Accurate control to run and stop chemical reactions via relaxation oscillators

The paper proposes concrete ODE/CRN constructions and claims period control, successive period halving, multi-module gating, and a counter that stops the system after n loops, but most claims are only argued heuristically or via simulation; key steps (e.g., uniqueness/stability of the limit cycle, rigorous complementarity and phase ordering, and the m-module Proposition 2) are left without proofs and even include a parameter-typing slip (text says “double η2” while equations double η1) . The candidate model supplies a substantially more rigorous geometric singular perturbation and contraction-based framework that supports most design claims, but it incorrectly asserts exact finite-time termination (x becoming identically zero) under the logistic gate; with x' = −κ(t) x^2 on the last cycle (when n − y = 0), solutions approach 0 asymptotically, not in finite time. Hence, both the paper and the model fall short of complete, fully correct proofs.