A CHARACTERIZATION OF QUASI-HOMOGENEITY IN TERMS OF LIFTABLE VECTOR FIELDS

The paper proves the advertised characterizations cleanly: for multiplicity 3 it establishes the full equivalence between quasi-homogeneity and (substantiality of the canonical OPSU) and (substantiality of every OPSU) via an explicit reduction to Id×G and computed liftables of G, and for the equidimensional corank‑1 minimal-stable case it shows that a substantial minimal unfolding forces weighted-homogeneity, while quasi-homogeneity guarantees weak substantiality of all minimal unfoldings via λ-equivalence. The candidate solution is correct for Problem B and gives some correct implications for Problem A, but it leaves a crucial gap for (1)⇒(3) in Problem A (transfer of the constructed lift to an arbitrary stable unfolding) and relies on unproven construction/diagonalization steps. Hence the model is incomplete relative to the paper’s complete treatment.