From 33b51b675b6f7eb1fb09cd115d65836a622eed0b Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 00:43:35 -0400 Subject: [PATCH] face_monochromatic_pairs: per-cycle refinement + Corollary 5.4 Empirical refinement of Lemma 5.3: h_phi is non-constant on V(K_b) alone (not just on the union) and likewise on V(K_c) alone, in every one of 142,812 chord-apex+Kempe colourings tested (n in [12, 20]). This is strictly stronger than what we previously reported. The proof of Lemma 5.3 already constructs the (F, e_1, e_2) witness from any consecutive same-Heawood failure on either Kempe cycle through merged -- never needing the other cycle. Pull that out into a separate Corollary 5.4 ("Per-cycle form"), which makes the empirical-to-conjecture path more direct. Update Remark 5.5 to: - Cite Corollary 5.4 instead of the contrapositive of Lemma 5.3. - Replace "non-constant on V(K_b) U V(K_c)" with the per-cycle form. - Extend the empirical table with separate columns for K_b and K_c non-constancy. Also commit experiments/check_constancy_obstruction.py, the script that produced these refined empirical findings. It additionally records that no single named vertex (v_n, A_i, ..., A_{i+4}) is structurally majority or minority -- the minority rates cluster in 31-39%, ruling out a single-vertex-mismatch identity. Co-Authored-By: Claude Opus 4.7 --- .../check_constancy_obstruction.py | 224 ++++++++++++++++++ papers/face_monochromatic_pairs/paper.aux | 11 +- papers/face_monochromatic_pairs/paper.log | 26 +- papers/face_monochromatic_pairs/paper.pdf | Bin 1085520 -> 1086630 bytes papers/face_monochromatic_pairs/paper.tex | 78 +++--- 5 files changed, 295 insertions(+), 44 deletions(-) create mode 100644 papers/face_monochromatic_pairs/experiments/check_constancy_obstruction.py diff --git a/papers/face_monochromatic_pairs/experiments/check_constancy_obstruction.py b/papers/face_monochromatic_pairs/experiments/check_constancy_obstruction.py new file mode 100644 index 0000000..5920807 --- /dev/null +++ b/papers/face_monochromatic_pairs/experiments/check_constancy_obstruction.py @@ -0,0 +1,224 @@ +"""For each chord-apex+Kempe colouring, dissect WHERE the constancy +hypothesis fails: + + (1) Is h_phi constant on V(K_b) alone? On V(K_c) alone? On the + intersection V(K_b) cap V(K_c)? + (2) Per colouring, distribution of (#+1, #-1) shared vertices. + (3) For each colouring, identify the "minority Heawood" shared + vertices (= those whose h differs from the more frequent value + on V(K_b) cup V(K_c)). How many are there? Are they + concentrated at specific structural positions (v_n, A_{i+1}, + A_{i+3}, A_{i+4})? + (4) Is h(v_n) always equal to / always different from the majority? + +Under the constancy hypothesis, all of these distributions would be +trivial (everything same Heawood, 0 minority vertices). Their +non-trivial empirical structure exposes where Path 4's "trap" +mechanism would have to apply. + +Run with: sage experiments/check_constancy_obstruction.py +""" +import os +import sys +import time + +from sage.all import Graph +from sage.graphs.graph_generators import graphs + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) + +from check_conj_3_8_scaled import ( + apply_reduction, + proper_3_edge_colorings, + matches_chord_apex_kempe, + trace_kempe_cycle, + edge_idx, + kempe_cycle_set, +) +from check_heawood_on_kempe import ( + dual_of, heawood_numbers, vertices_of_kempe, +) + + +def named_vertices(named, v_n=9999): + """Recover A_i, A_{i+1}, ..., A_{i+4}.""" + def other(fs, v): + return next(iter(fs - {v})) + A_i = other(named['side_0'], v_n) + A_i1 = other(named['spike'], v_n) + A_i2 = other(named['side_1'], v_n) + A_i3, A_i4 = sorted(named['merged']) + return A_i, A_i1, A_i2, A_i3, A_i4 + + +def test_one(D): + D.is_planar(set_embedding=True) + n_col = 0 + rec = { + 'const_kb': 0, 'const_kc': 0, 'const_cap': 0, + # (#+1, #-1) on V(K_b) cup V(K_c) histogram + 'plus_minus_dist': {}, + # # minority Heawood vertices on V(K_b) cup V(K_c) + 'minority_count': {}, + # Is v_n minority? majority? + 'v_n_pos': {'minority': 0, 'majority': 0, 'tie': 0}, + # Position-tallies of minority for