didericis
74030a5b8f
Adds experiments/test_conj_5_26_n_21_22.py, a clause-4 checker that re-uses find_all_36_witnesses + check_clause_4 from check_conj_final_scaled.py and runs them on n = 21, 22 with incremental JSONL output and a 10-minute PROGRESS heartbeat. Results (139 min wall, single thread): n=21: 192 tri, 392,370 colourings w/ clause-1–3 witness, all pass n=22: 651 tri, 1,786,314 colourings w/ clause-1–3 witness, all pass total at n ≤ 22: 2,321,496 / 2,321,496 (combined with the existing 142,812 at n ≤ 20 from check_conj_final_scaled.py) Paper edits: - Abstract: "|V(G)| ≤ 20 (142,812)" → "|V(G)| ≤ 22 (2,321,496)" for the strengthening; clauses-1–3 count unchanged at 535,182 / n ≤ 21. - Intro paragraph: matching update. - Remark rem:conj-3-8-empirical table: added n=21 and n=22 rows; new total ($n \le 22$) = 959 triangulations, 2,321,496 colourings. - Updated script reference in that remark to point at check_conj_final_scaled.py + test_conj_5_26_n_21_22.py. COMMENTARY.md summary table: Conjecture 5.26 row bumped to 2,321,496 / 2,321,496 (n ≤ 22). Also commits the test_*_results.jsonl artifacts (with per-tri records + n-summaries + grand summary) for reproducibility. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
90 lines
7.8 KiB
TeX
90 lines
7.8 KiB
TeX
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\newlabel{def:reduced-dual}{{3.2}{4}}
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\newlabel{def:edge-names}{{3.4}{4}}
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The four steps of Definition\nonbreakingspace 3.2\hbox {}, illustrated on $G' = $ the dodecahedron (dual of the icosahedron) with $F_v$ the inner pentagon and $i = 0$. Top left: delete the five boundary vertices of $F_v$, leaving five degree-$2$ vertices on a new face $F$. Top right: order them clockwise as $A_0,\dots ,A_4$. Bottom left: add $v_n$ joined to $A_0, A_1, A_2$. Bottom right: add the chord $A_3 A_4$, giving the cubic plane graph $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle G$}\mathaccent "0362{G}'_{v,0}$.}}{5}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The proof of Lemma\nonbreakingspace 3.7\hbox {}, illustrated for $i = 0$ on $G' = $ the dodecahedron. Top: under the assumption $W \neq Y$, propriety at $v_n$ forces $W \in \{X, Z\}$. Bottom: in either case the lift to $G'$ has externals satisfying the hypothesis of Lemma\nonbreakingspace 3.5\hbox {}, which colours $\partial F_v$ to extend $\psi $ to a proper $3$-edge-colouring of $G'$.}}{7}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Edge suppression (Definition\nonbreakingspace 4.1\hbox {}). Left: a fragment of a cubic plane graph with the suppressed edge $e = uv$ highlighted in red. Middle: deleting $e$ leaves $u$ and $v$ of degree\nonbreakingspace $2$. Right: smoothing $u$ and $v$ replaces each pair of incident edges by a single new edge, removing $u, v$ and giving a cubic plane graph again.}}{9}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces The recolouring used in the proof of Theorem\nonbreakingspace 4.2\hbox {}. Left: the $4$-face $f$ of $H$ under $\varphi $, with the forced colours $\varphi (e_0) = a$, $\varphi (e_1) = b$, $\varphi (e_2) = \varphi (e_3) = c$, $\varphi (w_0) = \varphi (w_1) = b$, and $\varphi (w_2) = \varphi (w_3) = a$. Right: the suppressed graph $H'$ under $\varphi '$. The smoothed-in edges $e_2', e_3'$ inherit the colour $b$ from $w_0, w_1$, and $e_1$ is recoloured from $b$ to $c$; every edge outside the face neighbourhood keeps its $\varphi $-colour (dotted in red: the five edges of $H$ removed by the suppression).}}{10}{}\protected@file@percent }
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\newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{5.1}{10}}
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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{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{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:conj-3-6-empirical}{{5.25}{21}}
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\newlabel{conj:face-monochromatic-pair-strengthened}{{5.26}{22}}
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\newlabel{rem:conj-3-8-empirical}{{5.27}{22}}
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