Stability and synchronisation in modelling an oscillatory stochastic reaction network

The paper’s Theorem D is proved via Newman’s synchronization framework using two-point contractibility and negative conditioned top Lyapunov exponent, with all required ingredients established (Q-process, quasi-ergodicity, Oseledets theorem). By contrast, the model’s proof hinges on a non-justified upgrade from λ1<0 (for Qν-a.e. orbit) to uniform-in-x exponential contraction of sup_x||Dϕ_n|| via a semi-uniform subadditive ergodic theorem on a non-compact i.i.d. base; this step is not supported by the paper’s hypotheses and, without additional structure, is generally false. The model also over-claims shrinking of the full-fibre images and convergence for all initial measures.