---
id: "2026-10-06-openai-kaplansky-zero-divisor-direct-finiteness-counterexamples-claim"
url: "https://postcutoff.com/e/2026-10-06-openai-kaplansky-zero-divisor-direct-finiteness-counterexamples-claim/"
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: Peer-reviewed
sources: 12
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-kaplansky-zero-divisor-direct-finiteness-counterexamples-claim/

# Kaplansky's zero-divisor and direct-finiteness conjectures refuted? (OpenAI, Lean)

Full title: OpenAI's math release claims counterexamples to Kaplansky's zero-divisor and direct-finiteness conjectures for group algebras, and with them to Gottschalk's surjunctivity conjecture and the group-ring Determinant Conjecture; Lean covers all but the torsion-free direct-finiteness example

Family 196 of OpenAI's 6 Oct 2026 release, "A Torsion-Free Group Algebra with Zero Divisors" (23 Sept, 26 pages): "We disprove Kaplansky's zero-divisor conjecture by constructing a finitely presented torsion-free group G for which F₂[G] has nonzero zero divisors. The group admits a finite two-dimensional classifying space." Family 197 claims counterexamples to Kaplansky's direct-finiteness conjecture (ab = 1 but ba ≠ 1 in K[G]) in characteristic 2 and in odd characteristic, both with torsion, and over F₂ for a torsion-free group (4 Oct, 31 pages); the groups are necessarily non-sofic, and the same elements give injective but non-surjective cellular automata, refuting Gottschalk's surjunctivity conjecture. A 9-page companion refutes the unrestricted group-ring Determinant Conjecture. The Lean challenges TorsionFreeZeroDivisors, KaplanskyFinitelyPresented, KaplanskyDirectFiniteness, OddKaplansky and GroupRingDeterminant were accepted in the oai-math-revised comparator run; the torsion-free direct-finiteness example and the non-soficity conclusion are not formalized. Family 207 claims, conversely, Kaplansky's idempotent conjecture in characteristic zero.

## Key facts
- Zero divisors (family 196): Theorem 1.1, 'There exist a finitely presented torsion-free group G and nonzero elements α, β ∈ F₂[G] such that αβ = 0. Moreover, G admits a finite two-dimensional classifying space.' The paper traces the question to Higman's 1940 thesis and Kaplansky's 1956 problem list, and contrasts it with Gardam's 2021 refutation of the related unit conjecture ('a nontrivial unit is not a zero divisor'). Lean challenge TorsionFreeZeroDivisors: 'accepted' (oai-math-revised comparator run, 25–26 Oct)
- Direct finiteness (family 197, four manuscripts): characteristic two, 23 Sept, 28 pages ('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 ∈ K[G] with ab=1 but ba≠1. The group G is nonsofic. The same elements define a cellular automaton on K^G that is injective but not surjective, disproving Gottschalk's surjunctivity conjecture'); odd characteristic, 26 Sept, 25 pages (a field of order p⁴ for one specified odd prime p); torsion-free over F₂, 4 Oct, 31 pages ('a finitely presented torsion-free counterexample to Kaplansky's direct finiteness conjecture over the field of two elements')
- Why non-sofic groups are needed (papers' history sections): Elek and Szabó (2004) proved stable finiteness of K[G] for all sofic G, and Gromov and Weiss proved sofic groups surjunctive, so any counterexample group is non-sofic. In characteristic zero group algebras are always directly finite (Kaplansky). OpenAI's Astra had claimed the first non-sofic group in August 2026 (see [2026-08-01-openai-astra-ten-advances](https://postcutoff.com/e/2026-08-01-openai-astra-ten-advances/index.md))
- Determinant Conjecture (23 Sept, 9 pages): 'We construct a finitely generated group G and a square matrix over Z[G] that is invertible over Q[G] and has Fuglede–Kadison determinant strictly between zero and one', refuting the unrestricted group-ring Determinant Conjecture
- Lean scope (lean/docs/197.md): the characteristic-two statements use a group with an element of odd prime order, and 'The further conclusion that the group is nonsofic is outside these statements'; the odd-characteristic statement uses a field of order p⁴, with p the smallest prime divisor of ((1200 choose 600)!)² + 1. Carlos Toledo's audit (finding G2) notes that the family headline is the torsion-free example while 'the finitely presented Lean statement REQUIRES the group to contain an element of odd prime order', and that the Kaplansky witnesses are prescriptions with no explicit instance published
- Reactions (Proofs and Prompts, 8 Oct): Michael Chapman listed 'the refutations to the Kaplansky conjectures' among problems 'I grew up on'. Johan Öinert observed that the torsion-free F₂ example gives ba as a nontrivial idempotent, hence a characteristic-2 counterexample to the idempotent conjecture: 'Yet, there is no mentioning of this obvious conclusion in the latter document! … This is NOT how human mathematicians write papers!' Formal Conjectures (Google DeepMind) restates the zero-divisor conjecture as false citing the paper, and merged PR #7440 (19 Oct) restating Gottschalk's conjecture as false
- Converse claim (family 207, 24 Sept): 'for every torsion-free group G and every commutative unital domain R of characteristic zero, the only idempotents in RG are 0 and 1. This proves Kaplansky's idempotent conjecture in characteristic zero', via the Bass trace conjecture (Lean: BassTrace, BassTorsionFree accepted). Indira Chatterji, who proved the amenable case with Berrick and Mislin: 'I don't understand the proof, and I still don't believe that this conjecture could be true in general.' Roman Sauer recognised a trick from his own work in OpenAI's 'deeply creative solutions of problems 207 and 335'

