Characteristic matrix functions for delay differential equations with symmetry

The paper’s Theorem 2.3 states precisely that Δ(z) = I − z h^{-1}F(Θ(h)p, z) is a characteristic matrix for Uh = h^{-1}U(Θ(h)p, 0), in the sense of Definition 2.2 where analytic E(z), F(z) are invertible for all z and satisfy the block factorization (Δ(z) ⊕ I) = F(z) (I ⊕ (I − zUh)) E(z). The proof proceeds by writing Uh as V + R with V Volterra and R finite rank, applying Theorem 3.5 to obtain Δ, and then identifying C(I − zV)^{-1}D = h^{-1}F(τ, z) (thus Δ(z) = I − z h^{-1}F(τ, z)) . By contrast, the model’s construction incorrectly asserts that I − zUh is invertible for all z (treating Uh as Volterra), which is false since the nonzero spectrum of Uh is precisely what Δ captures. It then defines F(z) with a lower-right block (I − zUh), contradicting the requirement that F(z) be invertible for all z in Definition 2.2 and leading to an inconsistency in the factorization step. Hence the paper is correct, while the model’s argument fails at the core operator-theoretic step.