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\label{def:causal-structure}
The \textbf{causal structure} on~$\tau^3$ is the
partial order on null-line segments determined by
the following data:
\begin{enumerate}
    \item[\textup{(C1)}] \textbf{Null directions.}
          The two characteristic families
          $\xi = \mathrm{const}$ and $\zeta = \mathrm{const}$
          from the split-complex wave equation (II.D21).

    \item[\textup{(C2)}] \textbf{Forward cone.}
          The B/C asymmetry from Prime Polarity
          (I.T05, Book~I) provides a preferred direction:
          the $\gamma$-channel ($B$-coordinate)
          and the $\eta$-channel ($C$-coordinate)
          are structurally ordered by
          the exponent--tetration hierarchy.
          $B$ precedes~$C$ in the peel-off sequence.
          This asymmetry selects a \textbf{forward} direction
          within each time-like sector.

    \item[\textup{(C3)}] \textbf{Causal ordering.}
          A point~$q$ is in the \textbf{causal future} of~$p$
          if $q$ can be reached from~$p$
          by following a curve that remains
          within the forward null cone at every point.
\end{enumerate}
A direction is \textbf{time-like} if it lies
strictly inside the null cone,
\textbf{space-like} if strictly outside,
and \textbf{null} (light-like) if on the cone boundary.