even_level: extend to n=25 -- second internally-6-connected core, also bridge-derived

Enumerate non-Hamiltonian cyclically-5-connected cubic planar graphs by
running plantri -c5 -d for n in {23,25,26} (n=24 already in the previous
commit) and filtering for non-Hamiltonian dual:
  n=23  -> 0 of 1970   (recomputes Faulkner-Younger minimality)
  n=24  -> 1 of 6833   (the Tutte/Fig 2.10 graph)
  n=25  -> 1 of 23384  (new; unique 46-vertex one)
  n=26  -> 0 of 82625

Both T (n=24) and T_25 (n=25) verified internally 6-connected by exhaustive
5-cut scan: every 5-cut is the neighborhood of a degree-5 vertex. This is
the strongest connectivity a planar triangulation can have and the level
at which Birkhoff-style reductions terminate, so both are genuinely
irreducible bases of any decomposition argument.

T_25 is also bridge-derived: witness Even Level Graph from source 24
(max level 4) at depth 2, orbit only 3114 states. Forward switches:
remove {21,23} add {22,24}; remove {3,5} add {1,6}. Both adds are bridges
of the even parity subgraph. Same witness signature as T (minimum total
Betti, tiny orbit, depth 2).

New subsection "Beyond n=24: enumeration and the next 5-connected core",
abstract extended, new Figure 7 (core_n25_dual.png). Reproducibility
scripts: draw_core_witness.py and verify_core_witness.py (both
parametrized so they work on any 5-conn non-Ham-dual core's g6).

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

papers/even_level_graph_generators/paper.tex

@@ -112,7 +112,15 @@ the unique $44$-vertex non-Hamiltonian \emph{cyclically $5$-connected}
 cubic planar graph -- settling a uniqueness question Holton--McKay left
 open -- whose $24$-vertex $5$-connected dual is the first test of the
 conjecture outside the $3$-cut family; it too is a bridge-derived level
-graph, two bridge switches from an Even Level Graph.
+graph, two bridge switches from an Even Level Graph. Iterating the same
+generation procedure up to $n = 26$ produces, at $n = 25$, a second
+non-Hamiltonian cyclically-$5$-connected cubic planar graph (likewise
+unique at its size); its dual is verified \emph{internally
+$6$-connected} -- the strongest connectivity possible for a planar
+triangulation, and the level at which Birkhoff-style reductions
+terminate -- and likewise bridge-derived, again at depth $2$. Both known
+internally-$6$-connected non-Hamiltonian-dual cores thus satisfy the
+conjecture, with identical witness signature.
 \end{abstract}
 
 \maketitle
@@ -586,8 +594,8 @@ non-Hamiltonian cyclically $5$-connected cubic planar graph on $44$
 vertices.
 
 Let $T$ be its dual: a $24$-vertex triangulation with vertex connectivity
-$5$ and no separating triangle, and -- since its dual is non-Hamiltonian
--- not an intertwining tree. We find that $T$ is nonetheless a
+$5$ (in fact internally $6$-connected, verified in the next subsection),
+and -- since its dual is non-Hamiltonian -- not an intertwining tree. We find that $T$ is nonetheless a
 bridge-derived level graph. Of its $333$ valid parity partitions most are
 useless: their backward bridge-orbits exceed $8 \times 10^5$ states with
 no Even Level Graph in sight. But one partition has a backward orbit of
@@ -622,6 +630,80 @@ in particular no odd cycle, is created.}
 \label{fig:n24-dual}
 \end{figure}
 
