Result 144, Dynamical systems and ergodic theory

Banach’s simple Lebesgue-spectrum problem

Resolves the probability-preserving form of Banach's simple Lebesgue-spectrum problem within smooth dynamics. A smooth volume-preserving diffeomorphism of the standard-volume three-torus has simple Lebesgue spectrum on its entire complex mean-zero L2 space: the bilateral iterates of one real observable form an orthonormal basis of that space.

Lean formalization Proof

The bigger picture

Why it matters

The manuscript reports a smooth, volume-preserving system in which one measurement and all its time shifts form a complete, mutually orthogonal set of signals. This connects smooth geometric motion with a particularly simple form of spectral behavior.

What changes?

The construction is an infinitely differentiable, invertible map with infinitely differentiable inverse on the three-torus, a space with three periodic coordinates. It preserves standard volume, viewed as a probability measure. Its Koopman operator, which evolves functions by composing them with the map, reportedly has simple Lebesgue spectrum on the entire complex mean-zero L2 space: all complex-valued functions with finite mean square and zero average. Concretely, the forward and backward iterates of one real-valued observable form an orthonormal basis.

What does that help mathematicians do?

Every finite-mean-square, zero-average observable can therefore be approximated in mean square by finite linear combinations of time shifts of that one observable. Distinct shifts have zero correlation, yet together they leave no such observable outside their closed span. The claimed example thus rules out smoothness and volume preservation, in this three-dimensional setting, as obstructions to this complete spectral behavior.

Are there practical applications?

The immediate value is foundational: the claimed construction answers the probability-preserving form of Banach's simple Lebesgue-spectrum problem within smooth dynamics. It gives researchers a concrete setting for studying how geometric evolution and spectral structure fit together. The supplied abstract does not establish a computational method or a practical application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A smooth three-torus diffeomorphism with simple Lebesgue spectrum

September 23, 2026 30 pages

We solve the probability-preserving form of Banach's simple Lebesgue-spectrum problem in smooth dynamics. We construct a C∞ diffeomorphism of the three-torus that preserves standard volume and whose Koopman operator has simple Lebesgue spectrum on the entire mean-zero L2 space.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026,
  author = {{OpenAI}},
  title = {{A smooth three-torus diffeomorphism with simple Lebesgue spectrum}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/paper.pdf}{OAI:A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/144.md.

Banach’s simple Lebesgue-spectrum problem

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Banach's simple Lebesgue-spectrum problem asks for a probability-preserving transformation with simple Lebesgue spectrum on its mean-zero L2L^2 space. The formalization constructs a smooth volume-preserving diffeomorphism of the three-torus with this property. It gives a real-valued cyclic vector whose integer iterates form an orthonormal basis of the entire mean-zero space, and an isometric identification under which the Koopman operator becomes multiplication by the coordinate function on the circle. The transformation is also ergodic.

Comparator links

Result Comparator statement
Smooth simple Lebesgue spectrum on the three-torus ThreeTorus.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.