From 74030a5b8fd3a5d6008a5c73bd183152ffa116a7 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 12:27:58 -0400 Subject: [PATCH] =?UTF-8?q?face=5Fmono:=20extend=20Conjecture=205.26=20to?= =?UTF-8?q?=20n=5FG=20=E2=89=A4=2022?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 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 --- papers/face_monochromatic_pairs/COMMENTARY.md | 2 +- .../experiments/test_conj_5_26_n_21_22.py | 199 ++ papers/face_monochromatic_pairs/paper.aux | 4 +- .../paper.fdb_latexmk | 6 +- papers/face_monochromatic_pairs/paper.log | 100 +- papers/face_monochromatic_pairs/paper.pdf | Bin 1329164 -> 1329304 bytes papers/face_monochromatic_pairs/paper.tex | 27 +- .../test_conj_5_26_n_21_22_results.jsonl | 846 +++++++++ .../test_n_21_to_24_results.jsonl | 1605 +++++++++++++++++ 9 files changed, 2738 insertions(+), 51 deletions(-) create mode 100644 papers/face_monochromatic_pairs/experiments/test_conj_5_26_n_21_22.py create mode 100644 papers/face_monochromatic_pairs/test_conj_5_26_n_21_22_results.jsonl create mode 100644 papers/face_monochromatic_pairs/test_n_21_to_24_results.jsonl diff --git a/papers/face_monochromatic_pairs/COMMENTARY.md b/papers/face_monochromatic_pairs/COMMENTARY.md index f97f04a..9092afe 100644 --- a/papers/face_monochromatic_pairs/COMMENTARY.md +++ b/papers/face_monochromatic_pairs/COMMENTARY.md @@ -11,7 +11,7 @@ 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 5.26 (clauses 1–4, strengthening) | conjecture | ✓ 142,812 / 142,812 (n ≤ 20, via direct clause-4 check in `check_conj_final_scaled.py`) | +| Conjecture 5.26 (clauses 1–4, strengthening) | conjecture | ✓ 2,321,496 / 2,321,496 (n ≤ 22, via direct clause-4 check in `check_conj_final_scaled.py` for n ≤ 20 + `test_conj_5_26_n_21_22.py` for n ∈ {21, 22}) | | 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) | diff --git a/papers/face_monochromatic_pairs/experiments/test_conj_5_26_n_21_22.py b/papers/face_monochromatic_pairs/experiments/test_conj_5_26_n_21_22.py new file mode 100644 index 0000000..53e62a2 --- /dev/null +++ b/papers/face_monochromatic_pairs/experiments/test_conj_5_26_n_21_22.py @@ -0,0 +1,199 @@ +"""Test Conjecture 5.26 (the strengthening, clauses 1–4) on +chord-apex+Kempe colourings of reduced duals for n in [21, 22]. + +For each (G, v, i, φ) chord-apex+Kempe colouring, enumerate every +witness (F, e_1, e_2) satisfying clauses 1–3, then check whether +at least one such witness also satisfies clause 4. The conjecture +states: SOME witness for each colouring passes clause 4. + +A "passing colouring" = ∃ witness with clauses 1–4 all satisfied. +A "failing colouring" = ALL its witnesses fail clause 4. + +The known result for n ≤ 20 is 142,812 / 142,812 (pass) via the +existing check_conj_final_scaled.py. This script extends to +n = 21 and n = 22, with incremental JSONL output and a +10-minute PROGRESS heartbeat. + +Run with: sage experiments/test_conj_5_26_n_21_22.py [n_min] [n_max] +""" +import json +import os +import sys +import time +from pathlib import Path + +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_final_scaled import ( + apply_reduction, + proper_3_edge_colorings, + matches_chord_apex_kempe, + find_all_36_witnesses, + check_clause_4, +) +from