The Tribonacci constant and finite automata

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