Verify Remark 5.8 mechanism; correct it to level-cycle conservation

Computational checks of the necessity of Kempe-balance (Remark 5.8):

- check_medial_face_parity.py shows the naive "even P-coloured vertices
  per medial face" claim is false (odd vertex-faces on the octahedron and
  stacked triangulations), so the original face-parity justification was
  wrong.
- check_remark58_bitefree.py builds genuine bite-free tire pieces (capped
  triangulated annuli) and confirms every proper 3-colouring of M(G)
  restricts to a Kempe-balanced colouring (|A(T)|=6,8,10,12, all
  colourings, zero failures).

Rewrite Remark 5.8 to cite the correct mechanism: the up/down apexes lie
on level cycles, and a P-Kempe cycle meets each level cycle in an even
number of P-coloured incidences (Lemma 5.6).  Note the bite case is not
yet checked end to end.

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

papers/medial_tire_decompositions_of_plane_triangulations/paper.tex

@@ -829,11 +829,24 @@ proper $3$-colouring of $M(G)$, then $\varphi$ is Kempe-balanced.
 Equivalently, a colouring of $\mathsf{M}(T)$ that fails the parity
 condition at some valid face and colour pair cannot extend to a proper
 $3$-colouring of $M(G)$.  This is an instance of Kempe-cycle
-conservation (Lemma~\ref{lem:kempe-conservation}): in the $4$-regular
-graph $M(G)$ each $P$-Kempe chain of $\mathsf{M}(T)$ closes up into a
-$P$-Kempe cycle, and such a cycle crosses the boundary separating a
-valid face from the rest of the sphere an even number of times, so the
-$P$-coloured apexes incident to that face occur in even number.
+conservation (Lemma~\ref{lem:kempe-conservation}).  The tooth apexes
+incident to a valid face are boundary medial vertices
+(Definition~\ref{def:boundary-medial-vertices}) lying on a single level
+cycle of the tire decomposition: the up-tooth apexes lie on the outer
+level cycle, and the singleton down-tooth apexes incident to an interior
+non-tooth face lie on the inner level cycle bounding that face.  In the
+$4$-regular graph $M(G)$ each $P$-Kempe chain of $\mathsf{M}(T)$ closes
+up into a $P$-Kempe cycle, which by Lemma~\ref{lem:kempe-conservation}
+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.
 \end{remark}
 
 More generally, let $T$ be a medial tire region with boundary