{"id":"openai-math/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two","pdf":"https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026","tags":["group-rings","direct-finiteness","cellular-automata"],"score":0,"title":"A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two","domain":"Algebra","family":"197","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.RA","summary":"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 K G that is injective but not surjective, disproving Gottschalk's surjunctivity conjecture.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-23","reactions":{},"discussion":[{"url":"https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/#:~:text=It%20follows%20trivially%20that%20the%20latter%20document%20provides%20a%20counterexample%20to%20the%20idempotent%20conjecture","date":"2026-10-08","kind":"web","site":"Proofs and Prompts","text":"It follows trivially that the latter document provides a counterexample to the idempotent conjecture in characteristic p = 2 p=2 . Indeed, they have a torsion-free group G G and the field with two elements K K , and find elements a,b in the group algebra K [ G ] K[G] satisfying a b = 1 ab=1 and b…","added":"2026-10-24","title":"100+ reactions to 100+ solutions"}],"formalized":true,"family_title":"A torsion-free group algebra that is not directly finite"}