Implicit Method for Degenerated Differential-Algebraic Equations and Applications

The paper’s Theorem 11 asserts that on a smooth connected component C where the top-block Jacobian J has constant rank r<n and the r×r leading minor is nonsingular, (i) the zero sets satisfy ZR(F(c)) ∩ C = π ZR(G) ∩ C, and (ii) the degree-of-freedom measure obeys δ(G) ≤ δ(F) − (n−r). The proof in the paper is coherent: it uses Lemma 3 to obtain a global y-independence of G(y,z)=g(φ(y,z),y,z), yielding the projection equality, and then constructs explicit offsets (c̄,d̄) (Table 2, Eq. (16)) together with the non-square δ-definition (Definition 2) and Proposition 3 to derive the δ-inequality. These steps are internally consistent and correctly cited in the paper (Definition 10 and Theorem 11; Lemma 3; Definition 2; Proposition 3) . By contrast, the candidate solution’s Step 2 relies on g being constant only on each connected component of the f=0 fiber Σz, then concludes g(s,y,z)=0 across all of Σz without proving these components coincide; this is a logical gap unless additional connectivity assumptions are supplied. The paper’s Lemma 3 avoids this gap by proving true y-independence of G(y,z) on C, not merely per component . Moreover, the candidate’s δ argument partitions G into a non-square “top block” A that is not square as required by the paper’s Definition 2 and incorrectly claims some rows contain “no derivatives,” conflicting with the precise signature bounds used in Table 2 and Eq. (16) . Hence, the paper’s argument is correct and complete, whereas the model’s proof has critical gaps in both the set-equality and δ parts.