Browse › Algebra › 197 A torsion-free group algebra that is not directly finite
A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
We disprove Kaplansky's direct-finiteness conjecture by constructing a finite field K of characteristic two, a finitely presented group G, and finite sums $a,b\in K[G]$ with $ab=1$ but $ba\ne1$. The group G is nonsofic. The same elements define a cellular automaton on KG that is injective but not surjective, disproving Gottschalk's surjunctivity conjecture.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI, Lean scope note for family 197 (
lean/docs/197.md) (openai/math release, 2026-10-06). States that the formalization covers the characteristic-two counterexample, including a finitely presented version with an element of odd prime order and a terminating search recipe; nonsoficity is outside the formalized statements. - mo271, KevorkianPhilippe and others, Wikipedia/SurjunctiveGroup:
gottschalk_surjunctivity_conjectureis false (formal-conjectures issue #6919, 2026-10-07 to 2026-10-19). Proposes restating Gottschalk's conjecture as false on the basis of this preprint and its odd-characteristic companion; the release's Lean proofs cover injectivity and non-surjectivity but not continuity or shift-equivariance, for which a commenter supplied a Lean bridge, and a commenter notes the finitely presented group appears only in the characteristic-two Lean statement. - KevorkianPhilippe, fix(Wikipedia/SurjunctiveGroup):
gottschalk_surjunctivity_conjectureis false (formal-conjectures PR #7440, merged 2026-10-19). Restates the conjecture as false, citing this preprint and the odd-characteristic companion and flagging that continuity and shift-equivariance are not part of the linked proof; the approving reviewer (bocowgill) says they checked the non-surjunctivity conclusions (Corollary 7.3) and "relied on the reported build and axiom checks without independently rerunning the external proof." - The Formal Conjectures Authors, Gottschalk’s surjunctivity conjecture — SurjunctiveGroup.lean (GitHub, accessed 21 October 2026). This Lean statement file cites both characteristic companions but retains
sorryin the counterexample declarations and notes that the linked odd-characteristic proof omits continuity and shift equivariance. [family-level]
Possibly relevant literature (found by the agent, not checked)
- OpenAI, Ten Advances in Mathematics and Theoretical Computer Science, Chapter 3 "Nonsofic groups exist" (OpenAI, updated 2026-08-06). An earlier claimed explicit nonsofic group; relevant to the introduction's statement that this construction "supplies a finitely presented counterexample to the assertion that every group is sofic."
- Francesco Fournier-Facio, A torsion-free non-sofic group (arXiv:2608.02025, August 2026). Further nonsofic examples, including finitely presented ones, from the same technical criterion as the August OpenAI result; relevant to the nonsoficity consequence of Theorem 1.1.
- Michal Doucha, Jakub Gismatullin, On dual surjunctivity and applications (arXiv:2008.10565, 2020). Proves that dual surjunctive groups satisfy direct finiteness over all fields of positive characteristic, so Theorem 1.1 would also imply that $G$ is not dual surjunctive; complements the surjunctivity discussion in Section 7.
- Ken Dykema, Timo Heister, Kate Juschenko, Finitely presented groups related to Kaplansky's Direct Finiteness Conjecture (arXiv:1112.1790, 2011). Introduces finitely presented "ULIE" groups that are universal for one-sided inverses in group rings, which may be compared with the finitely presented group of Theorem 1.1; not in this preprint's bibliography, although the torsion-free companion cites it.
- Henry Bradford, Francesco Fournier-Facio, Hopfian wreath products and the stable finiteness conjecture (arXiv:2211.01510, 2022). Relates Kaplansky's stable finiteness conjecture to the Hopf property of wreath products, so a counterexample may have consequences there.
Digests and expositions
- Ying Xie, What the Machine Invented: Mining New Mathematical Instruments from OpenAI's October 2026 Release of AI-Generated Mathematics (arXiv:2610.17090, 2026-10-12). Section 8.4.2 summarizes this construction: the incidence criterion (Thm 3.1), the polynomial pairing (Prop 2.3), the hole trades with the local lemma, and the HNN packing. It also lists caveats, for example that the prescription was not executed; per the paper, the first extraction for each family used AI readings.
- OpenAI, reasoning summary
kaplansky-direct-finiteness-characteristic-two.pdf(openai/math release, 2026-10-06). An abridged summary of the model's reasoning for this result, included in the release. [AI-generated per source] - AI Proved What?, One-sided inverse? Kaplansky’s direct-finiteness conjecture (Episode 197, October 2026). This illustrated exposition compares all four papers, explains their construction ideas, and distinguishes their claims and formalization scope. [AI-generated per source] [family-level]
News and blogs
- Kingy.ai, OpenAI's 722 Math Manuscripts: The Results, Proofs, Compute and Costs (October 2026). Discusses family 197 and contrasts the torsion-free headline with the Lean scope page, which covers the characteristic-two and odd-characteristic constructions with torsion. [family-level]
Elsewhere
- VibeMathed: Problem record
- VibeMathed: Problem record
- Emergent Mind: family 197
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026,
author = {{OpenAI}},
title = {{A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf}{OAI:A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/197.md. The Lean statements have not been checked by people against the paper's claims.
Family 197: A torsion-free group algebra that is not directly finite
Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽2 is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective cellular automata on all configurations, refuting Gottschalk's surjunctivity conjecture. Another counterexample is an integral group-ring matrix, invertible over the rational group ring, with Fuglede–Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture.
Other papers in this family:
- A Torsion-Free Group Algebra That Is Not Directly Finite · October 4, 2026
- A Counterexample to the Group-Ring Determinant Conjecture · September 23, 2026
- A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic · September 26, 2026