Scaffold Heawood restrictions on nested tire graph duals paper

Add a new paper stub referencing the nested tire decompositions paper,
with intro, Heawood bibliography entry, and an empty restrictions section.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex

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+%% filename: amsart-template.tex
+%% American Mathematical Society
+%% AMS-LaTeX v.2 template for use with amsart
+%% ====================================================================
+
+\documentclass{amsart}
+
+\usepackage{amssymb}
+\usepackage{graphicx}
+
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{conjecture}[theorem]{Conjecture}
+
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{example}[theorem]{Example}
+\newtheorem{xca}[theorem]{Exercise}
+
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+
+\numberwithin{equation}{section}
+
+\begin{document}
+
+\title{Heawood Restrictions on Nested Tire Graph Duals}
+
+%    author one information
+\author{Eric Bauerfeld}
+\address{}
+\curraddr{}
+\email{}
+\thanks{}
+
+\subjclass[2010]{Primary }
+
+\keywords{plane graph, triangulation, plane depth, level edge, dual graph,
+tire graph, Heawood number}
+
+\date{}
+
+\dedicatory{}
+
+\begin{abstract}
+%% TODO: abstract. Following \cite{bauerfeld-nested-tires}, which establishes
+%% the basic vocabulary of tire graphs and dual depth, we study the Heawood
+%% (mod-3 / face-sum) restrictions imposed on the duals of nested tire graphs.
+\end{abstract}
+
+\maketitle
+
+\section{Introduction}
+
+A classical theorem of Tait recasts the Four Colour Theorem in dual,
+edge-colouring terms: a plane triangulation $G$ is properly $4$-vertex-colourable
+if and only if its dual cubic graph $G'$ is properly $3$-edge-colourable. Thus a
+minimal counterexample to the Four Colour Theorem -- a smallest triangulation
+admitting no proper $4$-colouring -- corresponds to a smallest cubic plane graph
+admitting no proper $3$-edge-colouring.
+
+This paper continues the series studying that structure through the
+lens of \emph{nested level duals}.  The foundational vocabulary ---
+level sources, levels, the inner planar dual $G'$ and its dual depth,
+and tire graphs --- is developed in the companion paper
+\cite{bauerfeld-nested-tires}; we refer to that paper for those
+definitions and rely on them throughout.  In particular we use,
+without restating, the notions of:
+\begin{itemize}
+  \item \emph{level source} $S$ and $G$-vertex levels $\ell_G(v)$;
+  \item the inner planar dual $G'$
+        (\cite[Definition~1.3]{bauerfeld-nested-tires});
+  \item \emph{dual depth} $\delta_G(d_f)$
+        (\cite[Definition~1.4]{bauerfeld-nested-tires});
+  \item \emph{tire graph} $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$
+        with outer/inner boundaries and annular edges
+        (\cite[Definition~1.5]{bauerfeld-nested-tires});
+  \item the \emph{tire-component lemma}
+        (\cite[Lemma~1.8]{bauerfeld-nested-tires}); and
+  \item the \emph{tire-tread partition theorem}
+        (\cite[Theorem~1.9]{bauerfeld-nested-tires}).
+\end{itemize}
+
+Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation)
+with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$
+and $G$ has $2n - 4$ triangular faces.
+
+%% TODO: state the Heawood restriction this paper studies. The relevant
+%% classical input is Heawood's face-sum identity \cite{Heawood1898}; the goal
+%% here is to record what it forces on the dual of a (nested) tire graph.
+
+\section{Heawood restrictions on the tire dual}
+\label{sec:heawood-restrictions}
+
+%% TODO: main development.
+
+\begin{thebibliography}{9}
+
+\bibitem{Heawood1898}
+P.~J.~Heawood,
+\emph{On the four-colour map theorem},
+Quart. J.~Pure Appl. Math. \textbf{29} (1898), 270--285.
+
+\bibitem{bauerfeld-depth}
+E.~Bauerfeld,
+\emph{Plane Depth},
+manuscript (math-research repository), 2026.
+
+\bibitem{bauerfeld-nested-tires}
+E.~Bauerfeld,
+\emph{Nested Tire Decompositions of Plane Triangulations},
+manuscript (math-research repository), 2026.
+
+\bibitem{bauerfeld-nested-tire-duals}
+E.~Bauerfeld,
+\emph{Coloring Nested Tire Dual Graphs},
+manuscript (math-research repository), 2026.
+
+\end{thebibliography}
+
+\end{document}