Path extremality and exact growth of distance-three matchings in trees

This preprint proves path extremality for distance-three matchings in finite trees. For a tree T on n vertices, the number m_3(T) of distance-three matchings satisfies m_3(T) <= F_n, where F_0=1, F_1=1, F_2=2, F_3=3 and F_n=F_{n-1}+F_{n-4}; equality is unique to the path for n>=6, while the orders n<=5 have the stated small-tree ties. The proof combines an exact rooted-state graft recurrence with a finite rational upper-envelope certificate and a 49-entry promotion certificate repairing the missing universal induction step in the earlier source disposition. The package includes exact Python and symbolic checks, an independent Prüfer-sequence checker, exhaustive unlabelled-tree regression through order 16, negative controls, manuscript source, and reproducibility records. This is an AI-assisted preprint and has not undergone peer review; the finite computations are regression evidence and the bounded priority search makes no first-proof or worldwide-novelty claim. GitHub repository: https://github.com/gkamal666/distance-three-tree-matchings. GitHub release: https://github.com/gkamal666/distance-three-tree-matchings/releases/tag/v1.0.0.