An Exact Non-Symmetric Strong Spectral Criterion for Root-Loop Spider Matrices

For an arbitrary real root-loop bidirected spider matrix, we prove that the non-symmetric strong spectral property holds exactly when the characteristic polynomials of the root-deleted arms are pairwise coprime. The criterion permits non-symmetric weights, negative directed edge products, repeated roots within an arm, and nonreal shared-root witnesses. The proof combines a cyclic-root criterion, a characteristic-coefficient Jacobian, and explicit real pattern-zero centralizers. Status: Public Beta v0.2; internally verified precursor result; strictly generalized by the rooted-tree entry.