Einstein–Karahan Framework v122: A Geometric Torsion–Phase Extension of Galactic Dynamics

The Einstein–Karahan framework presents a geometric extension of spacetime dynamics by incorporating a coupled torsion–phase field system into general relativity. Instead of introducing dark matter as an additional particle component, the model interprets galactic rotation curves and large-scale structure as emergent effects of torsion-induced modifications to gravitational dynamics.In this formulation, spacetime is described by a metric-compatible geometry augmented with torsion degrees of freedom, while an additional phase field captures collective synchronization effects within the system. In the macroscopic limit, the framework reduces to an effective description in which an induced acceleration term modifies the standard baryonic rotation profile, naturally producing flat rotation curves without requiring dark matter halos.The present work provides a consistent minimal formulation that connects geometric field dynamics with astrophysical observables. While the theory remains at an effective level and requires further numerical validation and observational calibration, it establishes a testable pathway toward interpreting dark matter phenomena as manifestations of extended spacetime geometry rather than new particle species