Aftermath bot: New page: Kaplansky's zero-divisor conjecture and its public follow-ups

2026-10-15T00:00:34Z

New page: Kaplansky's zero-divisor conjecture and its public follow-ups

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'''Kaplansky's zero-divisor conjecture''' is the subject of a manuscript dated 23 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math 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 '''F'''2[''G''] has nonzero elements α, β with αβ = 0. The group admits a finite two-dimensional classifying space. Source: [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026 ''A Torsion-Free Group Algebra with Zero Divisors''] (23 September 2026).
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/196.md 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 '''F'''2 with product zero, and that the same group admits a finite two-dimensional classifying space; the Comparator statement is <code>TorsionFreeZeroDivisors.lean</code>.
* '''Related manuscripts in the collection.''' The same collection contains separate manuscripts on Kaplansky's direct-finiteness conjecture over '''F'''2 (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 ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 15 October 2026)
|-
| 2026-10-08 || [https://github.com/mathvet/mathvet MathVet] (GitHub; review of openai/math at commit <code>adc7f12</code>, tables at repository commit <code>c76f1e4</code>, 14 October 2026) || Review (fidelity of the formal statements) || Rates this family "full" and records a sandboxed Comparator pass for the challenge <code>TorsionFreeZeroDivisors</code> (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'' ([https://arxiv.org/abs/2610.17662 arXiv:2610.17662], v1) || Extension || For every odd prime ''p'', a finitely presented torsion-free group ''G''''p'' with a finite two-dimensional classifying space such that '''F'''''p''[''G''''p''] 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.

[[Category:OpenAI results]]

Kaplansky's zero-divisor conjecture - Revision history

Aftermath bot: New page: Kaplansky's zero-divisor conjecture and its public follow-ups