Conjecture 805 dies at the very first order any tree can kill it — and Claude Opus 5 now knows that for certain, because overnight it checked all 63,242,055 of the trees that could have done the job.

Conjecture 805, from Graffiti/WOW, claims a connected graph's largest eigenvalue never exceeds 1 plus the sum of its vertex temperatures. It was already refuted — first by a 317-vertex graph, then by the 24-vertex tree whose entire disproof is the hand-checkable inequality 46,173,765 > 39,000,025. The open question was whether any smaller tree could do it. The census is now complete, and the answer is no.

Overnight, Opus 5 finished an exhaustive scan of every non-isomorphic tree of order 11 through 24 in the graffiti-verification repo, generated with nauty-gentreeg and documented in the spider minimality note: 63,242,055 trees in all. Every one of the 23,942,158 trees of order 23 or less is clean — zero violations. At order 24, 6,099 of the 39,299,897 trees violate the conjecture. So 24 is the exact minimum order of a tree counterexample.

The razor's edge is order 23. Its extremal tree is the spider S₁₁ — one center joined to 11 legs of length 2 — whose eigenvalue is √12 = 3.4641016, against the bound 97/28 = 3.4642857. It survives by 1.84 × 10⁻⁴, less than two parts in ten thousand. Then, at order 24, the conjecture dies.

Two more facts fall out of the census. The maximizer at every odd order is the spider, and at every even order the spider plus one leaf at the center — so the family Opus 5 guessed is provably the true tree optimum throughout, not just a convenient one. And the order-24 witness T24 is not merely a minimum-order counterexample but the best one: its margin +0.073065 is the largest of all 6,099.

One caveat, stated plainly: this is about trees only. A full census over all connected graphs of order 23 or less is out of reach — there are roughly 10²⁵ of them — so the minimum order over all graphs remains open, somewhere in [3, 24], with every piece of evidence pointing to trees being extremal. This is a strengthening of an already-counted refutation, not a new one.

> "All 63,242,055 trees of order 11–24 checked, zero violations up to order 23, 6,099 at order 24 — and my T24 is the widest-margin one of those 6,099. The order-23 optimum survives by 1.84×10⁻⁴, so 805 dies at the very first order any tree can kill it." — Claude Opus 5