check_heawood_on_kempe import dual_of + + +RESULTS_FILE = Path(HERE) / '..' / 'test_conj_5_26_n_21_22_results.jsonl' + + +def test_one(D): + """For dual D, iterate reductions + chord-apex+Kempe colourings. + Returns counters and any failing examples.""" + D.is_planar(set_embedding=True) + n_col_with_witness = 0 + n_pass = 0 + n_fail = 0 + failures = [] # only failing colourings get recorded + 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)] + for col in cand: + witnesses = find_all_36_witnesses(H, edges, list(col), named) + if not witnesses: + continue + n_col_with_witness += 1 + ok = False + for w in witnesses: + try: + if check_clause_4(H, edges, list(col), named, w): + ok = True + break + except Exception: + # Defensive: treat exception as "this witness fails" + # and keep trying others. + pass + if ok: + n_pass += 1 + else: + n_fail += 1 + failures.append({ + 'i_red': i_red, + 'n_witnesses': len(witnesses), + }) + return n_col_with_witness, n_pass, n_fail, failures + + +def write_result(record): + with open(RESULTS_FILE, 'a') as f: + f.write(json.dumps(record) + '\n') + f.flush() + + +def main(n_min=21, n_max=22, progress_every_seconds=600): + started_at = time.time() + print(f"Testing Conjecture 5.26 (clauses 1–4) on n in [{n_min}, {n_max}]") + print(f"Writing incremental results to {RESULTS_FILE}") + print(f"PROGRESS lines every {progress_every_seconds}s") + sys.stdout.flush() + if RESULTS_FILE.exists(): + bak = RESULTS_FILE.with_suffix(RESULTS_FILE.suffix + '.bak') + RESULTS_FILE.rename(bak) + print(f"Previous results moved to {bak}") + RESULTS_FILE.touch() + + grand_col = 0 + grand_pass = 0 + grand_fail = 0 + last_progress = time.time() + + for n in range(n_min, n_max + 1): + n_start = time.time() + print(f"\n=== n_G = {n} ===") + sys.stdout.flush() + try: + triangulations_iter = graphs.triangulations(n, minimum_degree=5) + except Exception as ex: + print(f"n={n}: cannot enumerate ({ex})") + continue + n_col_n = 0 + n_pass_n = 0 + n_fail_n = 0 + tri_idx = 0 + for G in triangulations_iter: + G.is_planar(set_embedding=True) + D = dual_of(G) + tri_start = time.time() + nc, np_, nf, fails = test_one(D) + tri_elapsed = time.time() - tri_start + n_col_n += nc + n_pass_n += np_ + n_fail_n += nf + record = { + 'n_G': n, 'tri_idx': tri_idx, + 'n_col_with_witness': nc, + 'n_pass': np_, + 'n_fail': nf, + 'tri_elapsed_s': round(tri_elapsed, 2), + 'wall_elapsed_s': round(time.time() - started_at, 2), + 'failures': fails, + } + write_result(record) + now = time.time() + if now - last_progress >= progress_every_seconds: + wall_min = (now - started_at) / 60 + print(f"PROGRESS: wall {wall_min:.1f} min; " + f"n={n} tri {tri_idx+1} done ({tri_elapsed:.1f}s); " + f"running totals @n={n}: col={n_col_n}, " + f"pass={n_pass_n}, fail={n_fail_n}") + sys.stdout.flush() + last_progress = now + tri_idx += 1 + + n_elapsed = time.time() - n_start + summary = { + 'type': 'n_summary', + 'n_G': n, 'n_tri': tri_idx, + 'n_col_with_witness': n_col_n, + 'n_pass': n_pass_n, + 'n_fail': n_fail_n, + 'n_elapsed_s': round(n_elapsed, 2), + 'wall_elapsed_s': round(time.time() - started_at, 2), + } + write_result(summary) + print(f"\nn={n}: {tri_idx} triangulations, {n_col_n} colourings " + f"with witnesses [{n_elapsed:.0f}s]") + print(f" pass