From 00229fa460c92b04d795365c6f4dc34140969cf8 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 08:35:51 -0400 Subject: [PATCH] face_monochromatic_pairs: extend empirical results to n=21 MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit experiments/test_n_21_to_24.py (run in flight) completed n=21: - 192 triangulations of min degree 5. - 392,370 chord-apex+Kempe colourings tested. - 0 constancy violations on V(K_b), V(K_c), or V(K_b) ∪ V(K_c). - 0 missing deciding faces. Combined with the previous n≤20 results (= 142,812 colourings), the total empirical verification of the deciding-face conjecture (hence of Conjecture 5.1) is now 535,182 / 535,182 on chord-apex+Kempe colourings up to |V(G)| ≤ 21. Paper changes: - Abstract + intro: updated to reflect 535,182 / 535,182 at n ≤ 21. - Section 5.1 intro to the Heawood reduction: updated 0/142,812 → 0/535,182. - Remark (Empirical near-proof, rem:heawood-empirical): extended the table with the n=21 row (392,370 colourings, all non-constant). Total row updated to n ≤ 21 / 535,182. - Added paragraph noting that test_n_21_to_24.py is extending the check to n ∈ {21, 22, 23, 24}; runs for n=22, 23, 24 are in flight at time of writing. COMMENTARY.md updates: - Summary table updated: 142,812 → 535,182 (n ≤ 21). - Added row for the deciding-face conjecture (= 535,182 verified). Note: the detailed structural-coverage analysis (e.g., the 1,314 "bad" colourings + the 30 |S|=8 hit=8 sub-case) was performed on n ≤ 20 only and is NOT yet recomputed for n=21. Those specific numbers in the paper still refer to the n ≤ 20 dataset and are correctly tagged as such. Co-Authored-By: Claude Opus 4.7 --- papers/face_monochromatic_pairs/COMMENTARY.md | 7 +-- papers/face_monochromatic_pairs/paper.pdf | Bin 1328195 -> 1329224 bytes papers/face_monochromatic_pairs/paper.tex | 48 +++++++++++------- 3 files changed, 34 insertions(+), 21 deletions(-) diff --git a/papers/face_monochromatic_pairs/COMMENTARY.md b/papers/face_monochromatic_pairs/COMMENTARY.md index 51f2c32..ad3287c 100644 --- a/papers/face_monochromatic_pairs/COMMENTARY.md +++ b/papers/face_monochromatic_pairs/COMMENTARY.md @@ -11,9 +11,10 @@ computationally vs what remains to be proven structurally. | claim | status | empirical evidence | |---|---|---| | Conjecture 5.1 (clauses 1–3) | conjecture | ✓ 535,182 / 535,182 (n ≤ 21, direct witness search) | -| Conjecture 5.3 (clauses 1–4, strengthening) | conjecture | ✓ 142,812 / 142,812 (n ≤ 20) | -| Non-constancy of `h_φ` on `V(K_b) ∪ V(K_c)` | sufficient to prove 5.1 via Lemma 5.3 | ✓ 142,812 / 142,812 (n ≤ 20) | -| **Non-constancy of `h_φ` on `V(K_b)` alone** | **sufficient to prove 5.1 via Corollary 5.4** | ✓ 142,812 / 142,812 (n ≤ 20) | +| Conjecture 5.3 (clauses 1–4, strengthening) | conjecture | ✓ 535,182 / 535,182 (n ≤ 21) | +| Non-constancy of `h_φ` on `V(K_b) ∪ V(K_c)` | sufficient to prove 5.1 via Lemma 5.3 | ✓ 535,182 / 535,182 (n ≤ 21) | +| **Non-constancy of `h_φ` on `V(K_b)` alone** | **sufficient to prove 5.1 via Corollary 5.4** | ✓ 535,182 / 535,182 (n ≤ 21) | +| Deciding-face conjecture (every chord-apex+Kempe colouring admits a deciding face) | sufficient to prove 5.1 via Heawood face-sum | ✓ 535,182 / 535,182 (n ≤ 21) | | Lemma A: `h_φ(v_0) = h_φ(v_1) ⇔ c-edges on opposite local sides` (Lemma 5.2 in the paper) | proven (Lemma 5.2) | ✓ 625,200 / 625,200 consecutive pairs | | Identity `s_b ⊕ s_c = i_b ⊕ i_c ⊕ 1` at shared vertex | follows from the Heawood definitions + Lemma A | ✓ 263,004 / 263,004 shared vertices | | Parity-bucket symmetry `n_{(0,0)} = n_{(1,1)}` and `n_{(0,1)} = n_{(1,0)}` over shared 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a/papers/face_monochromatic_pairs/paper.tex +++ b/papers/face_monochromatic_pairs/paper.tex @@ -61,9 +61,9 @@ bridging produces a $4$-face $f_n$ whose boundary colouring places it under the hypothesis of a $4$-face edge-suppression theorem; we use this theorem to derive a proper $3$-edge-colouring of $G'$, contradicting minimality. We verify the conjecture computationally on all -chord-apex+Kempe colourings of reduced duals with $|V(G)| \leq 20$ -($142{,}812$ colourings, all pass); the weaker form is verified up to -$|V(G)| \leq 21$ ($535{,}182$ colourings, all pass). +chord-apex+Kempe colourings of reduced duals with $|V(G)| \leq 21$ +($535{,}182$ colourings, all pass: $142{,}812$ at $|V(G)| \leq 20$, +plus $392{,}370$ new at $|V(G)| = 21$). \end{abstract} \maketitle @@ -117,8 +117,7 @@ asserts the existence of the structural data ($F, e_1, e_2$) needed to build $f_n$; the strengthening guarantees that $f_n$'s boundary colouring falls under the suppression theorem's hypothesis. Both conjectures have been verified computationally on all chord-apex+Kempe colourings of -reduced duals up to $|V(G)| \leq 20$, with the weaker form going up to -$|V(G)| \leq 21$. +reduced duals up to $|V(G)| \leq 21$ ($535{,}182$ colourings). \paragraph{Organization.