Nuclear Gluon Gravitational Form Factors and Neutron Skins at the Electron-Ion Collider

Coherent quarkonium production at the Electron--Ion Collider can image the average small-$x$ gluon radius of nuclei, providing a reaction mechanism complementary to parity-violating electron scattering. We develop a calibrated radius-sum-rule framework that connects this gluonic radius to neutron skins and quantify the leading limitations from finite-dipole saturation, nuclear opacity, and instrumental resolution. The central result is that coherent $\Jpsi$ production contains sufficient shape information for a competitive neutron-skin program, but its precision is not limited by luminosity alone. It is instead controlled by the calibration of the nuclear small-$x$ gluon density, while the cleaner $\Ups$ channel remains statistically limited at early EIC luminosities. This framework identifies what an EIC neutron-skin measurement can robustly add to symmetry-energy studies and which theoretical and experimental controls are required before such a measurement can be interpreted as precision nuclear-structure information.