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 3d0fdea..85054d1 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 20:27 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 17 JUN 2026 21:32 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -291,7 +291,7 @@ onts/cm/cmsy8.pfb> -Output written on boundary_restriction_structure.pdf (3 pages, 217886 bytes). +Output written on boundary_restriction_structure.pdf (3 pages, 218332 bytes). PDF statistics: 104 PDF objects out of 1000 (max. 8388607) 62 compressed objects within 1 object stream diff --git a/papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.pdf b/papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.pdf index 8720fba..969d48a 100644 Binary files a/papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.pdf and b/papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.pdf differ diff --git a/papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.tex b/papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.tex index c17e939..1abe738 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.tex +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.tex @@ -102,10 +102,11 @@ realises $5$ sequences against the fan's $4$, since each interior vertex contributes two faces but only one constraint. Thus: \begin{obs} -The outer $n$-cycle cannot be constrained below $2^{n-2}$ achievable -sequences, and no nested structure is needed to reach the floor: a single -trivial tire is already maximal. Deep nesting only approaches the floor -from above. +The interior-free triangulation already attains $2^{n-2}$, no search disk +beats it, and deep nesting only approaches this value from above --- +suggesting it is a floor, with a single trivial tire already maximally +constraining. Whether $2^{n-2}$ is a genuine lower bound for \emph{all} +disks is the Conjecture below; it is \emph{not} a proven theorem. \end{obs} The achievability is transparent: in a fan from $v_0$, diff --git a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux index 525b3e2..84a3d03 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux @@ -34,13 +34,14 @@ \newlabel{rem:why-clusters}{{3.7}{5}} \newlabel{conj:heawood-chain-pigeonhole}{{3.8}{5}} \newlabel{conj:heawood-route-fct}{{3.9}{5}} -\bibcite{Heawood1898}{1} \@writefile{toc}{\contentsline {section}{\tocsection {}{4}{The constraint floor}}{6}{}\protected@file@percent } \newlabel{sec:constraint-floor}{{4}{6}} \newlabel{def:achievable-boundary-set}{{4.1}{6}} -\newlabel{prop:constraint-floor}{{4.2}{6}} -\newlabel{rem:freedom-positive}{{4.3}{6}} -\newlabel{rem:floor-consequences}{{4.4}{6}} +\newlabel{prop:attainment}{{4.2}{6}} +\newlabel{lem:unstack}{{4.3}{6}} +\newlabel{conj:constraint-floor}{{4.4}{6}} +\newlabel{rem:floor-status}{{4.5}{6}} +\bibcite{Heawood1898}{1} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tires}{3} \bibcite{bauerfeld-medial-tires}{4} @@ -50,5 +51,6 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} +\newlabel{rem:floor-consequences}{{4.6}{7}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{7}{}\protected@file@percent } \gdef \@abspage@last{7} diff --git a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.log b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.log index 45992b1..ffb00b8 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.log +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.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 02:23 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 17 JUN 2026 21:31 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -192,48 +192,37 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] -[2] [3] [4] [5] -Overfull \hbox (20.41568pt too wide) detected at line 494 -\OML/cmm/m/it/10 ^^U[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 v[]\OT1/cmr/m/n/10 ) = -\OML/cmm/m/it/10 ^^U[]; ^^U[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 v[]\OT1/cmr/m/n -/10 ) = \OML/cmm/m/it/10 ^^U[] \OT1/cmr/m/n/10 + \OML/cmm/m/it/10 ^^U[] \OT1/c -mr/m/n/10 (1 \OML/cmm/m/it/10 < i < n \OMS/cmsy/m/n/10 ^^@ \OT1/cmr/m/n/10 1)\O -ML/cmm/m/it/10 ; ^^U[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 v[]\OT1/cmr/m/n/10 ) = - \OML/cmm/m/it/10 ^^U[]; ^^U[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 