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 dominates over Newtonian acceleration and yields constant orbital velocity v² ≈ c² η² / r_c. This model resolves the mass‑discrepancy problem for spiral galaxies by replacing dark‑matter halos with intrinsic spacetime geometry. Observed scatter in asymptotic rotation velocities implies environment‑dependent effective parameters; quantitative data‑fitting will be reported in a follow‑up companion paper.