<p>We present a comprehensive theoretical and computational study of two-electron states in a GaAs quantum ring, integrating Mathieu equation modeling with configuration interaction (CI) calculations accelerated on GPUs. The model incorporates Rashba and Dresselhaus spin–orbit coupling (SOC), magnetic fields, and Coulomb interactions, enabling analysis of singlet–triplet transitions, SOC-driven anticrossings, and spin texture evolution. We derive the Mathieu equation from a well-defined radial confinement potential, validate its applicability to anisotropic rings, and benchmark our results against exact diagonalization. The CI solver achieves rapid convergence with large Hilbert spaces, revealing magnetic field-dependent entanglement entropy and tunable SOC effects. These results provide actionable predictions for transport spectroscopy experiments and a foundation for extending to excitonic states and strain-engineered nanostructures.</p> Graphical Abstract <p></p>