Kaplansky's zero-divisor conjecture
Kaplansky's zero-divisor conjecture is the subject of a manuscript dated 23 September 2026 in OpenAI's public mathematics collection (github.com/openai/math). The conjecture asserts that the group algebra of a torsion-free group over a field has no zero divisors. The manuscript claims a counterexample in characteristic two, and a human follow-up extends the construction to every odd characteristic. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
The original result
- Claim. There is a finitely presented torsion-free group G whose group algebra F2[G] has nonzero elements α, β with αβ = 0. The group admits a finite two-dimensional classifying space. Source: A Torsion-Free Group Algebra with Zero Divisors (23 September 2026).
- Formalization, as described by the repository. The scope note lean/docs/196.md states that the Lean formalization constructs the finitely presented torsion-free group and the two nonzero elements of its group algebra over F2 with product zero, and that the same group admits a finite two-dimensional classifying space; the Comparator statement is
TorsionFreeZeroDivisors.lean. - Related manuscripts in the collection. The same collection contains separate manuscripts on Kaplansky's direct-finiteness conjecture over F2 (a finitely presented torsion-free group whose group algebra is not directly finite) and on a finitely generated Eilenberg–Ganea counterexample; they are not covered on this page.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 15 October 2026) |
|---|---|---|---|---|
| 2026-10-08 | MathVet (GitHub; review of openai/math at commit adc7f12, tables at repository commit c76f1e4, 14 October 2026) |
Review (fidelity of the formal statements) | Rates this family "full" and records a sandboxed Comparator pass for the challenge TorsionFreeZeroDivisors (8 October 2026); its note records the formal statement as the existence of a finitely presented, torsion-free group with a finite two-dimensional classifying space and nonzero α, β in the group algebra over Z/2 with αβ = 0, with torsion-freeness in the usual sense. |
The review states that "full" means the Lean states the headline and "does not say the mathematics is correct beyond what Lean checked"; it describes itself as a model-assisted reading without a referee yet. |
| 2026-10-14 | S. V. Ivanov, Zero divisors in group algebras of torsion-free groups in odd characteristic (arXiv:2610.17662, v1) | Extension | For every odd prime p, a finitely presented torsion-free group Gp with a finite two-dimensional classifying space such that Fp[Gp] contains nonzero elements with product zero; the abstract describes a modified design with additional letters making every cross-intersection congruent to 1 modulo p, a squared-word decay estimate proved with an explicit test vector, and the geometric part taken from the characteristic-two construction. Together with the case p = 2, the conjecture therefore fails in every positive characteristic; the characteristic-zero case is stated to remain open. | arXiv v1. |
Notes
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.