From 43986f15db565f456ae0c2e9d3f5d395e1c0987e Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 01:23:37 -0400 Subject: [PATCH] face_monochromatic_pairs: add Theorem 5.5 (constant Heawood on edge-sharing Kempe cycles is impossible) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit If true, Theorem 5.5 + Lemma 5.3 → Conjecture 5.1: in the no-clause-3-witness world, both V(K_b) and V(K_c) have constant Heawood (Lemma 5.3), but K_b and K_c share the merged edge, contradicting Theorem 5.5. Proof sketch sets up the two Lemma-5.2 applications (c-edges at u,w on opposite sides of K_0; b-edges at u,w on opposite sides of K_1) and the theta-curve K_0 ∪ K_1, but the planar-orientation contradiction is left as "to be filled in". Co-Authored-By: Claude Opus 4.7 --- papers/face_monochromatic_pairs/paper.pdf | Bin 1086630 -> 1090606 bytes papers/face_monochromatic_pairs/paper.tex | 44 ++++++++++++++++++++++ 2 files changed, 44 insertions(+) diff --git a/papers/face_monochromatic_pairs/paper.pdf b/papers/face_monochromatic_pairs/paper.pdf index 902e42e6ce6c0d81622afadb720832d9c703b89a..63b80c7ea4f484c11c2c12cb5ec1dba80a3ef576 100644 GIT binary patch delta 73388 zcmX_{RZyP6l7?}22<~pd-5r9vyNBQo0X`gpySuvtcYfU6CAb84m(4l5wNv#}_03Gx z+;sQ6{Z2`_Sj~u7T_X~Nd1ga15(_w#)2n4d&XMj z2mC(}T27D=@%tg!S=mxSbFg%PzMjX11cu*q?G?d|&cdAm2Csk!7n!Yp9jHS94D}}J z41I)xVshZx1bEYQYpS_u<8e3UwK5V=tt`H%In`QR{MbDkrD2wEH`PfgyW2n8l~@VZ zLH_!EXbveOg+^G4SusR?_NY`=VmfDdtMlXGz9H4&>f)X|H`kJl@AU*&{cg(>+uh$W z-Q#^Lxas$CzI81oVa1%QHjenX)bmIAwqe=m^+$|>VUO3x?FfVGFdUZxjj;00;@YMYUT=I1d~o zeP=LXEMZsZnt~MnKGep8=N7mVd}$^9&baq|N$DFcQJU(Qs#&`bK;mj(`nyKb(}238 zp+=i7L|#o+8po}Hw#=g8glQ;hI8?IBp4)5M#m8Y@n0`6ZS%(Fm=Il?cKE<3T81A^9y5V%)piC>}!GxIqkqt z^7H7ZW8Sl!Un@@M%3<0ryh 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