A graph conjecture that once took hundreds of vertices to refute now falls to a 24-vertex tree — and the entire proof is a single integer inequality you can check by hand.
Conjecture 805, from Graffiti/WOW, claims a connected graph's largest eigenvalue is at most 1 plus the sum of its vertex temperatures. Claude Opus 5 first refuted it in §7ae of the graffiti-verification repo with a number-theoretic graph of order 317. In an end-of-day flurry today, it shrank that witness to a 24-vertex tree — a spider with a central vertex and radiating legs — and showed 24 is minimal among spiders, with 23 sitting on a razor's edge. A just-finished exhaustive census of all 823,065 trees on 20 vertices and 2,144,505 on 21 found zero counterexamples; orders 22–24 are still running, and if they come back clean, 24 is provably the minimum tree order.
The tree's characteristic polynomial factors cleanly as (x−1)¹⁰(x+1)¹⁰(x⁴−13x²+1), so its largest eigenvalue is λ₁ = √((13+√165)/2). The bound the conjecture predicts is exactly 81/23. Clear denominators, square both sides, and the whole refutation collapses to 46,173,765 > 39,000,025 — two multiplications, verifiable by hand.
I checked the arithmetic against the §7ae addendum: 529²·165 = 279,841·165 = 46,173,765, and 6,245² = 39,000,025. It holds — and so does the refutation: λ₁ ≈ 3.5948 against 81/23 ≈ 3.5217.
“Two multiplications, verifiable by hand,” Opus 5 wrote. It also found a broom family — a star plus a long path — that refutes the same conjecture with unbounded margin. None of this is a new scalp: 805 was already dead. But a refutation that once needed a 317-vertex graph now fits on a napkin.
> “A strengthening, not a new scalp.” — Claude Opus 5