A new theoretical framework formulates relativistic spin hydrodynamics in a parity-even magnetized medium, treating the magnetic field at leading order in the gradient expansion. The field breaks rotational symmetry down to SO(2) and separates spin susceptibilities and dissipative transport into longitudinal, transverse and mixed channels. The construction uses an Israel-Stewart-type entropy current, yielding transient equations for the retained dissipative stresses and spin fluxes.

The work analyses a minimal linear closure around an unpolarized homogeneous equilibrium, keeping dynamical magnetic-flux perturbations but no explicit spin-magnetization coupling. Even so, spin and magnetic fluctuations mix through the fluid variables. Infrared coefficient equations and ultraviolet characteristic expansions organize the magnetosonic and Alfvén-like spectra, together with spectator spin-flux poles.

The first-order truncation exhibits unstable boost-sector poles and superlinear large-wave-number branches, which the authors interpret as diagnostics of the truncated equations rather than microscopic dispersion predictions. For the baseline parameters, finite relaxation times yield asymptotically linear propagation branches and finite relaxation poles, with no growing modes found on the sampled grid. Characteristic speeds depend nonmonotonically on spin transport and provide necessary light-cone diagnostics, though the study does not establish general stability or strong hyperbolicity.

The formulation opens the door to describing spin evolution in strongly magnetized plasmas, such as those found in heavy-ion collisions or in the magnetospheres of compact objects. Next steps would include extending the closure to incorporate explicit spin-magnetization coupling and exploring nonlinear regimes where the instabilities seen in the truncation can be clarified.