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 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 Σ true /Σ obs ∝ (R true /R obs)-0.5 and M obs /M true ∝ r obs /r true. And reproduces observed rotation curves (RMS residual ∼ 34 km/s). The geometric projection recovers the empirical RAR and shifts the BTFR slope from ∼ 2.8 in the true frame to ∼ 3.7 in the observer frame, and eliminating the normalization offset. Furthermore, The Mezzi scale factor ζ governs mass and lensing corrections via distinct power laws: M true /M obs ∝ ζ-1.84 and α true /α obs ∝ ζ-1.26 , revealing that geometric scaling affects dynamical mass more strongly than lensing mass. These results indicate that geometric projection effects may offer a viable phenomenological explanation for galactic dynamics while remaining consistent with both Newtonian gravity and weak field general relativity. For reproducibility, the code used for this analysis is publicly available at https://github.com/Brahim-Benaissa/Zeta