From 1d981b4d015be2d730facf8e54f61e6b0ba6c28c Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 17 Jun 2026 02:24:04 -0400 Subject: [PATCH] Add the freedom-positive counting balance to the constraint floor Remark: a disk with k interior vertices has 2k+n-2 faces (Euler) but only k interior constraints, so each interior vertex adds two degrees of freedom against one constraint -- depth is freedom-positive and Phi can only retain or enlarge below the interior-free floor 2^(n-2). Motivates the lower bound and replaces the prior TODO sketch. Co-Authored-By: Claude Opus 4.8 --- .../paper.aux | 15 +++-- .../paper.log | 61 +++++++++--------- .../paper.pdf | Bin 266448 -> 267867 bytes .../paper.tex | 23 +++++-- 4 files changed, 56 insertions(+), 43 deletions(-) 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 7ff7c88..525b3e2 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux @@ -35,6 +35,12 @@ \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}} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tires}{3} \bibcite{bauerfeld-medial-tires}{4} @@ -44,10 +50,5 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@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:floor-consequences}{{4.3}{6}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{6}{}\protected@file@percent } -\gdef \@abspage@last{6} +\@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 167cd74..45992b1 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 02:14 +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 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -203,39 +203,40 @@ ML/cmm/m/it/10 ; ^^U[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 v[]\OT1/cmr/m/n/10 ) = n/10 ) = [][] \OML/cmm/m/it/10 ^^U[]: [] -[6] (./paper.aux) ) +[6] [7] (./paper.aux) ) Here is how much of TeX's memory you used: - 3021 strings out of 478268 - 42259 string characters out of 5846347 - 342330 words of memory out of 5000000 - 21067 multiletter control sequences out of 15000+600000 + 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 477578 words of font info for 59 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 - 69i,7n,76p,242b,290s stack positions out of 10000i,1000n,20000p,200000b,200000s -< -/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmss10.pfb> -Output written on paper.pdf (6 pages, 266448 bytes). + 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). 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b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex index 0a121ab..87afc21 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex @@ -499,12 +499,23 @@ the boundary sequence, so the map $\lambda \mapsto \lambda^{*}|_C$ is injective and $|\Phi(D)| = 2^{\,n-2}$. \end{proof} -%% TODO (lower bound): show |Phi(D)| >= 2^{n-2} for EVERY triangulated -%% disk D. Strategy: the n boundary-incident faces (one per boundary edge) -%% carry n-2 independent binary degrees of freedom after the interior -%% Heawood constraints are imposed; those constraints relate only -%% interior-incident faces and cannot collapse the boundary freedom below -%% 2^{n-2}. (See notes/boundary_restriction_structure.tex.) +\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. +\end{remark} \begin{remark} \label{rem:floor-consequences}