coloring_nested_tire_graphs: clarify uniqueness-failure figure

Previous figure shaded all of B in red, making it look like X
and Y were the same red region. New version distinguishes:

  - X (annulus between H_d and H_{d-1}) = light cyan.
  - Y (exterior of H_{d-1}) = light yellow.
  - A (inside H_d) = white.
  - B (low-side face of H_d) = everything outside red dashed
    boundary = cyan ∪ yellow.

Now visually clear:
  - X is the cyan annulus (proper subset of B).
  - Y is the yellow exterior (proper subset of B).
  - B = X ∪ Y, neither X nor Y contains all of B.

The two arrows from "face B" label point at both regions to
emphasize B spans both faces of H_{d-1}.

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

papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux

@@ -1,6 +1,6 @@
 \relax 
 \@writefile{toc}{\contentsline {paragraph}{Why low-side faces break uniqueness.}{1}{}\protected@file@percent }
-\@writefile{toc}{\contentsline {paragraph}{The coverage gap.}{1}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{The coverage gap.}{2}{}\protected@file@percent }
 \@writefile{toc}{\contentsline {paragraph}{Resolution.}{2}{}\protected@file@percent }
 \@writefile{toc}{\contentsline {paragraph}{Setup.}{2}{}\protected@file@percent }
 \@writefile{toc}{\contentsline {paragraph}{The low-side face of $H_1$.}{2}{}\protected@file@percent }

papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.log

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papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex

@@ -36,39 +36,49 @@ $H_{d-1}$ 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);
+    % Y (outside H_{d-1}): light yellow
+    \fill[yellow!22] (-4.4, -3.0) rectangle (4.4, 3.0);
+    % X (between H_{d-1} and H_d): light cyan
+    \fill[cyan!18] (0, 0) circle (2.4);
+    % A (inside H_d): white
     \fill[white] (0, 0) circle (1.0);
   \end{scope}
-  \node[blue] at (-1.05, 2.25) {\small $H_{d-1}$};
+  \draw[blue, very thick] (0, 0) circle (2.4);
-  \node[orange!90!black] at (-0.75, -0.55) {\small $H_d$};
+  \draw[orange!90!black, very thick] (0, 0) circle (1.0);
-  \node[blue, fill=white, inner sep=1.5pt] at (0, 0.4) {\small face $A$};
+  % Mark "face B = X ∪ Y" with a dashed red boundary just outside H_d
-  \node[blue, fill=white, inner sep=1.5pt] at (0, -0.0) {\small (high-side)};
+  \draw[red, thick, dashed] (0, 0) circle (1.08);
-  \node[red] at (-3.4, 0.2) {\small face $B$ of $H_d$};
+  \node[orange!90!black] at (0, 0) {\small $A$};
-  \node[red] at (-3.4, -0.1) {\small (low-side)};
+  \node[blue] at (1.05, 2.25) {\small $H_{d-1}$};
-  \node[gray] at (2.3, 0.0) {\small face $X$ of $H_{d-1}$};
+  \node[orange!90!black] at (-1.05, -0.45) {\small $H_d$};
-  \node[gray] at (2.3, -0.3) {\small (between cycles)};
+  \node[fill=cyan!18, inner sep=1pt] at (1.6, 1.35) {\small face $X$ of $H_{d-1}$};
-  \node[gray] at (3.1, 2.45) {\small face $Y$ of $H_{d-1}$};
+  \node[fill=yellow!22, inner sep=1pt] at (3.2, 2.55) {\small face $Y$ of $H_{d-1}$};
-  \node[gray] at (3.1, 2.15) {\small (outside)};
+  \node[red] at (-2.95, 1.45) {\small face $B$ of $H_d$};
-  \draw[->, gray, thick] (2.45, 2.25) -- (2.0, 1.85);
+  \node[red] at (-2.95, 1.15) {\small (= everything};
-  \draw[->, gray, thick] (1.8, 0.0) -- (1.55, -0.0);
+  \node[red] at (-2.95, 0.85) {\small outside red dashed)};
+  \draw[->, red, thick] (-1.95, 1.0) -- (-1.15, 1.0);
+  \draw[->, red, thick] (-1.95, 1.0) .. controls (-2.3, 1.6) and (-3.1, 2.4) .. (-3.5, 2.7);
 \end{tikzpicture}
 \end{center}
 
-The red region is face $B$ (low-side of $H_d$): a single connected
+Here face $B$ of $H_d$ (low-side) is \emph{everything outside the
-region of $\mathbb{R}^2 \setminus H_d$.  But $B$ is split by the
+dashed red boundary} --- i.e.\ the union of the cyan annulus and the
-$H_{d-1}$ cycle into:
+exterior yellow region.  This is a single connected face of
+$\mathbb{R}^2 \setminus H_d$ (the red boundary is just the inner
+edge of $H_d$; from $B$'s perspective there is no separating curve
+out at $H_{d-1}$).
+
+But the two \emph{faces of $H_{d-1}$} are distinct regions:
 \begin{itemize}
-\item Face $X$ of $H_{d-1}$: the annular region between $H_d$
+\item Face $X$ of $H_{d-1}$: the cyan annulus between $H_d$ and
-      and $H_{d-1}$.
+      $H_{d-1}$.  It is a proper subset of $B$ (the yellow exterior
-\item Face $Y$ of $H_{d-1}$: the exterior of $H_{d-1}$ (where the
+      lies outside $X$).
-      pendants live).
+\item Face $Y$ of $H_{d-1}$: the yellow exterior region.  Also a
+      proper subset of $B$ (the cyan annulus lies outside $Y$).
 \end{itemize}
-Neither $X$ nor $Y$ contains \emph{all} of $B$ --- they each contain
+So $X \subsetneq B$ and $Y \subsetneq B$, but $B \not\subseteq X$
-a piece.  If we tried to make $B$ a child of some unique
+and $B \not\subseteq Y$.  Neither $X$ nor $Y$ contains all of $B$,
-$H_{d-1}$ face in the tree, no such parent exists.  This is the
+so $B$ has no unique parent face in $H_{d-1}$.  This is the
 ``uniqueness step'' that fails for low-side faces.
 
 By contrast, face $A$ (high-side, inside the inner cycle) sits