OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves the ℓ1-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over \(\ell^1(G)\) are supported on finitely many finite-order conjugacy classes. The algebraic companion proves the integral Bass trace conjecture and Kaplansky’s idempotent conjecture for torsion-free groups over every commutative unital characteristic-zero domain. (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 independent assessments find that the Lean statements express only part of the result described above, or a narrower statement.Read the assessments
Consequently \tau_{G,*}=\operatorname{rk}_{\epsilon}, with \Z viewed as a subgroup of \C, and
\tau_{G,*}\bigl(K_0(\C G)\bigr)=\Z.
(ii)
For every commutative unital domain R of characteristic zero, every e\in RG satisfying e^2=e is either 0 or 1.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.
One more possible match
Theorem 1.1in The ℓ¹-Bass Conjecture for Discrete Groups, 5 October 2026.
Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We prove the ℓ1-Bass conjecture for every discrete group. The Hattori–Stallings trace of every idempotent matrix over the complex ℓ1 group algebra is supported on finitely many conjugacy classes of finite-order elements.
We prove the complex group-ring Bass trace conjecture for every discrete group: the Hattori–Stallings trace of a finitely generated projective module over its complex group ring is supported on conjugacy classes of finite-order elements. As a consequence, for every torsion-free group G and every commutative unital domain R of characteristic zero, the only idempotents in \(RG\) are 0 and 1. This proves Kaplansky’s idempotent conjecture in characteristic zero.
Two Lean statements in this family
Choose one to see it beside the paper theorem it is paired with.
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.
BassTrace.lean, the release lists it as a main result.
theorembassTraceModules_vanishing_and_support {G : Type u} [_root_.Group G] (x : ModuleK0.{u,v} (MonoidAlgebra ℂ G)) : (∀ g : G, ¬IsOfFinOrder g → HattoriStallings x (ConjClasses.mk g) = 0) ∧ ↑(HattoriStallings x).support ⊆ {c | ∃ g : G, IsOfFinOrder g ∧ ConjClasses.mk g = c} :=bysorry
This 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.
BassTorsionFree.lean, the release lists it as a supporting statement.
theoremtorsion_free_corollary (G : Type*) [Group G] (hG : TorsionFree G) : ((∀ (q : ℕ), 1 ≤ q → ∀ e : Matrix (Fin q) (Fin q) (MonoidAlgebra ℂ G), e ^ 2 = e → matrixTrace e = ((augmentedMatrix e).rank : ℂ)) ∧ traceK0 (G := G) = (Int.castAddHom ℂ).comp augmentationRank ∧ Set.range (traceK0 (G := G)) = Set.range (fun z : ℤ => (z : ℂ))) ∧ (∀ (R : Type*) [CommRing R] [IsDomain R] [CharZero R] (e : MonoidAlgebra R G), e ^ 2 = e → e = 0 ∨ e = 1) := bysorry
This 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.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
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 a weaker statement than the advertised result (their term: “weaker-statement”).
Their note:Headline is the ℓ¹-Bass conjecture over ℓ¹(G), but bassTraceModules_vanishing_and_support is only about the group algebra MonoidAlgebra ℂ G (Hattori-Stallings trace of every K0 class vanishes off finite-order classes and is supported on them); no ℓ¹(G) appears anywhere (docs: ‘proves the complex Bass trace conjecture for every group’; docs title says ℓ¹ but scope does not). The algebraic companion IS stated: torsion_free_corollary gives trace = augmentation rank on K0(ℂ[G]) and ∀ R comm. domain with CharZero, e^2 = e → e = 0 ∨ e = 1 for torsion-free G.
Definitions the group asks readers to compare with the paper:Hand-rolled K0: BassTrace.K0.Group R := FreeAbelianGroup (Idempotent R) ⧸ relations R (relations: e ~ f when a*b = e ∧ b*a = f, plus direct sums) and ModuleK0 of f.g. projective right modules, HattoriStallings into ConjClasses G →₀ ℂ; standard presentations, not Mathlib objects.
“weaker-statement”, in their words: The challenge statements do not imply the headline claim: they state a nontrivially weaker statement.
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 — UNDECLARED”).
The gap they name:The ℓ¹(G) version (the family’s titular result and the October 5 paper) and the integral Bass trace conjecture for K₀(ℤG) are not stated. Only the ℂ[G] Bass conjecture and the torsion-free consequences are.
“NARROWER — UNDECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The formalization proves the complex Bass trace conjecture for every group: the Hattori–Stallings trace of each virtual class of finitely generated projective right $\mathbb C[G]$-modules vanishes on conjugacy classes of infinite-order elements. No finiteness, countability, or geometric hypothesis on $G$ is imposed.
For torsion-free $G$, it identifies the trace on $K_0(\mathbb C[G])$ with the integral augmentation rank and proves the characteristic-zero Kaplansky idempotent conjecture: for every commutative unital domain $R$ of characteristic zero, an idempotent in $R[G]$ is $0$ or $1$.
AI-drafted comparisons, with no recorded human review
Complex Bass trace vanishing and supportBassTrace
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for every group G the Hattori–Stallings trace of projective C[G]-modules vanishes on conjugacy classes of infinite-order elements.
Trace rank and characteristic-zero Kaplansky idempotents for torsion-free groupsBassTorsionFree
Unreviewed AI draft: states a consequence of a paper’s theorem. States the paper’s torsion-free corollary: the trace on K₀ equals the augmentation rank, and Kaplansky’s idempotent conjecture holds in characteristic zero.
Who ran the proof-verification program?
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.
Complex Bass trace vanishing and supportBassTrace
Two 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.
What these papers cite in the release, and what cites them
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.
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
Explain a result, compare statements or propose a correction
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.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
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 “Major math journal”. The page says that nothing on it is verified.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 207The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
The challenge statements do not imply the headline claim: they state a nontrivially weaker statement.
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. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.