On the Observer-Frame Interpretation of Radial Coordinates in Gravitational Potentials

The discrepancy between galactic rotation curves and visible baryonic mass persists despite empirical scaling relations like the Radial Acceleration Relation (RAR) and Baryonic Tully-Fisher Relation (BTFR). We explore a phenomenological framework where this discrepancy arises from the geometric misinterpretation of observables. Inspired by Painlevé-Gullstrand coordinates, we model the vacuum as a radially infalling compliant medium that induces an apparent compression mapping of radial coordinates for distant observers, the "Mezzi effect". Assuming Newtonian dynamics govern an undistorted "true frame", we developed a discrete shell reconstruction method parameterized by a single universal compliance constant, tested against photometric and kinematic data from 175 late type galaxies in the SPARC database. This single parameter model yields universal scaling relations of \(\Sigma_{\text{true}}\text{/}\Sigma_{\text{obs}} \propto \left( R_{\text{true}}\text{/}R_{\text{obs}} \right)^{- 0.5}\) and \(M_{\text{obs}}\text{/}M_{\text{true}} \propto r_{\text{obs}}\text{/}r_{\text{true}}\). The model reproduces observed rotation curves (RMS residual \(\sim 34\) km/s). The geometric projection recovers the empirical RAR and shifts the BTFR slope from \(\sim 2.8\ \)in the true frame to \(\sim 3.7\ \ \)in the observer frame, and eliminats the normalization offset. Furthermore, The Mezzi scale factor \(\zeta\) governs mass and lensing corrections via distinct power laws: \(M_{\text{true}}\text{/}M_{\text{obs}} \propto \zeta^{- 1.84}\) and \(\alpha_{\text{true}}\text{/}\alpha_{\text{obs}} \propto \zeta^{- 1.26}\), revealing that geometric scaling affects dynamical mass more strongly than lensing mass. These results indicate that geometric projection effects may offer a phenomenological explanation for galactic dynamics, acting as a propagation level transfer function that leaves the baryonic stress-energy content and the weak field metric of the source frame untouched. For reproducibility, the code used for this analysis is publicly available at https://github.com/Brahim-Benaissa/Zeta.