Trackable species dynamics in reaction network models

The paper proves the L1-uniform approximation of the projected population dynamics by an ensemble of independent, Z-driven single‑molecule processes under Assumptions 3.1 and 5.1 (Theorem 5.5), with an explicit exponential-in-sqrt(V) tail bound given in Appendix A (Theorem A.4). The candidate solution reaches the same conclusion via a molecule‑wise maximal coupling argument based on labeled molecules and a Poisson‑thinning construction. The approaches are conceptually aligned but technically different: the paper works at the level of aggregated Poisson representations and derives explicit finite‑V bounds; the model works per molecule and aggregates mismatch counts. The only substantive caveat is that the model implicitly strengthens the initial condition requirement by using convergence of initial expectations (uniform integrability), whereas the paper only assumes convergence in distribution plus mild integrability and handles the initial discrepancy via probability bounds. Aside from this minor assumption gap, both arguments are sound and reach the same result.