Rotation Curves and Gravitational Lensing in Four Isolated Galaxies: A Comparison Between MOND and the κ-Model

Galaxy rotation curves alone do not provide a decisive test between dark matter and phenomenological alternatives. We examine four isolated SPARC galaxies for which resolved rotation curves and weak-lensing constraints provide a useful test of the κ-model. In this phenomenological construction, a galaxy is described in a mechanical base governed by standard Newtonian gravity in the present weak-field calculation, while reconstructed large-scale spatial quantities are related to the base by a density-dependent mapping to an observer bundle. The same density–κ relation is applied to all four galaxies. It reproduces the measured rotation curves with residuals comparable to those obtained with the MOND formulation used here and predicts a semi-Keplerian decline beyond the measured baryonic disc. A separate application of the same mapping to projected densities yields an outer weak-lensing trend. As an extended-baseline consistency test, the model is also applied to M31 without fitting an additional outer-halo parameter to the outer kinematic points. Because the main sample contains only four objects and the outer-halo and lensing constructions involve explicit phenomenological assumptions, the results constitute a proof of concept rather than evidence for a complete alternative to dark matter or modified gravity. The predicted outer decline and projected-density relation are falsifiable with larger samples combining resolved outer kinematics and galaxy–galaxy weak lensing.