Zero entropy does not guarantee a smooth positive-volume model
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
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.
Read the manuscripts
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
MANUSCRIPT 01
A zero-entropy system without a smooth positive-volume model
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…
Full source abstract
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.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.