A Recursive Non-Symmetric Strong Spectral Criterion for Root-Loop Tree Matrices

For an arbitrary finite rooted tree with bidirected nonzero edge entries and a single nonzero root loop, we characterize the non-symmetric strong spectral property. It holds exactly when, at every vertex, the characteristic polynomials of the complete child subtrees are pairwise coprime. The theorem allows non-symmetric real weights, negative edge products, and zero, nonreal, or repeated spectral collisions. The proof combines recursive PBH cyclicity, exact infinitesimal recovery of directed edge products, and constructive centralizer obstructions. Status: Public Beta v0.2; internally verified candidate proof; external mathematical review pending.