No smooth volume model?

The claim, in plain words

An abstract dynamical system moves the points of a probability space around, forever, so that every region keeps its probability; many are built from sequences of symbols with no geometry at all. The smooth realization problem, as the paper puts it, asks which of these can be represented by smooth dynamics on a compact manifold. The paper asks for a demanding kind of model: a C^∞ diffeomorphism of a compact manifold (of any finite dimension, possibly with boundary, orientable or not) preserving a smooth, strictly positive density, matched to the abstract system by a relabelling that need only be measurable, not continuous.

Finite entropy is necessary: in the paper’s words, “Ruelle’s differentiable entropy inequality … gives finite entropy from compact derivative bounds.” In our own example, a perturbed cat map on the torus, the natural logarithm of the stretch grows by about 0.96 per step, never faster than about 1.09. As the paper recounts, Lind and Thouvenot realized finite-entropy ergodic transformations by area-preserving homeomorphisms of the two-torus and showed that a hyperbolic torus automorphism represents every such system with entropy below the automorphism’s own, using an invariant measure of full support; Quas and Soo extended this. So every ergodic finite-entropy system has a smooth model, but its measure need not be a smooth density. Katok and Thouvenot built ℤ² actions with no Lipschitz realization on a compact space of finite box dimension, which, the paper notes, does not settle the case of one map; anti-classification results of Foreman and Weiss, Kunde, and Gerber and Kunde study equivalence inside families of smooth models.

Result 152 is one OpenAI preprint (42 pages, September 25, 2026). Its Theorem 1.1 claims 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 C^∞ diffeomorphism of a compact finite-dimensional manifold preserving a strictly positive smooth probability density. One example excludes every dimension: “the symbolic system is fixed first. Only then is a hypothetical smooth model, with any one finite dimension, considered.” The paper calls this “a negative answer to the smooth-volume realization question, already among systems of zero entropy.” (The repository folder is named “A-finite-entropy-system…”, but the paper’s title, abstract and theorem say zero entropy.)

What it doesn’t claim

The obstruction is about the smooth positive density. By the results the paper cites, this system does have a smooth model whose invariant measure has full support but no smooth density, and “we do not assert an obstruction for every equivalent-to-volume invariant measure”; Remark 7.4 adds that the proof “makes no analogous assertion for models carrying only a singular measure or a nonsmooth density equivalent to volume.” Nothing is said about finitely differentiable maps. The system is shown to exist by a probabilistic construction, not written down, and the paper does not say which systems do have smooth volume models. It does not call the problem famous or open or name who first posed it.

Where it stands

Posted as a preprint

OpenAI, 42 pages, dated September 25, 2026

Peer reviewed

Not yet

Checked in Lean

With finite entropy in place of zero. We have not run the check

Last checked October 19, 2026. The preprint has not been revised since it was added to the repository in its initial commit on October 6, 2026. OpenAI published a Lean formalization: a comparator statement that writes out standard nonatomic spaces, measurable conjugacy on invariant conull sets, smooth positive densities in charts (including charts with boundary), C^∞ diffeomorphisms and every finite dimension, but asks only for finite entropy: in the scope note’s words, “the paper’s stronger zero-entropy conclusion is outside this statement.” The proof, 53 files and 30,090 lines, formalizes the switch, the word construction, the window entropy and the final contradiction, with no unproved steps that we could find; its finite-entropy bound is the crude alphabet bound (about 80,000 bits per step), so the paper’s zero-entropy count is the paper’s alone. We have not run the check. The paper is not among those OSVerify reviewed, and we have seen no expert review.

How the proof goes

1. Splicing in a smooth volume model. At a time n = 2^k, cut the manifold into at most exp(CB_kn) cells, where ∑ B_k¹⁶ < ∞. Pick a cell with probability equal to its volume and two independent points x, y in it. For most pairs there is a point z whose next n steps follow x and whose last n steps follow y, and the law of z is at most twice the volume (Theorem 4.1). The paper builds z from a dyadic chord estimate for derivative growth, slow and fast directions on finite orbit segments (zero or positive slow rates allowed) and a boundary-value problem. It builds the mirror splice w as well; the Jacobian of (x, y) ↦ (z, w) is a ratio of four endpoint determinants in which the large stretch factors cancel, and injectivity turns this into the factor 2.
2. Words with built-in relations. The system is a hierarchy of words of words. Each level-i word carries a label, a nonzero vector in 𝔽₂^2s_i, and labels u, v pass the test when the symplectic form [u, v] is 0. Inside each group, labels a power of two apart must pass. This costs a sliver of each label’s information, so at high levels short lists of labels keep at least 99.6% of their full entropy for any invariant measure; words are recognizable from slightly corrupted copies; and the information per symbol tends to 0, so every invariant measure has zero entropy.
3. The clash. Compare y’s past list with x’s future list, a power-of-two number of words apart, and let Z be the fraction of passing pairs. Given the cell the lists are independent, and at well-chosen scales naming the cell costs few bits (because ∑ B_k¹⁶ < ∞ while the word rates make ∑ K_iR_i¹⁶ = ∞), so a character-sum bound gives E Z ≤ 0.62. But z is a point of the system whose symbols match x’s ahead and y’s behind in all but a tiny fraction of places, so recognition aligns its words and a legal point’s own lists pass: E Z ≥ (0.96 − 0.01) × 0.96 = 0.912. Both cannot hold.

The numbers in the video

• Our map S = A ∘ Sh on the torus, A = [[2, 1], [1, 1]], Sh(x, y) = (x, y + (0.3/2π) sin 2πx): average log-stretch 0.958 per step (1,000 orbits of 1,024 steps); ceiling ln sup ‖DS‖ = 1.091.
• The paper’s splice built on that map in two dimensions: every pair Jacobian, from the four-determinant identity, in [0.9967, 1.0027] (72 pairs, n = 8, 16, 32); in the worked pair the distances to x forward and to y backward shrink from about 0.02 to below 0.0001 in 8 steps; 300 splice points spread evenly over the cell.
• A small version of the words (labels in 𝔽₂²⁰, groups of 64, distances 1, 2, 4, 8): labels at the prescribed distances passed 100% of the time, labels from two unrelated lists 52%.
• The paper’s parameters at its smallest start (our arithmetic): an alphabet of 2⁶⁵⁵³⁸ − 1 labels and information per symbol of 65,538, then 81.9, then 0.051 bits, tending to 0.
• The paper itself contains no numerics; these are our illustrations, and none of them models the paper’s system.

Read the source

• A zero-entropy system without a smooth positive-volume model (PDF), 42 pages.
• The Lean scope note.
• The collection’s catalog, where this is result 152.

The paper

OpenAI groups related papers into one result; this one has a single paper.

• A zero-entropy system without a smooth positive-volume model

Transcript

Spot a mistake?

This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.