Add medial tire cut experiment and chaining section

New experiments/run_medial_tire_cut_experiment.py: generates a random
maximal planar graph (stacked seed + random diagonal flips), builds the
medial graph, takes the tire decomposition at a random vertex level
source, walk-depth labels and cuts each full medial tire graph chained
down the tire tree, and assembles one final cut graph of M(G) with a
global label map (data only; graphics go in a separate script).

Fix label_and_cut: the root face is None, which collided with the
next(..., None) sentinel, leaving teeth unlabelled when the entry up
tooth lay inside a bite gap; use a distinct sentinel so the ascent to
the root face runs.

Add a "Chaining across the tire tree" section to the paper, clarifying
that the candidate parent down teeth are the boundary (singleton) down
teeth only -- bite teeth are interior to the parent and shared with no
child, so a lower-walk bite is skipped.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

papers/medial_tire_cuts/experiments/medial_tire_cut_labelling.py

@@ -179,7 +179,9 @@ def label_and_cut(graph: FullMedialTireGraph, entry_edge: int,
     # Steps 1-3: the entry face.
     traverse(innermost_bite(entry_edge, graph.bites), entry_edge, is_entry=True)
 
-    # Steps 4-6: descend through bites, deepest first.
+    # Steps 4-6: descend (or ascend) through bites, deepest first.  The root
+    # face is ``None``, so we use a distinct sentinel for "no unlabelled face".
+    _MISSING = object()
     while len(depth) < graph.n:
         labelled_bite_teeth = sorted(
             (e for e in depth if door_bite(graph, e) is not None),
@@ -188,8 +190,8 @@ def label_and_cut(graph: FullMedialTireGraph, entry_edge: int,
         for t in labelled_bite_teeth:
             target = next((F for F in faces_bordered(graph, t)
                            if any(e not in depth for e in face_boundary(graph, F))),
-                          None)
-            if target is not None:
+                          _MISSING)
+            if target is not _MISSING:
                 traverse(target, t, is_entry=False)
                 break
         else: