%
\label{thm:dimension-four}
\[
    \boxed{\dim_\tau \;=\; 4.}
\]

\emph{Upper bound: $\dim_\tau \leq 4$.}
The ABCD chart (I.D17, Book~I) is a bijection
between $\tau$-admissible quadruples
and objects of Category~$\tau$
(Hyperfactorization Theorem, I.T04, Book~I).
Two points with identical ABCD coordinates
at every stage are the same point,
so the four rays jointly separate all points.

\emph{Lower bound: $\dim_\tau \geq 4$.}
We show that no triple of rays suffices.
For each of the four triples
$\{A,B,C\}$, $\{D,B,C\}$, $\{D,A,C\}$, $\{D,A,B\}$,
pairwise independence (I.P08, Book~I)
provides two distinct points
that agree on the three selected coordinates
at every stage but differ on the fourth:
\begin{itemize}
    \item \textbf{$\{A,B,C\}$} (missing~D):
          $D$ is independent of $A$, $B$, $C$,
          so fixing $(A,B,C)$ leaves $D$ free.
    \item \textbf{$\{D,B,C\}$} (missing~A):
          $A$ is independent of $D$, $B$, $C$.
    \item \textbf{$\{D,A,C\}$} (missing~B):
          $B$ is independent of $D$, $A$, $C$.
    \item \textbf{$\{D,A,B\}$} (missing~C):
          $C$ is independent of $D$, $A$, $B$.
\end{itemize}
Each triple fails to separate some pair.
Hence $\dim_\tau \geq 4$.