Isomorphism of the free group factors
The free group factor isomorphism problem asks whether the group von Neumann algebras L(𝔽n) of free groups of different ranks are isomorphic. A manuscript dated 23 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims an affirmative answer. This page summarizes the published claim and lists publicly available follow-up work.
The original result
- Claim. L(𝔽2) and L(𝔽3) are isomorphic as tracial von Neumann algebras. By the classical free group factor alternative, all interpolated free group factors, including L(𝔽∞), are then isomorphic and have fundamental group ℝ>0 (abstract).
- Source. An isomorphism of the free group factors in the openai/math repository (commit
1581d4c). - Formalization, as described by the repository. The scope note lean/docs/287.md states that the formalization proves that interpolated free group factors with any parameters r, s > 1, including the infinite parameter, are normally trace-preservingly isomorphic. The fundamental-group conclusion is described as a further consequence, not a separate selected statement. The Comparator statement is
InterpolatedFactors.lean.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 16 October 2026) |
|---|---|---|---|---|
| 2026-10-08 | MathVet (GitHub; review of openai/math, tables at repository commit c76f1e4, 14 October 2026) |
Review (fidelity of the formal statements) | Rates this family's challenge InterpolatedFactors "full" and records a sandboxed Comparator pass. |
MathVet states that it does not claim any theorem false and reports what was machine-checked as stated. |
| 2026-10-15 (listed) | D. Shlyakhtenko, On the Isomorphism of Semicircular Algebras, arXiv:2610.19590 | Extension | "Building on OpenAI's proof of isomorphism of free group factors", gives a self-contained proof of the corresponding result for C*-algebras: the isomorphism class of a C*-algebra generated by a free semicircular n-tuple does not depend on n ∈ {2, 3, …, ∞}. Reduced free products of n copies of the Jiang–Su algebra are likewise isomorphic for different n. Reduced free group C*-algebras of different ranks are not isomorphic (Pimsner–Voiculescu), but the paper shows strong free absorption properties for them. The key tool is an automorphism approximation property for inclusions of C*- and W*-algebras. | arXiv preprint. |
Related work
- D. Shlyakhtenko, arXiv:2609.11074 (September 2026), proves that surface-group factors are free group factors L(𝔽2g−1). The OpenAI manuscript cites it.
Notes
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.