It is important to note that we employ the full expression in Eq. ( ) rather than the commonly used approximation k ≈ lim t→t p C flux (t). The latter is equivalent to extracting the reaction rate from the slope of the time-dependent reactant population. However, this approximation only holds true for slow reactions when the molecule predominantly resides in the reactant regime at the time t p where the transient oscillatory dynamics subside, i.e., P r (t p ) ≈ 1. As such, the denominator in Eq. ( ) remains close to one and can be disregarded. This approximation does not hold for the model studied in the current and previous works. This observation is true across the entire cavity frequency regime, regardless of whether exact quantum-mechanical dynamics are used or the mixed quantum-classical approach. As an illustration, in Fig. (a), we compare the mixed quantum-classical k(t) and flux-side correlation function C flux (t) for a reaction inside the cavity with the optical frequency ω c = 1200cm -1 . It is evident that after the short-time transient dynamics have decayed, k(t) demonstrates a well-defined plateau, whereas C flux (t) does not.