Producing and Studying Rare Isotopes in $e+A$ Collisions at the Electron-Ion Collider
The Electron--Ion Collider (EIC) offers a unique environment to study kinematically controlled lepton--nucleus ($e{+}A$) reactions, where a primary hard scattering is followed by an intranuclear cascade and the subsequent statistical de-excitation of the nuclear remnant. Utilizing the \soft{BeAGLE} model, we demonstrate that event-by-event fluctuations in nucleon removal and energy deposition populate a diverse ensemble of excited remnants. Furthermore, we show that varying the target mass systematically shifts the distribution of these remnants across the $(N, Z)$ plane. Although this excited prefragment remnant is not directly observable, its properties are shown to be strongly correlated with final-state fragments; specifically, the largest nuclear residue and the intensity of evaporation yield serve as effective experimental proxies for event-level remnant characterization. We also evaluate photon observables essential for nuclear spectroscopy. While various photon sources overlap significantly in pseudorapidity, we find that in the nucleus-rest frame, the low-energy spectrum is dominated by de-excitation $γ$ rays and exhibits distinct discrete structures. These findings motivate an EIC research program that correlates rare-isotope production and de-excitation radiation with well-defined initial conditions, providing a collider-based approach to nuclear spectroscopy that is complementary to existing fixed-target facilities.