Absolutely continuous spectrum of the Anderson model in dimension at least three
From OpenAI Aftermath
Absolutely continuous spectrum of the Anderson model in dimension at least three concerns the weak-disorder Anderson operator on the integer lattice Zd, d ≥ 3, and Simon's Problem 1, which asks for a purely absolutely continuous energy range, a mathematical form of extended states. A manuscript dated 23 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims such a range for independent uniform site potentials at every sufficiently small fixed disorder in each fixed dimension. The collection has no Lean formalization of this manuscript. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- Anderson's model explains how spatial disorder can suppress quantum transport; the competing weak-disorder question is whether extended states survive on a lattice of dimension at least three. Fröhlich and Spencer, Fröhlich, Martinelli, Scoppola and Spencer, and Aizenman and Molchanov developed localization in strong-disorder and extreme-energy regimes.
- On the Bethe lattice, Klein proved purely absolutely continuous spectrum inside the free band at weak disorder, Aizenman, Sims and Warzel proved stability of the absolutely continuous component, Aizenman and Warzel found resonant delocalization, and Aggarwal and Lopatto obtained mobility edges on high-degree trees. Jakšić, Last and Warzel gave a conditional lattice criterion, and Erdős, Salmhofer and Yau proved a weak-coupling diffusion limit for the lattice model.
The original result
- Claim. Theorem 1.1 of Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three states that for every fixed d ≥ 3 there is λd > 0 such that, for every fixed coupling 0 < λ < λd, the operator with independent site potentials uniform on [−λ, λ] almost surely has the spectral restriction to a fixed energy interval Id purely absolutely continuous and nonzero. The interval lies just inside the lower edge −2d of the free spectrum and does not depend on λ; for d ≥ 4 the threshold is given through a finite-matrix specification, and for d = 3 its existence is proved without a numerical value. Corollary 1.2 combines the result with edge localization at the same coupling.
- The manuscript states that the result supplies the purely absolutely continuous energy range of Simon's Problem 1 for each fixed dimension, and that it does not locate the full mobility edge or resolve the diffusion question of Problem 3.
- Sources. The manuscript Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three at commit
1581d4c. - Formalization. The repository's scope note for this group of manuscripts concerns only a statement about the two-dimensional operator (see Pure-point spectrum of the two-dimensional Anderson model); this manuscript has no Lean formalization in the collection.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 19–20 October 2026) |
|---|---|---|---|---|
| 2026-10-08 | V. Jakšić, Y. Last, Extended States on the Bethe Lattice Revisited (arXiv:2610.11611, v1) | Remark | A short proof that the Anderson model on the Bethe lattice has purely absolutely continuous spectrum at weak disorder on compact intervals inside the free spectrum. The abstract observes that the same structural argument shortens the final passage from boundary positivity to purity in the OpenAI manuscript on Zd, d ≥ 3. | arXiv v1. |
| 2026-10-12 | P. Lopatto, Mobility Edge for Sparse Complex Gaussian Matrices (Hexagon 2610.00168v1) | Remark (citation) | In its discussion of recent advances, the paper cites both manuscripts. It describes them as preprints produced by an internal OpenAI large language model that announce two results: purely absolutely continuous spectrum on a nonempty energy interval at weak disorder on Zd for d ≥ 3, and pure-point spectrum throughout the spectrum at every positive disorder on Z2. The paper's own results concern a mobility edge for sparse Hermitian random matrices on Erdős–Rényi graphs. | Hexagon (version 1). |
| 2026-10-14 | Yao Xu, cadamcat/anderson-ac-lean4 (GitHub; first recorded at commit 38116e1, now 2724759) |
Formalization (Lean) | A Lean 4 formalization of the dimension-at-least-three result, stated with Mathlib definitions only: for every d ≥ 3 there is a width bound such that, for independent site potentials uniform on any interval shorter than that bound, almost surely an open energy interval lies in the spectrum, every spectral measure is absolutely continuous on it, and some vector has nonzero spectral weight in it. The README states that the final theorem uses only the standard axioms and that the work was developed with AI assistance. Later commits add a verification script and a non-vacuity check: independent uniform families exist for every interval, so for d ≥ 3 the theorem yields an operator with a spectral measure of nonzero mass on the energy interval. The README states that the recorded outputs, with Comparator and kernel replay, are in the repository's verification notes. | Author-reported. |
| 2026-10-16 | J. Jardine, Lattice24, referee review rank_085 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap. The reviewer lists the parts it did not fully verify: the long increment estimates of Section 4, Sections 5 and 7, and the end of Section 6. It states that a deeper audit should look there. | Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The review is published under CC BY 4.0. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
- Until 20 October 2026 these follow-ups were listed on Localization and delocalization in the Anderson model, which now gives an overview of both results.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
