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The smooth realization problem: does finite entropy suffice for a smooth volume-preserving model?

A smooth positive-volume model of an invertible ergodic transformation (X,μ,T)(X,\mu,T) of a standard nonatomic probability space is a C∞C^\infty diffeomorphism SS of a compact finite-dimensional manifold MM preserving a probability ν\nu with strictly positive smooth density, together with a measurable isomorphism FF between invariant conull sets with F∘T=S∘FF\circ T=S\circ F. Finite Kolmogorov-Sinai entropy is necessary (Ruelle's inequality), and Lind and Thouvenot realized every finite-entropy ergodic system by a homeomorphism of the two-torus, but a realization with a singular invariant measure does not give a smooth volume. The smooth realization problem, as discussed by Katok and Thouvenot, asks whether this entropy restriction is the only one. Does every ergodic invertible transformation with finite entropy admit a smooth positive-volume model?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Ergodic theory; smooth realization of measure-preserving systems
Posed by
The classical smooth realization problem, in the formulation discussed by Anatole Katok and Jean-Paul Thouvenot (Ann. IHP Probab. Stat. 1997, pp. 324-326)
Year posed
1997
Years open
29y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
42 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims Theorem 1.1: there is an ergodic invertible transformation of a standard nonatomic probability space with hμ(T)=0h_\mu(T)=0 that is not measurably conjugate on invariant conull sets to any C∞C^\infty diffeomorphism of a compact finite-dimensional manifold (boundary allowed, orientation not required) preserving a strictly positive smooth probability density. A single symbolic example excludes every dimension. So finite entropy, even zero entropy, does not suffice. It does not exclude models whose invariant measure is only equivalent to volume, and it does not characterize which systems are smoothly realizable.

What the AI did

Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1, read against the problem as the manuscript frames it. The proof was not refereed. The paper's own scope limits: the model must preserve a strictly positive smooth density; it does not claim an obstruction for invariant measures merely equivalent to volume (the Foreman-Weiss formulation), nor for Z^2 actions. Lean-checked on the release's own Comparator challenge SmoothObstruction together with its solution module OAI.Dynamics.SmoothObstruction.Main, both fetched at the pinned commit; the challenge is not listed in the release's formalization catalogue (lean/formalization.yaml), the statement was read here but not independently audited, and the development was not rebuilt here. The formal statement (OAI.SmoothObstruction.main) gives a standard nonatomic ergodic system with FINITE entropy and no smooth positive-density model on any compact manifold of any dimension, with or without boundary. That is a full negative answer to the posed question, but it is weaker than the paper's zero-entropy headline; the zero-entropy conclusion is not formalized (the release's own scope note for family 152 says so).

Sources

Changelog1 change

Discussion