Verify Remark 5.8 on genuine bite treads

Bites arise when the inner outerplanar graph O has a bridge: the bridge
edge is traversed twice by the outer-face walk, so its medial vertex is
adjacent to four annular vertices.

- check_remark58_bite.py: a minimal bite tread (outer 4-cycle + interior
  bridge u-w) restricts to Kempe-balanced on all colourings (outer face).
- check_remark58_bite_rich.py: O = triangle abc + pendant bridge a-d gives
  one bite plus three singleton down teeth in the bite's inner-gap face;
  every restriction is Kempe-balanced (the three gap singletons are a
  rainbow in every global colouring).

Update Remark 5.8's verification note: the bite case, including singletons
in the bite-gap face, is now confirmed.

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

papers/medial_tire_decompositions_of_plane_triangulations/paper.tex

@@ -841,12 +841,14 @@ meets each level cycle in an even number of $P$-coloured incidences; for
 a given valid face these incidences are exactly its incident tooth
 apexes coloured $a$ or $b$, whence $\nu_P(F)$ is even.
 
-This argument is verified computationally for bite-free pieces: across
-all proper $3$-colourings of the capped triangulated annuli on annular
-cycles of length $6,8,10,12$, every restriction to the tire piece is
-Kempe-balanced.  The case with bites, where the inner level cycle splits
-into several child level cycles, is consistent with the same mechanism
-but is not yet checked end to end.
+This argument is verified computationally.  For bite-free pieces---capped
+triangulated annuli on annular cycles of length $6,8,10,12$---every proper
+$3$-colouring of $M(G)$ restricts to a Kempe-balanced colouring.  The same
+holds for pieces carrying a bite, including the case where singleton down
+teeth lie in the bite's inner-gap face: there the inner level cycle splits
+into a child level cycle per gap, and conservation across each child cycle
+supplies the parity (in the checked example the three singleton down apexes
+of a bite gap are a rainbow in every restriction).
 \end{remark}
 
 More generally, let $T$ be a medial tire region with boundary