulk, Laplace functionals of the unfolded point process of a uniform simple labelled -regular graph converge to those of the GOE bulk process, along all admissible with no conditioning. The paper says this proves Bourgade-Huang Conjecture 2.11 in a fixed-energy Laplace-functional form; the 1999 prediction was phrased via empirical nearest-neighbor spacings. Uses Huang-Yau rigidity and delocalization as input. No Lean formalization.