\label{def:tau-tower-machine}
A \textbf{$\tau$-Tower Machine} (TTM) is a tuple
\[
    M \;=\; (Q,\, m,\, b_0,\, \Sigma,\, \delta_M,\, q_{\mathrm{start}},\, q_{\mathrm{acc}},\, q_{\mathrm{rej}})
\]
where:
\begin{enumerate}
    \item $Q$ is a finite set of \emph{control states},
          with $|Q| < \infty$.
    \item $m \in \mathbb{N}$, $m \geq 1$, is the number
          of \emph{registers}.
          Each register $r_i$ ($1 \leq i \leq m$)
          holds a $\tau$-address---an element
          of the NF tower
          $\hat{\mathbb{Z}}_{\tau}
           = \varprojlim_k \mathbb{Z} / \operatorname{Prim}(k) \mathbb{Z}$.
    \item $b_0 \in \mathbb{N}$ is the number
          of \emph{ports}---read-only input channels
          through which the machine receives
          its initial $\tau$-address arguments.
    \item $\Sigma$ is the \emph{instruction set},
          consisting of operations derived from the
          five generators $\{\alpha, \pi, \gamma, \eta, \omega\}$
          of Category~$\tau$:
          \begin{itemize}
              \item \textbf{Successor} $\rho$:
                    $r_i \;\mapsto\; r_i + 1$
                    \quad(the $\alpha$-step:
                    advance by one NF address).
              \item \textbf{Multiplication} $\times$:
                    $(r_i, r_j) \;\mapsto\; r_i \times r_j$
                    \quad(the $\pi$-operation:
                    primorial product).
              \item \textbf{Exponentiation} $\wedge$:
                    $(r_i, r_j) \;\mapsto\; r_i^{\,r_j}$
                    \quad(the $(\gamma, \eta)$-operation:
                    tower-building on the fiber).
              \item \textbf{Tetration step} $\sigma$:
                    $(r_i, r_j) \;\mapsto\;
                     r_i \mathbin{\uparrow\!\uparrow} r_j$
                    \quad(iterated exponentiation:
                    the $\omega$-absorber channel).
          \end{itemize}
          The instruction set also includes four \emph{predicates}
          that branch the control flow:
          \begin{itemize}
              \item \textbf{Equality}: $r_i = r_j$?
              \item \textbf{Divisibility}: $r_i \mid r_j$?
                    \quad(the $\KAxiom{3}$ test).
              \item \textbf{Set membership}: $r_i \in_{\tau} S$?
                    \quad(membership in a definable
                    $\tau$-subset, cf.\ Book~I, Part~VIII).
              \item \textbf{Orbit test}:
                    does $r_i$ lie on the orbit
                    of $r_j$ under the progression operator?
          \end{itemize}
    \item $\delta_M : Q \times \Sigma^m \to Q \times \Sigma^m$
          is the \emph{transition function},
          mapping the current state and register contents
          to a new state and updated register contents.
    \item $q_{\mathrm{start}}, q_{\mathrm{acc}}, q_{\mathrm{rej}} \in Q$
          are the start, accept, and reject states,
          with $q_{\mathrm{acc}} \neq q_{\mathrm{rej}}$.
\end{enumerate}
The machine is \emph{deterministic}:
$\delta_M$ is a total function.
A computation of $M$ on input
$(a_1, \ldots, a_{b_0})$
begins in state~$q_{\mathrm{start}}$
with ports loaded and registers initialised to~$0$,
and halts when the control enters
$q_{\mathrm{acc}}$ or $q_{\mathrm{rej}}$.