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Strong Ulam Stability Characterizes Amenability
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Abstract
A countable discrete group is amenable if and only if it is strongly Ulam stable: every sufficiently accurate unitary almost representation, on any complex Hilbert space, is uniformly close in operator norm to a genuine representation on the same space. We prove the converse to Kazhdan's amenable stability theorem, answering the question of Burger, Ozawa, and Thom. The inclusion of infinite-dimensional Hilbert spaces is essential.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- Alon Dogon, reaction in 100+ reactions to 100+ solutions (Proofs and Prompts, 8 October 2026; collection updated 19 October 2026). A short signed reaction singles out "the equivalence of strong Ulam stability and amenability (along with Dixmier's problem)", says the author has worked on the question for years, and remarks that the solution appears to combine Furstenberg boundary theory with quantum circuits. [family-level]
Formalization
- OpenAI math release, Lean scope note 251: Amenability, unitarizability, and strong Ulam stability (lean/docs/251.md, October 2026). The family's Lean scope note lists only the companion unitarizability paper, and its comparator statements concern Dixmier's problem rather than strong Ulam stability. [family-level]
Possibly relevant literature (found by the agent, not checked)
- M. Pichot and E. Séguin, Positive definite maps on amenable groups, II (arXiv:2210.10823, October 2022). Introduces an Ulam-type stability condition for positive definite maps on countable groups and shows that it characterizes amenability, a related characterization to compare with Theorem 1.1.
- P. Rolli, Quasi-morphisms on free groups (arXiv:0911.4234, November 2009). Extends quasi-morphism constructions on free groups to ε-representations, relating to the free-subgroup case of instability that the paper attributes to Burger–Ozawa–Thom.
News and blogs
- Carlos Toledo, OpenAI's math release: what it claims beyond what it checks (cert-machine audit of release commit fd4aeeb2, 10 October 2026). The audit lists family 251's strong Ulam stability assertion among claims from manuscripts not linked by their family's Lean scope note, distinguishing that assertion from the unitarizability formalization. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 251
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026,
author = {{OpenAI}},
title = {{Strong Ulam Stability Characterizes Amenability}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026/strong-ulam-stability.pdf}{OAI:Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/251.md. The Lean statements have not been checked by people against the paper's claims.
Family 251: Amenability, unitarizability, and strong Ulam stability
Resolves Dixmier's problem for all discrete groups: amenability is equivalent to every uniformly bounded Hilbert-space representation being similar to a unitary representation. For countable discrete groups, amenability is also equivalent to strong Ulam stability: sufficiently accurate unitary approximate representations are uniformly close in operator norm to genuine representations on the same, possibly infinite-dimensional, Hilbert space.
Other papers in this family:
- Unitarizability implies amenability for discrete groups · September 23, 2026