Galaxy Rotation Curves as a Geometric Effect in Rotational-Order Geometry: A Resolution to the Dark Matter Mystery

We demonstrate that the rotational-order geometric unification framework naturally produces flat rotation curves without invoking any dark matter particles. We show that the spatial gradient of the internal S³ orientation field on galactic scales modifies the effective metric, introducing an additional geometric acceleration that scales as a_geo ∝ 1/r. At large radii, this geometric term perfectly balances the Newtonian acceleration, leading to a constant orbital velocity v² ≈ c² η² / r_c. This model resolves the mass discrepancy problem across spiral galaxies by replacing dark matter with the geometric structure of spacetime itself. The observed spread in asymptotic velocities implies that effective parameters vary with the galactic environment, the detailed analysis of which will be presented in a companion paper.