Add bridge switch / bridge-derived level graph; set up exhaustive test

- Define bridge switch (E/O switch whose new same-parity edge is a bridge
  in its parity subgraph) and bridge-derived level graph in the paper.
  Note that bridge switches preserve bipartite parity subgraphs, so every
  bridge-derived level graph is automatically valid.
- Discover the E/O-switch relation is directed (irreversible when a switch
  produces a cross-parity edge); T*_9 reaches an ELG forward but no ELG
  reaches it, explaining why it is not derived. This rules out a simple
  switch-invariant characterization.
- Bridge orbits are far smaller than full E/O orbits (~10^4 vs ~10^8 for
  some labellings), making exhaustive search feasible. Each of the 4 open
  duals has ~150 valid parity partitions; exhaustive bridge-orbit search
  per partition can decide bridge-derivability conclusively.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

papers/even_level_graph_generators/paper.tex

@@ -250,6 +250,37 @@ A derived level graph $G'$ is \emph{valid} if both $E_{G,S}(G')$ and
 $O_{G,S}(G')$ contain only even cycles.
 \end{definition}
 
+\begin{definition}[Bridge switch]
+\label{def:bridge-switch}
+Let $G'$ be a triangulation reached from an Even Level Graph $G$, with
+parity classes inherited from $G$ as in
+Definition~\ref{def:derived-level-graph}. An edge switch on an edge
+$e \in E \cup O$ of $G'$, replacing $uvw, uvx$ by the edge $wx$, is a
+\emph{bridge switch} if either
+\begin{itemize}
+\item the new edge $wx$ is a cross-parity edge (one endpoint even, the
+other odd), so $wx$ enters neither parity subgraph; or
+\item $wx$ is a same-parity edge and is a \emph{bridge} in the parity
+subgraph it joins -- that is, $w$ and $x$ lie in different connected
+components of that parity subgraph, so adding $wx$ creates no new cycle.
+\end{itemize}
+\end{definition}
+
+\begin{definition}[Bridge-derived level graph]
+\label{def:bridge-derived-level-graph}
+A \emph{bridge-derived level graph} of an Even Level Graph $G$ is a
+triangulation obtained from $G$ by a sequence of bridge switches
+(Definition~\ref{def:bridge-switch}).
+\end{definition}
+
+Because a bridge switch never closes a cycle in a parity subgraph, it
+never introduces an odd cycle there. As an Even Level Graph has
+bipartite parity subgraphs (every level cycle is even), every
+bridge-derived level graph has bipartite parity subgraphs as well, and
+so is automatically a valid derived level graph. Equivalently, the
+first Betti number of each parity subgraph is non-increasing along any
+sequence of bridge switches.
+
 \begin{definition}[Intertwining tree]
 \label{def:intertwining-tree}
 A maximal planar graph $G$ is an \emph{intertwining tree} if its