coloring_nested_tire_graphs: extend even-cycle note with 8-cut question

Added section "Could the minimum non-trivial cyclic cut be 8?"

Answer: yes in principle.  Birkhoff gives "≥ 6"; nothing in the
condition pins the value to 6.  A planar cubic G^* with min
non-facial cyclic edge connectivity = 8 would have:
  - No non-facial 6-edge cut.
  - No non-facial 7-edge cut.
  - Some non-facial 8-edge cut.

By cut-parity lemma: 8-cuts have even sides; 7-cuts have odd
sides.

Heuristic when this happens: link vertices of degree ≥ 6
(rather than icosahedron-tight 5) push second-link length to
≥ 10, eliminating small non-facial separators.

The cut-tire framework adapts: chain DP and T_∂ construction
work for any cut size, with parameter changes.  Per-tire half
needs re-examining for larger structures.

Bottom line: min non-trivial cyclic cut is one of {6, 7, 8, 9,
...}; Birkhoff doesn't pin it down. Framework's natural domain
is whatever value it happens to take, with 6 being the simplest.

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

papers/coloring_nested_tire_graphs/notes/even_separating_cycle.aux

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papers/coloring_nested_tire_graphs/notes/even_separating_cycle.log

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papers/coloring_nested_tire_graphs/notes/even_separating_cycle.tex

@@ -185,4 +185,46 @@ hold, we'd need to either prove that the counterexample is not
 of the violating type, or extend the framework to higher-size
 cuts.
 
+\section*{Could the minimum non-trivial cyclic cut be $8$?}
+
+\paragraph{Yes, in principle.}  Birkhoff gives $\ge 6$; nothing in
+the condition pins the value to $6$.  A planar cubic $G^*$ with
+non-facial cyclic edge connectivity \emph{exactly $8$} would have:
+\begin{itemize}
+\item No non-facial cyclic cut of size $6$ (= no separating
+      $6$-cycle in $G$ with $\ge 2$ vertices each side).
+\item No non-facial cyclic cut of size $7$ (= no separating
+      $7$-cycle in $G$ with $\ge 2$ vertices each side).
+\item Some non-facial cyclic cut of size $8$.
+\end{itemize}
+
+By the cut-parity lemma, a size-$8$ cut would have even-sized
+sides.  Size-$7$ cuts would have odd-sized sides; for such cuts
+to not exist non-facially, the graph would need a structural
+parity barrier or just lack any odd-cardinality separations.
+
+\paragraph{What would force this?}  Looking at the second-link
+heuristic: if every vertex's link contains only vertices of degree
+$\ge 6$ rather than the icosahedron-tight degree $5$, the
+second-link length jumps to $\ge 5 + 5 \cdot 1 = 10$.  Such graphs
+exist (denser triangulations); whether such a graph is also a
+minimum $4$CT counterexample (= class-2 cubic dual + planar +
+internally $6$-connected) is unknown.
+
+\paragraph{The framework adapts.}  Even if the minimum non-trivial
+cyclic cut is $8$ (or some other value $> 6$), the cut-tire chain
+DP doesn't structurally depend on cut size $= 6$.  The same
+constructions (cut tires, boundary cut tire $T_\partial$, chain DP
+via shared edges) apply to $8$-edge cuts with minor parameter
+changes.  What \emph{does} change: per-tire enumeration size
+scales with cut size, and the per-tire half (Prop 1.13) was proved
+specifically for spoke-only cut tires with simple-cycle face
+boundaries --- it would need re-examining for larger structures.
+
+\paragraph{Bottom line.}  The minimum non-trivial cyclic cut size
+for a hypothetical $4$CT counterexample is one of
+$\{6, 7, 8, 9, \dots\}$, and Birkhoff alone doesn't pin it down.
+The framework's natural domain is whichever value it happens to
+take, with $6$ being the simplest case to enumerate and study.
+
 \end{document}