A torsion-free group algebra that is not directly finite

What OpenAI claims

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. (Lean)

Papers (4)

1. A Torsion-Free Group Algebra That Is Not Directly Finite

   4 October 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We construct a finitely presented torsion-free counterexample to Kaplansky's direct finiteness conjecture over the field of two elements. The group admits a finite two-dimensional classifying complex.

2. A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's 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 ∈ K[G] with ab=1 but ba ≠ 1. 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.

3. A Counterexample to the Group-Ring Determinant Conjecture

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We disprove the unrestricted group-ring Determinant Conjecture. 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. The logarithmic integral defining the determinant is finite.

4. A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic

   26 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's 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 ∈ K[G] with ab=1 but ba ≠ 1. The same elements define a cellular automaton on KG that is injective but not surjective.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.