View source for Thompson's group F is nonamenable

'''Thompson's group F is nonamenable''' is the title of a manuscript dated 23 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]). Thompson's group ''F'' is the group of piecewise linear homeomorphisms of the unit interval with finitely many breakpoints, all at dyadic rationals, and slopes that are integer powers of 2. Whether ''F'' is amenable had been a long-standing open problem. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== The original result ==
* '''Claim.''' Thompson's group ''F'' is not amenable. The abstract states that this confirms Geoghegan's conjecture and resolves the amenability problem for ''F''. Source: [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Thompsons-group-F-is-nonamenable-September-23-2026 ''Thompson's group F is nonamenable''] (23 September 2026).
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/248.md lean/docs/248.md] states that the Lean formalization rules out a positive normalized left-invariant mean on bounded real functions for the standard group of dyadic piecewise-linear homeomorphisms of the interval, proving nonamenability; no explicit boundary constant or prescribed generating set is given.
* '''Background.''' For the state of the problem before the manuscript, see V. Guba, ''Amenability problem for Thompson's group F: state of the art'' ([https://arxiv.org/abs/2305.07113 arXiv:2305.07113]).

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 15 October 2026)
|-
| 2026-10-08 (latest commit <code>a371d87</code>, 10 October 2026) || J. Guo, [https://github.com/aster2024/openai-math-followups/blob/a371d874a63222dd662a2ce4ce0562df0c210eb2/consequences/README.md ''Consequences of results in the OpenAI mathematics release''], entry 6, in the GitHub repository aster2024/openai-math-followups || Use (conditional consequences) || Combining the nonamenability of ''F'' with published results of Le Boudec and Matte Bon; Breuillard, Kalantar, Kennedy and Ozawa; and Bartholdi and Kielak: Thompson's group ''T'' is C*-simple (the list names this the Haagerup–Olesen question) and its reduced C*-algebra has a unique tracial state; ''F'', Aut(''F'') and Comm(''F'') are C*-simple; for every field ''K'' the group algebra ''K''[''F''] is a domain but not an Ore domain; and Myhill's property fails over ''F''. The entry notes that the C*-simplicity of ''T'' and of ''F'' also appears as entry GR-5 of the catalogue [https://github.com/atimics/math2/blob/fcaf37abb2a57c1aaace218546c6388c673838f1/docs/COROLLARIES.md atimics/math2] (8 October 2026). || The list states that each entry is conditional on the release theorem it names.
|-
| 2026-10-08 || [https://github.com/mathvet/mathvet MathVet] (GitHub; review of openai/math at commit <code>adc7f12</code>, tables at repository commit <code>cd0063d</code>, 13 October 2026) || Review (fidelity of the formal statements) || Rates this family "full": the Lean statement is the nonamenability of ''F'', with ''F'' defined by dyadic breakpoints and slopes that are powers of 2 and with composition as the group law. Its per-challenge table records a sandboxed Comparator pass for the challenge <code>ThompsonNonamenability</code> (8 October 2026). || The review states that "full" means the Lean states the headline and "does not say the mathematics is correct beyond what Lean checked"; it describes itself as a model-assisted reading without a referee yet.
|-
| 2026-10-13 || S.-h. Kim, Y. Lodha, ''A note on the proof of nonamenability of F'' ([https://arxiv.org/abs/2610.18360 arXiv:2610.18360], v1) || Exposition || The abstract states that the note provides "a human written exposition of OpenAI's recent proof that Thompson's group F is nonamenable". || arXiv v1.
|-
| 2026-10-13 || [https://arxiv.org/abs/2609.29399 arXiv:2609.29399] (v2), R. Arimoto; v2 titled ''C*-selflessness of groups of homeomorphisms'' || Use (conditional) || The main theorem implies that Thompson's group ''T'' is C*-selfless if and only if ''F'' is non-amenable. Version 2 (13 October 2026) notes that one of OpenAI's manuscripts states that ''F'' is non-amenable and, based on this result, concludes that ''T'' is C*-selfless. Version 1 was posted on 24 September 2026; the author discloses use of ChatGPT in developing ideas and proofs. || arXiv v2.
|}

== Notes ==
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]