even_level: add "Toward a characterization of bridge-derived graphs"

Record the partition sweep on the n=24 Fig 2.10 dual. New subsection +
experiments/bridge_partition_sweep.py.

Findings:
- A bridge switch is a constrained diagonal flip; bridge-derived via L
  means lying in an Even-Level-Graph component of the restricted flip
  graph. So the question is which flip-components contain an ELG.
- Identity: every 4-coloring of a triangulation has e_cross = 2n-4 (each
  face has one within-pair edge), so total parity-subgraph Betti =
  (c_A+c_B)-2; intertwining trees are the Betti-0 case.
- Of T's 333 valid partitions, total Betti splits 288/42/3 over 1/2/3;
  min is 1 (T not intertwining). All 27 partitions found bridge-derived
  (depth 2-3) have the minimum Betti 1 -> necessary.
- But not sufficient: only 27 of 288 Betti-1 partitions yield a witness;
  the rest have flip-orbits >1.5e5 with no ELG, and a 12x budget increase
  found none. The discriminator is flip-component structure (sharp
  orbit-size dichotomy), not a numerical invariant. Characterizing which
  Betti-minimal partitions sit in an ELG component is left open.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

papers/even_level_graph_generators/paper.tex

@@ -622,6 +622,58 @@ in particular no odd cycle, is created.}
 \label{fig:n24-dual}
 \end{figure}
 
+\subsection*{Toward a characterization of bridge-derived graphs}
+
+A bridge switch is a diagonal flip of the quadrilateral around a level
+edge, constrained so the flipped-in edge enters no parity subgraph as a
+cycle edge. Fixing a valid parity partition $L$, the bridge switches
+therefore act on the triangulations whose two $L$-parity subgraphs are
+both bipartite, and $G$ is a bridge-derived level graph via $L$ exactly
+when $G$ lies in the same connected component as an Even Level Graph in
+this restricted flip graph. So the question ``which triangulations are
+bridge-derived?'' is really ``which flip-components contain an Even Level
+Graph?'', quantified over $L$. We probed this on the $24$-vertex dual $T$
+of Figure~\ref{fig:n24-dual}, sweeping all of its valid parity partitions;
+the experiment is recorded in \texttt{experiments/bridge\_partition\_sweep.py}.
+
+One coordinate organises the picture. For any proper $4$-coloring of a
+triangulation, group the colours as $\{1,2\}\mid\{3,4\}$; then each
+triangular face has exactly one within-pair edge, and since there are
+$2n-4$ faces and each edge lies on two of them, the parity subgraphs carry
+exactly $n-2$ edges between them. Hence $e_{\mathrm{cross}} = 2n-4$ for
+\emph{every} valid partition (confirmed across all partitions of $T$), and
+the total first Betti number of the two parity subgraphs equals
+$(c_A + c_B) - 2$, where $c_A, c_B$ count their connected components. The
+intertwining-tree case is precisely total Betti $0$ -- both parts trees,
+$c_A = c_B = 1$ -- so for a triangulation that is not an intertwining tree
+the total Betti is at least $1$.
+
+The $333$ valid partitions of $T$ have total Betti $1$, $2$, $3$ for
+$288$, $42$, $3$ of them respectively; the minimum is $1$, consistent with
+$T$ not being an intertwining tree. A backward bridge-orbit search locates
+Even Level Graph witnesses (at depth $2$--$3$) for $27$ partitions, and
+\emph{every} one of them has the minimum total Betti $1$ -- one parity
+class a tree, the other a tree plus a single even cycle. Minimal total
+Betti is thus a necessary feature of the bridge-derived partitions we
+find.
+
+It is not sufficient, and the way it fails is informative. Of the $288$
+Betti-$1$ partitions only those $27$ yield a witness; the rest exhibit
+bridge-orbits exceeding $1.5\times 10^5$ states with no Even Level Graph,
+and increasing the search budget twelvefold produced no further witnesses.
+The bridge-derivable partitions are separated from the others not by any
+of the simple invariants we measured -- total Betti, component counts,
+class sizes, or BFS-level realizability (uniformly false here) -- but by a
+sharp dichotomy in flip-orbit size: a tiny component containing an Even
+Level Graph versus a vast one that appears not to. (We cannot yet certify
+the latter, as no large orbit was exhausted; but a twelvefold budget
+increase yielding nothing makes mere depth an unlikely explanation.) The
+evidence therefore points away from a numerical characterization and
+toward the component structure of the restricted flip graph: minimal total
+Betti is a clean necessary condition, but characterizing \emph{which}
+Betti-minimal partitions lie in an Even-Level-Graph component remains open
+and is, on this evidence, the crux of deciding bridge-derivability.
+
 \begin{thebibliography}{9}
 
 \bibitem{holton-mckay}