OpenAI claims Baum–Connes and Kadison–Kaplansky counterexamples
Awaiting review
Importance: historic (5 of 5)The takeaway
Three manuscripts in OpenAI’s 6 Oct 2026 math release (family 285, dated 23 Sept) claim the first counterexamples to the Baum–Connes conjecture without coefficients, one of them for a torsion-free group, and a torsion-free group whose reduced C*-algebra has a nontrivial projection, which would refute the Kadison–Kaplansky conjecture. Until now, counterexamples were known only for the version with coefficients. On 26 Oct Guoliang Yu, a leading specialist, posted his own construction of projections of every trace in (0,1), saying it develops arguments from OpenAI’s irrational-trace paper. None of the three papers has a Lean proof.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Historic (5 of 5)
- Last verified
- 29 October 2026
Your AI and this story
- GPT-6 Astra159 days after its cutoff
- Claude Opus 5.598 days after its cutoff
- Gemini 4 Argon189 days after its cutoff
- Grok 4.7128 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 98 days before it.
Key facts
- Injectivity paper (abstract, verbatim): ‘We disprove rational injectivity of the coefficient-free reduced Baum–Connes assembly map for torsion-free groups. Specifically, we construct a finitely generated torsion-free discrete group whose degree-zero assembly map has a kernel class of infinite order.’
- Kadison–Kaplansky paper (verbatim): ‘We construct a finitely generated torsion-free discrete group whose reduced group C∗-algebra contains a projection other than zero and the identity. This disproves the Kadison–Kaplansky projection conjecture.’
- Irrational-trace paper (verbatim): ‘We construct a finitely generated discrete group whose reduced group C∗-algebra contains a projection of irrational canonical trace. Its K0-class lies outside the image of the coefficient-free reduced Baum–Connes assembly map.’ Lück’s trace theorem puts every class in the image at a rational trace, so one such projection breaks surjectivity
- Context: the Baum–Connes conjecture (Baum, Connes, Higson, 1994 form) computes the K-theory of reduced group C*-algebras and implies the Novikov and, for torsion-free groups, the Kadison–Kaplansky conjectures; known counterexamples (Higson, Lafforgue and Skandalis, 2002) concern the version with coefficients
- Guoliang Yu, ‘Projections in reduced group C*-algebras with prescribed trace values in (0,1)’ (arXiv 2610.35674, 26 Oct, 57 pp.): for every r ∈ (0,1), a finitely generated group G_r and a projection of trace r; for irrational r its class ‘is not in the image of the reduced Baum–Connes map’; the groups have torsion and satisfy the maximal strong Novikov conjecture. Acknowledgement: ‘the geometric and the spectral techniques in this article are developed from arguments in the OpenAI preprint [27]… All proofs are solely developed by the author.’
- Erik van Erp (Dartmouth), who had run ChatGPT 5.6 in a loop searching for Baum–Connes counterexamples before the release, told The Dartmouth (20 Oct) he stopped after the introductions of the three papers, calling them ‘impenetrable’; unless mathematicians ‘can extract new insights’, the release is a ‘non-event’
- Status: no Lean scope note for family 285; no withdrawal or revision in history.md. Ying Xie’s survey of the release (arXiv 2610.17090) notes the three papers share one architecture, so a defect could affect the whole family
What happened
The Baum–Connes conjecture predicts that the K-theory of the reduced C*-algebra of a group can be computed from geometry, through an “assembly map”. It implies the Novikov conjecture on higher signatures and, for torsion-free groups, the Kadison–Kaplansky conjecture that the reduced C*-algebra has no projections other than 0 and 1. It is known for large classes of groups. Counterexamples were known only for the stronger version “with coefficients”, found by Higson, Lafforgue and Skandalis in 2002 using Gromov’s monster groups.
OpenAI’s release claims three counterexamples to the version without coefficients, all dated 23 Sept. One is a torsion-free group for which assembly fails to be rationally injective. Another is a group with a projection of irrational trace, which cannot lie in the image of assembly. The third is a torsion-free group with a nontrivial projection, which contradicts Kadison–Kaplansky.
On 26 Oct Guoliang Yu (Texas A&M), one of the main contributors to the Baum–Connes and Novikov programmes, posted a 57-page paper that builds groups with projections of every trace between 0 and 1. For irrational traces these are again outside the image of assembly. His groups have torsion, so his paper does not address the torsion-free or Kadison–Kaplansky claims. He credits OpenAI’s preprint for the geometric and spectral techniques and says the proofs are his own.
Why it matters
The coefficient-free Baum–Connes conjecture is a central conjecture of noncommutative geometry, and the Kadison–Kaplansky conjecture is a well-known open problem in the same area. If these papers hold, both are false, with direct consequences for the K-theory of group C*-algebras. Yu’s independent construction makes the surjectivity counterexample the best-supported part so far. The torsion-free claims still rest on unrefereed AI manuscripts without formal proofs.
What is disputed or not yet verified
| Verification | Unrefereed manuscripts without Lean proofs; surjectivity counterexample independently re-proved (stronger, groups with torsion) in an unrefereed expert preprint |
|---|
Sources
7 sources from 3 sites. Numbers match the chips in the text.
7 sources: 5 primary, 1 press, 1 reaction
Primary
- OpenAI: A torsion-free counterexample to reduced Baum–Connes injectivity (PDF)github.com, paper
- OpenAI: A torsion-free counterexample to the Kadison–Kaplansky projection conjecture (PDF)github.com, paper
- OpenAI: An irrational-trace counterexample to reduced Baum–Connes (PDF)github.com, paper
- Guoliang Yu: Projections in reduced group C*-algebras with prescribed trace values in (0,1) (arXiv 2610.35674)arxiv.org, paper
- Ying Xie: What the Machine Invented (arXiv 2610.17090, survey of the release)arxiv.org, paper
Press
- The Dartmouth: Dartmouth mathematicians weigh AI’s role (Erik van Erp on the Baum–Connes papers, 20 Oct)thedartmouth.com, press
Reactions
- oai-math-revised: community notes on the injectivity manuscriptgithub.com, discussion
Changes
- Related entry added for the Kadison similarity claim (family 288), science agent run 2
- Filed (families 251–378; three manuscripts and CONTENTS, Yu’s arXiv paper read, The Dartmouth article, oai-math-revised notes)