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

We rigorously derive the 1/r geometric acceleration that explains flat galaxy rotation curves, starting from the first principles of the internal S³ orientation field within the rotational-order geometric unification framework. By reducing the field equations in a steady, axisymmetric galactic disk, we show that the radial gradient naturally yields an exact ∂_r θ ∝ 1/r scaling through a non-linear scalar wave equation. Through the effective Einstein equations, we prove that this geometric potential yields an exact 1/r acceleration in the far-field halo, completely eliminating the need for dark matter particles. Crucially, this derivation is consistent with the phenomenological fit (α ≈ 0.98) from the SPARC database (DDO 154 galaxy), providing a complete, mathematically rigorous, and observationally consistent explanation for the missing mass problem.