%
\label{thm:parallel-preservation}
%   II.D71, II.T15--II.T18
In Stage-Finite Euclidean Geometry (II.D71):
\begin{enumerate}
    \item[\textup{(i)}]
          \textbf{Stage-$k$ preservation.}
          The Parallel Postulate holds on each $\tau^3_k$.
    \item[\textup{(ii)}]
          \textbf{Limit preservation.}
          The profinite limit preserves the Parallel Postulate:
          it holds on $\tau^3$.
    \item[\textup{(iii)}]
          \textbf{No light-cone obstruction.}
          The wave equation's characteristics
          operate between stages (inter-stage propagation)
          and do not generate intra-stage light cones.
\end{enumerate}

[Proof sketch]
(i) On each finite $\tau^3_k$,
betweenness is defined by
$B(x,y,z) \iff d(x,z) = \max(d(x,y),d(y,z))$
(ultrametric betweenness, II.D19).
Given a line $\ell$ and a point $P \notin \ell$
in $\tau^3_k$,
the ultrametric structure guarantees
exactly one line through $P$
that does not intersect $\ell$
within $\tau^3_k$
(because the clopen balls partition the space
into non-overlapping regions).
(ii) The Parallel Postulate is a universal statement:
``for all lines $\ell$, for all points $P$, \ldots''
Such statements are preserved under profinite limits
(inverse limits of surjections).
(iii) Light cones require continuous characteristics
in a manifold.
$\tau^3_k$ is discrete; there are no continuous curves.