+\subsection*{Beyond $n = 24$: enumeration and the next $5$-connected core}
+
+The graph $T$ of the previous subsection is one core; iterating the same
+generation procedure produces the rest. Below, the third column counts
+$5$-connected triangulations on $n$ vertices (\texttt{plantri -c5} $n$),
+and the fourth filters them by Hamiltonicity of the cubic dual:
+\begin{center}
+\begin{tabular}{cccc}
+$n$ & dual $|V|$ & $5$-connected triangulations & non-Hamiltonian dual \\\hline
+$23$ & $42$ & $1{,}970$ & $0$ \\
+$24$ & $44$ & $6{,}833$ & $1$ \quad (Holton--McKay Fig.~2.10) \\
+$25$ & $46$ & $23{,}384$ & $1$ \\
+$26$ & $48$ & $82{,}625$ & $0$ \\
+\end{tabular}
+\end{center}
+The $n = 23$ row recomputes Faulkner--Younger's minimality (no cyclically
+$5$-connected non-Hamiltonian cubic planar graph below $44$ vertices). At
+$n = 25$ we find a single new cyclically $5$-connected non-Hamiltonian
+cubic planar graph on $46$ vertices; its dual we call $T_{25}$, a
+$25$-vertex $5$-connected triangulation with degree sequence
+$5^{21}\,8^{3}\,9^{1}$. At $n = 26$ there are again none. (All counts
+depend on the correctness of \texttt{plantri} and of the Hamiltonicity
+test.)
+
+An exhaustive scan of all $\binom{n}{5}$ candidate $5$-cuts confirms that
+both $T$ and $T_{25}$ are \emph{internally $6$-connected}: every
+$5$-element vertex cut is the neighbourhood of a single degree-$5$
+vertex, so neither admits a nontrivial separation of size $\le 5$. This
+is the strongest connectivity a planar triangulation can have -- planar
+graphs are never $6$-connected, because every planar graph has a vertex
+of degree $\le 5$ -- and it is the level at which Birkhoff-style
+reductions terminate (a minimal counterexample to the four colour theorem
+can be assumed internally $6$-connected). Both cores are therefore
+genuinely irreducible bases of any decomposition-based argument: nothing
+in the cut-decomposition can simplify them further.
+
+The bridge-derivability test on $T_{25}$ follows the same recipe and
+yields the same conclusion. Enumerating valid parity partitions and
+ordering by total Betti, the minimum total Betti is $1$, and the very
+first Betti-$1$ partition encountered with a small backward bridge-orbit
+contains an Even Level Graph (source $s = 24$, maximum level $4$) at
+depth $2$ in an orbit of $3{,}114$ states. The two bridge switches
+carrying that Even Level Graph to $T_{25}$ are
+\[
+  \text{remove } \{21,23\},\ \text{add } \{22,24\}
+  \quad\text{and}\quad
+  \text{remove } \{3,5\},\ \text{add } \{1,6\},
+\]
+each adding a same-parity edge that is a bridge of the even parity
+subgraph; both steps are valid bridge switches
+(Figure~\ref{fig:n25-dual}). The witness signature -- minimum total
+Betti, a tiny bridge-orbit, a depth-$2$ Even Level Graph -- is the same
+that worked on $T$, suggesting it is not an accident of the $n = 24$
+example but the generic shape of bridge-derivability witnesses on
+internally-$6$-connected non-Hamiltonian-dual cores. The conjecture thus
+survives every irreducible small case the connectivity bound forces us
+to face.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=0.7\textwidth]{figures/core_n25_dual.png}
+\caption{The $25$-vertex dual $T_{25}$ of the unique $46$-vertex
+non-Hamiltonian cyclically $5$-connected cubic planar graph -- the only
+such cubic graph at $46$ vertices and the second internally
+$6$-connected core known. Drawn crossing-free and coloured by parity
+(blue even, orange odd) for its witness partition. $T_{25}$ is internally
+$6$-connected and not an intertwining tree, yet is a bridge-derived level
+graph: the two solid green edges $\{1,6\}$ and $\{22,24\}$ are the bridge
+edges introduced by the two bridge switches carrying its witness Even
+Level Graph (source $24$) to $T_{25}$. Each is a bridge of the even
+parity subgraph.}
+\label{fig:n25-dual}
+\end{figure}
+
 \subsection*{Toward a characterization of bridge-derived graphs}
 
 A bridge switch is a diagonal flip of the quadrilateral around a level