Redshift normalized finite multipole spectroscopy of a black hole in a relativistic dark matter halo

Black hole environments modify ringdown through a common gravitational redshift and smaller changes to the perturbation potential. For the exact Schwarzschild–Hernquist spacetime, we define a redshift normalization using the horizon surface gravity and show that it removes the leading compactness dependence of the near zone axial problem. The first local correction is quadratic in halo compactness at fixed halo to black hole mass ratio. We derive a redshift independent photon region invariant through cubic order and calculate fundamental axial quasinormal modes for multipoles two through eight using a global three domain spectral eigenvalue problem. The extracted quadrupole coefficients for oscillation and damping are 1.3537 and -1.8535, with observed numerical sensitivity envelopes of 0.0092 and 0.0234, respectively. A full profile eigenvalue perturbation check supports the direct frequency calculation. Intermode ratios cancel the common redshift and mass scale, yielding a model specific diagnostic of finite multipole environmental corrections.