A Second-Order Geometric Field Theory of Dark Matter

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.