%
\label{def:iterator-ladder}
The \textbf{iterator ladder} of Category~$\tau$ is the sequence
of operation levels:
\[
    \mathcal{L} = (L_0, L_1, L_2, L_3)
\]
defined abstractly as follows:
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{c l l l}
\toprule
\textbf{Level} & \textbf{Symbol} & \textbf{Name}
    & \textbf{Defining principle} \\
\midrule
$L_0$ & $\rho$ & Raw iteration
    & $\rho^n(x)$: apply $\rho$ a total of $n$ times \\
$L_1$ & $\mu$ & Multiplication
    & iterate $L_0$: $\mu(x,y) \sim \rho^y(x)$ ``$y$ times'' \\
$L_2$ & $\nu$ & Exponentiation
    & iterate $L_1$: $\nu(x,y) \sim \mu(x, \ldots)$ ``$y$ times'' \\
$L_3$ & $\theta$ & Tetration
    & iterate $L_2$: $\theta(x,y) \sim \nu(x, \ldots)$ ``$y$ times'' \\
\bottomrule
\end{tabular}
\end{center}
Each level is associated with a dedicated orbit channel:
\begin{align*}
    L_0 &\longleftrightarrow O_\alpha \quad\text{(counting scaffold)}, \\
    L_1 &\longleftrightarrow O_\pi \quad\text{(multiplicative channel)}, \\
    L_2 &\longleftrightarrow O_\gamma \quad\text{(exponential channel)}, \\
    L_3 &\longleftrightarrow O_\eta \quad\text{(tetration channel)}.
\end{align*}