%
\label{thm:euclidean-static-limit}
In the limit where the split-complex coupling vanishes---formally,
where the off-diagonal coefficient
$\partial v / \partial x$ in the split-complex
Cauchy--Riemann equations tends to zero---the
wave equation
$\partial^2 u / \partial x^2 - \partial^2 u / \partial y^2 = 0$
degenerates to the Laplace equation
$\partial^2 u / \partial x^2 + \partial^2 u / \partial y^2 = 0$.
In this limit:
\begin{enumerate}
    \item[\textup{(i)}]
          The null cone collapses:
          the two characteristic families merge,
          and no real characteristics survive.
    \item[\textup{(ii)}]
          The causal structure
          \textup{(}Definition~\ref{def:causal-structure}\textup{)}
          disappears:
          all directions become equivalent.
    \item[\textup{(iii)}]
          The surviving geometry is Euclidean:
          betweenness, congruence, Pasch,
          and the parallel postulate
          \textup{(}II.T15--II.T18\textup{)}
          remain valid.
\end{enumerate}

\emph{(i).}
The characteristic polynomial
$\xi^2 - c^2 \eta^2 = 0$
parametrizes the coupling strength by~$c$.
When $c \to 0$ (coupling vanishes),
the polynomial becomes $\xi^2 = 0$,
a double root---the two characteristics coalesce,
and the equation becomes parabolic at $c = 0$.
In the full degeneration to Laplace
($c^2 \to -1$, formally replacing $\jj \to i$),
the polynomial $\xi^2 + \eta^2 = 0$
has no real roots.

\emph{(ii).}
Without distinct characteristics,
conditions (C1)--(C3) of
Definition~\ref{def:causal-structure}
are vacuous.
No null cone exists; no forward direction
can be selected.

\emph{(iii).}
The Tarski axioms (II.T15--II.T18)
depend only on the ultrametric distance
$d(x,y) = 2^{-\delta(x,y)}$
and the cylinder structure,
neither of which involves~$\jj$.
They survive the $c \to 0$ limit unchanged.