%
\label{def:genealogical-decomposition}
For $X \in \tau\text{-Idx}$ with $X \geq \underline{2}$,
the \textbf{genealogical decomposition} (or \textbf{spine})
is the finite sequence of ABC-triplets obtained by
iterating the canonical peel along the remainder:
\begin{align*}
    X \;&\mapsto\; (\underline{A_1}, \underline{B_1},
    \underline{C_1}, \underline{D_1}), \\
    \underline{D_1} \;&\mapsto\; (\underline{A_2}, \underline{B_2},
    \underline{C_2}, \underline{D_2}), \\
    &\;\;\vdots \\
    \underline{D_{k-1}} \;&\mapsto\; (\underline{A_k}, \underline{B_k},
    \underline{C_k}, \underline{1}).
\end{align*}
The result is the ordered sequence
\[
    \mathcal{G}(X) \;=\;
    \bigl((\underline{A_1}, \underline{B_1}, \underline{C_1}),\;
    (\underline{A_2}, \underline{B_2}, \underline{C_2}),\;
    \ldots,\;
    (\underline{A_k}, \underline{B_k}, \underline{C_k})\bigr).
\]
The original $X$ is recoverable by evaluating and multiplying
the corresponding tower atoms:
\[
    X \;=\; T(\underline{A_1}, \underline{B_1}, \underline{C_1})
    \;\cdot\; T(\underline{A_2}, \underline{B_2}, \underline{C_2})
    \;\cdot\; \ldots \;\cdot\;
    T(\underline{A_k}, \underline{B_k}, \underline{C_k}).
\]
For $X = \underline{1}$, the spine is the empty sequence
$\mathcal{G}(\underline{1}) = ()$.