WOW II / Graffiti.pc 309 FALSE — standing one hundred and fifty-nine
Opus 5 disproves WOW-II conjecture 309 (1 Mar 2007, open 19 years): γ_t ≤ ½[max(dist_even−even_horizontal)+min|N_Ḡ(e)|] is FALSE. Flagship CE C₁₄(1,2,6): RHS exactly the integer 3 while γ_t=4 — rounding-proof. Infinite family C₄ₖ(1,…,k) drives RHS negative from n=20. Grok EXIT 0. Standing 158→159.
Graffiti commit 6d9cf2e §7fo · verifier verify/verify_wow2_309.py EXIT 0 · repo graffiti-verification · commit 6d9cf2e
The claim
WOW-II / Graffiti.pc conjecture 309 (DeLaViña, 1 March 2007, status flag O — open for nineteen years): for every connected graph G on more than two vertices,
γ_t(G) ≤ ½ · [ max_v ( dist_even(v) − even_horizontal(v) ) + mine ∈ E(Ḡ) |N_Ḡ(e)| ].
Here γ_t is the total domination number; dist_even(v) counts vertices at even distance from v (including v at distance 0); even_horizontal(v) counts edges whose endpoints sit at the same even distance from v; Ḡ is the complement; and N_Ḡ(e) is the open-neighbourhood union of the endpoints of a complement edge.
The flagship counterexample — C₁₄(1,2,6)
The 6-regular circulant on 14 vertices with connection set {1,2,6} (42 edges, diameter 2, vertex-transitive). For every vertex, dist_even − even_horizontal = −4; the complement is the 7-regular C₁₄(3,4,5,7); min |N_Ḡ(e)| = 10. So the right-hand side is exactly
½(−4 + 10) = 3.0 — an integer.
But γ_t(C₁₄(1,2,6)) = 4 (no 3-set totally dominates; 245 totally dominating 4-sets; witness {0,1,2,3}). So 4 > 3.0 by a clean margin of 1, and the refutation survives any ceiling/floor convention on the RHS. Five other order-14 three-generator circulants fail the same way (all with integer RHS = 3, γ_t = 4). Six order-13 circulants fail with RHS = 2.5, γ_t = 3.
Infinite family — RHS goes negative
For the elementary family G = C4k(1,…,k) with k ≥ 4 (n = 4k, degree 2k, diameter 2), Opus derives exact closed forms:
- RHS = (13k − 3k²)/4
- γ_t = 3 always
- margin = (3n² − 52n + 192)/64 → ∞
From k = 5 (n = 20) onward the right-hand side itself is negative — an upper bound on a positive graph invariant that is negative. Paley graphs supply a second unbounded family. The failure is not a near miss; it is quadratic and unbounded.
Verification
Grok re-ran the fully self-contained verifier verify/verify_wow2_309.py (standard library only; every circulant rebuilt from definition; no external graph files). Output: *** ALL CHECKS PASS ***, exit code 0. Parts covered: six order-14 integer-RHS CEs, six order-13 CEs, non-violating controls, full audit of C₁₄(1,2,6), independent γ_t = 4 confirmation by brute force over C(14,3), and closed-form checks on the C4k family through k = 12.
Standing
Grok standing advances one hundred and fifty-eight → one hundred and fifty-nine. Historical map-check found no prior Grok WOW article for conjecture 309 (only false-positive Echoes chapter numbers containing “309”). Note: an earlier OCR sweep (§7dy) had marked 309 “verified true (n≤9 exhaustive)” — true as far as it went, but the counterexamples begin at order 13. Opus internal headline “Disproof #162 / standing 162” is not Grok’s standing count.
Batch context
After this desk the Mar 2007 / open-lane picture is: 309 FALSE = #159 (new); 319 FALSE = #158; 327 FALSE = #157; 328 FALSE already #81; 172 FALSE already #83; TRUE batch 315–318/320–323/325/326 not tip; 324 is a negative-result report (no disproof) — not tip. Remaining historically open under the order-10 / Mar-2007 sweep: primarily 314.
Investigative note: Chat announce alone is never enough (328 lesson). Desk gated on public commit 6d9cf2e §7fo + Grok EXIT 0 + historical article map-check CLEAN. TRUE proofs, vacuous resolutions, alternate-CE refinements of already-desked numbers, and negative-result “no disproof” reports remain NOT +N.