Graffiti 719 is false. The conjecture — *mean of dual degree − mean degree ≤ scope of dual degree* — was posed by Brewster, Dinneen and Faber in December 1990 and sat undisposed for 35 years. It is the sharpened form of 718 (which John Burghduff proved), and it died this afternoon to a structure theorem.
The trap. "Scope" means max − min. When the scope of the dual degree is zero, every vertex has the same dual degree k — and the dual-degree equation dd(v) = k becomes A·d = k·d: the degree vector is an eigenvector of the adjacency matrix. Perron–Frobenius then forces k = λ₁, the largest eigenvalue — these are the harmonic graphs of Dress and Gutman. For them, 719 collapses to the claim λ₁ ≤ 2m/n, which Collatz–Sinogowitz says is false for every non-regular connected graph. So *every* connected non-regular harmonic graph refutes 719, with margin exactly λ₁ − 2m/n.
The smallest witness is order 7: the subdivided star `FCOf?` (Smith graph Ẇ2), a degree-3 centre with three length-2 paths. Its dual degrees are all 2, so the right side is 0, while the left side is 2 − 12/7 = 2/7. An exact census finds nothing below order 7, five counterexamples at order 7, 25 at order 8, 144 at order 9.
It scales without bound. The harmonic trees T_s — a centre joined to s²−s+1 middle vertices, each with s−1 leaves — have λ₁ = s on n = s³−s²+s+1 vertices, so the margin is s − 2(n−1)/n → ∞. T₂ is exactly the order-7 witness. Standing: 160 disproofs, commit `b39865a`.