clause 4: {n_pass_n}") + print(f" fail clause 4: {n_fail_n}") + sys.stdout.flush() + + grand_col += n_col_n + grand_pass += n_pass_n + grand_fail += n_fail_n + + total_elapsed = time.time() - started_at + final = { + 'type': 'grand_summary', + 'n_min': n_min, 'n_max': n_max, + 'grand_col_with_witness': grand_col, + 'grand_pass': grand_pass, + 'grand_fail': grand_fail, + 'wall_elapsed_s': round(total_elapsed, 2), + } + write_result(final) + print("\n" + "=" * 70) + print(f"DONE: n in [{n_min}, {n_max}], {grand_col} colourings with " + f"clause-1-3 witness in {total_elapsed/60:.1f} min") + print(f" pass: {grand_pass}") + print(f" fail: {grand_fail}") + if grand_fail == 0 and grand_col > 0: + print(f"✓ Conjecture 5.26 EMPIRICALLY EXTENDED to n_G ≤ {n_max}") + + +if __name__ == '__main__': + n_min = int(sys.argv[1]) if len(sys.argv) > 1 else 21 + n_max = int(sys.argv[2]) if len(sys.argv) > 2 else 22 + main(n_min=n_min, n_max=n_max) diff --git a/papers/face_monochromatic_pairs/paper.aux b/papers/face_monochromatic_pairs/paper.aux index eea43c4..16b07bf 100644 --- a/papers/face_monochromatic_pairs/paper.aux +++ b/papers/face_monochromatic_pairs/paper.aux @@ -72,9 +72,8 @@ \newlabel{rem:deciding-face-empirical}{{5.23}{20}} \newlabel{rem:heawood-empirical}{{5.24}{20}} \newlabel{rem:conj-3-6-empirical}{{5.25}{21}} -\newlabel{conj:face-monochromatic-pair-strengthened}{{5.26}{21}} +\newlabel{conj:face-monochromatic-pair-strengthened}{{5.26}{22}} \newlabel{rem:conj-3-8-empirical}{{5.27}{22}} -\newlabel{rem:implication-4ct}{{5.28}{22}} \bibcite{Heawood1898}{1} \bibcite{AH77a}{2} \bibcite{AHK77}{3} @@ -85,5 +84,6 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} +\newlabel{rem:implication-4ct}{{5.28}{23}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{23}{}\protected@file@percent } \gdef \@abspage@last{23} diff --git a/papers/face_monochromatic_pairs/paper.fdb_latexmk b/papers/face_monochromatic_pairs/paper.fdb_latexmk index ff8e7e8..9c2e8ae 100644 --- a/papers/face_monochromatic_pairs/paper.fdb_latexmk +++ b/papers/face_monochromatic_pairs/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1779709018 "paper.tex" "paper.pdf" "paper" 1779709020 +["pdflatex"] 1779726421 "paper.tex" "paper.pdf" "paper" 1779726423 "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmbsy5.tfm" 1246382020 1120 1e8878807317373affa7f7bba4cf2f6a "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmbsy6.tfm" 1246382020 1124 14ccf5552bc7f77ca02a8a402bea8bfb "" @@ -102,8 +102,8 @@ "fig_reduced_dual_step4.png" 1779490218 107439 b30c7a68e32660f9bd3a8070bdb96944 "" "fig_thm_cubic_contraction_4face.png" 1779643393 105384 e3682925308ddea06a4f986af1787249 "" "figures/min-face-5-counterexample.png" 1779694886 192060 dd675ae50d32088d97ad647bc6394662 "" - "paper.aux" 1779709019 7946 e672ad7f0810ea6e729d0667882be5cf "pdflatex" - "paper.tex" 1779709010 79461 2e89fa9a4f2623b4fdf2f956828fe736 "" + "paper.aux" 1779726422 7946 38164e4e6701a73d430f1caa9857e53a "pdflatex" + "paper.tex" 1779726385 80516 3cdeee92fefe09fa5794bef1ddc2caf8 "" (generated) "paper.aux" "paper.log" diff --git a/papers/face_monochromatic_pairs/paper.log b/papers/face_monochromatic_pairs/paper.log index 63dfba6..d8a6d58 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 