## What happened

Kaplansky's group-ring conjectures concern the algebra K[G] of finite formal sums of group elements. For a torsion-free group the
zero-divisor conjecture predicts no zero divisors. The direct-finiteness conjecture predicts, for every group and field, that a one-sided
inverse is two-sided. Gardam refuted the related unit conjecture in 2021. The other two stayed open, and direct finiteness was known for
all sofic groups. A counterexample to it therefore needs a non-sofic group, a kind of group whose existence was itself open until OpenAI's claim of
August 2026.

OpenAI's release claims both refutations. Family 196 builds a finitely presented torsion-free group with zero divisors over F₂.
Family 197 gives one-sided inverses over finite fields of characteristic 2 and of odd characteristic, using groups with torsion, and
over F₂ for a torsion-free group. These groups are non-sofic, and the elements also yield injective, non-surjective cellular automata,
refuting Gottschalk's 1973 surjunctivity conjecture. Lean covers the zero-divisor and torsion examples and the Determinant Conjecture
counterexample. The torsion-free direct-finiteness paper is not formalized.

## Why it matters

If correct, these are the first counterexamples to two of the best-known problems about group rings, with consequences in symbolic
dynamics and L²-invariants. The formal proofs make the torsion examples checkable. A reader pointed out that the torsion-free example
also refutes the idempotent conjecture in characteristic 2, a consequence the papers do not mention, while a companion family claims the
idempotent conjecture in characteristic 0. No specialist has published a check of the constructions.

## What is disputed or not yet verified

- Verification: Lean formalizations of the zero-divisor, torsion direct-finiteness and determinant counterexamples, accepted by an independent comparator run; the torsion-free direct-finiteness example and non-soficity are only in unrefereed manuscripts; no explicit witnesses published

## 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 (PDF): A Torsion-Free Group Algebra with Zero Divisors (23 Sept 2026)](https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026/paper.pdf) (github.com, paper)
2. [OpenAI (PDF): A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two (23 Sept)](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf) (github.com, paper)
3. [OpenAI (PDF): A Torsion-Free Group Algebra That Is Not Directly Finite (4 Oct)](https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026/direct-finiteness.pdf) (github.com, paper)
4. [OpenAI (PDF): A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic (26 Sept)](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026/paper.pdf) (github.com, paper)
5. [OpenAI (PDF): A Counterexample to the Group-Ring Determinant Conjecture (23 Sept)](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/paper.pdf) (github.com, paper)
6. [OpenAI Lean scope notes for families 196 and 197](https://github.com/openai/math/blob/main/lean/docs/197.md) (github.com, code)
7. [OpenAI reasoning summary: Kaplansky direct finiteness in characteristic two](https://github.com/openai/math/blob/main/reasoning_traces/kaplansky-direct-finiteness-characteristic-two.pdf) (github.com, paper)
8. [oai-math-revised comparator results (all five challenges accepted)](https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv) (github.com, code)
9. [formal-conjectures PR #7440: gottschalk_surjunctivity_conjecture is false (merged 19 Oct)](https://github.com/google-deepmind/formal-conjectures/pull/7440) (github.com, code)
10. [OpenAI (PDF): The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture (24 Sept)](https://github.com/openai/math/blob/main/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/paper.pdf) (github.com, paper)
11. [Carlos Toledo: OpenAI's math release, what it claims beyond what it checks (findings G2, G10)](https://carlostoledo.co/reports/openai-math.html) (carlostoledo.co, discussion)
12. [Proofs and Prompts: 100+ reactions to 100+ solutions (Chapman, Öinert, Chatterji, Sauer)](https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/) (proofsandprompts.com, discussion)

## Changes

- 2026-10-29 (filed): Created (CONTENTS entries of families 196, 197 and 207, the five PDFs (26, 28, 31, 25, 9 pp), Lean scope notes, comparator results, Proofs and Prompts reactions, formal-conjectures PR #7440)

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