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 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 activity 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 $\gamma$ 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.