\label{def:rank-as-tower-depth}
For each primorial depth $k \geq 1$, define the \emph{$k$-level rational group}
\begin{equation}
G_k \;=\; \bigl\{ a \in \hat{\mathbb{Z}}_{\T}^{\,\mathbb{Q}} : a \text{ stabilizes at depth} \leq k \bigr\},
\label{eq:ch45-k-level-group}
\end{equation}
and the \emph{rank function}
\begin{equation}
r(k) \;=\; \operatorname{rk}_{\mathbb{Z}} \bigl( G_k / G_k^{\mathrm{tor}} \bigr),
\label{eq:ch45-rank-function}
\end{equation}
where $G_k^{\mathrm{tor}}$ denotes the torsion subgroup of~$G_k$. The \emph{$\tau$-rank} of $\hat{\mathbb{Z}}_{\T}^{\,\mathbb{Q}}$ is the stabilization value
\begin{equation}
r_\infty \;=\; \lim_{k \to \infty} r(k),
\label{eq:ch45-tau-rank}
\end{equation}
provided this limit exists and is finite. Equivalently, $r_\infty$ is the minimal depth $k_*$ such that $r(k) = r(k_*)$ for all $k \geq k_*$. We call $k_*$ the \emph{rank-stabilization depth}.