Per user's request: try option 3 (push case analysis) as a sanity
check on the G'-pentagon fallback conjecture. Result: trivial
pigeonhole closes the |S| ≤ 1 case but no further.
Added:
- Lemma (Partial proof, |S| ≤ 1): a single uncovered vertex hits
at most 3 of the ≥ 6 G'-pentagons of the reduced dual, so 3
pentagons remain fully covered. Trivial pigeonhole.
- Remark "The pigeonhole stops at |S| = 1; the proof begins to
resemble discharging": Theorem deciding-face-partial-extended
+ Lemma gprime-pigeonhole cover ~91% of chord-apex+Kempe
configurations (73.87% with |S|=0 + 17% with |S|=1). The
remaining ~9% have |S| ≥ 2 and need finer graph-structural
input, exactly the discharging flavour Appel-Haken / RSST /
Gonthier used for 4CT. We stop the case-by-case route here.
So the sanity check confirms the user's intuition: continuing the
case analysis would replay the irreducible-configurations approach
in different vocabulary. Pivoting to option 2 (global argument)
next.
Paper stays at 22 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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