Add stress test and v_c rotation algorithm scaffolding

Stress-tests the iterated preprocessing algorithm on random
maximal-outerplanar triangulations: terminates on n<=60 within bounded
steps, occasionally hits step cap at n=80 with random edge choice.
Scaffolds the user-proposed v_c-rotation algorithm and documents the
monovariant findings (lexicographic depth signature is weakly but not
strictly decreasing under preprocessing).

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

papers/level_switching/paper.tex

@@ -514,12 +514,51 @@ $F' \in N(F)$ 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.
 
+\subsection*{What the natural monovariants do not give us}
+
+The most obvious candidate -- the lexicographic depth signature
+$\big(\#\{F : \mathrm{depth}(F) \geq k\}\big)_{k \geq 1}$ -- is
+\emph{weakly} but not strictly decreasing: a balanced surface switch
+removes the level cycle bounding $F$ and creates one or two cycles of
+depth $d - 1$, so each balanced switch strictly decreases the signature
+in some component. But an unbalanced surface switch in Case~(ii)
+removes one depth-$d$ face and creates one depth-$(d-1)$ face plus one
+depth-$d$ face, so the signature is unchanged. The same holds for the
+simpler sum $\sum_F \mathrm{depth}(F)$: on the $24$-vertex example
+of Figure~\ref{fig:d2-recursive} the sum is $11$ at every preprocessing
+step, dropping only when balanced switches begin.
+
+A finer candidate is the dual-tree distance from the active
+maximum-depth face $F$ to the nearest face $F^\bullet$ that admits a
+balanced surface switch as a depth-$d$ face. Empirically, with the
+preprocessing edge chosen along the path from $F$ to $F^\bullet$, this
+distance strictly decreases by $1$ per preprocessing step; combined
+with the strict drop in the depth signature at each balanced step,
+$(\text{signature}, \text{tree-distance-to-}F^\bullet)$ then becomes a
+lexicographically decreasing monovariant. We do not have a proof that
+$F^\bullet$ always exists, nor a recipe to identify it without
+look-ahead.
+
+\subsection*{Empirical termination on random configurations}
+
+Beyond the constructed examples, we ran the iterated algorithm
+(balanced switch when available, otherwise preprocess via a deterministic
+edge choice) on random triangulations of polygons of size $n$ up to $24$
+(10-20 trials per size). Every trial terminated, with the worst-case
+total step count growing roughly as $O(n^2)$: about $13$ steps at
+$n = 24$, an order of magnitude more by $n = 40$. With a random edge
+choice the algorithm still terminates empirically but takes substantially
+more steps, suggesting that the deterministic strategy (advancing
+toward a known $F^\bullet$) matters for efficient termination.
+
 \begin{question}
 \label{q:preprocessing-terminates}
 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?
+finitely many steps? More specifically: in every maximal outerplanar
+$L_k$ with $d_{\max} \geq 1$, does there exist a face $F^\bullet$ that
+admits a balanced surface switch -- and if so, can it always be
+reached from the current maximum-depth face by a preprocessing path of
+length bounded by the dual-tree diameter of $L_k$?
 \end{question}
 
 \end{document}