---
id: "2026-10-29-kervaire-conjecture-openai-kourovka-confirmed-bogachev"
url: "https://postcutoff.com/e/2026-10-29-kervaire-conjecture-openai-kourovka-confirmed-bogachev/"
as_of: "2026-10-30T13:42:00+01:00"
date: "2026-10-29"
date_precision: day
category: science
importance: 4
confidence: high
status: [Result confirmed]
verification: Lean-verified
sources: 6
editor: human editor
human_review: null
version: "2026-10-30"
---

As of: 2026-10-30 13:42 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-29-kervaire-conjecture-openai-kourovka-confirmed-bogachev/

# Kervaire's conjecture: OpenAI proof confirmed in the Kourovka Notebook

Full title: Kourovka Notebook marks OpenAI's proofs of Kervaire's conjecture and the Kervaire–Laudenbach conjecture as solved, 'AI solution confirmed by N. V. Bogachev'; its update lists 48 confirmed and 35 unverified AI solutions

The update file of the 21st edition of the Kourovka Notebook (headed "November 2026 update", at kourovkanotebookorg.wordpress.com, read 29–30 Oct) marks Problem 17.94 (Mycielski: can ℤ ∗ G be the normal closure of one element? — the negation of Kervaire's conjecture) answered "No, it cannot", and Problem 18.87(a) (Roman'kov: the Kervaire–Laudenbach conjecture) "proved", both with "AI solution confirmed by N. V. Bogachev" and OpenAI preprints hosted by the Notebook. These are the companion papers of OpenAI family 256 ('The Kervaire Theorem for Groups', 24 Sept; 'Nonsingular systems of equations over arbitrary groups', 5 Oct), whose Lean development covers the single-relator case. The Notebook's update marks unconfirmed AI solutions with ♯ and asks readers to check them.

## 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.

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 182 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 121 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 212 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 151 days after its cutoff

## Sources

1. [Kourovka Notebook, 21st edition update file (November 2026 update)](https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/10/21upd-3.pdf) (kourovkanotebookorg.wordpress.com, official)
2. [OpenAI preprint hosted by Kourovka: Problem 17.94](https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/10/17-94.pdf) (kourovkanotebookorg.wordpress.com, paper)
3. [OpenAI preprint hosted by Kourovka: Problem 18.87](https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/10/18-87.pdf) (kourovkanotebookorg.wordpress.com, paper)
4. [openai/math: The Kervaire Theorem for Groups (24 Sept 2026)](https://github.com/openai/math/blob/main/preprints/The-Kervaire-Theorem-for-Groups-September-24-2026/The-Kervaire-Theorem-for-Groups-September-24-2026.pdf) (github.com, paper)
5. [openai/math: Nonsingular systems of equations over arbitrary groups (5 Oct 2026)](https://github.com/openai/math/blob/main/preprints/Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026/nonsingular-systems-over-arbitrary-groups.pdf) (github.com, paper)
6. [openai/math Lean scope for family 256](https://github.com/openai/math/blob/main/lean/docs/256.md) (github.com, code)

## Changes

- 2026-10-30 (filed): Created from lead; Kourovka update PDF read and AI-solution phrases counted (science agent)

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