\label{def:observable-transition}
Let $M = (Q, m, b_0, \Sigma, \delta_M,
q_{\mathrm{start}}, q_{\mathrm{acc}}, q_{\mathrm{rej}})$
be a $\tau$-Tower Machine.
The \textbf{machine constant} of $M$ is
\[
    K_M \;=\; |Q| \cdot m.
\]
A \textbf{configuration} of $M$
is a pair $(q, \mathbf{r}) \in Q \times \hat{\mathbb{Z}}_{\tau}^{\,m}$,
consisting of the current control state
and the $m$-tuple of register values.
The \textbf{observable transition}
\[
    \delta_M^{\mathrm{obs}} :
    Q \times \hat{\mathbb{Z}}_{\tau}^{\,m}
    \;\longrightarrow\;
    Q \times \hat{\mathbb{Z}}_{\tau}^{\,m}
\]
is the restriction of $\delta_M$
to the information visible at a single step:
\begin{itemize}
    \item the current state $q \in Q$
          (one of $|Q|$ values),
    \item the $m$ register values
          $r_1, \ldots, r_m \in \hat{\mathbb{Z}}_{\tau}$,
\end{itemize}
for a total of $K_M = |Q| \cdot m$ \emph{observation channels}.
The control state selects which instruction
from~$\Sigma$ to execute,
and the instruction reads at most two registers
(source and target).
The observation window has width~$K_M$:
the transition depends on at most $K_M$
discrete pieces of information per step.