didericis/math-research
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

dual_decomposition: chord-apex proof + diagrams

Browse Source 
- Replace the chord-apex TODO with a full proof by contradiction: assume
  merged != spike, define X, Y, Z, W, lift to G' so that the externals
  inherit \psi(f) = (X, Y, Z, W, W), and split on W in {X, Z}. Either case
  meets the hypothesis of lem:pentagonal-externals, which extends \psi to a
  proper 3-edge-colouring of G' --- contradicting non-3-edge-colourability
  via Tait.
- Add fig:chord-apex-proof: the assumed reduced-dual colouring on top, and
  the two lifted-G' cases (W=Z, W=X) below, rendered on the dodecahedron.
- Add experiments/draw_chord_apex_proof.py.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-23 02:31:54 -04:00
parent 0303225f39
commit 409dea565a
8 changed files with 293 additions and 39 deletions
Download Patch File Download Diff File Expand all files Collapse all files
papers/dual_decomposition_minimal_counterexamples/paper.log
+45 -24
View File
@@ -1,4 +1,4 @@
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 22 MAY 2026 20:19
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 22 MAY 2026 20:45
entering extended mode
restricted \write18 enabled.
%&-line parsing enabled.
@@ -222,33 +222,54 @@ Package pdftex.def Info: fig_reduced_dual_step4.png used on input line 166.
LaTeX Warning: `h' float specifier changed to `ht'.
[2] [3 <./fig_reduced_dual_step1.png> <./fig_reduced_dual_step2.png> <./fig_red
uced_dual_step3.png> <./fig_reduced_dual_step4.png>] [4] (./paper.aux) )
uced_dual_step3.png> <./fig_reduced_dual_step4.png>]
<fig_chord_apex_step1.png, id=34, 505.03976pt x 502.06393pt>
File: fig_chord_apex_step1.png Graphic file (type png)
<use fig_chord_apex_step1.png>
Package pdftex.def Info: fig_chord_apex_step1.png used on input line 286.
(pdftex.def) Requested size: 251.9989pt x 250.5104pt.
<fig_chord_apex_step2.png, id=35, 490.16064pt x 499.51323pt>
File: fig_chord_apex_step2.png Graphic file (type png)
<use fig_chord_apex_step2.png>
Package pdftex.def Info: fig_chord_apex_step2.png used on input line 287.
(pdftex.def) Requested size: 172.79846pt x 176.08986pt.
<fig_chord_apex_step3.png, id=36, 490.16064pt x 499.51323pt>
File: fig_chord_apex_step3.png Graphic file (type png)
<use fig_chord_apex_step3.png>
Package pdftex.def Info: fig_chord_apex_step3.png used on input line 288.
(pdftex.def) Requested size: 172.79846pt x 176.08986pt.
LaTeX Warning: `h' float specifier changed to `ht'.
[4] [5 <./fig_chord_apex_step1.png> <./fig_chord_apex_step2.png> <./fig_chord_a
pex_step3.png>] [6] (./paper.aux) )
Here is how much of TeX's memory you used:
3020 strings out of 478268
42520 string characters out of 5846347
343128 words of memory out of 5000000
21064 multiletter control sequences out of 15000+600000
3042 strings out of 478268
43139 string characters out of 5846347
343136 words of memory out of 5000000
21083 multiletter control sequences out of 15000+600000
476364 words of font info for 55 fonts, out of 8000000 for 9000
1302 hyphenation exceptions out of 8191
69i,8n,76p,664b,298s stack positions out of 10000i,1000n,20000p,200000b,200000s
</usr/l
ocal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/lo
cal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/lo
cal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/loc
al/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/loca
l/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/
texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/local/te
xlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/local/texli
ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/texlive/
2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2
022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/local/texlive/202
2/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texlive/2022
/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive/2022/t
exmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
Output written on paper.pdf (4 pages, 501796 bytes).
</usr/local/texlive/2022/texmf-dist/fonts/ty
pe1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/typ
e1/public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/typ
e1/public/amsfonts/cm/cmex10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type
1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1
/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/p
ublic/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pub
lic/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public
/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
sfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/ams
fonts/cm/cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo
nts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfon
ts/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts
/symbols/msam10.pfb>
Output written on paper.pdf (6 pages, 750077 bytes).
PDF statistics:
91 PDF objects out of 1000 (max. 8388607)
50 compressed objects within 1 object stream
103 PDF objects out of 1000 (max. 8388607)
54 compressed objects within 1 object stream
0 named destinations out of 1000 (max. 500000)
21 words of extra memory for PDF output out of 10000 (max. 10000000)
36 words of extra memory for PDF output out of 10000 (max. 10000000)