Localization and delocalization in the Anderson model

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Resolves the predicted spectral contrast for the lattice Anderson model with independent uniform site potentials. In dimension two, every positive disorder strength gives almost surely pure-point spectrum. In every fixed dimension d ≥ 3, sufficiently weak positive disorder gives purely absolutely continuous spectrum on a fixed open interval with nonzero spectral weight. (OpenAI’s Lean note)

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What the papers state

Theorem 1.1

In Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three

The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.

There is a fixed sufficiently small number e_2\in(0,1/100) such that, for every fixed integer d\ge3, there is a number \lambda_d>0 with the following property. Set

I_d= \begin{cases} (-6+e_2/2,-6+e_2),&d=3,\\ -2d+(1/200,1/100),&d\ge4. \end{cases}

Then for every fixed 0<\lambda<\lambda_d, almost surely

I_d\subset\sigma(H_{\lambda,\omega}),\qquad \chi_{I_d}(H_{\lambda,\omega})\ell^2(\ZZ^d) \subset\mathcal H_{\mathrm{ac}}(H_{\lambda,\omega}).

In particular, the spectral restriction to I_d is purely absolutely continuous and \chi_{I_d}(H_{\lambda,\omega})\ne0. For d\ge4, one may take \lambda_d=\sqrt3\,2^{-n_*(d)}, where the finite-matrix specification of n_*(d) is given in Definition 8.2. For d=3, the proof supplies an existential positive threshold.

No Lean statement in the release is paired with this theorem.

Corollary 1.2 (Spectral coexistence)

In Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three

Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.

For every fixed d\ge3 and every fixed 0<\lambda<\lambda_d, there is a deterministic number \delta_{d,\lambda}\in(0,\lambda/2] such that the open interval

J_{d,\lambda} =(-2d-\lambda,-2d-\lambda+\delta_{d,\lambda})

is disjoint from I_d and, almost surely, both \chi_{I_d}(H_{\lambda,\omega}) and \chi_{J_{d,\lambda}}(H_{\lambda,\omega}) are nonzero. The restriction to I_d is purely absolutely continuous, whereas the restriction to J_{d,\lambda} is pure point with exponentially decaying eigenfunctions. The probability-one event may depend on d and \lambda.

No Lean statement in the release is paired with this theorem.

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.

One more possible match

Theorem 1.1 in Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder.

Reported proof verification: One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
Papers and explanations: Two current papers, each readable here as a page beside its PDF. One explanation by others.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three

Undated, current edition.

Read it here PDF

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. The interval may depend on d but is independent of the disorder strength. This settles the purely absolutely continuous energy-range question in Simon’s Problem 1 for this model.

Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder

Undated, current edition.

Read it here PDF

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.

One Lean statement in this family

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For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.

`PlanarAndersonSpectrum`

The paper states

Theorem 1.1

In Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder

For each fixed h>0, almost surely H_v has pure-point spectral type, admits a complete orthonormal eigenbasis, and has spectrum

\sigma(H_v)=[-4-h,4+h].

Its eigenvalues are dense in this interval. The probability-one event in this statement is allowed to depend on h.

The words they share: spectrum, interval, spectral, probability, disorder, anderson, pure, point, almost.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Almost-sure spectrum of the planar Anderson operator

PlanarAndersonSpectrum.lean, the release lists it as a supporting statement.

theorem anderson_spectrum_ae {h : ℝ} (hh : 0 < h) :    ∀ᵐ v ∂disorderLaw h, ∃ H : Hilbert →L[ℂ] Hilbert,      IsAndersonOperator v H ∧ IsSelfAdjoint H ∧ spectrum ℂ H = spectralInterval h := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program

cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.

Read the complete Lean statement

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express a weaker statement than the advertised result (their term: “weaker-statement”).

Their note: anderson_spectrum_ae (hh : 0 < h) : ∀ᵐ v ∂disorderLaw h, ∃ H, IsAndersonOperator v H ∧ IsSelfAdjoint H ∧ spectrum ℂ H = spectralInterval h identifies only the spectrum [-4-h, 4+h] of the 2D nearest-neighbor Anderson operator (iid Uniform[-h,h] potentials); docs: 'It does not assert the pure-point spectral type claimed in the accompanying paper'. The summary’s headline claims (almost-sure pure-point spectrum in d = 2 at every positive disorder, and purely absolutely continuous spectrum on an open interval for weak disorder in every d >= 3) are both unstated.

“weaker-statement”, in their words: The challenge statements do not imply the headline claim: they state a nontrivially weaker statement.

Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).

The gap they name: Both spectral-type claims: pure-point spectrum in d = 2 (declared) and absolutely continuous spectrum in d ≥ 3 (undeclared). The only formal content is the almost-sure spectrum as a set in d = 2.

“NARROWER — UNDECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The linked formalization proves a supporting spectral statement for the nearest-neighbor Anderson operator on $\mathbb Z^2$. For every disorder strength $h>0$ with independent site potentials uniform on $[-h,h]$, it constructs the bounded self-adjoint operator almost surely and identifies its spectrum as the real interval $[-4-h,4+h]$.
This selected statement identifies the spectral set. It does not assert the pure-point spectral type claimed in the accompanying paper.

AI-drafted comparisons, with no recorded human review

Almost-sure spectrum of the planar Anderson operator PlanarAndersonSpectrum
Unreviewed AI draft: states part of a paper’s theorem. States only the almost-sure spectrum [−4 − h, 4 + h] of the planar Anderson operator, one clause of the main theorem that the note calls supporting. The pure-point spectral type and eigenbasis, the paper’s claim, are not asserted.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Almost-sure spectrum of the planar Anderson operator `PlanarAndersonSpectrum`

Three reports, all at the release’s recorded version.

Reported by	Their word	Their account	Date	Original report
cert-machine	“CERTIFIED”	Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.	10 October 2026	Original report
MathVet	“comparator-sandboxed-pass”	Re-run through Comparator in a sandboxed virtual machine: accepted.	8 October 2026	Original report
Neal McBurnett, openai-math-lean-tooling	“accepted”	Re-run through Comparator with Lean’s kernel: accepted (the check took 228 s).	10 October 2026	Original report

Explanations by others

Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.

Yixin Ma, OpenAI mathematical physics, 25 families: a Chinese guided reading, 7 October 2026. Source
Chinese guided reading of this family that compares the manuscripts’ main theorems with their abstracts and flags boundary conditions and limit orders that change the physical reading; drafted with Codex and not a re-proof.

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Would win a Fields Medal”. The page says that nothing on it is verified.
Author not stated. Source

JohnDriscollAcademic, Significant Number of OpenAI Results Appearing to Resolve AIM Problems (issue 36), 7 October 2026. Source
AIM problems 015 and 016 (Anderson localisation in 3D and 2D) appear resolved or overlapped by family 261.

Questions awaiting review

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Read the Lean statement against the paper

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Identify the theorem the catalogue describes

Family 261 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

Compare the proposed paper and Lean matches

Family 261 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form