The smooth realization problem
From OpenAI Aftermath
The smooth realization problem asks which abstract measure-preserving systems can be represented by smooth dynamics on a compact manifold. In the version with smooth positive volume, an invertible probability-preserving transformation should be measurably conjugate to a C∞ diffeomorphism of a compact manifold that preserves a strictly positive smooth density. Finite Kolmogorov–Sinai entropy is necessary, by Ruelle's inequality, and the question is whether it suffices. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), listed as its result 152 and dated 25 September 2026, claims a negative answer already among systems of zero entropy: an ergodic zero-entropy transformation with no smooth positive-volume model on any compact manifold of any finite dimension. The collection includes a Lean formalization of a weaker statement, with finite entropy in place of zero entropy.
Background
- As recounted in the introduction of the manuscript:
- Ruelle's differentiable entropy inequality gives finite entropy from compact derivative bounds, so finite entropy is a necessary restriction.
- Lind and Thouvenot realized finite-entropy ergodic transformations by Lebesgue-preserving homeomorphisms of the two-torus, and showed that a hyperbolic toral automorphism represents every such system of smaller measure entropy, with an invariant Borel probability of full support; Quas and Soo extended the second result to toral automorphisms with no root-of-unity eigenvalue. So every finite-entropy system has a smooth model, but its invariant measure need not be a smooth positive density.
- Katok and Thouvenot discussed the classical formulation and constructed ergodic Z2 actions with no Lipschitz realization on a compact space of finite box dimension. Foreman and Weiss discussed finite-entropy realization with an invariant measure merely equivalent to volume, and anti-classification results of Foreman and Weiss, Kunde, and Gerber and Kunde concern measure isomorphism within families of smooth transformations.
The original results
- Claim. Theorem 1.1 of A zero-entropy system without a smooth positive-volume model states that there is an ergodic invertible measure-preserving transformation of a standard nonatomic probability space with zero entropy that is not measurably conjugate, on invariant conull sets, to any C∞ diffeomorphism of a compact finite-dimensional manifold preserving a strictly positive smooth probability density; the manifold may have smooth boundary and need not be orientable. The manuscript states that it does not assert an obstruction for every invariant measure equivalent to volume.
- Method, as described: a switching property of smooth positive volume that survives arbitrary measurable changes of coordinates. At dyadic times, for most pairs of points chosen from a small cell, a point is found whose future shadows that of the first point and whose past shadows that of the second, with law bounded by twice the original volume. A symbolic system is then built whose past and future tests are incompatible with this switching.
- Formalization, as described by the repository. The scope note states that the formalized supporting result constructs an ergodic invertible transformation with finite Kolmogorov–Sinai entropy and no smooth positive-volume model on any compact finite-dimensional manifold, including manifolds with smooth boundary, and that "the paper's stronger zero-entropy conclusion is outside this statement". The Comparator statement is
SmoothObstruction.lean. The repository folder name begins "A-finite-entropy-system", while the manuscript's title and theorem state zero entropy. - Source (openai/math repository, commit
7ef5c0c): A zero-entropy system without a smooth positive-volume model, dated 25 September 2026.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 30 October 2026) |
|---|---|---|---|---|
| 2026-10-06 | Zero entropy and the obstruction to smooth positive-volume models, an article of the Latent Seminar blog at lowrank.github.io | Reading (exposition) | Explains the claimed obstruction: finite entropy is necessary but proposed to be insufficient, zero entropy leaves room for an obstruction, smooth positive volume has a switching property that survives measurable changes of coordinates, and a symbolic system with incompatible past and future tests contradicts it after a selection of scales. | Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. |
| 2026-10-08 | MathVet (GitHub; review of openai/math, tables at repository commit c76f1e4, 14 October 2026) |
Review (fidelity of the formal statements) | Rates this family "weaker-statement" and records a sandboxed Comparator pass for SmoothObstruction. Its note states that the formal statement gives finite entropy, with all partition entropy rates bounded by one constant, and no smooth positive-density model, so that it is implied by the headline but does not imply the zero-entropy claim. It lists the finite-entropy and smooth-model definitions as custom encodings read as standard. |
MathVet states that it does not claim any theorem false and reports what was machine-checked as stated. |
| 2026-10-10 | cert-machine, openai/math lane K, run 38040214623 | Verification (two kernels) | Records that Lean's default kernel and the nanoda kernel accept the SmoothObstruction solution, in 737 s. C. Toledo's report lists the challenge among those certified by both kernels. |
Repository. The run concerns the finite-entropy formal statement. |
| 2026-10-16 | J. Jardine, Lattice24, referee review rank_065 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap. It states that it read all seven sections and checked, among other steps, the dyadic chord lemma, the scale hierarchy, the switching construction with its determinant identity and change of variables, and the word construction. It records two minor issues: the folder name and citation key say finite entropy while the title and theorem say zero entropy, and the endpoint fast-space comparison of Sections 3 and 4 is only sketched, which it calls a presentation-level thin spot rather than a gap. | Part of an audit of the collection's manuscripts that the audit describes as complete. The reviews are published under CC BY 4.0. |
| 2026-10-16 | AI Proved What?, episode 152 No smooth volume model?, remade version of 26 October 2026 | Reading (exposition) | Video explainer of the claim of an ergodic invertible transformation of a standard nonatomic probability space with zero Kolmogorov–Sinai entropy that is not measurably conjugate, on invariant conull sets, to any smooth diffeomorphism of a compact finite-dimensional manifold preserving a strictly positive smooth density, one example excluding every dimension. The episode recalls Ruelle's inequality, the Lind–Thouvenot and Quas–Soo realizations with invariant measures of full support, and Katok and Thouvenot's actions of the plane lattice, and states that the obstruction concerns the smooth positive density only, that the system is shown to exist by a probabilistic construction rather than written down, and that the paper does not say which systems do have smooth volume models. It notes that the repository folder name says finite entropy while the paper says zero entropy. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, 42 pages, dated September 25, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: with finite entropy in place of zero; the site has not run the check. The page states that the preprint has not been revised since the repository's initial commit of 6 October 2026, that the comparator statement asks only for finite entropy, and that the proof, 53 files and 30,090 lines with no unproved steps that the site could find, formalizes the switch, the word construction, the window entropy and the final contradiction, with a crude alphabet bound of about 80,000 bits per step for the entropy, so that the zero-entropy count is the paper's alone. |
Notes
- The background summarizes the manuscript's own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