v[]\OT1/cmr/m/ -n/10 ) = [][] \OML/cmm/m/it/10 ^^U[]: - [] - -[6] [7] (./paper.aux) ) +[2] [3] [4] [5] [6] [7] (./paper.aux) ) Here is how much of TeX's memory you used: - 3022 strings out of 478268 - 42281 string characters out of 5846347 - 342340 words of memory out of 5000000 - 21068 multiletter control sequences out of 15000+600000 + 3024 strings out of 478268 + 42307 string characters out of 5846347 + 342360 words of memory out of 5000000 + 21070 multiletter control sequences out of 15000+600000 477578 words of font info for 59 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 - 69i,7n,76p,242b,362s stack positions out of 10000i,1000n,20000p,200000b,200000s - -< -/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb> - -Output written on paper.pdf (7 pages, 267867 bytes). + 69i,7n,76p,242b,290s stack positions out of 10000i,1000n,20000p,200000b,200000s + +Output written on paper.pdf (7 pages, 269140 bytes). PDF statistics: 128 PDF objects out of 1000 (max. 8388607) 78 compressed objects within 1 object stream diff --git a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.pdf b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.pdf index bb794f0..e25440b 100644 Binary files a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.pdf and b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.pdf differ diff --git a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex index 87afc21..04fc7b4 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex @@ -448,9 +448,9 @@ A nested substructure constrains its outer interface through the set of Heawood boundary sequences it can realise. By the self-similarity of the tire decomposition (\cite{bauerfeld-nested-tires}), the region $G_T$ enclosed by a tire's outer cycle, away from the source, is itself a -triangulated disk; we record how tightly any such disk can constrain its -boundary. The bound below depends only on the disk triangulation, not on -a tire-tree labelling. +triangulated disk; we ask how tightly any such disk can constrain its +boundary. The achievable set below depends only on the disk +triangulation, not on a tire-tree labelling. \begin{definition}[Achievable boundary set of a disk] \label{def:achievable-boundary-set} @@ -468,62 +468,75 @@ is \] \end{definition} -\begin{proposition}[Constraint floor] -\label{prop:constraint-floor} -For every triangulated disk $D$ with boundary an $n$-cycle, -\[ - |\Phi(D)| \;\ge\; 2^{\,n-2}, -\] -and the bound is attained --- already by the triangulation of the -$n$-gon with no interior vertices. Consequently no nested structure -constrains the outer cycle below $2^{\,n-2}$ achievable Heawood -sequences; the trivial tire is already maximally constraining. +\begin{proposition}[Interior-free disks attain $2^{n-2}$] +\label{prop:attainment} +If $D$ has no interior vertices then $|\Phi(D)| = 2^{\,n-2}$. \end{proposition} -\begin{proof}[Proof of attainment] -Triangulate the $n$-gon as a fan from $v_0$, with faces -$\{v_0, v_i, v_{i+1}\}$ for $1 \le i \le n-2$ and labels -$\lambda_i := \lambda(\{v_0, v_i, v_{i+1}\})$; there are no interior -vertices, so every labelling is interior-valid. The induced boundary -values are -\[ - \lambda^{*}(v_1) = \lambda_1, \quad - \lambda^{*}(v_i) = \lambda_{i-1} + \lambda_i \ \ (1 < i < n-1), \quad - \lambda^{*}(v_{n-1}) = \lambda_{n-2}, \quad - \lambda^{*}(v_0) = \textstyle\sum_{j} \lambda_j . -\] -From $\lambda^{*}(v_1)$ and the relations -$\lambda_i = \lambda^{*}(v_i) - \lambda_{i-1}$ the tuple -$(\lambda_1, \dots, \lambda_{n-2}) \in \{+1,-1\}^{n-2}$ is recovered from -the boundary sequence, so the map $\lambda \mapsto \lambda^{*}|_C$ is -injective and $|\Phi(D)| = 2^{\,n-2}$. +\begin{proof} +A triangulation of the $n$-gon has an \emph{ear}: a face +$(v_{i-1}, v_i, v_{i+1})$ whose middle vertex $v_i$ has face-degree $1$, +so $\lambda^{*}(v_i)$ equals that face's label and is read directly off +the boundary sequence. Deleting the ear leaves a triangulation of the +$(n-1)$-gon inducing the restricted boundary sequence; inducting, the +$n-2$ face labels are recovered injectively, so $\lambda \mapsto +\lambda^{*}|_C$ is a bijection onto a set of size $2^{\,n-2}$. \end{proof} -\begin{remark}[Depth is freedom-positive] -\label{rem:freedom-positive} -The lower