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Math-2229 · mathematics · Dynamical systems

Rokhlin multiple-mixing problem

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Is every strongly mixing measure-preserving transformation strongly mixing of order three?

Formulation references: [S1], [S2].

Short English statement

Is every strongly mixing measure-preserving transformation strongly mixing of order three?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2229-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2229

Manuscript 1 (family 145, 2026-09-23): For the customary invertible Rokhlin mixing problem, the theorem establishes every finite order and hence the recorded order-three assertion; it imposes no quantitative mixing rate or standardness assumption.

Rokhlin multiple-mixing problem

OpenAI's claim

Every invertible probability-preserving mixing transformation is claimed to be mixing at every finite order. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
On any probability space $(X,\mathcal F,\mu)$, suppose $T$ is invertible, bimeasurable, measure preserving and mixing in the two-set sense. OpenAI claims that for every $k\ge3$ and fixed measurable $A_1,\ldots,A_k$, \[ \mu\left(A_1\cap T^{-n_1}A_2\cap\cdots\cap T^{-(n_1+\cdots+n_{k-1})}A_k\right) \longrightarrow\prod_{j=1}^k\mu(A_j) \] as the minimum of the positive integer gaps $n_1,\ldots,n_{k-1}$ tends to infinity. No countable-generation or standardness assumption is made.

Outline of the argument

Manuscript 1 (family 145, 2026-09-23): A hypothetical nonproduct joining yields moving labels coupled to a zero-entropy process. Dissociated label orbits are canceled by exponential estimates against subexponential word counts, while polynomial label orbits are reduced in degree by Hilbert-space averaging; ordinary mixing anchors the reduction and contradicts the preserved positive correlation.

Formal verification

For family 145, the scope page documents the following selections. The selected implication gives mixing of every finite order for one invertible probability-preserving mixing map, with arbitrary measurable-set inputs and divergent successive time gaps. The documented statement covers the main claim of manuscript 1. Its selected declaration also covers the clause used for this problem. Comparator files: Rokhlin.lean. Declarations: OAI.Rokhlin.mixing_all_finite_orders.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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