Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 197-04. A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic
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OpenAI@2026-10-07#rem:oai-p-197-04
Authors: OpenAI. Family 197.
Title: A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic.
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.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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