The Burger-Ozawa-Thom question: does strong Ulam stability characterize amenability?
A countable discrete group is strongly Ulam stable if for every there is such that every map into the unitary group of any complex Hilbert space, with and , is within uniformly of a genuine unitary representation on the same space. Kazhdan (1982) proved amenable groups are strongly Ulam stable; Burger, Ozawa and Thom (2013) showed groups with a nonabelian free subgroup are not, and Alpeev handled wreath products over nonamenable groups. Burger, Ozawa and Thom asked: is every strongly Ulam stable countable group amenable?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Group theory; stability of approximate representations
- Posed by
- Marc Burger, Narutaka Ozawa and Andreas Thom, On Ulam stability, Israel J. Math. 193 (2013), p. 111
- Year posed
- 2013
- Years open
- 13y
- Solved
- 2026-10-05
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: a countable discrete group is strongly Ulam stable if and only if it is amenable. For every nonamenable countable and there is a separable and a normalized with defect below that stays at uniform distance more than from every unitary representation on . Not shown: anything about finite-dimensional (ordinary) Ulam stability, or uncountable groups.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The manuscript (October 5, 2026) is self-contained apart from cited background (Furstenberg boundary theory, Kazhdan's theorem).
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 of 'Strong Ulam Stability Characterizes Amenability' was read against the question as quoted from Burger-Ozawa-Thom (p. 111). The proof (flags from a strongly proximal boundary action, layered unitary circuits, and an anticommuting-involution quadratic-form obstruction) was not refereed. No Lean formalization. Scope the paper itself states: infinite-dimensional Hilbert spaces are essential; the result says nothing about finite-dimensional Ulam stability, which some nonamenable groups have.