Reliable forecasting of geomagnetic storms remains a major challenge in space weather, especially at lead times beyond several days. Most models rely on near-Earth solar-wind data, which fundamentally limit lead time. Stealth coronal mass ejections with weak low-coronal signatures are often poorly constrained by traditional diagnostics. We extend prediction by exploiting galactic cosmic-ray (GCR) modulation as an upstream precursor and develop a physics-motivated long short-term memory model combining cosmic-ray flux, solar-wind, interplanetary magnetic-field, and geomagnetic parameters from a 1995–2020 dataset at 1 hr cadence. The model achieves RMSE of 5.106–14.788 nT for 2–48 hr lead times, with the 48 hr F1 score improving by 25.84% when cosmic-ray inputs are included. For interplanetary coronal mass ejection (ICME)-driven storms we add an event-based sub-model, predicting whether an approaching ICME will trigger a significant storm 24–48 hr before arrival; the 24 hr and 48 hr advance models achieve F1 scores of 0.55 and 0.48 respectively. Permutation-importance analysis indicates that cosmic-ray-related variables account for a substantial fraction of the model’s predictive skill, while additional statistical analysis shows that stronger cosmic-ray suppression is associated with more negative minimum Dst, supporting the interpretation of GCR modulation as a physically meaningful precursor.