each "named" vertex + 'minority_at': {'v_n': 0, 'A_i': 0, 'A_i1': 0, 'A_i2': 0, + 'A_i3': 0, 'A_i4': 0}, + # Is h(v_n) == h(A_{i+3}) == h(A_{i+4}) always? + 'v_n_eq_merged_endpts': 0, + # Is h constant on the 6 named vertices {v_n, A_0..A_4}? + 'const_on_named6': 0, + } + for face in D.faces(): + if len(face) != 5: continue + for i_red in range(5): + res = apply_reduction(D, face, i_red, 9999) + if res is None: continue + H = res['H']; named = res['named'] + H.is_planar(set_embedding=True) + edges, colorings = proper_3_edge_colorings(H) + cand = [c for c in colorings + if matches_chord_apex_kempe(edges, c, named)] + v_n = 9999 + A_i, A_i1, A_i2, A_i3, A_i4 = named_vertices(named, v_n) + for col in cand: + n_col += 1 + try: + h = heawood_numbers(H, edges, col) + except RuntimeError: + continue + merged_idx = edge_idx(edges, named['merged']) + a = col[merged_idx] + bs = [c for c in range(3) if c != a] + kc_b = kempe_cycle_set(edges, col, merged_idx, (a, bs[0])) + kc_c = kempe_cycle_set(edges, col, merged_idx, (a, bs[1])) + V_b = vertices_of_kempe(edges, kc_b) + V_c = vertices_of_kempe(edges, kc_c) + V_union = V_b | V_c + V_cap = V_b & V_c + h_b_vals = {h[v] for v in V_b} + h_c_vals = {h[v] for v in V_c} + h_cap_vals = {h[v] for v in V_cap} + if len(h_b_vals) == 1: rec['const_kb'] += 1 + if len(h_c_vals) == 1: rec['const_kc'] += 1 + if len(h_cap_vals) == 1: rec['const_cap'] += 1 + plus = sum(1 for v in V_union if h[v] == 1) + minus = sum(1 for v in V_union if h[v] == -1) + rec['plus_minus_dist'][(plus, minus)] = \ + rec['plus_minus_dist'].get((plus, minus), 0) + 1 + # Majority sign on union + if plus > minus: + maj = 1 + elif minus > plus: + maj = -1 + else: + maj = 0 + minority = sum(1 for v in V_union if h[v] != maj and maj != 0) + if maj == 0: + minority = min(plus, minus) + rec['minority_count'][minority] = \ + rec['minority_count'].get(minority, 0) + 1 + # v_n's position + hv = h.get(v_n) + if hv is not None and maj != 0: + if hv == maj: + rec['v_n_pos']['majority'] += 1 + else: + rec['v_n_pos']['minority'] += 1 + else: + rec['v_n_pos']['tie'] += 1 + # Named vertices: minority count + for name, vv in [('v_n', v_n), ('A_i', A_i), + ('A_i1', A_i1), ('A_i2', A_i2), + ('A_i3', A_i3), ('A_i4', A_i4)]: + if vv in V_union and maj != 0 and h[vv] != maj: + rec['minority_at'][name] += 1 + # h(v_n) == h(A_{i+3}) == h(A_{i+4}) ? + if h.get(v_n) == h.get(A_i3) == h.get(A_i4): + rec['v_n_eq_merged_endpts'] += 1 + # h constant on the 6 named vertices? + named6 = [v_n, A_i, A_i1, A_i2, A_i3, A_i4] + named6_vals = {h[vv] for vv in named6 if vv in h} + if len(named6_vals) == 1: + rec['const_on_named6'] += 1 + return n_col, rec + + +def merge_into(g, r): + for k in ('const_kb', 'const_kc', 'const_cap', + 'v_n_eq_merged_endpts', 'const_on_named6'): + g[k] += r[k] + for k in ('plus_minus_dist', 'minority_count'): + for kk, vv in r[k].items(): + g[k][kk] = g[k].get(kk, 0) + vv + for sub in ('v_n_pos', 'minority_at'): + for kk, vv in r[sub].items(): + g[sub][kk] = g[sub].get(kk, 0) + vv + + +def main(max_n=18, time_budget_per_n=1800): + print(f"Constancy obstruction analysis, n in [12, {max_n}]\n") + grand = { + 'const_kb': 0, 'const_kc': 0, 'const_cap': 0, + 'plus_minus_dist': {}, 'minority_count': {}, + 'v_n_pos': {'minority': 0, 'majority': 0, 'tie': 0}, + 'minority_at': {'v_n': 0, 'A_i': 0, 'A_i1': 0, 'A_i2': 0, + 'A_i3': 0, 'A_i4': 0}, + 'v_n_eq_merged_endpts': 0, + 'const_on_named6': 0, + } + grand_col = 0 + for n in range(12, max_n + 1): + start = time.time() + try: + triangulations = list(graphs.triangulations(n, minimum_degree=5)) + except Exception as ex: + print(f"n={n}: cannot enumerate ({ex})") + continue + n_col_n = 0 + for tri_idx, G in enumerate(triangulations): + if time.time() - start > time_budget_per_n: + print(f" n={n}: timeout at tri {tri_idx}/{len(triangulations)}") + break + G.is_planar(set_embedding=True) + D = dual_of(G) + ni, ri = test_one(D) + n_col_n += ni + merge_into(grand, ri) + elapsed = time.time() - start + print(f"n={n}: {n_col_n} col., [{elapsed:.0f}s]") + sys.stdout.flush() + grand_col += n_col_n + + print() + print("=" * 78) + print(f"Grand totals (n in [12, {max_n}], {grand_col} colourings)") + print(f"\n h constant on V(K_b): {grand['const_kb']}/{grand_col}") + print(f" h constant on V(K_c): {grand['const_kc']}/{grand_col}") + print(f" h constant on V(K_b) cap V(K_c): {grand['const_cap']}/{grand_col}") + print(f" h constant on {{v_n, A_0..A_4}}: " + f"{grand['const_on_named6']}/{grand_col}") + print(f" h(v_n) == h(A_{{i+3}}) == h(A_{{i+4}}): " + f"{grand['v_n_eq_merged_endpts']}/{grand_col}") + print(f"\n Minority count (= #vertices on V(K_b) cup V(K_c) with " + f"non-majority h) distribution:") + for k, v in sorted(grand['minority_count'].items()): + pct = 100 * v / max(1, grand_col) + print(f" {k:>3}: {v} ({pct:.2f}%)") + print(f"\n v_n status on V(K_b) cup V(K_c):") + print(f" in majority: {grand['v_n_pos']['majority']} " + f"({100*grand['v_n_pos']['majority']/max(1,grand_col):.2f}%)") + print(f" in minority: {grand['v_n_pos']['minority']} " + f"({100*grand['v_n_pos']['minority']/max(1,grand_col):.2f}%)") + print(f" tie: {grand['v_n_pos']['tie']} " + f"({100*grand['v_n_pos']['tie']/max(1,grand_col):.2f}%)") + print(f"\n How often each named vertex is in the minority:") + for name, c in grand['minority_at'].items(): + print(f" {name:>5}: {c}/{grand_col} " + f"({100*c/max(1,grand_col):.2f}%)") + + +if __name__ == '__main__': + main() diff --git a/papers/face_monochromatic_pairs/paper.aux b/papers/face_monochromatic_pairs/paper.aux index f6d49f7..4405665 100644 --- a/papers/face_monochromatic_pairs/paper.aux +++ b/papers/face_monochromatic_pairs/paper.aux @@ -40,10 +40,11 @@ \newlabel{lem:both-kempe-constant}{{5.3}{11}} \@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 } \newlabel{fig:lemma-kempe-heawood}{{5}{12}} -\newlabel{rem:heawood-empirical}{{5.4}{13}} -\newlabel{rem:conj-3-6-empirical}{{5.5}{13}} -\newlabel{conj:face-monochromatic-pair-strengthened}{{5.6}{14}} -\newlabel{rem:conj-3-8-empirical}{{5.7}{14}} +\newlabel{cor:single-cycle-non-constancy}{{5.4}{13}} +\newlabel{rem:heawood-empirical}{{5.5}{13}} +\newlabel{rem:conj-3-6-empirical}{{5.6}{13}} +\newlabel{conj:face-monochromatic-pair-strengthened}{{5.7}{14}} +\newlabel{rem:conj-3-8-empirical}{{5.8}{14}} \bibcite{Heawood1898}{1} \bibcite{AH77a}{2} \bibcite{AHK77}{3} @@ -54,6 +55,6 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{rem:implication-4ct}{{5.8}{15}} +\newlabel{rem:implication-4ct}{{5.9}{15}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{15}{}\protected@file@percent } \gdef \@abspage@last{15} diff --git a/papers/face_monochromatic_pairs/paper.log b/papers/face_monochromatic_pairs/paper.log index 8e050a1..34650c9 100644 --- a/papers/face_monochromatic_pairs/paper.log +++ b/papers/face_monochromatic_pairs/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) 25 MAY 2026 00:21 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 00:43 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -272,26 +272,28 @@ Package pdftex.def Info: fig_lemma_kempe_heawood.png used on input line 727. LaTeX Warning: `h' float specifier changed to `ht'. [11] [12 <./fig_lemma_kempe_heawood.png>] -Underfull \vbox (badness 10000) has occurred while \output is active [] +Overfull \hbox (45.67143pt too wide) in paragraph at lines 834--848 + [] + [] - [13] -Underfull \hbox (badness 1648) in paragraph at lines 917--923 -\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 5.7\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10 T +[13] +Underfull \hbox (badness 1648) in paragraph at lines 941--947 +\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 5.8\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10 T he strength-ened con-jec-ture was tested on the same chord- [] -Underfull \hbox (badness 1014) in paragraph at lines 917--923 -\OT1/cmr/m/n/10 apex+Kempe colour-ings as Re-mark 5.5[]; for each colour-ing we +Underfull \hbox (badness 1014) in paragraph at lines 941--947 +\OT1/cmr/m/n/10 apex+Kempe colour-ings as Re-mark 5.6[]; for each colour-ing we sought any [] [14] [15] (./paper.aux) ) Here is how much of TeX's memory you used: - 3110 strings out of 478268 - 44641 string characters out of 5846347 - 349419 words of memory out of 5000000 - 21142 multiletter control sequences out of 15000+600000 + 3111 strings out of 478268 + 44673 string characters out of 5846347 + 350430 words of memory out of 5000000 + 21143 multiletter control sequences out of 15000+600000 478386 words of font info for 63 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 69i,12n,76p,1047b,366s stack positions out of 10000i,1000n,20000p,200000b,200000s @@ -319,7 +321,7 @@ cal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb> -Output written on paper.pdf (15 pages, 1085520 bytes). +Output written on paper.pdf (15 pages, 1086630 bytes). 