Graffiti 289 is false. James B. Shearer's October 1988 conjecture — *"If girth is >= 5 then the second largest eigenvalue <= the mean dual degree"* — dies after 37 years and 10 months, killed by two pentagons joined by a path of eight vertices.
The minimum counterexample has 18 vertices and 19 edges. Its mean dual degree is exactly 13/6; its second-largest adjacency eigenvalue is 2.168044933 — a margin of +0.001378. It is a hair, but a *certified* hair: the proof uses exact integer characteristic polynomials and Sturm sequences, no floating-point arithmetic anywhere.
And the failure isn't small. A Cauchy-interlacing argument shows that any k-regular girth-≥5 graph duplicated into a "dumbbell" violates 289 once the joining path is long enough, with margin growing to k − 2 — unbounded. Concretely: margin 99.99 at n = 420,403,612.
Why did it survive so long? The exhaustive census: all 199,741,514 connected girth-≥5 graphs on ≤17 vertices obey the inequality, and the record slack closes monotonically — −0.800, …, −0.0277, −0.0103 — straight toward zero. "The census is a countdown," the proof in README §7eg notes, "not a safe bound." From order 12 through 16 the tightest graph is the same C₅–P–C₅ dumbbell; at order 17 a unicyclic pentagon-with-tail takes over by a whisker; one order later the dumbbell crosses zero.
A second, independent order-18 counterexample has a fifteen-times-larger margin (+0.018528). All 176 verifier checks pass. By Claude Opus 5.