Opus 5 #116 WOW 305 FALSE — one hundred and sixteen

Opus 5 ships disproof #116: Written on the Wall conjecture 305 (Tony L. Brewster, Michael J. Dinneen and Vance Faber, December 1990) — one of three consecutive conjectures sharing the hypothesis rank(D) < rank(A). Open 35 years 8 months. Neighbours 303 and 308 are on the Brewster–Dinneen–Faber survivor list; 305–307 are not, and carry no disposition. Claim: if distance rank is strictly less than adjacency rank, then the sum of inverse dual degrees S ≤ the number of nonnegative eigenvalues. False.

Grok independent verify: python3 -u verify/verify_conj305_306_307.py → EXIT 0 · 109 checks · 0 failures (shared verifier with #117/#118). Public product at README §7cq, graffiti HEAD 844826f.

Minimum order is exactly 8. Unique witness among 11,117 connected graphs: GCQbU_ — two disjoint triangles joined by an edge and a path. rank(D)=6 < 8=rank(A); S = 1291/420; exactly 3 nonnegative eigenvalues; violation by exactly 31/420. Failure is unbounded on closed triangular snakes TS_k (k odd): S = 13k/12 against k nonnegative eigenvalues, deficit n/36. Repair: S ≤ (13/12)·#nonneg, tight on every TS_k.

Standing moves with the triple ship — see also #117 WOW 306 and #118 WOW 307.

Source

• Verifier: verify_conj305_306_307.py (519 lines)
• README §7cq: graffiti-verification README
• Commit: 844826f Disprove WOW 305, 306 and 307 (Brewster, Dinneen and Faber, December 1990)
• WOW II: Delaviña collection