Universal $Z^{2/3}/\sqrt{Q_α}$ Scaling Law in Alpha Decay from Nuclei to Neutron-Star Mergers
We establish a universal scaling law for $α$-decay half-lives based on the variable $X_a = Z^a/\sqrt{Q_α}$, which collapses experimental and theoretical data spanning over twenty orders of magnitude onto a single linear correlation. Global optimization over the complete set of even-even $α$ emitters reveals a uniquely sharp optimum at $a \simeq 2/3$. This exponent is theoretically motivated by the leading geometric scaling of heavy nuclei and emerges quantitatively from the correlated nuclear systematics, driven by the correlated $(Z, Q_α)$ manifold of the nuclear chart. Strikingly, five structurally distinct semi-empirical models and an independent microscopic WKB calculation, when independently optimized with respect to the exponent $a$, yield values clustered around $2/3$, while the corresponding $a=2/3$ correlations remain highly linear without refitting the original model parameters. This emergent scaling law implies smooth variations of decay times along the heavy $r$-process path. We demonstrate analytically that, given a roughly uniform distribution of this variable, the scaling law naturally supports a quasi-power-law radioactive heating rate ($\dotε \propto t^{-1.24}$), consistent with full network calculations. Furthermore, integrating these scaling predictions directly into nuclear source terms for radiative-diffusion models yields multimessenger observables that accurately reproduce the kilonova AT2017gfo associated with the gravitational-wave event GW170817, demonstrating that the $Z^{2/3}/\sqrt{Q_α}$ scaling coordinate provides a robust tool for modeling the radioactive engines of neutron-star mergers.