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coloring_nested_tire_graphs: figure showing low-side face uniqueness failure

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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>
This commit is contained in:
didericis
2026-05-26 23:33:21 -04:00
parent 587d928736
commit a8af5612c4
4 changed files with 74 additions and 23 deletions
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papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux
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papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.log
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papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex
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@@ -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 forest's tree structure (low-side faces span multiple parent faces
of $H_{d-1}$, violating the uniqueness step in the proof). 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 \paragraph{The coverage gap.} Empirically
(\texttt{chain\_dp\_joint.py} on the dodecahedron, cut $\#0$, side (\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 $0$): when $|S_i|$ is small, $H_1$ on side $i$ can be a