Result 152, Dynamical systems and ergodic theory

Zero entropy does not guarantee a smooth positive-volume model

Constructs a zero-entropy ergodic invertible transformation of a standard nonatomic probability space that is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. One example rules out every finite dimension.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

A dynamical system can have zero average information production and still resist a smooth geometric description. The manuscript reports an example showing that zero entropy alone does not guarantee a model with smooth dynamics and a positive smooth density.

What changes?

The manuscript constructs an invertible, ergodic transformation of a standard nonatomic probability space: a reversible probability-preserving evolution with no invariant subset of intermediate probability and no points of positive probability. Its Kolmogorov-Sinai entropy, measuring average information production, is zero. Yet no measure-preserving relabeling, ignoring null sets, turns it into an infinitely differentiable diffeomorphism preserving a strictly positive smooth probability density on a compact manifold. One example excludes every finite dimension, including nonorientable manifolds and manifolds with smooth boundary.

What does that help mathematicians do?

Researchers studying when abstract probability-preserving dynamics can be realized on manifolds therefore need obstructions beyond entropy. For this example, increasing the manifold's dimension cannot rescue the proposed realization, nor can allowing nonorientability or a smooth boundary. The conclusion specifically concerns models with strictly positive smooth invariant densities; it does not rule out smooth realizations using other kinds of invariant probability measures.

Are there practical applications?

The immediate value is foundational: it identifies a limit on translating abstract probabilistic dynamics into finite-dimensional smooth geometry while preserving the system's measurable structure. Any proposed translation of this kind needs justification beyond zero entropy, with particular attention to whether the invariant probability density is smooth and strictly positive.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

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

September 25, 2026 42 pages

We construct an ergodic invertible transformation of a standard nonatomic probability space with zero Kolmogorov–Sinai entropy that has no smooth positive-volume model. More precisely, it is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. A single example excludes every finite dimension, including models on nonorientable manifolds and manifolds with smooth boundary.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026,
  author = {{OpenAI}},
  title = {{A zero-entropy system without a smooth positive-volume model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/paper.pdf}{OAI:A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/152.md.

Zero entropy does not guarantee a smooth positive-volume model

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The paper asks whether a measure-preserving system can be represented by a smooth diffeomorphism preserving positive smooth volume. The formalized supporting result constructs an ergodic invertible transformation of a standard nonatomic probability space with finite Kolmogorov–Sinai entropy that has no such model on any compact finite-dimensional manifold, including manifolds with smooth boundary. The comparison is measurable conjugacy after discarding null sets.

The linked statement gives finite entropy; the paper's stronger zero-entropy conclusion is outside this statement.

Comparator links

Result Comparator statement
Finite-entropy system without a smooth positive-volume model SmoothObstruction.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.