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A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic
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Abstract
We construct a counterexample to Kaplansky's direct-finiteness conjecture in odd characteristic. For one specified odd prime p, we obtain a field K of order p4, a finitely generated group G containing torsion, and finite sums $a,b\in K[G]$ with $ab=1$ but $ba\ne1$. The same elements define a cellular automaton on KG that is injective but not surjective.
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Notes
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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 constructs the field of order $p^4$, a finitely generated group with torsion and $a,b$ with $ab=1\ne ba$. It also covers the injective, non-surjective map on $K^G$ and some fixed-field transfer results. - mo271, KevorkianPhilippe and others, Wikipedia/SurjunctiveGroup:
gottschalk_surjunctivity_conjectureis false (formal-conjectures issue #6919, 2026-10-07 to 2026-10-19). Notes that the release's Lean proof for this preprint covers injectivity and non-surjectivity of the automaton only as a plain map. A commenter supplied a separate Lean bridge for continuity and shift-equivariance on top of the odd-characteristic main theorem; the resulting group is finitely generated, not necessarily finitely presented. - KevorkianPhilippe, fix(Wikipedia/SurjunctiveGroup):
gottschalk_surjunctivity_conjectureis false (formal-conjectures PR #7440, merged 2026-10-19). Restates Gottschalk's conjecture as false, linkingOAI.OddKaplansky.main_theoremand noting that continuity and shift-equivariance are not part of the linked proof; the approving reviewer (bocowgill) says they checked the non-surjunctivity conclusions (Section 8) 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)
- 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 cellular-automaton consequence in Section 8.
- 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; context for the introduction's remark that the group $G$ cannot be sofic.
- Ken Dykema, Timo Heister, Kate Juschenko, Finitely presented groups related to Kaplansky's Direct Finiteness Conjecture (arXiv:1112.1790, 2011). Finitely presented groups that are universal for one-sided inverses in group rings; may be compared with the finitely generated group of Theorem 1.1. Not in this preprint's bibliography.
- Brandon Seward, Krieger’s finite generator theorem for actions of countable groups II (arXiv:1501.03367, 2015; revised 2019). Its conditional implication from positive Rokhlin entropy for free ergodic actions of every countable group to surjunctivity and direct finiteness provides context for this paper’s claimed counterexample. [family-level]
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). The card for family 197 (Section 8.4.2) mainly covers the characteristic-two construction and says the repository formalizes this odd-characteristic companion; per the paper, the first extraction for each family used AI readings. [family-level]
- 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
- Emergent Mind: family 197
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026,
author = {{OpenAI}},
title = {{A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026/paper.pdf}{OAI:A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026}},
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 Kaplansky's Direct-Finiteness Conjecture in Characteristic Two · September 23, 2026
- A Counterexample to the Group-Ring Determinant Conjecture · September 23, 2026