From 36be3e14cb53abd3d2995cc78b16980dd99c3224 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 05:49:12 -0400 Subject: [PATCH] =?UTF-8?q?face=5Fmonochromatic=5Fpairs:=20sanity-check=20?= =?UTF-8?q?on=20G'-pentagon=20fallback=20=E2=80=94=20pigeonhole=20stops=20?= =?UTF-8?q?at=20|S|=3D1?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Per user's request: try option 3 (push case analysis) as a sanity check on the G'-pentagon fallback conjecture. Result: trivial pigeonhole closes the |S| ≤ 1 case but no further. Added: - Lemma (Partial proof, |S| ≤ 1): a single uncovered vertex hits at most 3 of the ≥ 6 G'-pentagons of the reduced dual, so 3 pentagons remain fully covered. Trivial pigeonhole. - Remark "The pigeonhole stops at |S| = 1; the proof begins to resemble discharging": Theorem deciding-face-partial-extended + Lemma gprime-pigeonhole cover ~91% of chord-apex+Kempe configurations (73.87% with |S|=0 + 17% with |S|=1). The remaining ~9% have |S| ≥ 2 and need finer graph-structural input, exactly the discharging flavour Appel-Haken / RSST / Gonthier used for 4CT. We stop the case-by-case route here. So the sanity check confirms the user's intuition: continuing the case analysis would replay the irreducible-configurations approach in different vocabulary. Pivoting to option 2 (global argument) next. Paper stays at 22 pages. Co-Authored-By: Claude Opus 4.7 --- papers/face_monochromatic_pairs/paper.pdf | Bin 1309881 -> 1312620 bytes papers/face_monochromatic_pairs/paper.tex | 47 ++++++++++++++++++++++ 2 files changed, 47 insertions(+) diff --git a/papers/face_monochromatic_pairs/paper.pdf b/papers/face_monochromatic_pairs/paper.pdf index 59cb7638c87501a73a930dabea0dcc097f48a564..4eb065be1679b5ec6f03299e3b30d93ca5bfcb7f 100644 GIT binary patch delta 76752 zcmV)EK)}Db`A_VQP_VKLe>%f$F1X-3MNEfPRWTIYaM!YbQg3K)E6TUw3Xh&$Kl<+@ zbcJ-KAh*ntEZ0u1TuNZ^V&mF`Ks*gJxB@2AS%3-O#!0H#)UIk95zyO(3OW#?yJ$D(Tpm9XwfmESK zUtIWxVt zGclL3QVuGATXP$^m45fH;EOA*vf0zP1LK;i&3Ns}B&phD#@T&v^3Wn%RFBD~!i#NH z<;U+iIOuMG0!2#JI6GDQBBGlhuIKWd4+r}G-Lr4rGB?qsGr~^ZT}>F*r8az`tt+jA zSM$mHg4v5HbFy;9o1cDseK8eQ72p5 zE1lAlDHo+<)XdMft5vhxw6l7-yoFYjF6v$Qsfd1RY^gY#RP?LghZ;N@r70zhamq^R z6xC{}Rx%e}T4wuZUdY@aWyyq?MD6^&Qq`yHc6PySaZO`vw_&Wy?K-^aa-)`|t5o0J ze`k7svts|FhdsZ=l^*!gJ? zQ{Ps-S?pI$hg-jc9|&12+K>3uY%?>`wsekv_fz^yM*eguoTe#BswJz;9>5TIMUx8E 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