diff --git a/papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.aux b/papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.aux index d490c35..9861c5c 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.aux +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.aux @@ -1,2 +1,6 @@ \relax -\gdef \@abspage@last{2} +\newlabel{lem:unstack}{{}{2}} +\newlabel{lem:base}{{}{3}} +\newlabel{prop:reduction}{{}{3}} +\newlabel{conj:irreducible}{{}{3}} +\gdef \@abspage@last{3} diff --git a/papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.log b/papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.log index 4ae5f36..3d0fdea 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.log +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_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) 17 JUN 2026 02:21 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 17 JUN 2026 20:27 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) (./boundary_restriction_structure.aux) \openout1 = `boundary_restriction_structure.aux'. -LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 18. -LaTeX Font Info: ... okay on input line 18. -LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 18. -LaTeX Font Info: ... okay on input line 18. -LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 18. -LaTeX Font Info: ... okay on input line 18. -LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 18. -LaTeX Font Info: ... okay on input line 18. -LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 18. -LaTeX Font Info: ... okay on input line 18. -LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 18. -LaTeX Font Info: ... okay on input line 18. -LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 18. -LaTeX Font Info: ... okay on input line 18. +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 19. +LaTeX Font Info: ... okay on input line 19. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 19. +LaTeX Font Info: ... okay on input line 19. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 19. +LaTeX Font Info: ... okay on input line 19. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 19. +LaTeX Font Info: ... okay on input line 19. +LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 19. +LaTeX Font Info: ... okay on input line 19. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 19. +LaTeX Font Info: ... okay on input line 19. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 19. +LaTeX Font Info: ... okay on input line 19. 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A proof programme for the lower bound} + +The lower bound $|\Phi(D)| \ge 2^{n-2}$ reduces, by an exact +$\Phi$-preserving reduction, to a single lemma about ``irreducible'' +disks. Two dead ends bound the search first: \emph{monotonicity is false} +--- inserting a degree-$4$ interior vertex can shrink $|\Phi|$ ($6\to5$, +$30\to28$; \texttt{monotonicity\_test.py}), so there is no reduce-to-base +proof by ``adding vertices only grows $\Phi$''; and \emph{universal +toggles are insufficient} --- a flip preserves feasibility for every +labelling only if it touches no interior vertex (a \emph{boundary-only} +face), and an irreducible disk can have none (the wheel has zero). What +does work: + +\begin{lem}[Un-stacking; degree-$3$ removal preserves $\Phi$] +\label{lem:unstack} +Let $v$ be a degree-$3$ interior vertex of $D$, with link triangle +$abc$ and incident faces $(vab),(vbc),(vca)$. Its constraint +$\lambda_{vab}+\lambda_{vbc}+\lambda_{vca}\equiv 0 \pmod 3$ over +$\{+1,-1\}$ forces the three to a common value $s$, so each of $a,b,c$ +receives $2s\equiv -s$ from $v$'s star. Let $D'$ delete $v$ and restore +$abc$ as one face. Then setting that face to $-s$ reproduces the +contribution $-s$ at $a,b,c$, and $s\mapsto -s$ is a bijection on +$\{+1,-1\}$. Hence the map is a bijection between feasible labellings of +$D$ and of $D'$ preserving every boundary value and interior constraint, +so +\[ + \Phi(D) = \Phi(D'), \qquad k(D') = k(D)-1 . +\] +\end{lem} + +\begin{verify}\textnormal{(\texttt{monotonicity\_test.py})} Degree-$3$ +insertion gave exact equality in $8884/8884$ trials.\end{verify} + +\begin{lem}[Base case; ear-peeling] +\label{lem:base} +If $D$ has no interior vertices ($k=0$) then $|\Phi(D)| = 2^{n-2}$. A +polygon triangulation has an \emph{ear} $(v_{i-1},v_i,v_{i+1})$ with +$v_i$ of face-degree $1$, so $\sigma_{v_i}=\lambda_{\mathrm{ear}}$ reads +the ear label directly; remove it and induct on the $(n-1)$-gon. The +boundary map is injective, giving $2^{n-2}$. +\end{lem} + +\begin{prop}[Reduction to the irreducible case] +\label{prop:reduction} +Iterating Lemma~\ref{lem:unstack} terminates ($k$ strictly decreases) at +a residue $D^{\ast}$ with no degree-$3$ interior vertex and the same +$n$, and $\Phi(D)=\Phi(D^{\ast})$. The residue is either $k=0$, where +$|\Phi|=2^{n-2}$ by Lemma~\ref{lem:base}, or \emph{irreducible}: $k\ge1$ +with every interior vertex of degree $\ge 4$. Hence +\[ + |\Phi(D)| \ge 2^{n-2} + \quad\Longleftarrow\quad + |\Phi(D^{\ast})| \ge 2^{n-2}\ \text{for every irreducible } D^{\ast}. +\] +\end{prop} + +\begin{conj}[Irreducible lemma --- the remaining content] +\label{conj:irreducible} +Every irreducible disk satisfies $|\Phi| \ge 2^{n-2}$; in fact +$|\Phi| \ge \tfrac54\cdot 2^{n-2} = 5\cdot 2^{n-4}$. +\end{conj} + +\begin{verify}\textnormal{(\texttt{irreducible\_floor.py}, +\texttt{wheel\_extremal.py})} Over $10^4{+}$ irreducible disks +($n=4,5,6$) there were $0$ floor violations and none sat on the floor. +The bound $\tfrac54\cdot 2^{n-2}$ is \emph{tight}, attained by a single +\textbf{minimal-degree} interior vertex (degree $4$ or $5$, which tie): +the ratio $|\Phi|/2^{n-2}$ rises monotonically with the interior vertex's +degree, $\tfrac54$ at $d\in\{4,5\}$, $\tfrac{21}{16}$ at $d\in\{6,7\}$, +$\ldots$, up to $\tfrac43$ at the wheel $d=n$, where +$|\Phi(W_n)|=\lfloor 2^n/3\rfloor$ exactly. So a proof of +Conjecture~\ref{conj:irreducible} should be stress-tested against the +degree-$4$ patch, the tight case --- \emph{not} the wheel.\end{verify} + +\noindent +\emph{Status.} Lemmas~\ref{lem:unstack}--\ref{lem:base} and +Proposition~\ref{prop:reduction} are proofs; they settle every disk that +un-stacks to $k=0$ (the entire Apollonian class). The whole open content +is Conjecture~\ref{conj:irreducible}, with guaranteed $25\%$ slack and a +single explicit extremal disk. + \section*{Consequence for the pigeonhole} Even a maximally-constraining child still presents $2^{n-2}$ outer