Band-matrix QUE and critical crossover, 1.2M lines of Lean by AxiomProver
Awaiting review
Importance: major (4 of 5)The takeaway
Random band matrices are the standard model for the Anderson localization transition. A Chicago–Axiom Math preprint proves delocalization (QUE) right down to W ≫ √N and the predicted critical law at W² ≍ N, and says AxiomProver formalized all of it in Lean, including the published results it relies on.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Lean-verified
- Importance
- Major (4 of 5)
- Last verified
- 13 November 2026
Your AI and this story
- GPT-6 Astra195 days after its cutoff
- Claude Opus 5.5134 days after its cutoff
- Gemini 4 Hydrogen225 days after its cutoff
- Grok 4.7164 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 134 days before it.
Key facts
- Conjecture (Fyodorov–Mirlin 1991, numerics of Casati et al. 1990): in 1-d, bulk eigenvectors have localization length ~W², so delocalization and GUE statistics for W² ≫ N, localization and Poisson for W² ≪ N, and a one-parameter family depending only on N/W² at the threshold
- Theorem (QUE): for the periodic block model of Yau and Yin, every bulk eigenvector at every deterministic rank spreads evenly whenever W²/N → ∞, ‘with no power or logarithmic separation between W² and N’
- Theorem (critical crossover): at N/W² → τ ∈ (0,∞), rescaled bulk eigenvalues and block-mass profiles converge jointly to a unique law encoded by a random entire function whose law is determined by the DMPK diffusion of the unitary class; profile variance is strictly positive; τ → 0 gives the sine process and τ → ∞ a Poisson process
- Lean certificate (Appendix A, verbatim): ‘Theorems 1.1, 1.3 and 1.4 have been formalized in Lean 4 using AxiomProver … The proofs assume no unproved hypothesis, and the only axioms used are the three standard axioms … about 1.0 million lines of Lean … in about 0.2 million further lines, the results from other papers that the proofs use, so that none of them is assumed’ — including Peled–Schenker–Shamis–Sodin, Cipolloni–Erdős–Henheik local laws and ETH, the Erdős–Riabov band local law, Johansson’s double-contour formula and Prékopa’s theorem
- AI statement (verbatim): ‘The results of this paper were obtained through a human–AI collaboration using AxiomProver and publicly available AI models … The core ideas behind the overall architecture of the results were provided by the authors. The writing of the proofs in this manuscript were drafted with LLMs and carefully edited by the authors … The correctness of Theorems 1.1, 1.3 and 1.4 is certified by the Lean formalization.’ Remy’s research was supported by Axiom Math
- Length: about 298 pages before the appendix; repository AxiomMath/BandMatrixQUE created 9 Nov
- Context: Manuel Lopatto’s two Hexagon volumes on a critical random band matrix, written with GPT-6 Astra, GPT-6.1 Sol and Claude Opus 5.5, were deposited on 12 Nov (see AI-assisted results, 12 Nov: Hexagon)
- Status: arXiv preprint; Lean-certified per the authors (statements to be checked against the paper by readers); no expert reaction found by 13 Nov
What happened
On 11 Nov 2026 Promit Ghosal (University of Chicago) and Guillaume Remy (Axiom Math) posted a long preprint on one-dimensional Gaussian random band matrices, the main matrix model for Anderson’s localization transition. They prove that eigenvectors are fully delocalized as soon as the band width W satisfies W² ≫ N, and that at the threshold W² ≍ N the joint law of eigenvalues and eigenvector profiles is a one-parameter family interpolating between the delocalized and localized phases, as physicists predicted in 1991. The paper says its main theorems, and the published results it uses, were formalized in Lean 4 by Axiom Math’s AxiomProver, about 1.2 million lines in all (repository AxiomMath/BandMatrixQUE).
Why it matters
The critical regime of band matrices has been a target of mathematical physics for decades; recent work by Yau, Yin and others reached delocalization near the threshold. The formalization is unusual in scope: it re-proves the recent local laws and other inputs instead of assuming them. The paper is an unrefereed preprint, and the AI disclosure says the written proofs were drafted with language models.
What is disputed or not yet verified
| Verification | Formal proof in Lean 4 (AxiomProver, ~1.2M lines including cited results); preprint |
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Sources
2 sources from 2 sites. Numbers match the chips in the text.
2 sources: 2 primary
Primary
Changes
- Filed (science agent; arXiv PDF and repository listing read)