Update Remark (gprime-pigeonhole-stop) with the empirical
characterization of S-vertex structure:
1. |S| is always EVEN.
2. S forms a 2-regular induced subgraph (= single cycle, or rare
|S|=4 disjoint pairs).
3. S-cycle is NEVER a face boundary of the reduced dual.
4. p_G ≥ 7 in every bad colouring (from bad-triple constraint).
Tabulate the max # pent hit and min p_G per |S| bucket:
|S|=2: hit≤2 < p_G≥7 ⇒ 5+ pentagons uncovered ✓
|S|=4: hit≤4 < p_G≥8 ⇒ 4+ uncovered ✓
|S|=6: hit≤7 < p_G≥8 ⇒ 1+ uncovered ✓
|S|=8: hit≤8, but empirically when hit=8 we have p_G≥9 (=
the combination hit=8 AND p_G=8 never occurs empirically)
⇒ 1+ uncovered. The "hit=8 AND p_G=8 don't co-occur" fact
is a structural property of chord-apex+Kempe colourings
we don't yet have a non-empirical proof of.
|S|=10: hit≤7 < p_G≥8 ⇒ 1+ uncovered ✓
Combined empirical verification: 1,314 / 1,314 (100%) of bad
chord-apex+Kempe colourings have at least one G'-pentagon with
boundary in V(K_b) ∪ V(K_c) ⇒ G'-pentagon fallback empirically
true on full 142,812 dataset.
Paper structure note: the proof now resembles discharging
(Appel-Haken/RSST/Gonthier style) but with ~8 structural buckets
instead of 633, because chord-apex+Kempe does most of the work
upfront. The structural ~95% coverage + empirical 100% closure is
the current state.
Paper grows from 22 to 23 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
b2aa14cefa