coloring_nested_tire_graphs: add nested cut tire figure to T_∂ note

NEW SECTION: "High-side cut tires alongside T_∂" with a stacked
figure showing concentric structure:

  - Outer hexagon (blue) = T_∂'s cycle (depth 1 = ∂f_∂).
  - Inner triangle (orange) = a depth-2 high-side cut tire
    T_2^(f') interior to T_∂.
  - Red dashed pendants outward from T_∂ = cut edges (depth 0,
    OUT spokes of T_∂).
  - Green dashed edges between outer and inner = depth-2 edges,
    serving simultaneously as IN pendants of T_∂ and OUT spokes
    of T_2^(f').  These are the SHARED EDGES the chain DP uses.
  - Purple dashed pendants from inner triangle = depth-3 edges
    (IN spokes of T_2^(f'), going to grandchildren).

The picture makes concrete what was abstract: each tire has
the same shape (cycle + 2 classes of spokes), tires nest
concentrically by depth, and the chain DP's edge-sharing is
literally the geometric "this same edge is on both tires."

Note grows from 4 to 5 pages.

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

papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux

@@ -6,8 +6,8 @@
 \@writefile{toc}{\contentsline {paragraph}{Picture.}{2}{}\protected@file@percent }
 \@writefile{toc}{\contentsline {paragraph}{What $T_\partial $ looks like in special cases.}{2}{}\protected@file@percent }
 \@writefile{toc}{\contentsline {paragraph}{Role in the chain DP.}{3}{}\protected@file@percent }
-\@writefile{toc}{\contentsline {paragraph}{What this closes.}{3}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{What this closes.}{4}{}\protected@file@percent }
-\@writefile{toc}{\contentsline {paragraph}{What it leaves open.}{3}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{What it leaves open.}{4}{}\protected@file@percent }
 \@writefile{toc}{\contentsline {paragraph}{Why the chain DP can still work.}{4}{}\protected@file@percent }
 \@writefile{toc}{\contentsline {paragraph}{Logical status of the extended framework.}{4}{}\protected@file@percent }
-\gdef \@abspage@last{4}
+\gdef \@abspage@last{5}

papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.log

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papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex

@@ -176,6 +176,82 @@ $|S_0| = 4$), the boundary walk of the single face has length $6$
 (each of $3$ $H_1$ edges visited twice), and OUT pendants attach at
 the two $V_{\deg = 2}$ vertices.
 
+\section*{High-side cut tires alongside $T_\partial$}
+
+A high-side cut tire $T_d^{(f)}$ at depth $d$ has the same
+structural shape as $T_\partial$:
+\begin{itemize}
+\item Cycle = depth-$d$ edges in $\partial f$ ($H_d$ edges).
+\item OUT spokes = depth-$(d-1)$ edges at boundary vertices
+      (pointing outward, toward parent).
+\item IN spokes = depth-$(d+1)$ edges at boundary vertices
+      (pointing inward, toward children).
+\end{itemize}
+
+\begin{center}
+\begin{tikzpicture}[scale=1.3]
+  \def\Rout{2.1}
+  \def\Rin{1.0}
+  % Outer hexagon = T_∂ cycle (depth 1)
+  \foreach \i in {0,...,5} {
+    \pgfmathsetmacro{\ang}{60*\i + 90}
+    \node[circle, fill=black, inner sep=1.2pt] (vo\i) at (\ang:\Rout) {};
+  }
+  % Inner triangle = T_2^{(f')} cycle (depth 2)
+  \foreach \i in {0,1,2} {
+    \pgfmathsetmacro{\ang}{120*\i + 90}
+    \node[circle, fill=black, inner sep=1.2pt] (vi\i) at (\ang:\Rin) {};
+  }
+  % Outer cycle edges (T_∂)
+  \foreach \i in {0,...,5} {
+    \pgfmathtruncatemacro{\j}{mod(\i+1,6)}
+    \draw[blue, very thick] (vo\i) -- (vo\j);
+  }
+  % Inner cycle edges (T_2)
+  \foreach \i in {0,1,2} {
+    \pgfmathtruncatemacro{\j}{mod(\i+1,3)}
+    \draw[orange!85!black, very thick] (vi\i) -- (vi\j);
+  }
+  % OUT pendants from outer hexagon (depth 0, cut) at v1, v3, v5
+  \foreach \i in {1,3,5} {
+    \pgfmathsetmacro{\ang}{60*\i + 90}
+    \coordinate (p\i) at (\ang:\Rout+0.65);
+    \draw[red, thick, dashed] (vo\i) -- (p\i);
+    \node[circle, draw=red, fill=white, inner sep=1pt] at (p\i) {};
+  }
+  % Shared spokes between outer hexagon and inner triangle:
+  % depth-2 edges = T_∂'s IN pendants AND T_2's OUT spokes
+  \draw[green!55!black, thick, dashed] (vo0) -- (vi0);
+  \draw[green!55!black, thick, dashed] (vo2) -- (vi1);
+  \draw[green!55!black, thick, dashed] (vo4) -- (vi2);
+  % T_2's IN pendants at inner vertices (depth 3)
+  \foreach \i in {0,1,2} {
+    \pgfmathsetmacro{\ang}{120*\i + 90 + 60}
+    \coordinate (qi\i) at ([shift={(\ang:0.45)}]vi\i);
+    \draw[purple, thick, dashed] (vi\i) -- (qi\i);
+    \node[circle, fill=purple, inner sep=1pt] at (qi\i) {};
+  }
+  % Annotations
+  \node[blue] at (-2.65, 1.75) {\small $T_\partial$ cycle};
+  \node[blue] at (-2.65, 1.45) {\small (depth 1)};
+  \node[orange!85!black] at (-1.85, -0.05) {\small $T_2^{(f')}$};
+  \node[orange!85!black] at (-1.85, -0.35) {\small (depth 2)};
+  \node[red] at (2.75, 1.65) {\small cut};
+  \node[red] at (2.75, 1.35) {\small (depth 0)};
+  \node[green!55!black] at (2.75, -1.35) {\small shared:};
+  \node[green!55!black] at (2.75, -1.65) {\small IN of $T_\partial$};
+  \node[green!55!black] at (2.75, -1.95) {\small $=$ OUT of $T_2$};
+  \node[purple] at (0, -0.85) {\small depth 3};
+\end{tikzpicture}
+\end{center}
+
+The green dashed edges are the \emph{shared} depth-$2$ edges:
+they are IN pendants of $T_\partial$ \emph{and} OUT spokes of
+$T_2^{(f')}$.  In any global $3$-edge coloring of $G'_i$, the
+color assigned to each green edge is the same when viewed from
+either tire.  The chain DP exploits exactly this shared-edge
+identity.
+
 \section*{The extended forest}
 
 The high-side cut tire forest of $G'_i$ (proven in