Banach's simple Lebesgue-spectrum problem
From OpenAI Aftermath
Banach's simple Lebesgue-spectrum problem asks, in its probability-preserving form, for an invertible measure-preserving transformation whose Koopman operator, on the whole space of square-integrable functions of mean zero, is unitarily equivalent to multiplication by the coordinate on the circle; equivalently, the iterates of a single function form an orthonormal basis of that space. A manuscript dated 23 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims a smooth volume-preserving diffeomorphism of the three-torus with this property, with a Lean formalization. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper or a formal proof is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- The manuscript distinguishes its probability-preserving formulation from the historical real-line question recorded by Ulam and attributed to Banach. Rokhlin asked in 1949 for ergodic automorphisms with simple, or at least finite-multiplicity, Lebesgue spectrum.
- Earlier results gave a Lebesgue component or related actions rather than one orbit spanning the whole space: Helson and Parry constructed cocycles with Lebesgue spectrum, Mathew and Nadkarni a transformation with a Lebesgue component of multiplicity two, and Guenais a group action with simple spectrum of mixed type. Fayad constructed smooth volume-preserving torus diffeomorphisms with simple spectrum, without Lebesgue type, and el Abdalaoui presents an infinite-measure conservative transformation with simple Lebesgue spectrum, from which the probability-space conclusion does not follow.
- In continuous time, Prikhod'ko constructed a finite-measure flow with simple Lebesgue spectrum; the manuscript notes that a nonzero time map of such a flow has infinite Lebesgue multiplicity, so these results do not give the discrete-time statement.
The original result
- Claim. Theorem 1.1 of A smooth three-torus diffeomorphism with simple Lebesgue spectrum states that there exist a smooth volume-preserving diffeomorphism T of the three-torus and a real square-integrable function f of mean zero such that the iterates f∘Tn, n ∈ ℤ, form an orthonormal basis of the mean-zero space. Consequently T is ergodic and its Koopman operator on that space is unitarily equivalent to multiplication by the coordinate on the circle. The invariant measure is the standard volume, and f is only claimed to be square-integrable. A final section derives mixing, zero entropy and vanishing Lyapunov exponents; mixing of every order uses, in addition, a separate multiple-mixing manuscript in the same collection.
- Method, as described: the approximating maps are integrable twists, whose finite angular Fourier expansions have explicit absolutely continuous spectral measures. Slow volume-preserving passages between successive twists, analysed by stationary phase, transport spectral densities to finer scales; small sinusoidal biases built from a positive Fourier construction and a finite prime sieve allow labels of the target observable to be predicted with little loss of spectral mass, a whitening step restores flat spectral density, and a diagonal limit gives an orthonormal orbit whose span is dense.
- Source (openai/math repository, commit
7ef5c0c): A smooth three-torus diffeomorphism with simple Lebesgue spectrum, dated 23 September 2026. - Formalization, as described by the repository. The scope note lean/docs/144.md states that the formalization constructs a smooth volume-preserving diffeomorphism of the three-torus with a real-valued cyclic vector whose integer iterates form an orthonormal basis of the mean-zero space, gives the isometric identification under which the Koopman operator becomes multiplication by the coordinate on the circle, and proves ergodicity. The Comparator statement is
ThreeTorus.lean.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 30 October 2026) |
|---|---|---|---|---|
| 2026-10-06 | Simple Lebesgue spectrum on a smooth three-torus, an article of the Latent Seminar blog at lowrank.github.io | Reading (exposition) | Explains Koopman operators and spectral multiplicity, the claimed smooth realization, mixing from the spectral model, integrable twists as approximating models, the prediction of labels while preserving spectral mass, and completeness in the limit, with proof outlines. | Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. |
| 2026-10-08 | MathVet (GitHub; review of openai/math, tables at repository commit 94c6630, 28 October 2026) |
Review (fidelity of the formal statements) | Rates this family "full", with no review note. It lists two hand-made definitions to check, smoothness on the torus through local smooth lifts and the mean-zero subspace of L2, and records a Comparator pass on cloud machines for ThreeTorus, finished on 8 October 2026. |
MathVet states that it does not claim any theorem false and reports what was machine-checked as stated. |
| 2026-10-10 | cert-machine, openai/math lane K, run 38040214623 | Verification (two kernels) | Records that Lean's default kernel and the nanoda kernel accept the ThreeTorus solution, in 368 s. The project reports two-kernel certification of 398 of 416 challenges of the collection. |
Repository. |
| 2026-10-10 to 2026-10-14 | con-leche development journal (task #348) | Verification (second kernel) | In its run over all Comparator challenges of the collection at commit fd4aeeb, the journal's table of exceptions does not list ThreeTorus, so it was accepted by con-leche and by Lean's official kernel. |
Development journal. |
| 2026-10-16 | J. Jardine, Lattice24, referee review rank_268 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap and no minor notes. The reviewer read all six sections and checked by hand the conjugation of the passages, the stationary-phase amplitudes and fine spectral densities, the Fourier moments and sieve argument of the signal lemma, the refresh error bookkeeping and the moment identities of the prediction step. | Part of an audit of the collection's manuscripts that the audit describes as complete. The reviews are published under CC BY 4.0. |
| 2026-10-16 | AI Proved What?, episode 144 One signal, every pattern?, remade version of 26 October 2026 | Reading (exposition) | Video explainer of the claim and its history. The episode states that no formula is claimed, since the map is a smooth limit of infinitely many steps; that the function f is not claimed smooth or continuous; that analyticity is not claimed; that the paper does not address the historical real-line question; and that Lebesgue spectrum here comes with zero entropy and zero Lyapunov exponents. It notes that the paper does not call the problem open. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, 30 pages, dated September 23, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: the main theorem, not run by the site. The page states that the corollary on mixing, entropy and Lyapunov exponents is not formalized and that it has seen no expert review. |
Notes
- The background summarizes the manuscript's own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
