coloring_nested_tire_graphs: figure showing low-side face uniqueness failure

Concrete picture demonstrating "low-side faces span multiple
parent faces of H_{d-1}":

  - H_{d-1} drawn as the outer blue circle.
  - H_d as a smaller nested orange circle inside.
  - Face A of H_d (high-side, inside inner cycle): a small disk,
    sits entirely inside face X of H_{d-1}. Unique parent. ✓
  - Face B of H_d (low-side, outside inner cycle): RED REGION
    spanning across the H_{d-1} cycle. It is one connected face
    of ℝ² \ H_d, but it intersects BOTH face X (annulus between
    cycles) and face Y (exterior of H_{d-1}). Neither X nor Y
    contains all of B → no unique parent. ✗

This makes the uniqueness step's failure visible: the forest
proposition's containment argument works for high-side (= face A,
nested inside) but fails for low-side (= face B, spanning across).

The motivation section of the note now has 3 figures:
  1. Low-side spans uniqueness failure (this commit, page 1)
  2. T_∂ thick-H_1 hexagon (page 2)
  3. T_∂ thin-H_1 tree (page 2)
Plus the nested-cut-tires figure on page 3.

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

papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux

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papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex

@@ -27,6 +27,54 @@ pendants of $G'_i$.  This omission is essential to prove the
 forest's tree structure (low-side faces span multiple parent faces
 of $H_{d-1}$, violating the uniqueness step in the proof).
 
+\paragraph{Why low-side faces break uniqueness.}  Concretely:
+suppose $H_{d-1}$ is the outer (larger) cycle and $H_d$ a nested
+cycle inside.  The two faces of $H_d$ are face $A$ (inside the
+inner cycle, high-side, contains depth-$>d$ stuff) and face $B$
+(outside the inner cycle, low-side, contains pendants and
+$H_{d-1}$ edges).
+
+\begin{center}
+\begin{tikzpicture}[scale=0.95]
+  \draw[blue, very thick] (0, 0) circle (2.4);
+  \draw[orange!90!black, very thick] (0, 0) circle (1.0);
+  \begin{scope}[on background layer]
+    \fill[red!10] (-4.2, -3.0) rectangle (4.2, 3.0);
+    \fill[white] (0, 0) circle (1.0);
+  \end{scope}
+  \node[blue] at (-1.05, 2.25) {\small $H_{d-1}$};
+  \node[orange!90!black] at (-0.75, -0.55) {\small $H_d$};
+  \node[blue, fill=white, inner sep=1.5pt] at (0, 0.4) {\small face $A$};
+  \node[blue, fill=white, inner sep=1.5pt] at (0, -0.0) {\small (high-side)};
+  \node[red] at (-3.4, 0.2) {\small face $B$ of $H_d$};
+  \node[red] at (-3.4, -0.1) {\small (low-side)};
+  \node[gray] at (2.3, 0.0) {\small face $X$ of $H_{d-1}$};
+  \node[gray] at (2.3, -0.3) {\small (between cycles)};
+  \node[gray] at (3.1, 2.45) {\small face $Y$ of $H_{d-1}$};
+  \node[gray] at (3.1, 2.15) {\small (outside)};
+  \draw[->, gray, thick] (2.45, 2.25) -- (2.0, 1.85);
+  \draw[->, gray, thick] (1.8, 0.0) -- (1.55, -0.0);
+\end{tikzpicture}
+\end{center}
+
+The red region is face $B$ (low-side of $H_d$): a single connected
+region of $\mathbb{R}^2 \setminus H_d$.  But $B$ is split by the
+$H_{d-1}$ cycle into:
+\begin{itemize}
+\item Face $X$ of $H_{d-1}$: the annular region between $H_d$
+      and $H_{d-1}$.
+\item Face $Y$ of $H_{d-1}$: the exterior of $H_{d-1}$ (where the
+      pendants live).
+\end{itemize}
+Neither $X$ nor $Y$ contains \emph{all} of $B$ --- they each contain
+a piece.  If we tried to make $B$ a child of some unique
+$H_{d-1}$ face in the tree, no such parent exists.  This is the
+``uniqueness step'' that fails for low-side faces.
+
+By contrast, face $A$ (high-side, inside the inner cycle) sits
+entirely inside face $X$ of $H_{d-1}$.  Unique parent.  This is
+why the forest proposition restricts to high-side faces.
+
 \paragraph{The coverage gap.}  Empirically
 (\texttt{chain\_dp\_joint.py} on the dodecahedron, cut $\#0$, side
 $0$): when $|S_i|$ is small, $H_1$ on side $i$ can be a