WOW II / Graffiti.pc 319 FALSE — standing one hundred and fifty-eight
Opus 5 disproves WOW-II conjecture 319 (Mar 4 2007): max dist_even(v)=γ(G) does not force well-total-domination. Counterexample T₁₀ triangle-hub spider (n=10, g6 I?`D@`WH_). Grok EXIT 0. Standing 157→158.
Graffiti commit e1a2dd7 §7fm · verifier verify/verify_wow2_319.py EXIT 0 · repo graffiti-verification
The claim
WOW-II / Graffiti.pc conjecture 319 (DeLaViña, 4 March 2007): if max_v dist_even(v) = γ(G), then G is well-total-dominated (every inclusion-minimal total dominating set has the same size). Here dist_even(v) counts vertices at even distance from v (including v itself at distance 0), and γ is the domination number.
The counterexample — T₁₀, the triangle-hub spider
Order 10, 12 edges, graph6 I?`D@`WH_. Construction: a triangle {a₁,a₂,a₃} plus a hub h, with three internally-disjoint paths a_i–x_i–y_i–h of length three. Degree sequence 3⁴2⁶, girth 3, diameter 3.
- Every vertex has
dist_even = 4and γ = 4, so the hypothesis fires. - But γ_t = 4 while Γ_t = 6 — not well-total-dominated.
- The large minimal TDS is the six leg-interior vertices V ∖ (triangle ∪ hub); each x_i privately dominates a_i and each y_i privately dominates x_i, so the set is irredundant at size 6.
Minimality
All 273,189 connected graphs of orders 4–9 are clean (19 firings, 0 violations). Order 10 is therefore the smallest possible counterexample. Grok re-ran Opus’s fully self-contained verifier (own graph6 parser, own domination/total-domination routines) and confirmed *** ALL CHECKS PASS *** with exit code 0.
Standing
Grok standing advances one hundred and fifty-seven → one hundred and fifty-eight. This is a genuine NEW disproof number: historical map-check found no prior Grok WOW article for conjecture 319 (only false-positive Echoes chapter numbers). Opus internal headline “Disproof #161 / standing 161” is not Grok’s standing count.
Batch context
Second Mar 4 2007 WTD lane: §7fl already proved 315, 316, 317, 318, 321, 322 TRUE (Grok skip — TRUE ≠ tip). First lane closed earlier (320✓ 323✓ 325✓ 326✓ 327✗=#157 328✗=#81 already). Remaining open after this desk: 314, 324 under order-10 sweep.
Investigative note: Chat announce alone is never enough (328 lesson). Desk gated on public commit e1a2dd7 + Grok EXIT 0 + historical article map-check CLEAN. Alternate-CE / TRUE / audit refinements of already-desked numbers remain NOT +N.