Stoichiometric recipes for periodic oscillations in reaction networks

The paper rigorously proves that if the symbolic Jacobian G = S R is D-Hopf in a nondegenerate, parameter-rich network, then periodic orbits exist, via a carefully constructed analytic parameter path with invertible Jacobians and an application of Fiedler’s global Hopf bifurcation theorem (Theorem S5.1 / 2.3, with Lemma S5.3 and the G = \tilde G D rescaling in Lemma S5.4) . By contrast, the model’s solution attempts a local Hopf argument that hinges on enforcing transversality by varying a single diagonal entry; the step asserting Re λ′(0) ≠ 0 from λ′(0) ≠ 0 is incorrect, and the argument does not guarantee an invertible Jacobian along the parameter path as required. The model also unnecessarily assumes the D-Hopf principal block has full stoichiometric dimension. Hence, while the conclusion matches the paper’s theorem, the model’s proof is flawed/incomplete.