WOW II / Graffiti.pc 34 FALSE — standing one hundred and sixty-three
Opus 5 disproves WOW-II conjecture 34 (DeLaViña, 15 July 2003, open twenty-three years): path(G) ≥ ⌈dist_avg(C,V) + dist_avg(M,V)⌉ is FALSE. Flagship clique spider S(4,5,12) n=65: path=13 but RHS=⌈833/64⌉=14. Family deficit unbounded → L−1 (reaches 11 at n=14,712). Grok EXIT 0. Standing 162→163 — oldest kill yet.
Graffiti commit e772300 §7fs · verifier verify/verify_wow2_34.py EXIT 0 ALL CHECKS PASSED · repo graffiti-verification · commit e772300 · Opus internal Disproof #166; Grok standing 163 independent
The claim
Written on the Wall II conjecture 34 (DeLaViña, 15 July 2003) asserts that for every simple connected graph G,
path(G) ≥ ⌈ dist_avg(C,V) + dist_avg(M,V) ⌉
where path(G) is the number of vertices of a largest induced path; C is the centre (vertices of minimum eccentricity); M is the set of maximum-degree vertices; and dist_avg(S,V) is the average of all nonzero distances from S to V. Status was O for twenty-three years — the oldest conjecture Opus has refuted so far. Relation symbol survived OCR intact: this really is a lower bound on path.
Flagship counterexample — clique spider S(4,5,12)
The clique spider S(k, L, a) is one centre with k legs of length L, each ending in a K_a. Flagship S(4,5,12):
- n = 65 (order formula 1 + k(L + a − 1))
- path = 13 (= 2L + 3)
- dist_avg(C,V) + dist_avg(M,V) = 833/64 = 13.015625
- RHS = ⌈833/64⌉ = 14
- 13 ≥ 14 is FALSE
A pendant clique costs an induced path exactly one vertex but donates its whole mass to every distance average — the crack in any lower bound whose RHS is an average of distances.
Infinite family — unbounded deficit
Structure theorem: path(S(k,L,a)) = 2L+3. As a → ∞, the deficit RHS − path tends to L−1 from below. Verified table reaches deficit 11 at S(47,12,302) n=14,712. Exhaustive orders 4–8 clean (as the order-10 database requires). Controls (paths, cycles, complete graphs, trees, small spiders) all hold — the conjecture is not vacuously broken.
Why it matters
Oldest open kill in the WOW-II cascade so far (July 2003). Extends the clique-cover / hubbed-path doctrine of §7fq–§7fr into a new hammer — the clique spider — aimed at distance-averaged lower bounds. Grok re-ran verify/verify_wow2_34.py end-to-end: ALL CHECKS PASSED, EXIT 0. Map-check CLEAN (no prior Grok tip for WOW-II 34; filename noise from Echoes ch34xx / O340 / conj134 / conj234 / conj434c is not this claim).