From b2aa14cefae5acbcd2324aea571ace1b70e664fe Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 07:23:31 -0400 Subject: [PATCH] face_monochromatic_pairs: write up strengthened Lemma for G'-pentagon fallback MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Update Remark (gprime-pigeonhole-stop) with the empirical characterization of S-vertex structure: 1. |S| is always EVEN. 2. S forms a 2-regular induced subgraph (= single cycle, or rare |S|=4 disjoint pairs). 3. S-cycle is NEVER a face boundary of the reduced dual. 4. p_G ≥ 7 in every bad colouring (from bad-triple constraint). Tabulate the max # pent hit and min p_G per |S| bucket: |S|=2: hit≤2 < p_G≥7 ⇒ 5+ pentagons uncovered ✓ |S|=4: hit≤4 < p_G≥8 ⇒ 4+ uncovered ✓ |S|=6: hit≤7 < p_G≥8 ⇒ 1+ uncovered ✓ |S|=8: hit≤8, but empirically when hit=8 we have p_G≥9 (= the combination hit=8 AND p_G=8 never occurs empirically) ⇒ 1+ uncovered. The "hit=8 AND p_G=8 don't co-occur" fact is a structural property of chord-apex+Kempe colourings we don't yet have a non-empirical proof of. |S|=10: hit≤7 < p_G≥8 ⇒ 1+ uncovered ✓ Combined empirical verification: 1,314 / 1,314 (100%) of bad chord-apex+Kempe colourings have at least one G'-pentagon with boundary in V(K_b) ∪ V(K_c) ⇒ G'-pentagon fallback empirically true on full 142,812 dataset. Paper structure note: the proof now resembles discharging (Appel-Haken/RSST/Gonthier style) but with ~8 structural buckets instead of 633, because chord-apex+Kempe does most of the work upfront. The structural ~95% coverage + empirical 100% closure is the current state. Paper grows from 22 to 23 pages. Co-Authored-By: Claude Opus 4.7 --- papers/face_monochromatic_pairs/paper.pdf | Bin 1312620 -> 1327812 bytes papers/face_monochromatic_pairs/paper.tex | 105 +++++++++++++++++----- 2 files changed, 81 insertions(+), 24 deletions(-) diff --git a/papers/face_monochromatic_pairs/paper.pdf b/papers/face_monochromatic_pairs/paper.pdf index 4eb065be1679b5ec6f03299e3b30d93ca5bfcb7f..65041e922f2514b4b03fc43d413ccb1cf643941a 100644 GIT binary patch delta 181175 zcmV(;K-<6Uj!?vfQGkR2gaU*Egam{Iga(8Mgb0KQgbIWUgbaiYv<^aL12j1`mw|E+ zDSz!;Ta(*1mVWoI(2Hx>6^p_B5?88r;;}QCnX1Xgy}J+2JkS!mo3TWCD9Ltj&5z%6 z0FVSEsHJXqoJrNbv_J|C902F?oeT8c<&&>pQ9Dzm)!fW3uV$2~QfW3*#+JsytL5xX zLCwXST0v~_@~0nOT+F$V#dm+a;I#O!mw!LKdil3MUHs$nFJHf6WM)gNm~sQr(n>X( zGhSNihk1RwUf0{cSyZdlEsP>nQEkJ&^5|c+DJ7#b;{WT9p$Cho)TIC#=d=`7`d*Fi zm8^x8hMHlRuSAYWQBuxl(KtU7Nj}~*iwkCo8&6}i3AC;@U3k;xs1~Iqa+vPFGk>$0 zl0bt@6JrO{&O^ViJMO(_&jmi{&~l_&{ed+Va1pNjgal5sndM7hY2WMM3QYF+X% z)|DRVBEK$a4%X!km`2i3|=T!Y;S7+&8Aw{I4E$X1cKn?=YdV#SXF<$ 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