Dynamical systems and ergodic theory·OpenAI Math·Result 145·Lean formalisation listed
Rokhlin’s multiple-mixing problem
One of the 372 results in OpenAI's public mathematics catalogue, filed under dynamical systems and ergodic theory: 1 preprint, the latest dated 23 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports an answer to Rokhlin’s multiple-mixing problem (dynamical systems and ergodic theory) in a preprint dated 23 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More, is on YouTube.
Chapter 9 of 84 in Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · starts at 2:45 · watch from there on YouTube →
What OpenAI claims
Proves that every invertible mixing probability-preserving transformation is mixing of all finite orders, resolving Rokhlin's multiple-mixing problem for a single transformation. Correlations among any finite collection of measurable sets converge to the product of their measures whenever all pairwise time separations diverge. (Lean)
Paper
Rokhlin's multiple-mixing problem for one transformation
OpenAI's abstract Every invertible mixing probability-preserving transformation is mixing of every finite order. Thus Rokhlin's multiple-mixing problem for a single transformation has an affirmative answer.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 23 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- Proves that every invertible mixing probability-preserving transformation is mixing of all finite orders, resolving Rokhlin's multiple-mixing problem for a single transformation. Correlations among any finite collection of measurable sets converge to the product of their measures whenever all pairwise time separations diverge. (Lean)
- Where is the proof?
- In the paper, Rokhlin's multiple-mixing problem for one transformation (PDF).
- Is there a video?
- Yes, on YouTube (29:10, chapter 9).
Also in
- Chapter 2 of 9 in Birkhoff Conjecture, Zero Entropy and 7 More in Dynamical Systems and Ergodic Theory · starts at 0:21 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · Compilation · 29:10 · chapter 9 from 2:45 · description, sources and chapters →
Related results in Dynamical systems and ergodic theory
- 146Positive metric entropy for the standard mapLean
- 147The near-boundary Birkhoff conjectureVideo
- 148The entropy-rate dimension formula for self-similar measuresVideoLean
- 149Classwise permanence for weakly reversible mass-action systemsVideoLean
- 150Weak mixing of triangular billiards with an irrational angleVideoLean
- 151A C1 counterexample to the entropy conjectureVideoLean
