From 626a837a0670cdf52f899259297384e1d51b68d5 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 02:12:33 -0400 Subject: [PATCH] face_monochromatic_pairs: Theorem 5.5 Step 4 closes Case A (lune face contradiction) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Case A: K_1 visits the four shared vertices {p, p', q, q'} in the same cyclic order as K_0. The K_0-arc A_1 from p' to q and the K_1-arc B_1 from p' to q share both endpoints, so they bound a "lune" face Φ* of K_0 ∪ K_1 with two (b, c)-corners. - Planarity: B_1's interior is non-shared, so B_1 lies on one side of K_0; hence c-edges at p' and q (endpoints of B_1) point to the SAME side of K_0. - Lemma 5.2 along A_1 (m odd): c-edges at p' and q alternate, so they point to OPPOSITE sides of K_0. → Contradiction. This is clean and uses only Lemma 5.2 + planarity (does not require the mod 3 face-sum from Step 3). Case B: K_1 visits the shared vertices in opposite order {p, p', q', q}. Then K_0 ∪ K_1's faces are 3-corner triangles instead of lunes; both Lemma 5.2 alternations and the mod 3 face-sum hold consistently. Listed as the explicit open case. Co-Authored-By: Claude Opus 4.7 --- papers/face_monochromatic_pairs/paper.pdf | Bin 1109009 -> 1116449 bytes papers/face_monochromatic_pairs/paper.tex | 61 +++++++++++++++++----- 2 files changed, 49 insertions(+), 12 deletions(-) diff --git a/papers/face_monochromatic_pairs/paper.pdf b/papers/face_monochromatic_pairs/paper.pdf index beaf4fc47df3f08820d932c815cfd03bb9313cd5..1ec1cf71618467cf10b9cdbff22f9f1ecb9a29cb 100644 GIT binary patch delta 89067 zcmV)MK)An=*h8U`M1X_=gaU*Egam{Iga(8Mgb0KQgbK6^2IT`ZHaL^v8z_I>Tv=}; z#}R(_uMqNL3h|8lKD;oH-DC~tKwvnVAb=epBXT5-Ly>o{dt!g3w)Dbv4Sr;r z5xvOLS=F`qV{DM^pevOT7X>esPm^K9PzqZ4i%}yOjuT5WrFQ+Cf85)>-*4M-|v?;A~$7tdLP0Sumc6Ofe zNR!lX+0=1lm( zw~|;_Hr!932ZTV0r6|Qg0+sCA@+hnI&kxlyXbgq+HonBAt%FWkETWplw?Vr++~MD- zm+DRY9cY@+V7^(mZL_**zlhH_%}015CS5qB2GGEkO8din=6=ZyIg43k&Ob$GysG?& zQ_Si~>&k}viRgbQTX>Uge+!Ne`%!*)#>dF@MK zYoKoCrwwXIz|O)< zqql$7Y~#+yjTuwLg0Chys8t16DRl8&qKp`ab)sD?;EFO4+I=MbIwAG0`5bqoOF3wr 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