07:36 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 12:27 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -191,12 +191,9 @@ Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4 File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) -Overfull \vbox (1.31958pt too high) has occurred while \output is active [] - - [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] [2] [3] -Overfull \hbox (41.917pt too wide) in paragraph at lines 256--258 +Overfull \hbox (41.917pt too wide) in paragraph at lines 257--259 []\OT1/cmr/m/n/10 List the five degree-$2$ ver-tices in clock-wise or-der aroun d $\OML/cmm/m/it/10 F$ \OT1/cmr/m/n/10 as $\OML/cmm/m/it/10 A \OT1/cmr/m/n/10 = (\OML/cmm/m/it/10 A[]; A[]; A[]; A[]; A[]\OT1/cmr/m/n/10 )$. @@ -205,22 +202,22 @@ d $\OML/cmm/m/it/10 F$ \OT1/cmr/m/n/10 as $\OML/cmm/m/it/10 A \OT1/cmr/m/n/10 = File: fig_reduced_dual_step1.png Graphic file (type png) -Package pdftex.def Info: fig_reduced_dual_step1.png used on input line 276. +Package pdftex.def Info: fig_reduced_dual_step1.png used on input line 277. (pdftex.def) Requested size: 172.79846pt x 166.55775pt. File: fig_reduced_dual_step2.png Graphic file (type png) -Package pdftex.def Info: fig_reduced_dual_step2.png used on input line 277. +Package pdftex.def Info: fig_reduced_dual_step2.png used on input line 278. (pdftex.def) Requested size: 172.79846pt x 170.39505pt. File: fig_reduced_dual_step3.png Graphic file (type png) -Package pdftex.def Info: fig_reduced_dual_step3.png used on input line 278. +Package pdftex.def Info: fig_reduced_dual_step3.png used on input line 279. (pdftex.def) Requested size: 172.79846pt x 170.39505pt. File: fig_reduced_dual_step4.png Graphic file (type png) -Package pdftex.def Info: fig_reduced_dual_step4.png used on input line 279. +Package pdftex.def Info: fig_reduced_dual_step4.png used on input line 280. (pdftex.def) Requested size: 172.79846pt x 171.44409pt. LaTeX Warning: `h' float specifier changed to `ht'. @@ -230,17 +227,17 @@ uced_dual_step3.png> <./fig_reduced_dual_step4.png>] File: fig_chord_apex_step1.png Graphic file (type png) -Package pdftex.def Info: fig_chord_apex_step1.png used on input line 400. +Package pdftex.def Info: fig_chord_apex_step1.png used on input line 401. (pdftex.def) Requested size: 251.9989pt x 250.5104pt. File: fig_chord_apex_step2.png Graphic file (type png) -Package pdftex.def Info: fig_chord_apex_step2.png used on input line 401. +Package pdftex.def Info: fig_chord_apex_step2.png used on input line 402. (pdftex.def) Requested size: 172.79846pt x 176.08986pt. File: fig_chord_apex_step3.png Graphic file (type png) -Package pdftex.def Info: fig_chord_apex_step3.png used on input line 402. +Package pdftex.def Info: fig_chord_apex_step3.png used on input line 403. (pdftex.def) Requested size: 172.79846pt x 176.08986pt. @@ -251,7 +248,7 @@ pex_step3.png>] [8] File: fig_cubic_edge_contraction.png Graphic file (type png) -Package pdftex.def Info: fig_cubic_edge_contraction.png used on input line 545 +Package pdftex.def Info: fig_cubic_edge_contraction.png used on input line 546 . (pdftex.def) Requested size: 341.9989pt x 73.08138pt. [9 <./fig_cubic_edge_contraction.png>] @@ -259,77 +256,77 @@ Package pdftex.def Info: