Kaplansky's direct finiteness conjecture

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This event attributed to Unreleased internal OpenAI model

Input: Kaplansky counterexample in characteristic two evidence for this event

Companion: Counterexample in Odd Characteristic (Lean-checked automaton) evidence for this event

Comparator statement: KaplanskyFinitelyPresented.lean evidence for this event

Lean proof (OAI.KaplanskyCounterexample.finitelyPresented_counterexample) evidence for this event

VibeMathed record: Kaplansky's direct finiteness conjecture evidence for this event

Lean proof (OAI.OddKaplansky.main_theorem) evidence for this event

Comparator statement: KaplanskyDirectFiniteness.lean evidence for this event

Companion: A Torsion-Free Group Algebra That Is Not Directly Finite evidence for this event

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. parent of this event

Kaplansky's direct finiteness conjecture parent of this event