The Tribonacci constant and finite automata

We show that there is no automaton accepting the Tribonacci representations of $n$ and $x$ in parallel, where $\psi =1.839\cdots$ is the Tribonacci constant, and $x=\lfloor n\psi \rfloor$. Similarly, there is no Tribonacci automaton generating the Sturmian characteristic word with slope $\psi -1$.