Remove the Empirical status subsection (small-n table)

Drop the n<=9 bridge-derived classification table and its surrounding
discussion; the n=21 boundary case now follows directly from the
trivial-below-21 observation.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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2026-05-22 11:27:55 -04:00
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@@ -334,38 +334,6 @@ $n = 21$ and there are exactly $6$ of them. Below $n = 21$ every
maximal planar graph is an intertwining tree, which is why the maximal planar graph is an intertwining tree, which is why the
disjunction holds trivially in that range. disjunction holds trivially in that range.
\subsection*{Empirical status}
For each isomorphism class of maximal planar graphs on $n$ vertices,
we ask whether (i) some isomorphic representative is a bridge-derived
level graph of some Even Level Graph, and/or (ii) it is an intertwining
tree. The conjecture holds for the class iff at least one of (i), (ii)
holds. Below $n = 21$ condition (ii) holds for \emph{every} class, so
the table mainly records how far the bridge-derived disjunct (i) reaches
on its own. We classified bridge-derivability exhaustively for
$n \le 9$, where every backward bridge-orbit can be enumerated in full.
\begin{center}
\begin{tabular}{rcccccc}
$n$ & \# iso & bridge only & inter.\ only & both & missing & status \\\hline
$6$ & $2$ & $0$ & $0$ & $2$ & $0$ & holds \\
$7$ & $5$ & $0$ & $1$ & $4$ & $0$ & holds \\
$8$ & $14$ & $0$ & $2$ & $12$ & $0$ & holds \\
$9$ & $50$ & $0$ & $14$ & $36$ & $0$ & holds \\
\end{tabular}
\end{center}
\noindent
Here ``bridge only'' counts classes that are bridge-derived but not
intertwining trees, ``inter.\ only'' the reverse, and ``both'' the
intersection; ``missing'' counts classes that are neither (a
counterexample). The ``bridge only'' column is $0$ throughout this range
precisely because every class is an intertwining tree for $n \le 20$;
the ``inter.\ only'' counts ($1,2,14$) are the classes that the
bridge-derived disjunct alone does not yet reach, showing that
bridge-derivability is strictly weaker than ``intertwining tree'' here
and that the two disjuncts genuinely complement one another.
\subsection*{The boundary case $n = 21$} \subsection*{The boundary case $n = 21$}
The first triangulations that are \emph{not} intertwining trees are the The first triangulations that are \emph{not} intertwining trees are the