Rokhlin's multiple-mixing problem for a single transformation
Let be an invertible measure-preserving transformation of a probability space . is mixing if as , and mixing of order if whenever and . Rokhlin introduced higher-order mixing in 1949 and asked whether ordinary mixing already implies mixing of the next order. Partial answers needed extra structure: Kalikow (rank one, threefold), Host (singular spectrum), Ryzhikov (finite rank), while Ledrappier's mixing action that is not threefold mixing shows the question is specific to one transformation. Is every mixing transformation mixing of all orders?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Ergodic theory; mixing and joinings
- Posed by
- V. A. Rokhlin, On endomorphisms of compact commutative groups (Izv. Akad. Nauk SSSR, 1949)
- Year posed
- 1949
- Years open
- 77y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 62 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
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 , 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 . Not shown: anything for or other group actions (where Ledrappier's example shows the analogue fails), or any rate of convergence.
What the AI did
The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, using on average about three hours of ChatGPT Pro thinking compute per result, with outputs grouped into families and manuscripts. The README's two exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region write-up, which was human-edited for readability) do not concern this family. The manuscript is credited to OpenAI alone, names no human author and has no acknowledgements. A Lean formalization of the main theorem is in the release's lean/ library (lean/docs/145.md). The release does not say how much human review happened before publication.
Verification
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.