\label{def:minkowski-extension}
The \emph{Minkowski extension} of $M_{\tau^3}$ is the $3{+}1$-dimensional spacetime
\begin{equation}\label{eq:ch75-minkowski}
\mathcal{M}_\tau \;=\; \tau^1 \times M_{\tau^3},
\end{equation}
where:
\begin{enumerate}
\item\emph{(Time.)}
The base $\tau^1$ provides the time direction; the progression structure (Book~I, Part~II) provides causal ordering.

\item\emph{(Space.)}
The fiber at each $x \in \tau^1$ is the three-dimensional bulk $U_x \subset M_{\tau^3}$ constructed by Theorem~\ref{thm:global-cartesian-gluing}.

\item\emph{(Signature.)}
The split-complex boundary ring $H_\tau = \mathbb{Z}[\,\jj\,]/(\jj^2 - 1)$ provides the metric signature.  The decomposition $H_\tau \cong \mathbb{Z} \oplus \mathbb{Z}$ via idempotents $e_\pm = \tfrac{1}{2}(1 \pm \jj)$ separates the time-like component ($e_-$) from the space-like components ($e_+$), yielding the hyperbolic signature $(-, +, +, +)$.

\item\emph{(Light cones.)}
The null directions are the zero divisors of $H_\tau$: elements $a + b\jj$ with $a^2 = b^2$.  Their global assembly via the Coherent Force produces the causal structure of spacetime.
\end{enumerate}