From 415c33cfc3c89aff15da891cdba8893850885852 Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 22:32:29 -0400 Subject: [PATCH] coloring_nested_tire_graphs: rigorous proof of cut tire forest proposition (high-side refinement) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Replaces the informal Stage 2 argument with a rigorous one, achieved by refining the proposition to high-side faces only. KEY INSIGHT: the original (unrestricted) proposition was problematic because the LOW-SIDE face of H_{d+1} (= face containing pendants) also contains all depth ≤ d edges in its interior, including H_d edges. Hence low-side H_{d+1} faces span multiple H_d faces. The fix: restrict to HIGH-SIDE faces only. For a high-side face f' of H_{d+1}: by Lemma 2 (level-set), f''s interior contains only depth-> d+1 edges = depth ≥ d+2. Since depth-d edges are NOT in this range, no H_d edge sits inside f'. Therefore f' is contained in a unique H_d face (by partition). This H_d face is also high-side (contains f', which contains depth-≥d+2 edges, hence depth->d). Result: high-side cut tires form a forest, rigorously. The proof uses only Lemma 1 (BFS-adj) and Lemma 2 (level-set), no rotation system case analysis needed. Low-side cut tires are not relevant for chain pigeonhole; the single low-side face is identified with the cut C itself as the forest's "virtual root." Note grows to 5 pages. Co-Authored-By: Claude Opus 4.7 --- .../notes/cut_tire_tree_structure.aux | 2 +- .../notes/cut_tire_tree_structure.log | 109 ++++------------ .../notes/cut_tire_tree_structure.pdf | Bin 211436 -> 211700 bytes .../notes/cut_tire_tree_structure.tex | 123 ++++++++++-------- 4 files changed, 95 insertions(+), 139 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.aux b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.aux index d2df511..c843d2f 100644 --- a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.aux +++ b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.aux @@ -2,6 +2,6 @@ \newlabel{prop:tree}{{}{1}} \newlabel{lem:bfs-adj}{{}{1}} \newlabel{lem:level-set}{{}{1}} -\@writefile{toc}{\contentsline {paragraph}{Caveat on Stage 2.}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Remark on the proof.}{2}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Reformulated chain half (tree DP form).}{5}{}\protected@file@percent } \gdef \@abspage@last{5} diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.log b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.log index 33e26fc..132016b 100644 --- a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.log +++ b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.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 22:24 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 22:32 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -178,20 +178,20 @@ File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX) (./cut_tire_tree_structure.aux) \openout1 = `cut_tire_tree_structure.aux'. -LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 15. -LaTeX Font Info: ... okay on input line 15. -LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 15. -LaTeX Font Info: ... okay on input line 15. -LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 15. -LaTeX Font Info: ... okay on input line 15. -LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 15. -LaTeX Font Info: ... okay on input line 15. -LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 15. -LaTeX Font Info: ... okay on input line 15. -LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 15. -LaTeX Font Info: ... okay on input line 15. -LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 15. -LaTeX Font Info: ... okay on input line 15. +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 16. +LaTeX Font Info: ... okay on input line 16. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 16. +LaTeX Font Info: ... okay on input line 16. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 16. +LaTeX Font Info: ... okay on input line 16. