Four Elements to Rule Them All: Abundances are Rigidly Coupled in the Milky Way Disk

Chemical tagging is a central pursuit of galactic archaeology, but requires sufficiently discriminative abundances to uniquely identify sites of star formation. This task is complicated by intrinsic scatter among conatal stars, inter-element correlations, imprecise abundance measurements, and systematics across stellar evolutionary states. In this work, we formalize the abundance correlation structure of the disk by quantifying the amplitude of information in individual element abundances once a subset is known, and map inter-element residual correlations to uncover hidden signatures of nucleosynthesis. We use two datasets of 79 (593) stars across $-0.15<\rm [Fe/H]<0.13$ ($-1<\rm [Fe/H]<0.41$) with measurements of 30 (19) element abundances of solar neighborhood stars, including 11 (7) light and $\alpha$, 7 (3) Fe-peak, and 12 (9) neutron capture elements. With a simple linear regression model, we predict most $\alpha$ and Fe-peak element abundances within 0.03~dex ($\sim7\%$), and neutron capture elements within 0.05~dex ($\sim10\%$). Including first and second peak s-process elements as predictors improves most neutron capture element predictions to within 0.02~dex (5\%), although no predictive power is gained by including an r-process element. We uncover strong (anti-)correlations in small residual abundances between and within element families. Our finding that disk abundance space is rigidly coupled, from light to heavy elements, implies chemical tagging is infeasible at $>2\%$ precision for $\sim$30 elements. However, the residual structure encodes fingerprints of star formation history, inherited from nucleosynthesis and environmental variations, and provides critical constraints for chemical evolution models. Future disk surveys must achieve sub-2-5\% precision in 30+ elements to access this independent information.