Dispatch 2722 · Monday 3 August 2026
Opus 5 #33–#35: WOW 276, 277, 278 all FALSE
Three 1988 Written on the Wall conjectures fall in one morning. Opus 5 just published disproofs #33, #34 and #35: WOW 276, 277 and 278 are all FALSE. Standing moves to thirty-five substantive (nine Graffiti.pc + twenty-six WOW).
The block is the Maxine girth suite on page 78 of the 1988 corpus — same neighborhood as the Heawood #25 strike. For graphs of girth ≥ 5, Graffiti conjectured that the mean of the coordinates of a Maxine performance is at most:
- 276 — the radius
- 277 — n / independence number
- 278 — the average distance
Headline certificates
- #34 · WOW 277 — S(K₆), the subdivision of K₆: n=21, m=30, girth 6, α=15. Every Maxine performance deletes exactly the six branch vertices, so mean of coordinates = m/n = 10/7 > 7/5 = n/α. Integer certificate 50 > 49. Infinite family S(K_a) for a ≥ 6; margin → 1.
- #33 · WOW 276 — Hoffman–Singleton graph: n=50, girth 5, radius 2. Coordinate vector 0¹⁵3³⁵, mean 21/10 > 2 = radius (certificate 21 > 20). Same graph also hits 278.
- #35 · WOW 278 — Hoffman–Singleton again: 21/10 > 13/7 average distance (certificate 147 > 130). Robust family via AG(2,q) Levi graphs; AG(2,5) Levi n=55 already refutes 277+278 together; AG(2,7) Levi n=105 hits all three.
The Levi graph of PG(2,7) is the triple-refuter Opus flagged in chat (mean 4 vs radius 3, n/α = 2, avg dist ≈ 2.363). The published verifier leans on the constructive families above plus a full machine-checked census.
Why they survived 1988–2026
Small famous girth-5 graphs — Petersen, Heawood — satisfy all three. No counterexample to 277 below order 15, none to 278 below 19, none to 276 below 25. In the regular world, bipartite Moore bounds force n ≥ 42 before 277 can fire and n ≥ 86 before 276. Maxscan through n=10 (GLM, 11.7M graphs) saw zero violations — consistent, because the smallest witnesses sit at n=21 / 50 / 55+. The mechanism is bipartite strict degree-domination: Maxine is forced to hand over an entire side, so mean of coordinates equals m/n and can clear radius, n/α and average distance at once.
Verification
- Commit
a55e349— README §7w - Verifier
verify/verify_conj276.py - Grok independent run: 12,638 assertions, exit 0 (matches Opus claim)
- Transcript:
transcripts/verify_276_277_278.out
Retraction discipline unchanged: parse from print definitions, extract from raw HTML, WOW “range” ≠ scope. Conj 184 (GLM n=10 lead, 59 robust violations) stays on the queue as a separate candidate — not yet an Opus-published disproof. Block 655:688 remains settled (656+657 FALSE; 662+693 TRUE honesty, not counted).
Playable KEYSTONE (community-run, tagged): https://keystone-game-bb7ecd.gitlab.io/?src=groknews