bound is plausible from a counting balance. A triangulated -disk with $k$ interior vertices has $2k + n - 2$ faces (Euler) and -imposes exactly $k$ interior Heawood constraints, one per interior -vertex. So each interior vertex contributes \emph{two} faces --- two new -$\{+1,-1\}$ degrees of freedom --- against only \emph{one} constraint, -and the free dimension $(2k + n - 2) - k = k + n - 2$ \emph{grows} with -depth. Going deeper is freedom-positive on balance: the boundary -projection $\Phi(D)$ can only retain or enlarge its options, never drop -below the interior-free value $2^{\,n-2}$. (Empirically $|\Phi(D)|$ does -grow with $k$; e.g.\ on the $4$-cycle the central-apex wheel realises $5$ -sequences against the fan's $4$.) The constraints relate only -interior-incident faces and cannot collapse the $n-2$ degrees of freedom -carried by the boundary-incident faces --- which is the content the lower -bound must make precise. +\begin{lemma}[Un-stacking] +\label{lem:unstack} +If $v$ is a degree-$3$ interior vertex of $D$, deleting it and restoring +its link triangle as a single face yields a disk $D'$ with one fewer +interior vertex and $\Phi(D') = \Phi(D)$. +\end{lemma} + +\begin{proof} +The constraint at $v$ forces its three faces to a common value $s$, +contributing $2s \equiv -s$ to each of the three link vertices; the +restored triangle, labelled $-s$, reproduces that contribution at each, +and $s \mapsto -s$ is a bijection on $\{+1,-1\}$. The resulting map is a +bijection between interior-valid labellings of $D$ and of $D'$ preserving +every boundary value, hence $\Phi(D) = \Phi(D')$. +\end{proof} + +\begin{conjecture}[Constraint floor] +\label{conj:constraint-floor} +For every triangulated disk $D$ with boundary an $n$-cycle, +$|\Phi(D)| \ge 2^{\,n-2}$. Equivalently, no nested structure constrains +the outer cycle below $2^{\,n-2}$ achievable Heawood sequences. +\end{conjecture} + +\begin{remark}[Status of Conjecture~\ref{conj:constraint-floor}] +\label{rem:floor-status} +Iterating Lemma~\ref{lem:unstack} reduces any disk, $\Phi$-faithfully and +at fixed $n$, to one with no degree-$3$ interior vertex: either +interior-free, where Proposition~\ref{prop:attainment} gives exactly +$2^{\,n-2}$, or \emph{irreducible} (every interior vertex of degree +$\ge 4$). Thus Conjecture~\ref{conj:constraint-floor} holds for the +entire stacked (Apollonian) class and reduces to the irreducible case. +Empirically it holds without exception over more than $10^4$ disks, and +\emph{strictly}: every irreducible disk satisfies $|\Phi(D)| \ge +\tfrac54 \cdot 2^{\,n-2}$, with equality at a single minimal-degree +interior vertex; the wheel $W_n$ gives $|\Phi(W_n)| = \lfloor 2^n/3 +\rfloor$ and is \emph{not} the minimiser. A counting balance makes the +floor plausible --- a disk with $k$ interior vertices has $2k+n-2$ faces +(Euler) but only $k$ interior constraints, so the linear free dimension +$k+n-2$ grows with depth --- but this is only heuristic: $|\Phi(D)|$ is +\emph{not} monotone in $k$ (inserting a degree-$4$ vertex can shrink it), +it merely never drops below $2^{\,n-2}$. Two natural elementary proofs, +a $\Phi$-non-increasing vertex reduction and a direct $(n{-}2)$-face +transversal, both provably fail; a proof appears to need a global +argument on the Boolean / mod-$3$ structure of $\Phi$. \end{remark} \begin{remark} \label{rem:floor-consequences} Two consequences. First, $\Phi(D)$ is a $\mathbb{Z}/3$ zonotope --- a projected cube, sign-closed but not a $\mathrm{GF}(3)$ subspace --- and at -the floor it has size $2^{\,n-2}$ with affine hull of dimension $n-2$. -Second, since the floor is exponential in the interface length $n$, a -maximally-constraining child still offers $2^{\,n-2}$ outer options, so +the interior-free value it has size $2^{\,n-2}$ with affine hull of +dimension $n-2$. Second, granting Conjecture~\ref{conj:constraint-floor}, +the floor is exponential in the interface length $n$, so a +maximally-constraining child still offers $2^{\,n-2}$ outer options, and the gluing of Conjecture~\ref{conj:heawood-chain-pigeonhole} has the least slack at \emph{short} interfaces (e.g.\ $n = 4$ leaves $4$ options) and is easy at long ones; the difficulty of the programme is concentrated at