**Three consecutive conjectures from Graffiti.pc's *Written on the Wall II* — open for seventeen and a half years — were all disproved at once, by just two trees.**
Conjectures 352, 358 and 359, posed by Ermelinda DeLaViña on 18 February 2009 and still carrying status O (open) ever since, are three lower bounds on γ_T(T) — the total domination number of a tree, the size of the smallest set whose members are each adjacent to a member (a dominating vertex does not dominate itself). Claude Opus 5 checked them against a complete census of every tree of order 3 through 19 — all 522,957 of them, generated with nauty-gentreeg — and found that all three fail.
The disproofs are absurdly economical. Conjecture 352 (γ_T ≥ components of ⟨N(D₂) ∪ D₂⟩ + ⌈½·ecc_avg(M)⌉) is refuted by a single order-18 tree T₁₈ whose total domination number is 7 while the bound demands 8 — a margin of exactly 1. Conjectures 358 and 359 (γ_T ≥ ½·ecc(C) + a count of isolated supports / support-plus-leaf components) are both refuted by the *same* order-19 tree T₁₉, a perfectly symmetric centre joined to two identical 9-vertex branches: γ_T is 9 while both bounds demand 9.5.
That half-unit margin is the smallest failure that counts as failure: γ_T is an integer and ecc(C) is odd, so the right-hand side is a half-integer and a deficit smaller than ½ cannot exist without vanishing. And the census proves both trees are exhaustively minimal — 352 has zero counterexamples below order 18 and T₁₈ is its only one at 18; 358 and 359 have zero below 19 and T₁₉ is their only one at 19.
That is precisely why they survived: a conjecture can only be emitted if it holds on every graph in Graffiti.pc's database, so any counterexample has to be larger than anything the database contained — and there is nothing at all to find below order 18. The three share a structural cause: each bound adds a component count (growing like the number of leaves) to a distance term (growing like the diameter), but total domination pays for both *jointly* — a long spine and scattered supports can share dominating vertices. The counterexamples are the trees that maximise that overlap, and lengthening the branches doesn't help, which is why order 19 is not merely the first place they appear but essentially the only one.
The census doubled as a control: the seven statements of this block recorded as theorems (347, 349, 350, 355, 357, 366, 371) produced zero violations across all 522,957 trees, validating the encodings. A self-contained verifier rebuilds both trees, recomputes every invariant from first principles, and prints all three verdicts in about a second. The standing count of disproved Graffiti.pc conjectures rises from 179 to 182 — and this is the first time a single section of the wall has yielded three kills from two counterexamples.