From d28896be1295937bf0b8d60c77106d1829abf700 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 02:21:13 -0400 Subject: [PATCH] face_monochromatic_pairs: demote Theorem 5.5 to a (disproved) Conjecture MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit The user produced a concrete counterexample (whiteboard photo) showing that h_φ can be constant on both an {a,b}-Kempe cycle K_0 and an {a,c}-Kempe cycle K_1 sharing a colour-a edge. Changes: - theorem → conjecture environment, header marked **FALSE** - New Remark records the disproof and identifies which step of the proof attempt breaks: in the counterexample, no pair of shared a-edges is consecutive on both cycles, so the lune-face premise (Step 4 / Case A) doesn't apply - Proof attempt re-tagged as "Partial proof attempt (now superseded)"; Steps 1-2 remain unconditional, Step 4 closes the sub-case where some shared-a-edge pair is consecutive on both K_0 and K_1 (e.g. automatically when |E(K_0) ∩ E(K_1)| = 2) - Figure placeholder added referencing figures/no-two-constant-kempe-counterexample.{png,pdf} - COMMENTARY.md updated with a "Failed proof route" section so future readers don't retread this path Impact on Conjecture 5.1: the "Theorem 5.5 + Lemma 5.3 → 5.1" route is closed; a structural proof of Conjecture 5.1 needs a different angle. Lemma 5.3, Corollary 5.4, and the 142,812/142,812 empirical near-proof all stand. Co-Authored-By: Claude Opus 4.7 --- papers/face_monochromatic_pairs/COMMENTARY.md | 30 +++++++ papers/face_monochromatic_pairs/paper.pdf | Bin 1116449 -> 1119834 bytes papers/face_monochromatic_pairs/paper.tex | 73 +++++++++++++++--- 3 files changed, 91 insertions(+), 12 deletions(-) diff --git a/papers/face_monochromatic_pairs/COMMENTARY.md b/papers/face_monochromatic_pairs/COMMENTARY.md index de48fcf..4a4395d 100644 --- a/papers/face_monochromatic_pairs/COMMENTARY.md +++ b/papers/face_monochromatic_pairs/COMMENTARY.md @@ -107,6 +107,36 @@ or revealed why they don't yield a contradiction on their own: `experiments/check_min_flip_structure.py`, `experiments/check_minority_location.py`. +## Failed proof route via edge-sharing Kempe constancy + +A natural-looking strategy to prove Conjecture 5.1 was: + +> **Conjecture (now disproved):** If $K_0$ is an $\{a,b\}$-Kempe cycle +> of $\varphi$ and $K_1$ is an $\{a,c\}$-Kempe cycle of $\varphi$ that +> shares an edge with $K_0$, then $h_\varphi$ cannot be constant on +> both $V(K_0)$ and $V(K_1)$ simultaneously. + +Combined with Lemma 5.3 (no clause-3 witness $\Rightarrow$ constancy +on both $V(K_b)$ and $V(K_c)$), this would have closed Conjecture 5.1. +It is **false** by a concrete counterexample (Figure in `paper.tex` at +`\ref{fig:no-two-constant-kempe-counterexample}`). A partial proof +attempt is preserved in the paper alongside the disproof: + +- Step 1 (local CW structure) — unconditional. +- Step 2 (forced odd-crossing $\Rightarrow |E(K_0) \cap E(K_1)|$ even + and $\geq 2$) — unconditional. +- Step 3 (Heawood face-sum mod 3) — unconditional but does not yield + a contradiction on its own. +- Step 4 (lune-face Case A) — closes the sub-case where two shared + a-edges are consecutive on \emph{both} cycles (automatic when + $|E(K_0) \cap E(K_1)| = 2$). +- Step 5 (general case) — open / now known false via the + counterexample. + +The counterexample shows the general case of the conjecture is +unsalvageable; the search for a structural proof of Conjecture 5.1 +will need a different angle. + ## Snapshot date This commentary is current as of commit `e688037`. diff --git a/papers/face_monochromatic_pairs/paper.pdf b/papers/face_monochromatic_pairs/paper.pdf index 1ec1cf71618467cf10b9cdbff22f9f1ecb9a29cb..0c6d559edef035d46092c59aeb032c0c4131a58b 100644 GIT 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