NUMBER OF MODULI FOR AN UNION OF SMOOTH CURVES IN (C2, 0).

The paper introduces the linear system (H) on the resolution tree, proves that any Saito vector field X determines an admissible solution via ES = (εi(X,E)) and ΔS encoding invariance (Theorem 10), using the global identity ν(X)+1 = Σ ρEi εi(X,E) and the invertibility relation with the proximity matrix (the inverse of the multiplicity matrix), and establishes existence/uniqueness of the admissible decoration ΔS for unions of smooth curves (Proposition 1). It then computes the generic moduli dimension as a sum of local contributions σ(C) with the four case-by-case formulas (E), (O), (Ed), (Od) (Section 4) . The candidate solution mirrors this structure: (1) it derives E^S = P·S^S from the weighted proximity identity and index–tangency bookkeeping; (2) it proves uniqueness of ΔS by a backward, parity-based induction on the resolution tree (a different argument from the paper’s Lemma-2-based induction, but consistent with Proposition 1); and (3) it computes σ(C) by a Čech jet count, reproducing the same four formulas and the same algorithmic summation over intermediate tangency points. Definitions of Saito bases, relative strict transforms, and Saito vector fields match the paper’s setup . No essential hypotheses are dropped: both the paper and the model restrict uniqueness and the algorithm to unions of smooth components and discuss the generic moduli dimension. Accordingly, both the paper and the model solution are correct, though their uniqueness arguments differ.