From a8af5612c4fca01a37626d5f0dfe7a0bf373074f Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 23:33:21 -0400 Subject: [PATCH] coloring_nested_tire_graphs: figure showing low-side face uniqueness failure MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 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 --- .../notes/boundary_cut_tire.aux | 19 +++---- .../notes/boundary_cut_tire.log | 30 ++++++----- .../notes/boundary_cut_tire.pdf | Bin 233743 -> 249957 bytes .../notes/boundary_cut_tire.tex | 48 ++++++++++++++++++ 4 files changed, 74 insertions(+), 23 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux index 01e7d6a..06009b0 100644 --- a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux +++ b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux @@ -1,13 +1,14 @@ \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}{Resolution.}{1}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{Setup.}{1}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{The low-side face of $H_1$.}{1}{}\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 } \@writefile{toc}{\contentsline {paragraph}{Picture.}{2}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{What $T_\partial $ looks like in special cases.}{2}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{Role in the chain DP.}{3}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{What this closes.}{4}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{What it leaves open.}{4}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{Why the chain DP can still work.}{4}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{Logical status of the extended framework.}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{What $T_\partial $ looks like in special cases.}{3}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Role in the chain DP.}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{What this closes.}{5}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{What it leaves open.}{5}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Why the chain DP can still work.}{5}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Logical status of the extended framework.}{5}{}\protected@file@percent } \gdef \@abspage@last{5} diff --git a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.log b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.log index 9cc0185..884f36a 100644 --- a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.log +++ b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 23:27 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 23:33 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -587,10 +587,10 @@ File: umsb.fd 2013/01/14 v3.01 AMS symbols B [4] [5] (./boundary_cut_tire.aux) ) Here is how much of TeX's memory you used: - 14724 strings out of 478268 - 300879 string characters out of 5846347 - 568561 words of memory out of 5000000 - 32679 multiletter control sequences out of 15000+600000 + 14739 strings out of 478268 + 301145 string characters out of 5846347 + 571661 words of memory out of 5000000 + 32694 multiletter control sequences out of 15000+600000 482413 words of font info for 79 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 84i,6n,89p,452b,713s stack positions out of 10000i,1000n,20000p,200000b,200000s @@ -609,16 +609,18 @@ mr7.pfb> - -Output written on boundary_cut_tire.pdf (5 pages, 233743 bytes). + +Output written on boundary_cut_tire.pdf (5 pages, 249957 bytes). 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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