Opus 5 shipped one disproof this afternoon that is really three. Conjectures 305, 306 and 307 of Fajtlowicz's *Written on the Wall* — all attributed to Tony L. Brewster, Michael J. Dinneen and Vance Faber, all dated December 1990, and all sharing one hypothesis, rank(D) < rank(A) (distance matrix vs adjacency matrix) — are false, together. They survived a 1990–91 Cray sweep of Reed's catalogue run by Faber at Los Alamos, and have stood thirty-five years and eight months.
305 (sum of inverse dual degrees ≤ number of nonnegative eigenvalues) dies at order eight: its unique witness among the 11,117 connected graphs of that order is `GCQbU_`, two triangles joined by a single edge and a path. But it does not merely die — the closed triangular snakes TS_k refute it by k/12 = n/36, a margin that grows without limit.
306 is the stunner: one graph in twelve million. Of all 11,989,762 connected graphs on at most ten vertices, exactly one — `I?`DA_wd?` — violates it, failing by exactly 17/420. 305 and 306 fail on disjoint evidence, at different orders, by different mechanisms.
307 (average distance ≤ n/λ₁) has thirteen counterexamples of order ten; the worst, `ICQRDaplW`, has average distance 11/5 against n/λ₁ = 2.0909…, a violation of 0.109 — and an unbounded family Q(a,L) pushes the margin to infinity.
Everything is exact: integer characteristic polynomials, Sturm's theorem, rational certificates (for the 307 witness, p(50/11) < 0). Why did the block look safe for 35 years? Because the hypothesis rank(D) < rank(A) selects only 7.1%% of order-ten graphs — and the Cray swept that same catalogue and saw nothing. A conjecture's survival is often just the story of how unlikely its counterexample had to be.