---
id: "2026-10-06-openai-baum-connes-kadison-kaplansky-counterexamples"
url: "https://postcutoff.com/e/2026-10-06-openai-baum-connes-kadison-kaplansky-counterexamples/"
as_of: "2026-10-30T13:42:00+01:00"
date: "2026-10-06"
date_precision: day
category: science
importance: 5
confidence: medium
status: [Awaiting review]
verification: Preprint only
sources: 7
editor: human editor
human_review: null
version: "2026-10-29"
---

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/2026-10-06-openai-baum-connes-kadison-kaplansky-counterexamples/

# OpenAI claims Baum–Connes and Kadison–Kaplansky counterexamples

Full title: OpenAI's release claims counterexamples to the Baum–Connes conjecture without coefficients and to the Kadison–Kaplansky conjecture; Guoliang Yu builds on the irrational-trace construction

Family 285 of OpenAI's 6 Oct math release contains three manuscripts dated 23 Sept 2026: a finitely generated torsion-free group whose coefficient-free reduced Baum–Connes assembly map has an infinite-order kernel class (failure of rational injectivity); a finitely generated group whose reduced C*-algebra contains a projection of irrational trace, outside the image of the assembly map (failure of surjectivity); and a torsion-free group with a projection other than 0 and 1 in its reduced C*-algebra (against Kadison–Kaplansky). No Lean formalization. On 26 Oct Guoliang Yu (arXiv 2610.35674) constructed projections with every trace value in (0, 1), giving further surjectivity counterexamples (groups with torsion), and wrote that his techniques "are developed from arguments in the OpenAI preprint". The torsion-free and Kadison–Kaplansky claims have no published expert check.

## 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

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 159 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 98 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 189 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 128 days after its cutoff

## Sources

1. [OpenAI: A torsion-free counterexample to reduced Baum–Connes injectivity (PDF)](https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/paper.pdf) (github.com, paper)
2. [OpenAI: A torsion-free counterexample to the Kadison–Kaplansky projection conjecture (PDF)](https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026/paper.pdf) (github.com, paper)
3. [OpenAI: An irrational-trace counterexample to reduced Baum–Connes (PDF)](https://github.com/openai/math/blob/main/preprints/An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026/paper.pdf) (github.com, paper)
4. [Guoliang Yu: Projections in reduced group C*-algebras with prescribed trace values in (0,1) (arXiv 2610.35674)](https://arxiv.org/abs/2610.35674) (arxiv.org, paper)
5. [Ying Xie: What the Machine Invented (arXiv 2610.17090, survey of the release)](https://arxiv.org/abs/2610.17090) (arxiv.org, paper)
6. [The Dartmouth: Dartmouth mathematicians weigh AI's role (Erik van Erp on the Baum–Connes papers, 20 Oct)](https://www.thedartmouth.com/article/2026/10/rodriguez-nagle-dartmouth-mathematicians-weigh-artificial-intelligences-role-in-the-future-of-their-work) (thedartmouth.com, press)
7. [oai-math-revised: community notes on the injectivity manuscript](https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/AGENT_NOTES.md) (github.com, discussion)

## Changes

- 2026-10-29: Related entry added for the Kadison similarity claim (family 288), science agent run 2
- 2026-10-29 (filed): Created (families 251–378; three manuscripts and CONTENTS, Yu's arXiv paper read, The Dartmouth article, oai-math-revised notes)

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