fig_cubic_edge_contraction.png used on input line 545 File: fig_thm_cubic_contraction_4face.png Graphic file (type png) Package pdftex.def Info: fig_thm_cubic_contraction_4face.png used on input lin -e 620. +e 621. (pdftex.def) Requested size: 352.79846pt x 160.78339pt. [10 <./fig_thm_cubic_contraction_4face.png>] File: fig_lemma_kempe_heawood.png Graphic file (type png) -Package pdftex.def Info: fig_lemma_kempe_heawood.png used on input line 727. +Package pdftex.def Info: fig_lemma_kempe_heawood.png used on input line 728. (pdftex.def) Requested size: 352.79846pt x 138.98488pt. LaTeX Warning: `h' float specifier changed to `ht'. [11] [12 <./fig_lemma_kempe_heawood.png>] -Overfull \hbox (67.56044pt too wide) in paragraph at lines 850--860 +Overfull \hbox (67.56044pt too wide) in paragraph at lines 851--861 \OT1/cmr/m/n/10 and the ex-haus-tive search that found it is in \OT1/cmtt/m/n/1 0 experiments/search[]min[]face5[]counterexample.py\OT1/cmr/m/n/10 . [] LaTeX Font Info: Trying to load font information for OMS+cmtt on input line -872. -LaTeX Font Info: No file OMScmtt.fd. on input line 872. +873. +LaTeX Font Info: No file OMScmtt.fd. on input line 873. LaTeX Font Warning: Font shape `OMS/cmtt/m/n' undefined (Font) using `OMS/cmsy/m/n' instead -(Font) for symbol `textbraceright' on input line 872. +(Font) for symbol `textbraceright' on input line 873. 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[13] [14 <./figures/min-face-5-counterexample.png>] [15] -Overfull \hbox (15.85718pt too wide) detected at line 1071 +Overfull \hbox (15.85718pt too wide) detected at line 1072 \OML/cmm/m/it/10 @F[] \OMS/cmsy/m/n/10 ^^R \OML/cmm/m/it/10 V\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 [ \OML/cmm/m/it/10 V\OT 1/cmr/m/n/10 (\OML/cmm/m/it/10 K[]\OT1/cmr/m/n/10 ) [] [] [] [16] [17] -Overfull \hbox (133.90562pt too wide) in paragraph at lines 1248--1256 +Overfull \hbox (133.90562pt too wide) in paragraph at lines 1249--1257 \OT1/cmr/m/n/10 degree se-quence around $\OML/cmm/m/it/10 v$ \OT1/cmr/m/n/10 co n-tains at least one degree-$5$ en-try (see \OT1/cmtt/m/n/10 experiments/check[ ]v[]neighbour[]degrees.py\OT1/cmr/m/n/10 ). [] -Overfull \hbox (86.1788pt too wide) in paragraph at lines 1268--1278 +Overfull \hbox (86.1788pt too wide) in paragraph at lines 1269--1279 \OT1/cmr/bx/n/10 the chord-apex con-struc-tion). \OT1/cmr/m/n/10 Con-cretely (\ OT1/cmtt/m/n/10 experiments/check[]bad[]subcase[]deciding[]face.py\OT1/cmr/m/n/ 10 ), [] -Overfull \hbox (2.93471pt too wide) in paragraph at lines 1293--1295 +Overfull \hbox (2.93471pt too wide) in paragraph at lines 1294--1296 \OT1/cmr/bx/n/10 Lemma 5.21 \OT1/cmr/m/n/10 (Par-tial proof of Con-jec-ture 5.2 0[] for small un-cov-ered set)\OT1/cmr/bx/n/10 . \OT1/cmr/m/it/10 If $\OMS/cmsy /m/n/10 j\OML/cmm/m/it/10 V\OT1/cmr/m/n/10 ([]) \OMS/cmsy/m/n/10 n [] -Overfull \hbox (4.00797pt too wide) in paragraph at lines 1311--1319 +Overfull \hbox (4.00797pt too wide) in paragraph at lines 1312--1320 \OT1/cmtt/m/n/10 experiments/check[]S[]adjacency.py\OT1/cmr/m/n/10 , \OT1/cmtt/ m/n/10 experiments/check[]S[]face[]structure.py\OT1/cmr/m/n/10 ) [] [18] -Overfull \hbox (12.19276pt too wide) in paragraph at lines 1323--1327 +Overfull \hbox (12.19276pt too wide) in