Kaplansky's direct finiteness conjecture

A unital ring is directly finite if $ab=1$ implies $ba=1$ . For a field $K$ and a group $G$ , the group algebra $K[G]$ consists of finite formal sums $\sum c_g g$ . In characteristic zero $K[G]$ is directly finite (and stably finite) for every group, by Kaplansky's trace argument. In positive characteristic it was known for free-by-amenable groups (Ara-O'Meara-Perera) and for all sofic groups (Elek-Szabo). Kaplansky asked whether the characteristic-zero theorem persists in positive characteristic. For every field $K$ , every group $G$ and all $a,b\in K[G]$ , does $ab=1$ imply $ba=1$ ?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Group rings; direct and stable finiteness
Posed by: Irving Kaplansky, Fields and Rings (Chicago Lectures in Mathematics), Part II, Section 3, Problem 2, as the manuscripts cite
Year posed: 1969
Years open: 57y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 52 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1 (September 23): a finite field $K$ of characteristic two, a finitely presented group $G$ containing odd-order torsion, and $a,b\in K[G]$ with $ab=1$ , $ba\ne1$ . By Elek-Szabo, $G$ is not sofic. Companions: for a specified odd prime $p$ (the least prime factor of $(\binom{1200}{600}!)^2+1$ ), a field of order $p^4$ and a finitely generated group with torsion (September 26); and a finitely presented torsion-free group with a finite two-dimensional classifying complex and $a,b,c\in\mathbb F_2[G]$ with $ab=1$ , $ac=0$ , $c\ne0$ (October 4). No counterexample in characteristic zero is possible, and none is claimed.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. A reasoning summary for this result (family 197, characteristic two) is published with the release. Companions give an odd-characteristic counterexample (September 26) and a torsion-free counterexample over $\mathbb F_2$ (October 4).

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the September 23 manuscript was read against Kaplansky's question as the companions cite it; it claims a finite field $K$ of characteristic two, a finitely presented group $G$ with an element of odd prime order, and $a,b\in K[G]$ with $ab=1\ne ba$ , specified by a terminating prescription that has not been executed. The proof was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator KaplanskyFinitelyPresented, declaration OAI.KaplanskyCounterexample.finitelyPresented_counterexample); ComparatorChallenges/KaplanskyFinitelyPresented.lean was read here and asserts exactly that: a finite field of characteristic 2, a finitely presented group with an element of odd prime order, and $a,b$ in the monoid algebra with $ab=1$ , $ba\ne1$ . This states the headline claim. A second challenge, KaplanskyDirectFiniteness (finitely generated group version), is not in the catalogue but its solution module exists. Not rebuilt here. The nonsoficity of $G$ is outside the Lean statements. The torsion-free companion (October 4) is not formalized.

Kaplansky's direct finiteness conjecture

What was actually shown

What the AI did

Verification

Sources

Discussion