%
\label{thm:label-convergence}
For every prime~$p_i$,
the sequence $\mathrm{Label}_n(p_i)$ for $n \geq i$
is eventually constant.
More precisely:
\begin{enumerate}
    \item[\textup{(i)}]
          \textbf{Immediate stabilisation.}
          $\mathrm{Label}_n(p_i) = \mathrm{Label}_i(p_i)$
          for all $n \geq i$.
          The label stabilises at the depth
          where the prime first enters.
    \item[\textup{(ii)}]
          \textbf{Limiting classifier.}
          The function
          \begin{equation}\label{eq:ch18-label-infty}
              \mathrm{Label}_\infty \;:\;
              \{\text{all primes}\}
              \;\longrightarrow\;
              \{B, C, X\},
              \qquad
              \mathrm{Label}_\infty(p_i) \;=\; \mathrm{Label}_i(p_i),
          \end{equation}
          is well defined.
          It is the unique classifier
          compatible with all finite-depth classifiers
          under the primorial projections.
    \item[\textup{(iii)}]
          \textbf{Explicit formula.}
          For odd primes $p > 2$:
          \begin{equation}\label{eq:ch18-explicit-label}
              \mathrm{Label}_\infty(p) \;=\;
              \begin{cases}
                  B & \text{if } p \equiv \pm 1 \pmod{8}, \\
                  C & \text{if } p \equiv \pm 3 \pmod{8}.
              \end{cases}
          \end{equation}
          For $p = 2$:
          $\mathrm{Label}_\infty(2) = X$.
\end{enumerate}