From 351ae0cdfe6d868947587230c637c20c4c13896e Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 17 Jun 2026 01:33:07 -0400 Subject: [PATCH] Account for the outer face in the Heawood face-sum identity The bounded-face sum omits the outer face at outer-boundary vertices, so restrict the gluing identity to interior vertices (where all cluster interfaces live) and recover a colouring by carrying a single +/-1 label on the unbounded face f_inf, giving Heawood's identity on the full cubic dual for the Tait step. Co-Authored-By: Claude Opus 4.8 --- .../paper.aux | 11 ++-- .../paper.log | 54 +++++++++--------- .../paper.pdf | Bin 258014 -> 259543 bytes .../paper.tex | 33 +++++++---- 4 files changed, 56 insertions(+), 42 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 fc8149c..15db1e2 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux @@ -25,11 +25,15 @@ \newlabel{rem:no-interior-constraint}{{3.2}{3}} \newlabel{def:boundary-sequences}{{3.3}{3}} \newlabel{def:heawood-compatible}{{3.4}{3}} +\citation{Heawood1898} \newlabel{rem:compat-is-heawood}{{3.5}{4}} \newlabel{eq:heawood-face-sum-dual}{{3.1}{4}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Why the programme runs between nested clusters}}{4}{}\protected@file@percent } \newlabel{prop:two-sided-decomposition}{{3.6}{4}} \bibcite{Heawood1898}{1} +\newlabel{rem:why-clusters}{{3.7}{5}} +\newlabel{conj:heawood-chain-pigeonhole}{{3.8}{5}} +\newlabel{conj:heawood-route-fct}{{3.9}{5}} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tires}{3} \bibcite{bauerfeld-medial-tires}{4} @@ -39,8 +43,5 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{rem:why-clusters}{{3.7}{5}} -\newlabel{conj:heawood-chain-pigeonhole}{{3.8}{5}} -\newlabel{conj:heawood-route-fct}{{3.9}{5}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{5}{}\protected@file@percent } -\gdef \@abspage@last{5} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{6}{}\protected@file@percent } +\gdef \@abspage@last{6} 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 b30ce1f..345706b 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 01:10 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 17 JUN 2026 01:32 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -192,7 +192,7 @@ 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] (./paper.aux) ) +[2] [3] [4] [5] [6] (./paper.aux) ) Here is how much of TeX's memory you used: 3017 strings out of 478268 42161 string characters out of 5846347 @@ -200,32 +200,32 @@ Here is how much of TeX's memory you used: 21063 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,272s stack positions out of 10000i,1000n,20000p,200000b,200000s - - -Output written on paper.pdf (5 pages, 258014 bytes). + 69i,7n,76p,242b,340s stack positions out of 10000i,1000n,20000p,200000b,200000s + +Output written on paper.pdf (6 pages, 259543 bytes). 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