Episode 251, October 2026

Average it, straighten it?

Two AI preprints claim Dixmier’s problem for discrete groups and strong Ulam stability for countable groups

Covers result 251 in OpenAI’s catalog: all 2 papers. See its connections to other results and the work it cites.

Play Episode 251: Average it, straighten it? A stretched ellipse and the round circle an averaged inner product gives.

4:36. English captions available. The video loads from YouTube when you press play.

Chapters

The claim, in plain words

Let a group act by linear maps on a Hilbert space. The action is unitary if every map preserves lengths, and uniformly bounded if no map stretches by more than a fixed factor. Can one change of ruler, a new inner product, turn every move into a rotation? For amenable groups, yes: Day and Dixmier showed in 1950 that you can average the inner product over the group. Dixmier asked whether this characterizes amenability. Groups containing a free group on two letters were known to fail, and Pisier’s survey restated the question in 2004. Partial answers for nonamenable groups without such subgroups followed, by Epstein and Monod, Osin, Monod and Ozawa, and others.

A parallel question goes back to Ulam: is a map that is almost a representation near a true one? Kazhdan showed that amenable groups can correct such maps. Burger, Ozawa and Thom showed that groups with nonabelian free subgroups fail, and asked whether strong Ulam stability, on any Hilbert space, characterizes amenability.

Result 251 is two OpenAI preprints. The first, 19 pages, dated September 23, 2026, claims that a discrete group is amenable if and only if it is unitarizable. Every nonamenable discrete group would then have a representation with norms at most 1 + ε that is not similar to a unitary one. The second, 19 pages, dated October 5, claims that a countable discrete group is strongly Ulam stable exactly when it is amenable.

What it doesn’t claim

Dixmier’s question for locally compact or general topological groups.
The easy direction, which is classical.
Stability in finite dimensions, where some nonamenable groups are stable.
Explicit representations: both constructions only show that they exist.

Where it stands

Posted as preprints: OpenAI, 2 papers, 38 pages, dated September 23 and October 5, 2026
Peer reviewed: Not yet
Formalized in Lean: In part: the first paper (docs)

Last checked October 19, 2026. Neither preprint has been revised. In Lean, the first paper’s theorem is formalized for all discrete groups: about 4,100 lines with no unproved steps (we did not build it). The second paper is not formalized. OSVerify has not reviewed the papers. Two days after the release, Vadim Alekseev, working with an AI assistant, posted a geometric retelling of the first proof. No error has been reported.

How the proof goes

The first paper. In a nonamenable group, some average of translations contracts. Hall’s marriage theorem assigns the long steps of the walk to their endpoints. Random sign vectors pack many nearly orthogonal directions into far fewer dimensions. The result is a sequence of operators whose commutators with translations stay at most 100, while their distance from everything translation-invariant grows without bound. A triangular representation built from them cannot be straightened by any change of ruler.
The second paper. A boundary of the group marks each step. The layers of a layered Hilbert space are tilted against each other by anticommuting operators borrowed from quantum physics, and unitary circuits glue the layers together. A quadratic form reads 0 for every true representation, but 1/2 for the construction.

The numbers in the video

Ours: averaging over windows of the integers, the error fell from 2.1 to 0.0003 as the window grew from 1 to 3,000, like 1/N.
Ours: for a lazy random walk, the return probability shrinks by a factor of about 0.893 per step on the free group on two letters. On the integers the per-step factor tends to 1.
Ours: Kazhdan’s averaging corrected an almost-representation of a group of 24 elements, taking the error from 0.047 to 2×10⁻¹⁵ in three steps.
Ours: with the first paper’s constants and the simplest steps on the free group, the lower bound first turns positive between stages 140 and 150, with about 10¹⁰³ directions. A toy version of the second paper’s quadratic form, in 4,096 dimensions, gives exactly 1/2.

Read the source

Unitarizability implies amenability for discrete groups (PDF), 19 pages.
Strong Ulam Stability Characterizes Amenability (PDF), 19 pages.
The Lean formalization notes (first paper only).
The collection’s catalog, where this is result 251.

The 2 papers in this result

OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.

Unitarizability implies amenability for discrete groups main paper
Strong Ulam Stability Characterizes Amenability companion

Transcript

Read the full script

0:00The question

Let a group act by linear maps on a Hilbert space. If every map preserves lengths, the action is unitary. If the maps only never stretch by more than a fixed factor, it is uniformly bounded. Can one change of ruler, a new inner product, turn every move into a rotation? For amenable groups, yes, as Day and Dixmier showed: average the inner product over the group. Our computer did this for the integers: the error fell from 2 to 0.0003 as the window grew from 1 to 3,000, like 1/N. In 1950, Dixmier asked whether this characterizes amenability. Groups containing a free group on two letters were known to fail. Pisier’s survey restated the question in 2004, and partial answers for nonamenable groups without such subgroups followed, by Epstein and Monod, Osin, Monod and Ozawa, and others. What separates the two worlds is averaging: on the free group, the chance that a lazy random walk is back home shrinks by a factor of about 0.89 per step; on the integers, by less than any fixed factor. A parallel question, going back to Ulam: if a map is almost a representation, is it near a true one? Kazhdan showed amenable groups can correct such maps; our computer corrected one on a group of 24 elements, the error falling from 0.05 to 2×10⁻¹⁵ in three steps. Burger, Ozawa and Thom showed groups with nonabelian free subgroups fail, and asked whether this strong Ulam stability, on any Hilbert space, characterizes amenability.

1:40The claim

Result 251 is two OpenAI preprints. The first, 19 pages, dated September 23, 2026, claims a discrete group is amenable if and only if it is unitarizable: every nonamenable discrete group has a representation with norms at most 1 + ε that is not similar to a unitary one. The second, 19 pages, dated October 5, claims a countable discrete group is strongly Ulam stable exactly when it is amenable. Not claimed: Dixmier’s question for locally compact or general topological groups; the easy direction, which is classical; stability in finite dimensions, where some nonamenable groups are stable; or explicit representations, since both constructions only show existence.

2:31How the proof goes

The first paper: in a nonamenable group, some average of translations contracts. Hall’s marriage theorem assigns the long steps of the walk to their endpoints; random sign vectors pack many nearly orthogonal directions into far fewer dimensions; and the result is a sequence of operators whose commutators with translations stay at most 100, while their distance from everything translation-invariant grows without bound. A triangular representation built from them cannot be straightened by any change of ruler. These are existence proofs. With the paper’s own constants and the simplest choice of steps on the free group, our computer found the lower bound first turns positive between stages 140 and 150, where the number of directions is about 10¹⁰³. The second paper uses a different engine: a boundary of the group marks each step, a layered Hilbert space has its layers tilted against each other by anticommuting operators borrowed from quantum physics, and unitary circuits glue the layers. A quadratic form reads zero for every true representation, but one half for the construction. Our toy version of that form, in 4,096 dimensions, gives exactly one half.

3:47Where it stands

Both preprints are unrefereed and unchanged since release. In Lean, the first paper’s theorem is formalized for all discrete groups, about 4,100 lines with no unproved steps; we did not build it. The second paper is not formalized. OSVerify has not reviewed them. Vadim Alekseev, working with an AI assistant, posted a geometric retelling of the first proof two days after release, and no error has been reported. If they hold, averaging is exactly the dividing line for discrete groups: the groups you can average over are the ones whose bounded representations can be straightened, and, for countable groups, whose almost-representations can be corrected on every Hilbert space. Seventy-six years after Dixmier asked.

Spot a mistake?

This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.