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b/papers/face_monochromatic_pairs/paper.tex index b0702b6..af32925 100644 --- a/papers/face_monochromatic_pairs/paper.tex +++ b/papers/face_monochromatic_pairs/paper.tex @@ -795,43 +795,67 @@ cycles, so its two endpoints --- which lie on $V(K_b) \cap V(K_c)$ --- force the two constants to coincide. \end{proof} -\begin{remark}[Empirical near-proof of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} via Lemma~\ref{lem:both-kempe-constant}] +\begin{corollary}[Per-cycle form] +\label{cor:single-cycle-non-constancy} +Let $G$, $\widehat{G}'_{v,i}$, $\varphi$ be as in +Lemma~\ref{lem:both-kempe-constant}, and let $K$ be either of the two +Kempe cycles of $\varphi$ through the merged edge. If $h_\varphi$ is not +constant on $V(K)$, then a triple $(F, e_1, e_2)$ satisfying +clauses~(1)--(3) of +Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} on +$(G, \widehat{G}'_{v,i}, \varphi)$ exists. +\end{corollary} + +\begin{proof} +This is precisely the case analysis used to prove +Lemma~\ref{lem:both-kempe-constant}: applied to any consecutive pair of +vertices on $K$ with differing Heawood numbers, the construction in +that proof produces a clauses-(1)--(3) witness without ever needing to +inspect the other Kempe cycle. +\end{proof} + +\begin{remark}[Empirical near-proof of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} via Corollary~\ref{cor:single-cycle-non-constancy}] \label{rem:heawood-empirical} \sloppy -The contrapositive of Lemma~\ref{lem:both-kempe-constant} reduces -Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} to -the following structural claim: +By Corollary~\ref{cor:single-cycle-non-constancy}, +Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} +follows from the (a~priori weaker) structural claim: \emph{for every chord-apex+Kempe colouring $\varphi$ of every reduced -dual $\widehat{G}'_{v,i}$, $h_\varphi$ is not constant on -$V(K_b) \cup V(K_c)$.} 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}. +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}. \begin{center} \small \renewcommand{\arraystretch}{1.15} -\begin{tabular}{r|r|r|l} -$n$ & \#col.\ tested & \#non-constant on $V(K_b)\cup V(K_c)$ & status \\ +\begin{tabular}{r|r|r|r|l} +$n$ & \#col.\ tested + & \#non-const. on $V(K_b)$ + & \#non-const. on $V(K_c)$ & status \\ \hline -$14$ & $216$ & $216$ & all non-constant \\ -$16$ & $864$ & $864$ & all non-constant \\ -$17$ & $4{,}650$ & $4{,}650$ & all non-constant \\ -$18$ & $8{,}070$ & $8{,}070$ & all non-constant \\ -$19$ & $21{,}138$ & $21{,}138$ & all non-constant \\ -$20$ & $107{,}874$ & $107{,}874$ & 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 \\ \hline -total ($n \le 20$) & $142{,}812$ & $142{,}812$ & \\ +total ($n \le 20$) & $142{,}812$ & $142{,}812$ & $142{,}812$ & \\ \end{tabular} \end{center} -\noindent Since $h_\varphi$ on $V(K_b) \cup V(K_c)$ was non-constant in -every tested colouring, Lemma~\ref{lem:both-kempe-constant}'s -contrapositive supplies a Conjecture-\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} -witness in each case --- giving an empirical near-proof of the -conjecture for $|V(G)| \le 20$ that is 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 $V(K_b) \cup V(K_c)$ would convert this into a proof -of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} +\noindent In particular, $h_\varphi$ is non-constant on $V(K_b)$ alone +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 +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. \end{remark}