face_monochromatic_pairs: rename check_conj_3_8_scaled → check_conj_final_scaled; add n=21-24 test
Rename the shared helper module to a number-resistant name. Update all 26 dependent scripts via sed. Add experiments/test_n_21_to_24.py — extends the empirical check beyond |V(G)| ≤ 20 to n_G ∈ [21, 24]. Checks per chord-apex+Kempe colouring: (1) h_φ constant on V(K_b)? (counterexample to Corollary 5.4) (2) h_φ constant on V(K_b) ∪ V(K_c)? (counterexample to Conj 5.1) (3) Deciding face exists? Writes results incrementally to test_n_21_to_24_results.jsonl (one JSON line per triangulation, plus n-level and grand summaries). Emits PROGRESS lines every 10 minutes (default) to stdout for live monitoring. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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@@ -40,21 +40,50 @@
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\newlabel{lem:both-kempe-constant}{{5.3}{11}}
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The two cases in the proof of Lemma\nonbreakingspace 5.2\hbox {}. Vertices $v_0, v_1$ are consecutive on the $\{a, b\}$-Kempe cycle $K$, joined by an edge $e$, with the lemma's hypothesis $h_\varphi (v_0) = h_\varphi (v_1) = +1$ --- so both vertices share the clockwise colour order $(a, b, c)$. \emph {Left (Case\nonbreakingspace A):} when $\varphi (e) = a$, the colour-$b$ edge at $v_0$ lies south of $e$ (on $\partial F_R$) and the colour-$b$ edge at $v_1$ lies north of $e$ (on $\partial F_L$); the two would-be witness edges are on opposite faces, so no face of $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle G$}\mathaccent "0362{G}'_{v,i}$ contains both. \emph {Right (Case\nonbreakingspace B):} when $\varphi (e) = b$, the colour-$a$ edges at $v_0, v_1$ are likewise on opposite sides of $e$. In either case the clause-$(3)$ arc of Conjecture\nonbreakingspace 5.1\hbox {} cannot be realised at $e$.}}{12}{}\protected@file@percent }
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\newlabel{fig:lemma-kempe-heawood}{{5}{12}}
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\citation{Heawood1898}
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\newlabel{cor:single-cycle-non-constancy}{{5.4}{13}}
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\newlabel{rem:heawood-empirical}{{5.5}{13}}
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\newlabel{rem:conj-3-6-empirical}{{5.6}{13}}
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\newlabel{conj:face-monochromatic-pair-strengthened}{{5.7}{14}}
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\newlabel{rem:conj-3-8-empirical}{{5.8}{14}}
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\newlabel{conj:no-two-constant-kempe-cycles}{{5.5}{13}}
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\newlabel{rem:no-two-constant-kempe-cycles-counterexample}{{5.6}{13}}
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\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Smallest counterexample to Conjecture\nonbreakingspace 5.5\hbox {}: a $C_{28}$ fullerene-style cubic plane graph (12 pentagons + 4 hexagons) with a proper $3$-edge-colouring on which $h_\varphi $ is simultaneously constant ($\equiv -1$) on the red/blue $12$-cycle and the red/green $12$-cycle, which share the colour-red edge $(0, 1)$. Light-shaded nodes are on $V(K_0) \cap V(K_1)$; medium-shaded on $V(K_0) \cup V(K_1) \setminus V(K_0) \cap V(K_1)$; grey on neither.}}{14}{}\protected@file@percent }
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\newlabel{fig:no-two-constant-kempe-counterexample}{{6}{14}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{A reduction of Conjecture 5.1\hbox {} via Heawood's face-sum identity}}{14}{}\protected@file@percent }
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\newlabel{eq:heawood-face-sum}{{5.1}{14}}
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\newlabel{conj:deciding-face}{{5.7}{14}}
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\newlabel{thm:deciding-face-implies-conj-5-1}{{5.8}{14}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{A partial structural proof of Conjecture\nonbreakingspace 5.7\hbox {}}}{15}{}\protected@file@percent }
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\newlabel{def:flank-face}{{5.9}{15}}
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\newlabel{lem:flank-length}{{5.10}{15}}
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\newlabel{lem:flank-covering-base}{{5.11}{15}}
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\newlabel{lem:flank-covering-hex}{{5.12}{16}}
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\newlabel{thm:deciding-face-partial}{{5.13}{17}}
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\newlabel{rem:n-i-6-flank-fails}{{5.14}{17}}
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\newlabel{def:outer-face}{{5.15}{17}}
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\newlabel{lem:outer-face-length}{{5.16}{17}}
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\newlabel{lem:outer-face-covering-base}{{5.17}{17}}
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\newlabel{thm:deciding-face-partial-extended}{{5.18}{18}}
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\newlabel{rem:deciding-face-remaining-case}{{5.19}{18}}
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\newlabel{conj:gprime-pentagon-fallback}{{5.20}{18}}
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\newlabel{lem:gprime-pigeonhole}{{5.21}{18}}
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\newlabel{rem:gprime-pigeonhole-stop}{{5.22}{18}}
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\citation{AH77a}
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\citation{AHK77}
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\citation{RSST97}
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\citation{Gonthier08}
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\newlabel{rem:deciding-face-empirical}{{5.23}{20}}
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\newlabel{rem:heawood-empirical}{{5.24}{20}}
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\newlabel{rem:conj-3-6-empirical}{{5.25}{21}}
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\newlabel{conj:face-monochromatic-pair-strengthened}{{5.26}{21}}
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\newlabel{rem:conj-3-8-empirical}{{5.27}{22}}
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\newlabel{rem:implication-4ct}{{5.28}{22}}
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\bibcite{Heawood1898}{1}
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\bibcite{AH77a}{2}
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\bibcite{AHK77}{3}
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\bibcite{RSST97}{4}
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\bibcite{Gonthier08}{5}
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\newlabel{tocindent-1}{0pt}
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\newlabel{tocindent0}{12.7778pt}
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\newlabel{tocindent0}{14.69437pt}
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\newlabel{tocindent1}{17.77782pt}
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\newlabel{tocindent2}{0pt}
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\newlabel{tocindent3}{0pt}
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\newlabel{rem:implication-4ct}{{5.9}{15}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{15}{}\protected@file@percent }
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\gdef \@abspage@last{15}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{23}{}\protected@file@percent }
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\gdef \@abspage@last{23}
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