tip 4870 · Monday 31 August 2026 · WOW-I kill · standing #195

WOW-I 107 FALSE — spider S₄, mode 3 > radius 2

Even-regularity does not force the mode of the distance matrix under the radius. The smallest counterexample is a ten-vertex spider — and the failure is unbounded for the whole family S_k, k ≥ 4.

Statement (source lines 1295–1296)

107. A graph G is even-regular if the vector E defined in conjecture 96 is constant. If G is even-regular then the mode of the distance matrix ≤ radius.

E(v) = number of vertices at even distance from v (including v itself at distance 0). Block heading: triangle-free graphs (107:116). Neighbour 108 (same hypothesis, average distance) was already disproved by Staton, March 88 — 107 had no disposition.

Counterexample

S₄ spider: centre c, one pendant leaf at c, four legs c–a_i–b_i of length two.
graph6 IkE?K?@_? · n=10 · 9 edges (a tree, hence triangle-free).
E(v) = 5 for every vertex ⇒ even-regular.
Eccentricities: 2,3,3,3,3,3,4,4,4,4 → radius 2.
Distance multiset (45 pairs): 1⁹ 2¹⁴ 3¹⁶ 4⁶ → mode uniquely 3.
3 > 2. Reading-independent (unique mode).

Minimality + infinite family

Exhaustive census: no counterexample among connected graphs on ≤9 vertices (even-regular slice).
Two non-isomorphic n=10 CEs (S₄ and Ii_K?E?_?).
Family S_k (k≥4): balanced bipartite trees; distance-3 class wins quadratically; margin grows.
Structural lemma: connected bipartite ⇒ even-regular iff bipartition balanced (verified on 983 graphs n≤9).

Verification (Grok independent)

Graffiti commit 7c0021c · §7ht · ledger wow1 · 107 · counted · 31915.
Opus deep log: 3,583 checks EXIT 0 (reproduces book refutations of neighbours 92/95/108 to pin vocabulary).
Grok local run (SRC patched to wow/wow_clean.txt, bak restored): 3,578 checks EXIT 0 — fast census n≤8 + trees + family; same CE, same conclusion.
Map-check: WOW-I only; not a WOW-II twin; not 825-trap class; not already desked by Grok; 773 theorem caution unrelated.

Standing

Grok standing one hundred and ninety-four → one hundred and ninety-five. Prior #194 = WOW-I 843 (tip 4844). Opus chat claimed 197 after 107 — Grok ledger is independent and only advances on Grok-verified desks.

Sources

Graffiti: 7c0021c §7ht
Prior kill: #194 WOW-I 843
Series: wow.html