Result 145, Dynamical systems and ergodic theory

Rokhlin’s multiple-mixing problem

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 formalization Proof

The bigger picture

Why it matters

The unreviewed manuscript claims that, for one invertible probability-preserving dynamical system, loss of dependence between pairs of events forces the same loss for every finite group. This connects a two-event condition to much richer long-term behavior.

What changes?

A probability-preserving transformation is a rule for evolving a system without changing its probability distribution; invertibility means that each step can be reversed. Mixing means that any two measurable events become statistically independent in the limit of a growing time gap. The manuscript reports that this implies mixing of every finite order: for any fixed finite collection of measurable sets, their joint occurrence probability tends to the product of their probabilities whenever all pairwise time separations diverge.

What does that help mathematicians do?

Under this conclusion, a single invertible transformation cannot erase all long-range two-event dependence while retaining dependence among three or more events at mutually diverging times. Researchers could therefore deduce every finite-order mixing property from ordinary mixing alone. In particular, any fixed finite collection of positive-probability events would have positive joint occurrence probability once all observation times are sufficiently far apart, connecting decorrelation to simultaneous recurrence.

Are there practical applications?

The immediate value is foundational: the claimed result would simplify how ergodic theorists establish higher-order independence in probability-preserving dynamics, replacing separate arguments for each finite order with a two-event mixing condition. The supplied material gives no convergence rates or computational procedure, so it does not establish practical bounds for finite-time prediction or simulation.

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

Rokhlin's multiple-mixing problem for one transformation

September 23, 2026 40 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026,
  author = {{OpenAI}},
  title = {{Rokhlin's multiple-mixing problem for one transformation}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026/paper.pdf}{OAI:Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/145.md.

Rokhlin’s multiple-mixing problem

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

Scope

Rokhlin's multiple-mixing problem asks whether ordinary mixing of one invertible probability-preserving transformation implies mixing of every finite order. The formalization proves this implication. For each k≥3k\ge3 and every kk measurable sets, the measure of their translated intersection tends to the product of their measures as all successive time gaps tend to infinity.

Comparator links

Result Comparator statement
Mixing of every finite order for one mixing transformation Rokhlin.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.