View source for Isomorphism of the free group factors

The '''free group factor isomorphism problem''' asks whether the group von Neumann algebras ''L''(𝔽<sub>''n''</sub>) of free groups of different ranks are isomorphic. A manuscript dated 23 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math 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''(𝔽<sub>2</sub>) and ''L''(𝔽<sub>3</sub>) are isomorphic as tracial von Neumann algebras. By the classical free group factor alternative, all interpolated free group factors, including ''L''(𝔽<sub>∞</sub>), are then isomorphic and have fundamental group ℝ<sub>&gt;0</sub> (abstract).
* '''Source.''' [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/An-isomorphism-of-the-free-group-factors-September-23-2026 ''An isomorphism of the free group factors''] in the openai/math repository (commit <code>1581d4c</code>).
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/287.md lean/docs/287.md] states that the formalization proves that interpolated free group factors with any parameters ''r'', ''s'' &gt; 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 <code>InterpolatedFactors.lean</code>.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 16 October 2026)
|-
| 2026-10-08 || [https://github.com/mathvet/mathvet MathVet] (GitHub; review of openai/math, tables at repository commit <code>c76f1e4</code>, 14 October 2026) || Review (fidelity of the formal statements) || Rates this family's challenge <code>InterpolatedFactors</code> "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'', [https://arxiv.org/abs/2610.19590 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, [https://arxiv.org/abs/2609.11074 arXiv:2609.11074] (September 2026), proves that surface-group factors are free group factors ''L''(𝔽<sub>2''g''−1</sub>). 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.

[[Category:OpenAI results]]