Understanding Calcium Dynamics via Process-Oriented Geometric Singular Perturbation Theory

The paper states and proves Theorem 5.9: for system (5.1) there is, for small ε, a locally unique relaxation cycle Γε that is O(ε^{1/3})-close to the singular orbit Γ = Γc ∪ Γh ∪ Γl and is exponentially attracting with Floquet exponent bounded above by −κ/ε^2. The statement and proof outline appear verbatim in the main text and Appendix A, including the multi-regime analysis (R1/R2), Krupa–Szmolyan fold blow-up giving the O(ε^{1/3}) closeness, and a Poincaré map with contractions of order e^{−a/ε^2} along the infra-slow manifold, which yields the −κ/ε^2 Floquet bound (Theorem 5.9 and Lemma 5.8; see also the explicit e^{−a/ε^2} estimates in Appendix A) . By contrast, the candidate solution reproduces the existence argument and the O(ε^{1/3}) proximity near the fold, but explicitly leaves the sharp Floquet bound −κ/ε^2 “open,” only obtaining a weaker e^{−κ1/ε} estimate; it also assumes (rather than proves) the existence/uniqueness of the fast ‘drop’ Γl and relies on heuristic monotone-wedge arguments near the degenerate axes instead of the paper’s cylindrical-plus-spherical blow-ups and foliations. Since the sharp Floquet rate and the rigorous construction across the degenerate line Sh are central parts of the theorem as stated and proved in the paper, the model solution is incomplete relative to the problem, while the paper’s analysis is correct and complete on these points .