The Scale-Lens Mechanism for Flat Rotation Curves: Part I — A Geometric Derivation of the tanh Profile

Title: The Scale-Lens Mechanism for Flat Rotation Curves: Part I — A Geometric Derivation of the tanh Profile This paper introduces the Scale-Lens mechanism as a geometric explanation for flat galaxy rotation curves. We propose that observed velocities include a projection from an extended scale geometry, leading to the velocity-norm projection equation: V_obs² = V_bar² + V_∞² tanh(r/r_g) Key contributions: Derivation of the tanh saturation profile from the Symmetric Saturation Principle Alternative derivation via φ⁴ kink soliton (topological domain wall) Physical interpretation: outer stars appear faster due to scale projection, not additional mass Two free parameters per galaxy (V_∞, r_g) comparable to dark matter models Framework for understanding "missing mass" as geometric projection rather than invisible matter This is Part I of the Galaxy Rotation Curve Series, focusing on theoretical foundations.