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 16. +LaTeX Font Info: ... okay on input line 16. +LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 16. +LaTeX Font Info: ... okay on input line 16. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 16. +LaTeX Font Info: ... okay on input line 16. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 16. +LaTeX Font Info: ... okay on input line 16. 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We call $f$ a \emph{high-side} face if its +interior contains only depth-$>d$ edges, and a \emph{low-side} face +otherwise. The low-side face is the unique face of $H_d$ that +contains the pendants (depth $0$ edges). + +\begin{prop}[Cut tires form a forest, refined] \label{prop:tree} -For each side $i$ of a $6$-edge cut of $G'$, the cut tires of $G'_i$, -parameterised by pairs $(d, f)$ with $d \ge 1$ and $f$ a face of -$H_d$, form a \emph{forest} under the parent--child relation +For each side $i$ of a $6$-edge cut of $G'$, the high-side cut tires +of $G'_i$, parameterised by pairs $(d, f)$ with $d \ge 1$ and $f$ a +\emph{high-side} face of $H_d$, form a \emph{forest} under the +parent--child relation \[ \mathrm{parent}\bigl(T_{d+1}^{(f')}\bigr) := T_d^{(f)} \] -where $f$ is the unique face of $H_d$ in whose planar interior $f'$ -lies in the inherited embedding of $G'_i$. +where $f$ is the unique high-side face of $H_d$ in whose planar +interior $f'$ lies in the inherited embedding of $G'_i$. -The forest's roots are the cut tires at depth $1$ (one per face of -$H_1$); their ``virtual parent'' is the cut $C$ itself. +The forest's roots are the high-side cut tires at depth $1$ (one per +high-side face of $H_1$); their ``virtual parent'' is the cut $C$ +itself. + +\emph{Remark.} The restriction to high-side faces is what makes the +geometric containment clean. A low-side face of $H_{d+1}$ contains +$H_d$ edges in its interior, so the literal ``face-contained-in-face'' +relation is not well-defined for low-side faces. In the cut-tire +framework, only the high-side faces give the ``concentric'' cut +tires we care about for chain pigeonhole; the low-side face is the +``outside pendant region'' identified with the cut. \end{prop} \begin{proof} @@ -93,51 +111,31 @@ interior). \end{proof} \medskip -\noindent\textbf{Stage 2: faces of $H_{d+1}$ embed in faces of $H_d$.} +\noindent\textbf{Stage 2: high-side faces of $H_{d+1}$ embed in +high-side faces of $H_d$.} -Pendants (depth $0$ edges) lie in some specific face of $H_d$; that -face is low-side. All other faces of $H_d$ are high-side and -contain depth-$> d$ edges, which includes all of $H_{d+1}$'s edges. +Let $f'$ be a high-side face of $H_{d+1}$. By definition, every edge +of $G'_i$ in the open interior of $f'$ has depth $> d + 1$. -Let $f'$ be a face of $H_{d+1}$. We claim $f'$ is contained in -exactly one face of $H_d$. +In particular, no edge of depth $d$ lies in the open interior of +$f'$: every depth-$d$ edge of $G'_i$ has depth $d \neq > d+1$, so +depth-$d$ edges are not in $f'$'s open interior. -\emph{Containment in at least one face:} $f'$ is an open connected -region of $\mathbb{R}^2 \setminus H_{d+1}$. In particular it is -connected. By Lemma~\ref{lem:level-set}, each face of $H_d$ is -either entirely low-side or entirely high-side, and the two types -are separated topologically by $H_d$. Suppose for contradiction -$f'$ intersects two distinct faces $g_1, g_2$ of $H_d$. Then a -path in $f'$ from a point in $g_1$ to a point in $g_2$ crosses some -edge of $H_d$ (since faces of $H_d$ are separated by $H_d$ edges). -But $H_d \subset E(G'_i) \setminus E(H_{d+1})$, so $H_d$ edges are -in $\mathbb{R}^2 \setminus E(H_{d+1})$; they could in principle lie -within $f'$ \emph{except} that $f'$ is a maximal connected open -component of that complement, which already includes the $H_d$ -edges. This is where the elementary topological argument