The analytic construction of Sec. II and Appendix A may be useful in its own right, and the perturbation equations (25)–(27) are correct. However, the claimed 10−4 correction is not established by the present analysis: the background is internally inconsistent and improperly normalised, the poten tial as printed fails the Schwarzschild limit, the continued-fraction scheme contains no dark-matter dependence, and the reported shifts are both far too large and scale with rb in the wrong direction. Addressing these issues would require a rederivation of the background with proper matching, a validated and independently cross-checked numerical calculation, and a source-specific detector analysis. I would be glad to look at a revised version.

The analytic construction of Sec. II and Appendix A may be useful in its own right, and the perturbation equations (25)–(27) are correct. However, the claimed 10−4 correction is not established by the present analysis: the background is internally inconsistent and improperly normalised, the poten tial as printed fails the Schwarzschild limit, the continued-fraction scheme contains no dark-matter dependence, and the reported shifts are both far too large and scale with rb in the wrong direction. Addressing these issues would require a rederivation of the background with proper matching, a validated and independently cross-checked numerical calculation, and a source-specific detector analysis. I would be glad to look at a revised version.