From b20c8122da66ff78ab5c044027bc0c4680f1d3c8 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 07:49:42 -0400 Subject: [PATCH] face_monochromatic_pairs: write final status summary in COMMENTARY.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit After several rounds of structural attempts and audits, the picture has stabilized. Updated COMMENTARY.md with a "Final status" section: - What we have proved: the Heawood-face-sum reduction (tight), the tight structural cases (a', b', c) covering 94.97% of (G, v, i) configurations, and the refined pigeonhole + S-cycle arguments closing most of the residual ~5%. - What's open structurally: full G'-pentagon fallback, and the Kempe-cycle structural regularity "|S|=8 + hit=8 ⇒ p_G=11" (30/30 empirical). - Empirical closure: 100% across 142,812 chord-apex+Kempe colourings up to |V(G)| ≤ 20. - Why this isn't Appel-Haken in disguise: ~4 main steps + ~8 case-style sub-lemmas, vs RSST's 633. The chord-apex+Kempe restriction does most of the work upfront. Co-Authored-By: Claude Opus 4.7 --- papers/face_monochromatic_pairs/COMMENTARY.md | 78 ++++++++++++++++++- 1 file changed, 76 insertions(+), 2 deletions(-) diff --git a/papers/face_monochromatic_pairs/COMMENTARY.md b/papers/face_monochromatic_pairs/COMMENTARY.md index 2e06609..51f2c32 100644 --- a/papers/face_monochromatic_pairs/COMMENTARY.md +++ b/papers/face_monochromatic_pairs/COMMENTARY.md @@ -205,6 +205,80 @@ This suggests the contradiction in the chord-apex+Kempe setting is spike = merged) then forgotten. Possibly minimality has a stronger consequence we haven't extracted. -## Snapshot date +## Final status (commit `2d8c679`) -This commentary is current as of commit `e688037`. +### What we have proved + +1. **Theorem (Conjecture 5.1 ⇐ Deciding face)**: clean reduction via Heawood's + classical face-sum identity. Tight. + +2. **Tight structural cases for the Deciding face conjecture**: + - (a') $n_i = 5$ → flank face $F^\flat_{i, i+1}$ of length 4. + - (b') $n_{i+1} = 5$ → flank face $F^\flat_{i+1, i+2}$. + - (c) $n_{i+2} = n_{i+4} = 5$ → outer face $F^\flat_{\text{outer}}$ of length 7. + + Empirical coverage: 7,531 / 7,930 (G, v, i) configurations = **94.97%**. + +3. **Refined pigeonhole + S-cycle structure**: for bad colourings + (= where Lemma flank-covering-hex fails), the uncovered vertex set + $S$ is even, forms a 2-regular subgraph (= a single cycle), and is + bounded so that # G'-pentagons hit by S < $p_G$. This closes most + bad cases structurally. + +4. **Empirical closure**: the G'-pentagon fallback conjecture is true on + 1,314 / 1,314 bad colourings, hence on all **142,812 / 142,812** + chord-apex+Kempe colourings up to $|V(G)| \leq 20$. + +### What's open structurally + +1. **G'-pentagon fallback in full generality** — proven via pigeonhole for + $|S| \leq 1$, sketched for higher $|S|$ via empirical bounds (max hit, + min $p_G$), but no fully structural proof of the empirical bounds. + +2. **The "|S| = 8 + hit = 8 ⇒ $p_G = 11$" regularity** — striking + empirical fact (30/30 cases) but no Kempe-cycle structural argument + for it. The marginal $|S|$ ↔ # pent $F_k$ coupling doesn't hold; + the regularity is a joint structural property of specific + $(|S|, \text{hit})$ pairs. + +### What the path is + +The structural proof has stabilized at the following form: + +``` +Conjecture 5.1 (face-monochromatic-pair) + ⇐ Theorem (Conj 5.1 ⇐ Deciding face) [tight] +Conjecture (Deciding face) + ⇐ Theorem (Extended partial proof) [tight on 7,531/7,930] + ⇐ Conjecture (G'-pentagon fallback) [empirical 1,314/1,314] + ⇐ Lemma (G'-pentagon pigeonhole, |S| ≤ 1) [tight on |S| ≤ 1 ≈ 91%] + + open: structural |S|-cycle arguments for |S| ≥ 2. +``` + +So Conjecture 5.1 is reduced to **two open structural conjectures**: + + (i) The G'-pentagon fallback conjecture (empirically 100%). + (ii) Kempe-cycle-structural lemmas about chord-apex+Kempe colourings + sufficient to close the |S| ≥ 2 cases of the fallback. + +Closing (i) and (ii) structurally would give a full structural proof. +The empirical 100% across 142,812 colourings is strong evidence both +are true. + +### Why this isn't Appel-Haken in disguise + +The structural proof has 4 main steps + ~8 case-style sub-lemmas (vs +RSST's 633 cases). The compression comes from the chord-apex+Kempe +restriction doing most of the heavy lifting upfront — specifically, +Lemma 5.3 (no-witness ⇒ constancy on V(K_b) ∪ V(K_c)) and Lemma 5.X +(Kempe-spike, forced edge containments) pre-restrict the problem to +a very constrained colour configuration. The Heawood face-sum +identity then converts the local constancy into a mod-3 obstruction, +and the case analysis closes specific structural sub-buckets. + +The remaining open conjectures are at the chord-apex+Kempe class +level, not at the level of arbitrary minimal counterexamples to 4CT. + +### Snapshot date + +This commentary is current as of commit `2d8c679` (2026-05-25).