We propose a geometric theory in which dark matter is not a new particle species but a dynamical tensor field built from the second-order derivative structure of spacetime itself. The field couples exclusively to spacetime curvature and carries no Standard Model charges, so it interacts with ordinary matter only through gravity. Starting from a ghost-free Lagrangian, we derive the field equations and show that around any concentration of baryonic matter the field develops a halo whose density falls as the inverse cube of the distance, with an enclosed mass growing logarithmically with radius. The characteristic scale of the halo is not a free parameter but is dynamically fixed by the mass of the central supermassive black hole. The theory reproduces quasi-flat galactic rotation curves, predicts a weak-lensing convergence profile with a slope distinctly steeper than the standard NFW dark matter profile, and yields a single dimensionless coupling constant consistent with constraints from rotation curves, the Bullet Cluster, and gravitational lensing. We describe an explicit observational test using weak-lensing data from the Euclid satellite, stratified by galaxy morphology and central black hole mass, that will either confirm or falsify the theory by around 2030.
Dario de Judicibus (Fri,) studied this question.