Extend Level Switching paper with d>=2 preprocessing analysis

Add 21-vertex and 24-vertex examples showing recursive lopsidedness
at d=2. Empirically confirm that the iterated algorithm (balanced
switch when available, preprocess otherwise) drives every face to
depth 0 on all tested configurations. Frame the remaining open
question as identifying a strictly-decreasing monovariant under
unbalanced preprocessing switches.

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

papers/level_switching/paper.tex

@@ -442,12 +442,84 @@ unbalanced surface switch -- and a corresponding statement for $d \geq
 2$, where balancedness depends on depth-$(d-2)$ structure rather than
 just spans -- remains open.
 
+\subsection*{The $d \geq 2$ analog and recursive lopsidedness}
+
+For $d \geq 2$ the obstruction to a balanced surface switch is no longer
+"$F$ has no edge of span 1": it is recursive. We say a depth-$(d-1)$
+neighbour $F' = uvx$ of $F$ is \emph{lopsided} if exactly one of its
+non-$F$ neighbours has depth $d-2$ (the other being deeper or an
+interior face of depth $d-1$). $F$ admits a balanced surface switch
+iff at least one depth-$(d-1)$ neighbour is not lopsided.
+
+The analog of the $9$-vertex example at $d = 2$ is a $21$-vertex
+configuration where the unique depth-$2$ face $F = (0, 7, 14)$ has
+three depth-$1$ neighbours $(0,3,7), (7,10,14), (14,17,0)$, each
+lopsided: their depth-$1$ "deep side" is a degree-$3$ face
+$(3,5,7), (10,12,14), (17,19,0)$ that itself reaches depth $0$ via
+two ears. So the obstruction at $F$ is one layer of lopsidedness;
+after a single preprocessing step the new depth-$2$ face $(3,7,14)$
+sees the previously-hidden balanced descender as a direct neighbour
+and the algorithm terminates immediately.
+
+Stacking lopsidedness yields a $24$-vertex example
+(Figure~\ref{fig:d2-recursive}) where every depth-$1$ neighbour of $F$
+is lopsided \emph{and} the depth-$1$ degree-$3$ face inside each arm
+($G_i$) is itself lopsided. Two preprocessing steps are needed before a
+balanced switch becomes available: the active depth-$2$ face migrates
+from $(0,8,16)$ to $(2,8,16)$ to $(4,8,16)$, at which point the
+\emph{innermost} depth-$1$ face $(4,6,8)$ -- whose two non-$F$ neighbours
+$(4,5,6)$ and $(6,7,8)$ are both ears -- becomes a direct neighbour and
+the balanced condition is satisfied. After the balanced switch, $10$
+further balanced switches drive every face to depth $0$.
+
+\begin{figure}[h]
+\centering
+\includegraphics[width=\textwidth]{fig_d2_recursive.png}
+\caption{Recursive lopsidedness at $d = 2$. Left: $F = (0,8,16)$ depth
+$2$, every arm doubly-lopsided. Middle: one preprocessing switch
+$(0,8) \mapsto (2,16)$ exposes the first lopsided layer; the new
+depth-$2$ face $(2,8,16)$ still has no balanced switch. Right: a
+second preprocessing switch $(8,2) \mapsto (4,16)$ reaches the inner
+balanced face $K_0 = (4,6,8)$, whose two non-$F$ neighbours are both
+ears; the depth-$2$ face $(4,8,16)$ now admits a balanced surface
+switch on edge $(4,8)$.}
+\label{fig:d2-recursive}
+\end{figure}
+
+\subsection*{Empirical termination}
+
+On every tested configuration, iterated preprocessing terminates and
+the algorithm
+\[
+\text{while max-depth face $F$ has $\mathrm{depth}(F) > 0$: }
+\text{do a balanced switch if available, else preprocess}
+\]
+drives every face to depth $0$. The observed step count is
+
+\begin{center}
+\begin{tabular}{lccc}
+configuration & $n$ & $d_{\max}$ & total switches \\\hline
+no-balanced $d=1$ (Figure~\ref{fig:no-balanced}) & 9 & 1 & 4 \\
+singly-lopsided $d=2$ (Figure~\ref{fig:d2-recursive} left only) & 21 & 2 & 8 \\
+doubly-lopsided $d=2$ (Figure~\ref{fig:d2-recursive}) & 24 & 2 & 13 \\
+\end{tabular}
+\end{center}
+
+Each preprocessing step appears to advance the active maximum-depth
+face one vertex along the lopsided arm of the chosen depth-$(d-1)$
+neighbour, peeling off one layer of recursive lopsidedness. The
+remaining open question is to identify the monovariant that captures
+this: a candidate is the total number of triples $(F, F', F'')$ where
+$F' \in N(F)$ is lopsided and $F'' \in N(F')$ is its depth-$d-1$
+"deep side". We do not yet have a proof that this strictly decreases
+under every unbalanced surface switch on a maximum-depth face.
+
 \begin{question}
 \label{q:preprocessing-terminates}
-Does iterated preprocessing reach a balanced surface switch in finitely
-many steps from every initial configuration? Equivalently, is there a
-monovariant on the inner-face structure of $L_k$ that strictly decreases
-at every unbalanced surface switch on a maximum-depth face?
+Does iterated preprocessing always reach a balanced surface switch in
+finitely many steps? Equivalently, is there a monovariant on the
+inner-face structure of $L_k$ that strictly decreases at every
+unbalanced surface switch on a maximum-depth face?
 \end{question}
 
 \end{document}