---
id: "2026-10-06-openai-anderson-localization-2d-delocalization-3d"
url: "https://postcutoff.com/e/2026-10-06-openai-anderson-localization-2d-delocalization-3d/"
as_of: "2026-10-30T13:42:00+01:00"
date: "2026-10-06"
date_precision: day
category: science
importance: 5
confidence: medium
status: [Awaiting review]
verification: "Lean proof, not yet reviewed"
sources: 6
editor: human editor
human_review: null
version: "2026-10-29"
---

As of: 2026-10-30 13:42 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-06-openai-anderson-localization-2d-delocalization-3d/

# OpenAI claims Anderson localization in 2D and extended states in 3D

Full title: OpenAI's release claims both halves of the Anderson localization picture: localization in 2D at every disorder and extended states in 3D and up at weak disorder (Simon's Problems 1 and 2)

Family 261: "Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder" and "Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three" (23 Sept 2026, released 6 Oct), both for independent uniform site potentials. Known before: localization at strong disorder or band edges in any dimension, localization at all disorder in one dimension, and extended states on tree (Bethe) lattices, but no extended states on ℤ^d and no full 2D localization. Vojkan Jakšić and Yoram Last (arXiv 2610.11611, appendix of 7 Oct) give a shorter passage from the 3D paper's boundary-positivity statement to purity. Yao Xu's GitHub project anderson-ac-lean4 (14–15 Oct) states a sorry-free Lean proof of the 3D theorem. OpenAI's Lean scope note covers only the 2D spectrum as a set, "not… the pure-point spectral type".

## Key facts
- 2D (abstract, verbatim): 'For each fixed positive disorder strength, we prove that the nearest-neighbor Anderson operator on the square lattice with independent uniform site potentials almost surely has pure-point spectral type throughout its spectrum. This resolves the pure-point assertion of the two-dimensional Anderson localization conjecture for the uniform single-site law.'
- 3D and up (verbatim): 'For each fixed dimension d ≥ 3 and each fixed sufficiently small positive disorder strength, we prove that the Anderson operator on ℤ^d with independent uniform site potentials almost surely has purely absolutely continuous spectrum with nonzero weight on an open energy interval… This settles the purely absolutely continuous energy-range question in Simon's Problem 1 for this model.'
- Context: Anderson's 1958 model of electrons in a disordered crystal; physics (scaling theory, 1979) predicts localization at any disorder in 2D and a mobility edge in 3D. Rigorous results covered strong disorder and band edges (Fröhlich–Spencer, Aizenman–Molchanov), all disorder in 1D, and weak disorder on the Bethe lattice (Klein), but not the two central cases, which Barry Simon listed as Problems 1 and 2 in 2000
- Expert engagement: Jakšić (Politecnico di Milano) and Last (Hebrew University), 'Extended States on the Bethe Lattice Revisited' (arXiv 2610.11611, 8 Oct). Appendix B, 'written on October 7, 2026… one day after OpenAI publicly released the preprint [10], claiming to prove purely absolutely continuous spectrum', shows that their structural argument gives purity directly from the preprint's boundary-positivity estimate (its Eq. 177)
- Third-party Lean: cadamcat/anderson-ac-lean4 (Yao Xu, 'Developed with AI assistance'): 'The final theorem uses only the axioms propext, Classical.choice and Quot.sound. No proof in the repository uses sorry or an added axiom.' The statement file imports only Mathlib, with a non-vacuity check; not independently reviewed by us
- OpenAI's Lean scope note 261 (verbatim): the formalization 'identifies its spectrum as the real interval [−4−h, 4+h]… It does not assert the pure-point spectral type claimed in the accompanying paper.' The challenge PlanarAndersonSpectrum was accepted in oai-math-revised's Comparator run

## What happened

In 1958 Philip Anderson showed that disorder can trap electrons: in a sufficiently random crystal, wave functions stay localized
instead of spreading. Physics predicts that in two dimensions any amount of disorder localizes every state, while in three dimensions
weak disorder leaves a band of extended, conducting states. Mathematicians proved localization at strong disorder, at band edges and in
one dimension, and extended states on tree-like lattices. The two central cases on the ordinary lattice stayed open. Barry Simon put
them first and second on his 2000 list of open problems for Schrödinger operators.

OpenAI's release claims both, for the standard model with uniformly distributed random potentials. Two days after the release, Vojkan
Jakšić and Yoram Last, who had just written a new proof of the tree-lattice result, added an appendix showing that their method shortens
the 3D paper's final step from a boundary estimate to pure absolute continuity. They describe the preprint as "claiming to prove" the
result and do not assess the rest of it. A separate developer, Yao Xu, has published a Lean 4 formalization of the 3D theorem that he
says uses no unproved steps. OpenAI's own formal proof for the family covers only the location of the 2D spectrum.

## Why it matters

Anderson localization underlies the physics of disordered metals and insulators, and these two statements are its best-known open
mathematical problems. If correct, the release settles both for the uniform model. The 3D claim has more support (an expert appendix
and a third-party formal proof). The 2D localization claim rests on the manuscript alone, and we found no public assessment by
specialists as of 29 Oct.

## What is disputed or not yet verified

- Verification: Unrefereed manuscripts; 3D theorem formalized in a third-party Lean project (statement fidelity unreviewed); 2D pure-point claim unformalized

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 159 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 98 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 189 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 128 days after its cutoff

## Sources

1. [OpenAI: Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder (PDF)](https://github.com/openai/math/blob/main/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/paper.pdf) (github.com, paper)
2. [OpenAI: Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three (PDF)](https://github.com/openai/math/blob/main/preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026/paper.pdf) (github.com, paper)
3. [Jakšić, Last: Extended States on the Bethe Lattice Revisited (arXiv 2610.11611, Appendix B on the OpenAI preprint)](https://arxiv.org/abs/2610.11611) (arxiv.org, paper)
4. [cadamcat/anderson-ac-lean4: Lean 4 formalization of the 3D theorem (Yao Xu)](https://github.com/cadamcat/anderson-ac-lean4) (github.com, code)
5. [OpenAI Lean scope note 261 (2D spectrum only)](https://github.com/openai/math/blob/main/lean/docs/261.md) (github.com, code)
6. [oai-math-revised: community notes on the 3D manuscript](https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026/AGENT_NOTES.md) (github.com, discussion)

## Changes

- 2026-10-29 (filed): Created (families 251–378; both abstracts, Lean scope note, Jakšić–Last paper and appendix read, anderson-ac-lean4 README, oai-math-revised notes)

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