Kervaire’s conjecture: OpenAI proof confirmed in the Kourovka Notebook
Result confirmed
Importance: major (4 of 5)The takeaway
Kervaire’s conjecture says that adding one generator and one relation to a nontrivial group never gives the trivial group. It was known for torsion-free groups (Klyachko 1993) and residually finite groups (Gerstenhaber–Rothaus 1962). OpenAI’s 23–24 Sept release (family 256) claimed it in general, and also the Kervaire–Laudenbach conjecture on independent systems of equations (Howie’s form). The Kourovka Notebook’s update file now marks problems 17.94 and 18.87(a) solved: ‘AI solution confirmed by N. V. Bogachev’. The same file lists 48 AI solutions confirmed by named mathematicians and 35 still unverified, including OpenAI’s claim of infinite finitely presented periodic groups (8.52).
Status
- Claim
Result confirmed
- Our reporting
- High confidence
- Verification
- Lean-verified
- Importance
- Major (4 of 5)
- Last verified
- 30 October 2026
Your AI and this story
- GPT-6 Astra182 days after its cutoff
- Claude Opus 5.5121 days after its cutoff
- Gemini 4 Argon212 days after its cutoff
- Grok 4.7151 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 121 days before it.
Key facts
- Problem 17.94 (Kourovka, ‘Well-known problem’, J. Mycielski): ‘Can the free product ℤ ∗ G of the infinite cyclic group ℤ and a nontrivial group G be the normal closure of a single element? Editors’ comments: the negation is also known as Kervaire’s conjecture. It was proved for torsion-free groups (A. Klyachko … 1993) and for residually finite groups (M. Gerstenhaber, O. S. Rothaus … 1962).’ Entry now: ‘∗ No, it cannot (AI solution confirmed by N. V. Bogachev: OpenAI, Preprint, 2026, …/2026/10/17-94.pdf)’
- Problem 18.87(a) (V. A. Roman’kov), the Kervaire–Laudenbach conjecture: ‘every independent system of equations with coefficients in an arbitrary group G has a solution in some overgroup’; entry: ‘∗ a) The conjecture is proved (AI solution confirmed by N. V. Bogachev: OpenAI, Preprint, 2026, …/2026/10/18-87.pdf)’. Parts b) (nilpotent) and c) (solvable) are not marked
- OpenAI family 256 (openai/math): ‘Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie’s conjecture … A companion proves Kervaire’s conjecture’; Lean scope lean/docs/256.md covers the single-relator case
- Policy of the Notebook (update file, verbatim): ‘Problems are marked with ♯ in those cases where some Artificial Intelligence (AI) system proposed a solution that has not yet been confirmed by mathematicians. Such a problem should be regarded as unsolved … in the first case a solution will be marked as a proper solution with ∗, and in the second altogether deleted’
- Tally in the same file (our count of the phrases): 48 ‘AI solution confirmed by …’ and 35 ‘unverified AI solution’. Confirmed contributors: Achyuth Jayadevan 23 (Zenodo preprints, confirmed by e.g. A. A. Buturlakin, O. G. Kharlampovich, M. Dixon, J. S. Wilson), P. Monticone 9, Nilradical 5, OpenAI 3 (17.94, 18.87(a), and 12.20 Thompson’s group F nonamenable, confirmed by S.-H. Kim and Y. Lodha), others 1–2 each. Unverified include OpenAI on 8.52 (Olshanskii’s ‘well-known problem’: infinite finitely presented periodic groups exist) and Jayadevan on 8.54(b)
- Verification status: Kervaire and Kervaire–Laudenbach confirmed by a named group theorist and accepted by the Notebook’s editors; OpenAI preprints not refereed; Lean only for the single-relator case
What happened
Kervaire’s conjecture asks whether a nontrivial group can be killed by adding one new generator and one relation. It was proved in the 1960s for residually finite groups and in 1993 for torsion-free groups, but stayed open in general. OpenAI’s September release claimed it, together with a stronger statement about systems of equations over groups.
The Kourovka Notebook, the standard collection of open problems in group theory, now lists both as solved. Its update file credits OpenAI preprints that the Notebook hosts and says the AI solutions were confirmed by N. V. Bogachev. The same file marks 35 other AI claims as unverified and invites readers to check them.
Why it matters
It is one of the few famous claims from OpenAI’s release that an outside expert has confirmed in public, and the Notebook gives the confirmation an institutional form. The update also shows how much AI work the problem book now processes: 83 AI solutions in one file, most of them from individual users rather than labs.
Sources
6 sources from 2 sites. Numbers match the chips in the text.
6 sources: 6 primary
Primary
- Kourovka Notebook, 21st edition update file (November 2026 update)kourovkanotebookorg.wordpress.com, official
- OpenAI preprint hosted by Kourovka: Problem 17.94kourovkanotebookorg.wordpress.com, paper
- OpenAI preprint hosted by Kourovka: Problem 18.87kourovkanotebookorg.wordpress.com, paper
- openai/math: The Kervaire Theorem for Groups (24 Sept 2026)github.com, paper
- openai/math: Nonsingular systems of equations over arbitrary groups (5 Oct 2026)github.com, paper
- openai/math Lean scope for family 256github.com, code
Changes
- Filed from lead; Kourovka update PDF read and AI-solution phrases counted (science agent)