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Remove the Empirical status subsection (small-n table)

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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
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 $n = 21$}
The first triangulations that are \emph{not} intertwining trees are the