\label{def:tau-rational-point}
An element $a = (a_k)_{k \geq 1} \in \hat{\mathbb{Z}}_{\T}$ is a \emph{$\tau$-rational point} if it satisfies two conditions:
\begin{enumerate}
\item[\emph{(R1)}] \textbf{Stabilization.} There exists a finite depth $k_0 \in \mathbb{N}$ such that for all $k \geq k_0$,
\begin{equation}
a_k \;\equiv\; a_{k_0} \pmod{\operatorname{Prim}(k_0)}.
\label{eq:ch45-stabilization-condition}
\end{equation}
That is, the address at $\operatorname{Prim}(k_0)$ determines the address at all deeper levels: no new information enters the tower beyond depth~$k_0$.

\item[\emph{(R2)}] \textbf{Rationality.} Each ABCD coordinate of $a$, viewed as an element of the $p$-adic completion $\mathbb{Z}_p$ for each prime $p \leq p_{k_0}$, actually lies in $\mathbb{Q} \cap \mathbb{Z}_p$. Equivalently, there exist $\alpha, \beta \in \mathbb{Z}$ with $\beta \neq 0$ and $\gcd(\beta, \operatorname{Prim}(k_0)) = 1$ such that $a_{k_0} = \alpha / \beta$ in $\mathbb{Z}/\operatorname{Prim}(k_0)\mathbb{Z}$.
\end{enumerate}
We write $\hat{\mathbb{Z}}_{\T}^{\,\mathbb{Q}}$ for the set of all $\tau$-rational points. This set inherits a group structure from $\hat{\mathbb{Z}}_{\T}$ (componentwise addition), making $\hat{\mathbb{Z}}_{\T}^{\,\mathbb{Q}}$ a subgroup.