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\label{def:abcd-chart}
The \textbf{ABCD coordinate chart} is the map
\[
    \boxed{\Phi : \Obj(\tau) \longrightarrow
    \tau\text{-Idx}^4,
    \qquad
    x \longmapsto
    (\operatorname{coord}_A(x),\;
     \operatorname{coord}_B(x),\;
     \operatorname{coord}_C(x),\;
     \operatorname{coord}_D(x))}
\]
where, for $x \in \Obj(\tau)$ with index
$\underline{n} = \mathrm{idx}(x) \geq \underline{2}$:
\begin{itemize}
    \item $\operatorname{coord}_A(x) = \underline{A}$:
          the \textbf{largest prime divisor} of $\underline{n}$
          ($\pi$-channel).
    \item $\operatorname{coord}_B(x) = \underline{B}$:
          the \textbf{maximal exponent} at tower level
          ($\gamma$-channel).
    \item $\operatorname{coord}_C(x) = \underline{C}$:
          the \textbf{maximal tetration height}
          ($\eta$-channel).
    \item $\operatorname{coord}_D(x) = \underline{D}$:
          the \textbf{remainder} after tower extraction
          ($\alpha$-channel).
\end{itemize}
For $x$ with $\mathrm{idx}(x) \in \{\underline{0}, \underline{1}\}$,
we set $\Phi(x) = (\underline{1}, \underline{0},
\underline{0}, \mathrm{idx}(x))$
(degenerate addresses).