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Run Heawood pigeonhole between nested connected tire clusters

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Add the two-sided cluster decomposition proposition: a vertex's full
Heawood face-sum splits as exactly one child-cluster contribution plus
one parent-cluster contribution (the at-most-two-clusters bound makes the
pairing binary and complete). Explain why this fails per-tire -- a vertex
on many same-depth tires has only a fragment of its face-star in any one
tire -- and recast the chain-pigeonhole and 4CT conjectures to nested
clusters with a cluster restriction relation.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-06-17 01:10:30 -04:00
parent 646cf9d12f
commit c5f81842c7
4 changed files with 155 additions and 55 deletions
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papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux
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@@ -25,19 +25,22 @@
\newlabel{rem:no-interior-constraint}{{3.2}{3}}
\newlabel{def:boundary-sequences}{{3.3}{3}}
\newlabel{def:heawood-compatible}{{3.4}{3}}
\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}
\bibcite{bauerfeld-depth}{2}
\bibcite{bauerfeld-nested-tires}{3}
\bibcite{bauerfeld-medial-tires}{4}
\bibcite{bauerfeld-nested-tire-duals}{5}
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{12.7778pt}
\newlabel{tocindent0}{14.69437pt}
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt}
\newlabel{rem:compat-is-heawood}{{3.5}{4}}
\newlabel{eq:heawood-face-sum-dual}{{3.1}{4}}
\newlabel{conj:heawood-chain-pigeonhole}{{3.6}{4}}
\newlabel{conj:heawood-route-fct}{{3.7}{4}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent }
\gdef \@abspage@last{4}
\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}