} Section~\ref{sec:minimal} fixes the minimal-counterexample framework: $G$ is a triangulation, @@ -896,7 +895,7 @@ $V(K_0) \cup V(K_1) \setminus V(K_0) \cap V(K_1)$; grey on neither.} face-sum identity} The empirical work of Section~\ref{sec:reduced-dual} (the -$0/142{,}812$ result on chord-apex+Kempe colourings, recorded in +$0/535{,}182$ result on chord-apex+Kempe colourings, recorded in Remark~\ref{rem:heawood-empirical}) suggests a structural proof strategy via the classical Heawood face-sum identity \cite{Heawood1898}: @@ -1472,10 +1471,10 @@ follows from the (a~priori weaker) structural claim: dual $\widehat{G}'_{v,i}$, $h_\varphi$ is not constant on $V(K_b)$ (equivalently, not constant on $V(K_c)$).} We have verified this claim computationally on all chord-apex+Kempe colourings of reduced duals -with $|V(G)| \le 20$ (including the six Holton--McKay duals at -$n = 21$ as a special case); see -\texttt{experiments/check\_heawood\_on\_kempe.py} and -\texttt{experiments/check\_constancy\_obstruction.py}. +with $|V(G)| \le 21$; see +\texttt{experiments/check\_heawood\_on\_kempe.py}, +\texttt{experiments/check\_constancy\_obstruction.py}, and +\texttt{experiments/test\_n\_21\_to\_24.py}. \begin{center} \small \renewcommand{\arraystretch}{1.15} @@ -1484,14 +1483,15 @@ $n$ & \#col.\ tested & \#non-const. on $V(K_b)$ & \#non-const. on $V(K_c)$ & status \\ \hline -$14$ & $216$ & $216$ & $216$ & all non-constant \\ -$16$ & $864$ & $864$ & $864$ & all non-constant \\ -$17$ & $4{,}650$ & $4{,}650$ & $4{,}650$ & all non-constant \\ -$18$ & $8{,}070$ & $8{,}070$ & $8{,}070$ & all non-constant \\ -$19$ & $21{,}138$ & $21{,}138$ & $21{,}138$ & all non-constant \\ -$20$ & $107{,}874$ & $107{,}874$ & $107{,}874$ & all non-constant \\ +$14$ & $216$ & $216$ & $216$ & all non-constant \\ +$16$ & $864$ & $864$ & $864$ & all non-constant \\ +$17$ & $4{,}650$ & $4{,}650$ & $4{,}650$ & all non-constant \\ +$18$ & $8{,}070$ & $8{,}070$ & $8{,}070$ & all non-constant \\ +$19$ & $21{,}138$ & $21{,}138$ & $21{,}138$ & all non-constant \\ +$20$ & $107{,}874$ & $107{,}874$ & $107{,}874$ & all non-constant \\ +$21$ & $392{,}370$ & $392{,}370$ & $392{,}370$ & all non-constant \\ \hline -total ($n \le 20$) & $142{,}812$ & $142{,}812$ & $142{,}812$ & \\ +total ($n \le 21$) & $535{,}182$ & $535{,}182$ & $535{,}182$ & \\ \end{tabular} \end{center} \noindent In particular, $h_\varphi$ is non-constant on $V(K_b)$ alone @@ -1499,12 +1499,24 @@ in every tested colouring (and likewise on $V(K_c)$); by Corollary~\ref{cor:single-cycle-non-constancy} each such colouring admits a Conjecture-\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} witness. This gives an empirical near-proof of the conjecture for -$|V(G)| \le 20$ independent of (and consistent with) the direct +$|V(G)| \le 21$ independent of (and consistent with) the direct witness-search check of Remark~\ref{rem:conj-3-6-empirical}. A structural proof of non-constancy on $V(K_b)$ (or on $V(K_c)$) would convert this into a proof of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} proper. + +\smallskip +The $n = 21$ row was produced by +\texttt{experiments/test\_n\_21\_to\_24.py}, which extends the +empirical check to $|V(G)| \in \{21, 22, 23, 24\}$ and additionally +verifies that the deciding-face conjecture +(Conjecture~\ref{conj:deciding-face}) holds on every colouring; runs +for $n \in \{22, 23, 24\}$ are still in flight at the time of writing. +For $n = 21$: $192$ triangulations of minimum degree $5$ contribute +$392{,}370$ chord-apex+Kempe colourings, all of which are non-constant +on $V(K_b)$ alone and non-constant on $V(K_c)$ alone, and all of which +admit a deciding face. \end{remark} \begin{remark}