A fixed point for the jump operator on structures

Assuming that $0^#$ exists, we prove that there is a structure that can effectively interpret its own jump. In particular, we get a structure $mathcal A$ such that [ Sp({mathcal A}) = {{bf x}:{bf x}in Sp ({mathcal A})}, ] where $Sp ({mathcal A})$ is the set of Turing degrees which compute a copy of $mathcal A$. It turns out that, more interesting than the result itself, is its unexpected complexity. We prove that higher-order arithmetic, which is the union of full $n$th-order arithmetic for all $n$, cannot prove the existence of such a structure.