VibeMathedMath problems solved with AI
Sign in

Simon's Problem 1: absolutely continuous spectrum for the weakly disordered Anderson model in dimension three and higher

Let Hλ=−Δ+λVH_\lambda=-\Delta+\lambda V on ℓ2(Zd)\ell^2(\mathbb{Z}^d) with iid site potentials VxV_x uniform on [−1,1][-1,1]. Anderson (1958) predicted localization at strong disorder; physics predicts that for d≥3d\ge 3 and weak disorder extended states survive. Absolutely continuous spectrum was proved on the Bethe lattice and on trees (Klein, Aizenman-Sims-Warzel, Aizenman-Warzel), but never on Zd\mathbb{Z}^d. Simon's Problem 1 asks: for d≥3d\ge 3 and suitable small disorder, does HλH_\lambda almost surely have an energy range of purely absolutely continuous spectrum?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Random Schrodinger operators; delocalization
Posed by
Barry Simon (Problem 1 in 'Schrodinger operators in the twenty-first century'), formalizing the extended-states prediction going back to Anderson
Year posed
2000
Years open
26y
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
65 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For each d≥3d\ge 3 there is λd>0\lambda_d>0 such that for every 0<λ<λd0<\lambda<\lambda_d, almost surely a fixed open interval IdI_d near the bottom band edge (for d≥4d\ge 4, −2d+(1/200,1/100)-2d+(1/200,1/100)) lies in the spectrum and the spectral restriction to IdI_d is purely absolutely continuous and nonzero. IdI_d does not depend on λ\lambda. Not shown: a.c. spectrum in the bulk of the band, the location of a mobility edge, quantum diffusion, uniformity in dd, or other laws.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. 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). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: the main theorem was read against Simon's Problem 1 as the manuscript paraphrases it; Simon's text itself was not opened. The proof was not refereed. No Lean formalization exists for this manuscript. The paper itself says it does not locate the mobility edge and does not resolve Simon's diffusion question (Problem 3), and that for d = 3 the disorder threshold is only existential.

Sources

Changelog1 change

Discussion