Add Level Switching paper with surface-switch framework

Defines level cycles, edge switches, surface switches, and facial depth
on level components of plane triangulations. Proves outerplanarity of
level components and a depth-descent lemma. Introduces balanced surface
switches and proves they remove a depth-d level cycle while creating
1-2 new depth-(d-1) cycles. Documents the 9-vertex counterexample where
no balanced switch exists and sketches preprocessing toward
balancedness, leaving general termination open.

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

papers/level_switching/experiments/counterexample_balanced_existence.py

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+"""9-vertex L_k where the unique depth-1 face has NO balanced surface switch.
+
+Outer cycle: 0..8. Triangulated with chords 0-2, 0-3, 3-5, 3-6, 0-6, 6-8.
+Central triangle F = (0,3,6) has depth 1; its three neighbours
+(0,2,3), (3,5,6), (6,8,0) are all depth 0 but each has only ONE
+outer-cycle edge (not two), so none is an "ear" of F.
+
+For d = 1, balancedness requires F' to be an ear of uv (both non-uv
+edges on the outer cycle). No neighbour of F qualifies.
+"""
+import os
+import math
+import networkx as nx
+import matplotlib.pyplot as plt
+from matplotlib.patches import Polygon
+
+OUT_DIR = os.path.join(os.path.dirname(os.path.abspath(__file__)), os.pardir)
+
+n = 9
+POS = {i: (math.cos(math.radians(90 - i * 360 / n)),
+           math.sin(math.radians(90 - i * 360 / n))) for i in range(n)}
+OUTER_EDGES = [(i, (i + 1) % n) for i in range(n)]
+CHORDS = [(0, 2), (0, 3), (3, 5), (3, 6), (0, 6), (6, 8)]
+
+FACES = [
+    (0, 1, 2),       # ear
+    (0, 2, 3),       # 1 outer edge, depth 0
+    (3, 4, 5),       # ear
+    (3, 5, 6),       # 1 outer edge, depth 0
+    (6, 7, 8),       # ear
+    (6, 8, 0),       # 1 outer edge, depth 0
+    (0, 3, 6),       # central, depth 1 -- the troublemaker
+]
+
+
+def face_edges(f):
+    return {frozenset((f[0], f[1])), frozenset((f[1], f[2])),
+            frozenset((f[0], f[2]))}
+
+
+outer_set = {frozenset(e) for e in OUTER_EDGES}
+
+D = nx.Graph()
+D.add_nodes_from(range(len(FACES)))
+for i, fi in enumerate(FACES):
+    for j, fj in enumerate(FACES):
+        if i < j and face_edges(fi) & face_edges(fj):
+            D.add_edge(i, j)
+
+B = [i for i, f in enumerate(FACES)
+     if len(face_edges(f) & outer_set) >= 1]
+depth = {i: min(nx.shortest_path_length(D, i, b) for b in B)
+         for i in range(len(FACES))}
+
+palette = {0: '#86efac', 1: '#fde68a', 2: '#fca5a5'}
+edge_pal = {0: '#16a34a', 1: '#d97706', 2: '#dc2626'}
+
+fig, ax = plt.subplots(figsize=(7, 7))
+for i, f in enumerate(FACES):
+    d = depth[i]
+    poly = Polygon([POS[v] for v in f], closed=True,
+                   facecolor=palette[d], edgecolor=edge_pal[d],
+                   linewidth=1.6, alpha=0.7, zorder=0)
+    ax.add_patch(poly)
+    cx = sum(POS[v][0] for v in f) / 3
+    cy = sum(POS[v][1] for v in f) / 3
+    ax.text(cx, cy, rf'$\mathrm{{depth}}={d}$',
+            ha='center', va='center', fontsize=10,
+            color=edge_pal[d], fontweight='bold')
+
+# Mark the three "bad" chord edges (would-be-switched edges of F that
+# fail balancedness because the chord side has no outer-cycle edge to
+# pair with).
+F_edges = [(0, 3), (3, 6), (0, 6)]
+for (a, b) in OUTER_EDGES + CHORDS:
+    color = '#333'; lw = 1.2
+    if (a, b) in F_edges or (b, a) in F_edges:
+        color = '#dc2626'; lw = 3.0
+    ax.plot([POS[a][0], POS[b][0]], [POS[a][1], POS[b][1]],
+            color=color, linewidth=lw, zorder=1)
+
+for i, (x, y) in POS.items():
+    ax.scatter([x], [y], s=300, c='#1f2937', edgecolors='black',
+               linewidths=1.0, zorder=2)
+    ax.text(x, y, str(i), ha='center', va='center',
+            fontsize=10, color='white', fontweight='bold', zorder=3)
+
+ax.set_aspect('equal'); ax.axis('off')
+ax.set_xlim(-1.3, 1.3); ax.set_ylim(-1.3, 1.3)
+ax.set_title('Depth-1 face with no balanced surface switch',
+             fontsize=12)
+fig.tight_layout()
+out = os.path.join(OUT_DIR, 'fig_no_balanced_switch.png')
+fig.savefig(out, dpi=180, bbox_inches='tight')
+plt.close(fig)
+print(f'wrote {out}')
+for i, f in enumerate(FACES):
+    print(f'  {f} -> depth {depth[i]}')