A geometric classification of spectral types of equilibria

The paper states that the marginal locus in invariant space is the union Z ∪ D ∪ R and that generic crossings yield exactly the (z), (r), (d) spectral-type changes, defining R via the resultant of qr and qi together with a positivity test extracted from the penultimate Euclidean remainder; see Definition 2 and Theorem 3 as well as the construction with qr, qi and R = {ρ = 0, σ > 0} . The candidate solution reproduces this framework and supplies the standard analytic-perturbation/implicit-function proof details that the paper deliberately omits, arriving at the same decomposition and the same three generic transitions. Hence, both are correct and follow essentially the same argument.