%
\label{def:first-disagreement-depth}
For $x, y \in \tau^3$, the \textbf{first disagreement depth}
is the function
\[
    \boxed{\delta(x, y)
    \;:=\;
    \max\bigl\{\, k \geq 0
    \;\big|\;
    \pi_k(x) = \pi_k(y) \,\bigr\}}
\]
with the conventions:
\begin{itemize}
    \item If $x = y$, then $\pi_k(x) = \pi_k(y)$
          for all~$k$, and we set $\delta(x,y) = \infty$.
    \item If $x \neq y$, then by the CRT separation property
          (I.T04, Book~I),
          there exists a finite stage~$k_0$
          such that $\pi_{k_0}(x) \neq \pi_{k_0}(y)$.
          The supremum is then finite:
          $\delta(x,y) \in \mathbb{N}_{\geq 0}$.
\end{itemize}
Equivalently, $\delta(x,y)$ is the \emph{last} stage
at which $x$ and~$y$ have identical CRT reductions---the
depth of the deepest cylinder that contains both.