Banach’s simple Lebesgue-spectrum problem

What OpenAI claims

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)

Paper

1. A smooth three-torus diffeomorphism with simple Lebesgue spectrum

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.