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

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.