---
title: "A counterexample to Kaplansky’s zero-divisor conjecture"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 196; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "algebra"
  - "lean4"
  - "formalized"
seed_rank: 2051
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 8
lineage: ai-mathematical-reasoning
cites:
  - title: "A Torsion-Free Group Algebra with Zero Divisors"
    url: "https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# A counterexample to Kaplansky’s zero-divisor conjecture

## One-sentence takeaway

OpenAI's result family 196 (Algebra) claims: 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.

## Why it matters here

Pure mathematics with no direct engine use; it is in the library as part of the openai/math corpus, which records what an AI research model claims to have proved. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems. The Lean development makes the claim machine-checkable, which is the pattern Broadside cares about for agentic verification loops.

## Key ideas

- Family summary (repo): The group has a finite two-dimensional classifying space.
- *A Torsion-Free Group Algebra with Zero Divisors*: We disprove Kaplansky's zero-divisor conjecture by constructing a finitely presented torsion-free group G for which $\mathbb F_2[G]$ has nonzero zero divisors. The group admits a finite two-dimensional classifying space.
- Lean scope (lean/docs/196.md): 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.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: formalized in part. The repo has a scope note listing 1 Comparator challenge statement(s), and that note defines which statements are covered. Do not read the Lean coverage as the full content of every manuscript in the family.
- A grep of the repo's `lean/OAI` tree found no `sorry` and no top-level `axiom` declarations. The library was not compiled for this card, and Comparator was not run.

## Links

- Manuscript: [A Torsion-Free Group Algebra with Zero Divisors](https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026/paper.pdf)
- Lean scope: https://github.com/openai/math/blob/main/lean/docs/196.md
- Comparator statement (Torsion-free group-algebra counterexample): https://github.com/openai/math/blob/main/lean/ComparatorChallenges/TorsionFreeZeroDivisors.lean
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math