Gibbs Factorials Become Kinetic in History-Dependent Reactions
Gibbs factorials are usually regarded as equilibrium counting factors. We show that they can also appear directly in a measurable kinetic observable when products formed at different stages are statistically distinguished in the history ensemble. In a model designed to isolate the essential ingredients, transient AB$_2$ complexes are stabilized as C molecules either in a single operation or through a two-stage procedure. The mean-waiting-time ratio is governed by a kinetic advantage factor $\mathcal{A}_C$ defined from equilibrium probabilities. A fluctuation-theorem argument identifies the dominant contribution $n_{\mathrm C}!/[n_{\mathrm m}!(n_{\mathrm C}-n_{\mathrm m})!]$, where $n_{\mathrm m}$ is the intermediate product number and $n_{\mathrm C}$ is the final target. This Gibbs factorial arises because products formed before and after the intermediate operation are statistically distinguished in the history ensemble, although the final molecules are macroscopically identical. Molecular dynamics simulations confirm the predicted combinatorial scaling of the mean-waiting-time ratio.