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Rokhlin's multiple-mixing problem for a single transformation

Theorem 1.1: every invertible mixing probability-preserving transformation, on an arbitrary probability space with no standardness assumption, is mixing of every finite order. The proof assumes a least failing order $k$ , reduces to a zero-entropy finite-alphabet process, builds saturated (sated) array extensions of the limit joinings, extracts orthogonal lines of modulus-one functions from row and column conditional-expectation operators, and cancels against them. Corollary: the same for mixing endomorphisms and for mixing flows whose Koopman operators are strongly continuous on $L^1$ . Not shown: anything for $\mathbb Z^d$ or other group actions (where Ledrappier's example shows the analogue fails), or any rate of convergence.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Rokhlin's multiple-mixing problem for a single transformation · Rokhlin's multiple-mixing problem

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: Rokhlin's multiple-mixing problem for a single transformation

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the TeX source against Rokhlin's question, and the Lean statement. The result is not in the release's formalization catalogue (formalization.yaml), but lean/docs/145.md links ComparatorChallenges/Rokhlin.json, whose solution_module OAI.Dynamics.MultipleMixing.Main exists at the pinned commit; theorem OAI.Rokhlin.mixing_all_finite_orders, permitted axioms propext, Quot.sound and Classical.choice. The comparator statement, read here, takes an arbitrary measurable space with a probability measure, a measurable equivalence T preserving it, and mixing defined over integer times with |n| to infinity, and concludes for every k >= 3 that the measure of the intersection of the k translated sets tends to the product of measures as the vector of positive successive gaps tends to infinity (atTop on the product order, so every gap diverges). That is the headline claim, on arbitrary rather than standard probability spaces. Not rebuilt here; the comparator run and axiom check were not reproduced.

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Rokhlin's multiple-mixing problem for a single transformation

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lean proof: OAI.Rokhlin.mixing_all_finite_orders

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Lean comparator statement: Rokhlin.lean

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lean-proof link supplied by the VibeMathed registry.