Ultra-high energy cosmic rays impinging on the Earth’s atmosphere offer a unique opportunity to probe hadronic interactions at the highest energies through measurements of the extensive air showers they generate. In this work, we introduce a semi-analytic universal mapping between new macroscopic variables describing the first ultra-high-energy proton-air interaction and the shower-to-shower values of the depth of the shower maximum, $X_{\max}$, and the shower muon number. The distributions of these new production variables are highly dependent on the hadronic interaction model. These are combined into the primary interaction variables $(\alpha_1, \xi_1)$ which determine most of the features of the joint distribution of $X_{\max}$ and $N_\mu$. We characterize this universal causal relation and show that it can be used to constrain the shape of the distributions of multi-particle production variables in proton-air interactions beyond current accelerator capabilities.