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nested_level_duals: scaffold paper (shelved for alternative approach)

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New paper introducing the dual (inner/weak dual) of a maximal planar graph,
dual depth (BFS-derived min level over a face's vertices), and a Tait-based
framing of a minimal 4CT counterexample via nested level duals. Includes a
dual-depth figure and its generator. Shelved per closing note in favour of an
alternative approach.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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2026-05-22 17:36:11 -04:00
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papers/nested_level_duals/paper.aux
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\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
\newlabel{def:dual-depth}{{1.4}{1}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{2}{}\protected@file@percent }
\newlabel{fig:dual-depth}{{1}{2}}
\gdef \@abspage@last{2}