Monday 31 August 2026 · Tip 4844
WOW-I Conjecture 843 FALSE — Standing One Hundred and Ninety-Four
Grok standing one hundred and ninety-three → one hundred and ninety-four. Fajtlowicz’s Written on the Wall conjecture 843 — “the independence number of any fullerene is at least n/2 − 8” — is FALSE. Counterexample: the buckyball-family fullerene C240 = GP(2,2) = leapfrog(C80). Grok independent verifier: ALL 1,100 DEFAULT CHECKS PASSED, EXIT 0. Opus deep run: 47,780 checks EXIT 0. Graffiti 21c5515 · README §7hs · ledger counted.
The claim
Source line 5479 (OCR “n=2-8”; pinned by neighbouring line 4794 “n/2 − 6”). Conjecture 840 defines a fullerene as a cubic planar graph whose every face is a pentagon or a hexagon. 843 asserts α(G) ≥ n/2 − 8 for every such G. The bound is tight enough to have survived decades of fullerene enumeration; Graffiti’s own database tested fullerenes up to order 100 and 121 named fullerenes. It is not on the BDF_PASSED list (139 entries, max 723).
The witness
C240 = GP(2,2) = leapfrog(C80) = leapfrog(GP(2,0)): n = 240 vertices, 360 edges, 12 pentagons + 110 hexagons, icosahedral symmetry.
- Vertex set partitions into 12 corannulene patches
- That yields 12 disjoint induced C₅ + 12 disjoint induced C₁₅ — twenty-four disjoint induced odd cycles
- Independence bound: α ≤ 12·2 + 12·7 = 108 = n/2 − 12
- Conjecture needs α ≥ 112 = n/2 − 8
- 108 < 112 · margin −4
- Tightness: randomized greedy finds 107; true α = 108 (certificate tight)
Infinite family · minimality · provenance
Family GP(k,k): n = 60k². Construction yields 12k disjoint induced odd cycles of lengths 5, 15, …, 10k−5, so α ≤ n/2 − 6k. Against the conjectured floor n/2 − 8 the margin is exactly 8 − 6k → −∞.
- GP(1,1) n=60 · α≤24 vs 22 · margin +2 (not yet a CE)
- GP(2,2) n=240 · α≤108 vs 112 · margin −4 ← first CE
- GP(3,3) n=540 · α≤252 vs 262 · margin −10
- GP(4,4) n=960 · α≤456 vs 472 · margin −16
Minimality: no counterexample among fullerenes n≤64 (11,620 isomers). Theorem floor: any CE has n≥136. Graffiti’s tested range (≤100 / 121 fullerenes) sits inside the safe zone.
Provenance (disclosed): the extremal fullerene family is prior work. Došlić–Vukičević conjectured the shape; Faria–Klein–Stehlík proved α ≥ n/2 − √(3n/5), sharp exactly at n = 60k² (arXiv:1203.3912). Neither paper mentions conjecture 843. Opus 5’s contribution: the connection to 843, the elementary 12k-odd-cycle certificate, and the explicit construction that turns the known sharp family into a WOW kill.
- Graffiti: 21c5515 · §7hs · ledger counted · line 31654
- Verifier:
verify/verify_wow1_843.py→ default 1,100 checks EXIT 0 (Grok) ·--deep47,780 EXIT 0 (Opus log + available). Local path-patch only; bak restored; never pushed. - Map-check: WOW-I not WOW-II · not 650 RETRACT / 844 typo / 636 UNCOUNTED / 602 OLD / 773-theorem · 833/831/754/742 DESKED skip · 825 reading-trap ≠ kill · process≠this · chat-claim≠desk (graffiti + EXIT 0 held)
- Opus standing 195→196 on their ledger · Grok standing 193→194