Non-Abelian Dirac oscillator in a uniform Yang–Mills background: exact spectrum and spin–isospin mixing

We study a planar Dirac oscillator in a uniform gauge background with an Abelian magnetic component and two non-Abelian amplitudes. The noncommuting spatial potentials generate a chromomagnetic curvature along T3 = τ3/2, while the temporal potential generates chromoelectric components proportional to their product. We derive the second-order stationary identity without discarding the isospin–orbit, diamagnetic, or scalar-potential terms. The first-order Hamiltonian, with minimal coupling, a Pauli interaction, or both, admits a conserved oscillator–spin–isospin operator and reduces to finite blocks. We obtain the exact secular equations, the exceptional lowest signed levels, and the leading avoided-crossing gaps. Minimal coupling produces transverse mixing already when the scalar amplitude vanishes; a gap proportional to the product of the amplitudes instead occurs at resonance in the pure Pauli model. We also solve a separately defined field-strength comparison operator. Its spin-dependent backbone does not commute with the transverse internal coupling, so its spectrum is not obtained by adding the isolated internal eigenvalues to the backbone energies. The mixed branches have quadratic weak-coupling shifts away from degeneracy. We distinguish gauge invariants from internal-basis labels and give conditional, spectrally resolved parameter examples, without claiming demonstrated graphene or atomic realizations.