The free group factor isomorphism problem

For a discrete group $G$ let $L(G)=\lambda(G)''$ be its group von Neumann algebra on $\ell^2(G)$ , and let $\mathbb F_n$ be the free group on $n$ generators. Murray and von Neumann showed the free group factors are not hyperfinite but could not distinguish their ranks. Voiculescu, Radulescu and Dykema built interpolated factors $L(\mathbb F_r)$ , $1<r\le\infty$ , and proved they are either all isomorphic or pairwise non-isomorphic. Are $L(\mathbb F_m)$ and $L(\mathbb F_n)$ isomorphic for distinct $m,n\ge 2$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Operator algebras: II_1 factors and free probability
Posed by: Richard V. Kadison, as credited in Dykema (1994); the factors go back to Murray and von Neumann (1943)
Year posed: —
Years open: —
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 68 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Claims that $L(\mathbb F_2)\cong L(\mathbb F_3)$ , by an explicit iterative construction inside $L(\mathbb F_{n+1})$ of a freely generating Haar $n$ -tuple, built from small trace-preserving automorphisms driven by a polynomial flow. With the classical interpolated-factor dichotomy, all $L(\mathbb F_r)$ , $1<r\le\infty$ , are isomorphic and their fundamental group is $\mathbb R_{>0}$ . It also shows free entropy dimension depends on the generating tuple. It does not identify $L(\mathbb F_2)$ with other non-free group factors.

What the AI did

The OpenAI math release (github.com/openai/math) states that its results were produced by an unreleased internal OpenAI model, with on average about three hours of ChatGPT Pro thinking compute per result, under one fixed procedure applied to roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. The manuscript is credited to OpenAI alone and names no human author. The release also publishes an abridged summary of the model's reasoning for this family (reasoning_traces/free-group-factor-isomorphism.pdf).

Verification

No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 of the TeX source read against the posed problem; a trace-preserving normal *-isomorphism $L(\mathbb F_n)\cong L(\mathbb F_{n+1})$ for $n\ge3$ , hence $L(\mathbb F_2)\cong L(\mathbb F_3)$ , answering it affirmatively with the Radulescu-Dykema dichotomy. Lean-checked on the release's own Comparator challenge InterpolatedFactors (OAI.FreeGroupFactorMain.Interpolation.allInterpolatedIsomorphic) together with its solution module, both present at the pinned commit; the challenge is not listed in the release's formalization catalogue, the statement was read here but not independently audited, and the development was not rebuilt here.

The free group factor isomorphism problem

What was actually shown

What the AI did

Verification

Sources

Discussion