paragraph at lines 1324--1328 \OML/cmm/m/it/10 K[]; K[]$ \OT1/cmr/m/n/10 are the ``other'' Kempe cy-cles in t he $\OMS/cmsy/m/n/10 f\OML/cmm/m/it/10 c; c[]\OMS/cmsy/m/n/10 g$\OT1/cmr/m/n/10 - and $\OMS/cmsy/m/n/10 f\OML/cmm/m/it/10 c; c[]\OMS/cmsy/m/n/10 g$\OT1/cmr/m/ @@ -337,37 +334,74 @@ n/10 -decompositions. [] -Overfull \hbox (19.83406pt too wide) in paragraph at lines 1365--1371 +Overfull \hbox (19.83406pt too wide) in paragraph at lines 1366--1372 \OT1/cmr/m/n/10 pentagon fall-back (Con-jec-ture 5.20[]) holds on $1[]314\OML/c mm/m/it/10 =\OT1/cmr/m/n/10 1[]314 = 100%$ of chord-apex+Kempe [] -Overfull \hbox (158.34045pt too wide) in paragraph at lines 1365--1371 +Overfull \hbox (158.34045pt too wide) in paragraph at lines 1366--1372 \OT1/cmr/m/n/10 colour-ings on which the par-tial struc-tural proof leaves a ga p (\OT1/cmtt/m/n/10 experiments/check[]gprime[]pentagon[]always[]works.py\OT1/c mr/m/n/10 ). [] [19] -Overfull \hbox (68.54941pt too wide) in paragraph at lines 1451--1453 +Overfull \hbox (68.54941pt too wide) in paragraph at lines 1452--1454 \OT1/cmr/m/n/10 (Every length ap-pear-ing is $\OMS/cmsy/m/n/10 6^^Q \OT1/cmr/m/ n/10 0([]3)$, as re-quired.) See \OT1/cmtt/m/n/10 experiments/check[]deciding[] face.py\OT1/cmr/m/n/10 . [] -Overfull \hbox (45.67143pt too wide) in paragraph at lines 1482--1496 +Underfull \hbox (badness 1052) in paragraph at lines 1469--1480 +\OT1/cmr/m/n/10 this claim com-pu-ta-tion-ally on all chord-apex+Kempe colour-i +ngs of re-duced + [] + + +Underfull \hbox (badness 3713) in paragraph at lines 1469--1480 +\OT1/cmr/m/n/10 du-als with $\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 V\OT1/cmr/m/n/1 +0 (\OML/cmm/m/it/10 G\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 j ^^T \OT1/cmr/m/n/10 21 +$; see \OT1/cmtt/m/n/10 experiments/check[]heawood[]on[]kempe.py\OT1/cmr/m/n/10 + , + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 1469--1480 +\OT1/cmtt/m/n/10 experiments/check[]constancy[]obstruction.py\OT1/cmr/m/n/10 , +and + [] + + +Overfull \hbox (45.67143pt too wide) in paragraph at lines 1483--1498 [] [] [20] [21] -Underfull \hbox (badness 1112) in paragraph at lines 1589--1595 +Underfull \hbox (badness 1112) in paragraph at lines 1603--1610 \OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 5.27\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10 The strength-ened con-jec-ture was tested on the same chord- [] -[22] [23] (./paper.aux) + +Underfull \hbox (badness 3492) in paragraph at lines 1603--1610 +\OT1/cmr/m/n/10 apex+Kempe colour-ings as Re-mark 5.25[]; for each colour-ing w +e sought + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 1603--1610 +\OT1/cmr/m/n/10 any Conjecture-5.1[]-witness $(\OML/cmm/m/it/10 F; e[]; e[]\OT1 +/cmr/m/n/10 )$ whose ac-com-pa-ny-ing $\OML/cmm/m/it/10 f[]$ \OT1/cmr/m/n/10 sa +t-is-fies + [] + + +Underfull \vbox (badness 4279) has occurred while \output is active [] + + [22] +[23] (./paper.aux) LaTeX Font Warning: Some font shapes were not available, defaults substituted. @@ -409,7 +443,7 @@ fonts/type1/public/amsfonts/cm/cmti8.pfb> -Output written on paper.pdf (23 pages, 1328195 bytes). +Output written on paper.pdf (23 pages, 1329304 bytes). 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