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Simon's Problem 2: pure-point spectrum for the two-dimensional Anderson model at every disorder

Let H=Δ+vH=\Delta+v on ℓ2(Z2)\ell^2(\mathbb{Z}^2), with nearest-neighbor hopping and independent site potentials vxv_x uniform on [−h,h][-h,h]. The scaling theory of Abrahams, Anderson, Licciardello and Ramakrishnan (1979) predicts no metallic behavior in two dimensions, at any disorder. Rigorous localization was known only at large disorder or at extreme energies (Frohlich-Spencer, Frohlich-Martinelli-Scoppola-Spencer, Aizenman-Molchanov) and near the band edge (Ding-Smart). Simon's Problem 2 asks: for every h>0h>0, does HH almost surely have pure-point spectrum throughout its spectrum?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Random Schrodinger operators; Anderson localization
Posed by
Barry Simon (Problem 2 in 'Schrodinger operators in the twenty-first century'), formalizing the two-dimensional localization conjecture of Abrahams, Anderson, Licciardello and Ramakrishnan
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
55 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For each fixed h>0h>0, the operator Hv=Δ+vH_v=\Delta+v on ℓ2(Z2)\ell^2(\mathbb{Z}^2) with iid vxv_x uniform on [−h,h][-h,h] almost surely has pure-point spectral type with a complete orthonormal eigenbasis, spectrum [−4−h,4+h][-4-h,4+h] and dense eigenvalues. The null set may depend on hh. Not shown: one event for all hh, bounds uniform in energy, exponential decay of eigenfunctions, dynamical localization, or other single-site laws such as Bernoulli.

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 2 as the manuscript quotes it; Simon's text itself was not opened. The proof was not refereed. Lean: lean/docs/261.md describes a comparator challenge (PlanarAndersonSpectrum.lean) that states only that the spectrum is [-4-h, 4+h], explicitly not the pure-point claim; it is not in formalization.yaml and its solution module is absent at the pinned commit, so it counts for nothing. Scope stated by the paper: uniform single-site law only; no dynamical localization or eigenfunction decay is claimed.

Sources

Changelog1 change

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