A smooth three-torus diffeomorphism with simple Lebesgue spectrum

Formalization

• OpenAI, Banach’s simple Lebesgue-spectrum problem: Lean scope note (lean/docs/144.md, OpenAI Math Release, 6 October 2026). The release's scope note says the Lean development constructs the smooth volume-preserving three-torus diffeomorphism, gives a real cyclic vector whose iterates form an orthonormal basis of the mean-zero space, provides the unitary model as multiplication by the coordinate on the circle, and proves ergodicity. [AI-generated per source]
• OpenAI math release, ThreeTorus.lean (comparator challenge statement) (lean/ComparatorChallenges, 2026). This is the comparator statement ThreeTorus.simple_lebesgue_spectrum, left as sorry in this file by design: smooth T with smooth inverse, Haar measure preservation, ergodicity, a real mean-zero f with orthonormal iterates spanning a dense subspace of H₀, and a unitary model in which the Koopman operator becomes multiplication by the first character.

Possibly relevant literature (found by the agent, not checked)

• el Houcein el Abdalaoui, $L^1$-flat polynomials and simple Lebesgue spectrum for conservative maps exist: A simple proof (arXiv:2210.15480, Oct 2022, rev. Jun 2023). This gives a conservative ergodic map and flow with simple Lebesgue spectrum, a companion to the infinite-measure result the paper cites from arXiv:1508.06439v8 (Theorem 2.2), which the paper says does not imply the probability-space conclusion.
• Valery V. Ryzhikov, Spectral properties of dynamical tensor powers, and tensor factorizations of simple Lebesgue spectrum (arXiv:2406.08409, Jun 2024). Its text calls the question "is there an automorphism with simple Lebesgue spectrum?" Banach's problem and treats the probability-space case separately, the formulation Theorem 1.1 addresses.
• Valery V. Ryzhikov, Unsolved problems on joinings, multiple mixing, spectrum, and rank (arXiv:2405.03641, May 2024). This survey links a hypothetical simple Lebesgue-spectrum automorphism to "the problems of Banach, Rokhlin and del Junco-Rudolph", which is relevant to the mixing consequences derived in Section 6.
• A. A. Prikhod'ko, On group actions with simple Lebesgue spectrum (arXiv:1111.0230, Nov 2011). This constructs measure-preserving $\mathbb R^d$-actions with simple Lebesgue spectrum, which relates to the introduction's comparison with continuous-time results (it cites Prikhod'ko's 2020 flow paper but not this one).
• Bassam Fayad, Anatole Katok, Constructions in elliptic dynamics (arXiv:math/0501362, Jan 2005; Ergodic Theory Dynam. Systems 24 (2004)). This is a survey of the Anosov–Katok approximation-by-conjugation method, which the introduction names as background for its successive smooth coordinate changes.

Digests and expositions

• AI Proved What?, One signal, every pattern? A smooth three-torus map with simple Lebesgue spectrum (episode 144, October 2026). This illustrated exposition and transcript explain the claimed spectral model and the twist, passage, density-bias, whitening, and diagonal-limit construction, with numerical illustrations and a statement that its authors have not run the Lean check. [AI-generated per source]