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 \(\mathbb R_{\gt 0}\). (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
The two independent assessments disagree on how much of the result described above is expressed by the Lean statements. Read the assessments
In An isomorphism of the free group factors, 23 September 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
There exists a unital normal trace-preserving *-isomorphism
No Lean statement in the release is paired with this theorem.
In An isomorphism of the free group factors, 23 September 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For every integer n\geq3 there is a unital trace-preserving normal *-isomorphism
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. The statements above share the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.
Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.
OpenAI released an abridged summary of the model’s reasoning for this family: Isomorphism of free group factors
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
InterpolatedFactorsThe paper states
In An isomorphism of the free group factors, 23 September 2026
Let M=L(\F_2) with its canonical trace. For every integer n\geq2 there are a self-adjoint n-tuple X^{(n)} and a self-adjoint 2n-tuple Y^{(n)} such that
and
Consequently, none of \delta,\delta_0,\delta^*,\delta^\star is invariant under changing a finite self-adjoint W^*-generating tuple of a tracial von Neumann algebra. Each takes unbounded values on such generating tuples of the single factor L(\F_2).
The words they share: canonical, star, single, tracial, adjoint, self, von, algebra, neumann, integer, trace, factor.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
InterpolatedFactors.lean, the release lists it as a supporting statement.
theorem allInterpolatedIsomorphic (r s : ℝ≥0∞) (hr : 1<r) (hs : 1<s) : Nonempty (NormalTracialEquiv (interpolatedTrace r) (interpolatedTrace s) (interpolatedTopology r) (interpolatedTopology s)) := by sorryThis file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
MathVet and cert-machine disagree on how much of the advertised result the Lean statements express. MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”); cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
Their note: allInterpolatedIsomorphic (r s : ℝ≥0∞) (hr : 1 < r) (hs : 1 < s) : Nonempty (NormalTracialEquiv (interpolatedTrace r) (interpolatedTrace s) (interpolatedTopology r) (interpolatedTopology s)): every pair of interpolated free group factors L(F_r), L(F_s) with r,s ∈ (1,∞] (integer ranks via groupVonNeumann (FreeGroup (Fin n)), FreeGroup ℕ for ∞, and non-integer r as the corner of L(F_2) ⊗ B(ℓ²) by a projection of trace 1/√(r-1)) is related by a *-isomorphism that preserves the canonical trace and is ultraweakly bicontinuous. The summary’s statement L(F_2) ≅ L(F_3) and all interpolated factors; the fundamental-group conclusion is a consequence, not stated.
Definitions the group asks readers to compare with the paper: Hand-built II_1 factor models: groupVonNeumann G := centralizer (centralizer (leftTranslations G)) (double commutant of left translations), canonicalTrace, ultraweakOperatorTopology := ⨅ ξ η, induced (T ↦ Σ' k ⟨η k, T (ξ k)⟩), corners Corner p hp and selectedProjection a := Classical.epsilon (stabilizedProjectionTrace p = ofReal a) (junk default projection 0 if none existed, which does not happen in L(F_2)⊗B(ℓ²)); standard but not Mathlib’s VonNeumannAlgebra-with-trace.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
The gap they name: The fundamental-group clause is not stated; it is a (short, unformalized) consequence of the stated isomorphisms.
“NARROWER — DECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note says so.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
OpenAI’s own note on what its Lean covers
The free group factor problem asks whether the von Neumann algebras of free groups of different ranks are isomorphic. The formalization proves that interpolated free group factors with any parameters $r,s>1$, including the infinite parameter, are normally trace-preservingly isomorphic. The paper’s fundamental-group conclusion is a further consequence rather than a separate selected statement here.
Isomorphism of all interpolated free group factors InterpolatedFactors
Unreviewed AI draft: states a paper’s theorem as the paper does. States that all interpolated free group factors L(F_r), 1 < r ≤ ∞, are isomorphic, which contains the main theorem L(F₂) ≅ L(F₃) and the rank-step theorem.
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
InterpolatedFactorsThree reports, all at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original report |
|---|---|---|---|---|
| cert-machine | “CERTIFIED” | Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
| Neal McBurnett, openai-math-lean-tooling | “accepted” | Re-run through Comparator with Lean’s kernel: accepted (the check took 287 s). | 10 October 2026 | Original report |
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source
Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
This family’s papers cite no paper elsewhere in the release.
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Would win a Fields Medal”. The page says that nothing on it is verified.
Author not stated. Source
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Why this passage is shown
Theorem 1.2
In An isomorphism of the free group factors, 23 September 2026
Why this passage is shown
Theorem 1.1 (Isomorphism of consecutive ranks)
In An isomorphism of the free group factors, 23 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, InterpolatedFactors
Proof-verification report
MathVet, InterpolatedFactors
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, InterpolatedFactors
Lean statement
InterpolatedFactors