OpenAI claim summary for Math-2228

Banach's simple Lebesgue spectrum problem

OpenAI's claim

A smooth volume-preserving diffeomorphism of the three-torus is claimed to have simple Lebesgue spectrum on its entire mean-zero square-integrable space. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.

With $\mathbb T^3=(\mathbb R/\mathbb Z)^3$ and its standard normalized smooth volume $\mu$, OpenAI claims a smooth volume-preserving diffeomorphism $T$ and a real mean-zero $f\in L^2(\mathbb T^3,\mu)$ such that \[ \{f\circ T^n:n\in\mathbb Z\}\text{ is an orthonormal basis of }L^2_0(\mathbb T^3,\mu). \] Consequently $T$ is ergodic and the Koopman operator $U_Tg=g\circ T$ on the complement of constants is unitarily equivalent to $g(w)\mapsto wg(w)$ on $L^2(S^1,m)$, with normalized Haar measure $m$. This is simple Lebesgue spectrum in a smooth compact-manifold model.

Outline of the argument

Manuscript 1 (family 144, 2026-09-23): Finite blocks of small density-weighted biases learn observable labels while losing only a quadratic amount of spectral mass. Whitening restores density one with a small physical-space change; a diagonal construction preserves an orthonormal orbit and makes that orbit span a dense family of observables.

Formal verification

For family 144, the scope page documents the following selections. The torus construction is selected on the full mean-zero square-integrable space, including a real cyclic vector with an orthonormal integer orbit and the circle-multiplication model. The documented statement covers the main claim of manuscript 1. Its selected declaration also covers the clause used for this problem. Comparator files: ThreeTorus.lean. Declarations: OAI.ThreeTorus.simple_lebesgue_spectrum.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

• OpenAI, A smooth three-torus diffeomorphism with simple Lebesgue spectrum, 2026-09-23; citation key OAI:A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 144: Lean scope.
• Comparator ThreeTorus.json.
• Comparator ThreeTorus.lean.
• OpenAI catalogue entry.