  For an ordinal $\alpha$, an $\alpha$-ITRM is a machine model of transfinite
computability that operates on finitely many registers, each of which can
contain an ordinal $\rho<\alpha$; they were introduced by Koepke in \cite{KM}.
  In \cite{alpha itrms}, it was shown that the $\alpha$-ITRM-computable subsets
of $\alpha$ are exactly those in a level $L_{\beta(\alpha)}$ of the
constructible hierarchy. It was conjectured in \cite{alpha itrms} that
$\beta(\alpha)$ is the first limit of admissible ordinals above $\alpha$. Here,
we show that this is false; in particular, even the computational strength of
$\omega^{\omega}$-ITRMs goes far beyond $\omega_{\omega}^{\text{CK}}$.
