Constructs a finitely presented torsion-free group G whose group algebra \(\mathbb F_2[G]\) has nonzero zero divisors, disproving Kaplansky’s zero-divisor conjecture. The group has a finite two-dimensional classifying space. (OpenAI’s Lean note)
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Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments
The paper states
In A Torsion-Free Group Algebra with Zero Divisors, 23 September 2026
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
There exist a finitely presented torsion-free group G and nonzero elements \alpha,\beta\in\mathbb F_2[G] such that \alpha\beta=0. Moreover, G admits a finite two-dimensional classifying space.
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TorsionFreeZeroDivisorsThe paper states
In A Torsion-Free Group Algebra with Zero Divisors, 23 September 2026
There exist a finitely presented torsion-free group G and nonzero elements \alpha,\beta\in\mathbb F_2[G] such that \alpha\beta=0. Moreover, G admits a finite two-dimensional classifying space.
The words they share: finite, classifying, dimensional, presented, two, group, finitely, space, moreover, torsion, free, algebra, nonzero, elements, beta, alpha.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
TorsionFreeZeroDivisors.lean, the release lists it as a main result.
def MainTheorem : Prop := ∃ (G : Type) (inst : Group G), letI : Group G := inst Group.IsFinitelyPresented G ∧ TorsionFree G ∧ HasFiniteTwoDimensionalClassifyingSpace G ∧ ∃ α β : MonoidAlgebra (ZMod 2) G, α ≠ 0 ∧ β ≠ 0 ∧ α * β = 0theorem main : MainTheorem := 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.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
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: MainTheorem : ∃ G, Group.IsFinitelyPresented G ∧ TorsionFree G ∧ HasFiniteTwoDimensionalClassifyingSpace G ∧ ∃ α β : MonoidAlgebra (ZMod 2) G, α ≠ 0 ∧ β ≠ 0 ∧ α * β = 0. TorsionFree is the usual g^n = 1 → g = 1; the K(G,1) clause asks for a finite 2-dimensional Hausdorff path-connected Mathlib CWComplex X with G ≃* FundamentalGroup X x and a contractible covering space, which is the standard meaning.
“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 or imply the whole advertised result (their term: “MATCHES”).
“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.
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
Kaplansky’s zero-divisor conjecture asserts that the group algebra of a torsion-free group over a field has no zero divisors. The formalized result constructs a finitely presented torsion-free group $G$ and nonzero elements $\alpha,\beta\in\mathbb F_2[G]$ with $\alpha\beta=0$, giving a counterexample.
The same group admits a finite two-dimensional classifying space $K(G,1)$.
Torsion-free group-algebra counterexample TorsionFreeZeroDivisors
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem including its second clause: a finitely presented torsion-free group with zero divisors in F₂[G] that admits a finite two-dimensional classifying space.
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TorsionFreeZeroDivisorsTwo 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 |
These reports examine claims in the papers using HOL Light, a different proof assistant and exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
The paper’s two exact rationals and both bounds < 149/150
HOL Light card 196
A Torsion-Free Group Algebra with Zero Divisors, 23 September 2026
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This family’s papers cite no paper elsewhere in the release.
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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 counterexample and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
Author not stated. Source
Moritz Firsching, anxkhn, Paul Lezeau, Formal Conjectures: mark problems solved in the OpenAI math release and link formal proofs, 11 October 2026. Source
A Formal Conjectures maintainer, after source inspection, links the group algebra counterexample to Kaplansky’s zero-divisor conjecture, stated as a negation.
Pico, openai-math-counterexamples: do the counterexamples in openai/math hold when someone else computes them?, 9 October 2026. Source
The group is built from random matchings for sufficiently large parameters, so the two group-algebra elements are never written down.
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Why this passage is shown
Theorem 1.1
In A Torsion-Free Group Algebra with Zero Divisors, 23 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, TorsionFreeZeroDivisors
Proof-verification report
MathVet, TorsionFreeZeroDivisors
Report using another proof or calculation method
The paper’s two exact rationals and both bounds < 149/150
Proofs of selected steps in HOL Light, Ben Knill, openai-math-hol-cards
68a1228Report using another proof or calculation method
A Torsion-Free Group Algebra with Zero Divisors, 23 September 2026
Exact calculations for finite claims, cert-machine
9814e4cLean statement
TorsionFreeZeroDivisors