Axis–Collimated 1/r Gravity: A Disk–Plane "Hyperbolic Top" Model for Galaxy Dynamics

We propose a phenomenological modification of gravity tied to a system’s angular momentum: an axis–collimated contribution that falls as 1/r in the equatorial plane but is suppressed towardthe spin axis. The total acceleration is atot(r, θ) = ab(r) + k r f (θ), with ab = GM (r)/r2, polar angle θ measured from the spin axis, and f (θ) a smooth angular weight peaked in the disk (f ( π2 ) = 1, f (0) ≈ 0). In the disk plane this yields flat rotation curves and a baryonic Tully–Fisherscaling v4 ∞ ∝ Mb. Anisotropy leads to distinct lensing and kinematic signatures versus spherical halos. We sketch near–term tests with rotation curves, vertical dispersions, and weak lensing