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An isomorphism of the free group factors
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Abstract
We solve the free group factor isomorphism problem affirmatively by proving that $L(\mathbb F_2)$ and $L(\mathbb F_3)$ are isomorphic as tracial von Neumann algebras. The classical free group factor alternative then implies that all interpolated free group factors, including $L(\mathbb F_\infty)$, are isomorphic and have fundamental group $\mathbb R_{\gt 0}$.
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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.
Follow-up works
- D. Shlyakhtenko, On the Isomorphism of Semicircular Algebras (arXiv:2610.19590 [math.OA], v1 14 Oct 2026). Gives a self-contained C-algebraic version of the result, showing that the C-algebra of a free semicircular n-tuple and the n-fold reduced free product of Jiang–Su algebras do not depend on n for 2 ≤ n ≤ ∞, and recovers $L(\mathbb F_n)\cong L(\mathbb F_2)$; it follows "the same strategy as OpenAI's paper [9], although implemented differently" without relying on its results. [AI-generated per source]
- Jizhou Guo, Consequences of results in the OpenAI mathematics release, §7: Group von Neumann algebras of surface groups (GitHub research notes, 8 October 2026). Conditionally combines this paper's isomorphisms with Shlyakhtenko's Fuchsian-group-factor preprint to deduce that the factors of all closed orientable surface groups of genus at least two are isomorphic to $L(\mathbb F_2)$.
Formalization
- OpenAI math release, Lean scope note for family 287 (mirror on mathcommunity/oai-math-revised) (lean/docs/287.md, 6 October 2026). States that the formalization covers normal trace-preserving isomorphism of interpolated free group factors for all parameters r, s > 1, including ∞, and links the Comparator statement; the fundamental-group conclusion is not a separately selected statement.
- OpenAI math release, InterpolatedFactors.lean comparator statement (lean/ComparatorChallenges). Lean/Mathlib statement
allInterpolatedIsomorphic, together with its own definitions of group von Neumann algebras, amplifications, corners and interpolated factors; in this file the proof is given assorry.
Related prior literature mentioned by others
- Dima Shlyakhtenko, On the II$_1$ Factors of Fuchsian Groups (arXiv:2609.11074, 10 September 2026). Guo's §7 uses its surface-group-factor identification as the bridge to this paper, and Shlyakhtenko's October follow-up identifies its relative automorphism approximation method as a predecessor; the abstract acknowledges ChatGPT Pro use.
Possibly relevant literature (found by the agent, not checked)
- Kenley Jung, Strongly 1-bounded von Neumann algebras (arXiv:math/0510576, 2005; GAFA 2007). Shows that for strongly 1-bounded algebras the microstates free entropy dimension does not depend on the generating set, which is relevant to the paper's negative answer to Voiculescu's invariance question for $\delta,\delta_0,\delta^*,\delta^\star$ on $L(\mathbb F_2)$.
- Ken Dykema, Free products of hyperfinite von Neumann algebras and free dimension (Duke Math. J. 69 (1993), 97–119). Identifies free products of finite hyperfinite algebras as interpolated free group factors with a free-dimension parameter; these identifications are among those affected by the paper's claim that all interpolated free group factors are isomorphic.
Digests and expositions
- OpenAI Math Digest, An isomorphism of the free group factors (online reading guide, undated). Gives proof diagrams and source references for the polynomial flow, small cocycle, adaptive generating limit, amplification step and free entropy dimension consequences, distinguishing its checks from independent referee certification. [AI-generated per source]
News and blogs
- Azmat, Free Group Factor Isomorphism Problem: How OpenAI’s Proof Absorbs an Extra Generator (BinaryVerse AI, 10 October 2026). Explains the manuscript's generator-absorption construction, including why preserving the limiting Haar distribution and proving generation require separate arguments.
Elsewhere
- VibeMathed: Problem record
- VibeMathed: Problem record
- Emergent Mind: family 287
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:An-isomorphism-of-the-free-group-factors-September-23-2026,
author = {{OpenAI}},
title = {{An isomorphism of the free group factors}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/An-isomorphism-of-the-free-group-factors-September-23-2026/An-isomorphism-of-the-free-group-factors-September-23-2026.pdf}{OAI:An-isomorphism-of-the-free-group-factors-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/287.md. The Lean statements have not been checked by people against the paper's claims.
Family 287: Isomorphism of the free group factors
Resolves the free group factor isomorphism problem: $L(\mathbb F_2)\cong L(\mathbb F_3)$, and hence all interpolated free group factors, including $L(\mathbb F_\infty)$, are isomorphic. Their common factor has fundamental group $\mathbb R_{\gt 0}$.
This is the only paper in this family.