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\label{thm:primorial-cofinality}
The primorial ladder $\{M_k\}_{k \geq 1}$
is cofinal in the full modular tower.
Specifically:
\begin{enumerate}
    \item[\textup{(i)}] \textbf{Squarefree divisibility.}
          For every squarefree integer~$N$,
          there exists $k_0$ such that
          $N \mid M_k$ for all $k \geq k_0$.
          The cutoff is $k_0 = \pi(p_{\max}(N))$,
          where $p_{\max}(N)$ is the largest prime factor of~$N$
          and $\pi$ is the prime-counting function.
    \item[\textup{(ii)}] \textbf{General divisibility.}
          For every positive integer~$N$,
          there exists $k_0$ such that the reduction map
          $\Z / M_k\Z \to \Z / \mathrm{rad}(N)\Z$
          is well-defined for all $k \geq k_0$,
          where $\mathrm{rad}(N) = \prod_{p \mid N} p$
          is the radical of~$N$.
    \item[\textup{(iii)}] \textbf{Inverse limit equivalence.}
          The canonical map
          \begin{equation}\label{eq:ch14-cofinal-iso}
              \widehat{\Z}_\tau
              \;=\;
              \varprojlim_{k} \Z / M_k\Z
              \;\longrightarrow\;
              \varprojlim_{N,\, \mathrm{sqfree}} \Z / N\Z
          \end{equation}
          is an isomorphism of profinite rings.
\end{enumerate}