Two-Term Scaling Law for Nuclear Charge and Stability Across Isotopes

The structure of the atomic nucleus, of over 5,800 isotopes and isomers from the NuBase 2020 dataset, is traditionally explained through a patchwork of complex models. We demonstrate that a two-term law, Q(A) = c₁A²/³ + c₂A, captures the global charge-mass relationship of all known isotopes—stable and unstable alike—with a coefficient of determination R² ≈ 0.979. This "Core Compression Law," derived from a physical model of competing surface and volume effects, relating the charge distribution and competing mass density gradients defines a universal "backbone" for the entire chart of nuclides. The contribution of this study is the analysis of the residuals (ΔQ) from this law. Deviations from the backbone are not random but systematically classify nuclides into "Overcharged" (ΔQ > 0) and "Starved" (ΔQ < 0) bins, which directly correspond to known decay pathways. Crucially, the magnitude of the residual, |ΔQ|, exhibits a strong correlation with the logarithm of the half-life. This reframes nuclear instability not as a collection of separate rules but as a single, predictable principle: a nuclide's stability is determined by its geometric "charge stress" relative to the universal compression law.