Certified SAT Verification of the Vertex Formulation of Erdős Problem #580 for Orders at Most 19

We give a certified finite verification of the literal vertex formulation currently displayed as Erd\H{o}s Problem \#580. Namely, for every $1\le n\le 19$, an $n$-vertex graph having at least $\lceil n/2\rceil$ vertices of degree at least $\lceil n/2\rceil$ contains every tree on at most $\lfloor n/2\rfloor$ vertices. The only order requiring new computer-assisted analysis is $n=18$. An edge-minimal counterexample is reduced to a host partition $V(G)=L\sqcup S$ with $|L|=|S|=9$, degree exactly $9$ on $L$, and $S$ independent. Exact embedding theorems cover $42$ of the $47$ non-isomorphic trees on nine vertices. The five remaining trees reduce, by deleting their leaves, to four rooted cores. For each core we construct a Boolean formula whose models are precisely the reduced hosts avoiding that rooted core. All four formulas are unsatisfiable. The deposited data include complete CNF instances and DRUP refutations. The archived traces were validated by reverse unit propagation, all four formulas are independently solved as unsatisfiable by a second SAT solver, a standalone semantic audit reconstructs the complete formulas---including all $3735$ base clauses and every avoidance clause---without importing the production generator, the complete $47$-tree witness table is generated from the classifier, and regeneration from the published encoder reproduces all four CNFs byte for byte. This is a finite partial result concerning trees on at most $n/2$ vertices; it does not settle the stronger classical formulation asking for trees with at most $n/2$ edges.