is -subtle: we need the additional constraint that no $H_d$ edge -sits strictly inside $f'$. +Therefore $f' \cap H_d = \emptyset$ (where $H_d$ is treated as the +topological union of its vertices and edges in $|\Pi|$). +Equivalently, $f' \subseteq \mathbb{R}^2 \setminus H_d$. -\emph{No $H_d$ edge sits strictly inside $f'$:} suppose an $H_d$ edge -$e$ is strictly inside $f'$. Then $e$'s endpoints are inside $f'$ -(or on $\partial f'$). An endpoint $v$ of $e$ is also incident to -$H_{d+1}$ edges (since $V(H_d) \cap V(H_{d+1})$ contains vertices -where depth-$d$ and depth-$(d+1)$ edges meet; in cubic $G'_i$, $v$ -has $3$ edges with various depths). The $H_{d+1}$ edges incident -to $v$ are on $\partial f'$ (the boundary walk of $f'$), so $v \in -\partial f'$. Then $e$'s other endpoint $w$ is also on or inside -$f'$. But moving from $v$ along $e$ into $w$: this curve segment -is inside $f'$ until it reaches $w$. If $w$ is on $\partial f'$, -the entire edge $e$ lies on the boundary closure $\overline{f'}$, -not strictly inside. If $w$ is strictly inside $f'$, then $w$'s -incident edges (including $e$) project into $f'$ in a way that -should appear on $\partial f'$ --- but $e$ is not in $H_{d+1}$, -contradiction. +Since $f'$ is an open connected region (= face of $H_{d+1}$), and +$\mathbb{R}^2 \setminus H_d$ partitions into the disjoint open +faces of $H_d$, the connected $f'$ is contained in exactly one +face of $H_d$. Call this face $f$. Then $f' \subseteq f$. -\medskip -The careful case analysis shows: no $H_d$ edge sits strictly inside -$f'$, hence $f'$ is contained in a single face of $H_d$ (the unique -face whose interior contains $f'$). +Furthermore, $f$ is high-side: it contains $f'$, which contains +depth-$\ge d + 2$ edges, which are $> d$ depth. So $f$ is in the +``high-side'' classification of Lemma~\ref{lem:level-set}. + +Hence $\mathrm{parent}(T_{d+1}^{(f')}) := T_d^{(f)}$ is well-defined +and unique among high-side cut tires. \medskip \noindent\textbf{Conclusion: forest structure.} @@ -150,15 +148,26 @@ at the ``cut'' for the depth-$1$ roots' virtual parent). No cycles can form. Hence the parent relation defines a forest. \qed \end{proof} -\paragraph{Caveat on Stage 2.} The argument that ``no $H_d$ edge sits -strictly inside $f'$'' uses an informal topological case analysis on -how an $H_d$ edge inside $f'$ would have to interact with $f'$'s -boundary. A fully rigorous proof would set up the topological -framework more carefully (e.g.\ via the rotation system of the -planar embedding, tracing the boundary walk of $f'$ around an -``intruder'' $H_d$ edge to show it must already lie in -$\partial f'$). Empirically, the conclusion holds across -\textbf{$1486$ tested cases, $0$ failures} (see broader sweep below). +\paragraph{Remark on the proof.} Stage 2 is now fully rigorous, +thanks to the refinement to \emph{high-side} faces of $H_{d+1}$. +The key step is: a high-side face of $H_{d+1}$ contains, by +definition, only depth-$\ge d + 2$ edges in its interior. Depth-$d$ +edges (= $H_d$ edges) are not in this depth range, so they cannot +sit inside $f'$. No rotation-system case analysis is needed for the +high-side case; the level-set lemma does all the work. + +The original (unrestricted) proposition was problematic for the +\emph{low-side} face of $H_{d+1}$, which contains the pendants +(depth $0$) plus all edges of depth $\le d$ in $G'_i$'s ``outside'' +region. This low-side face can contain $H_d$ edges in its interior +and therefore spans multiple $H_d$ faces. By restricting to +high-side faces, this difficulty is avoided. + +For the cut-tire chain pigeonhole framework, only the high-side +cut tires are relevant: they form the ``concentric layers'' going +inward from the cut. The low-side face is the unique outside face +containing the pendants and is identified with the cut $C$ itself +(playing the ``virtual root'' role in the forest). \section*{Why this matters for the chain half}