Bridging MOND and $\Lambda$CDM by Cosmological Jerk and the $v^3$ Scaling

We propose an adjusted framework for Modified Newtonian Dynamics (MOND) that derives its low-acceleration behavior from a cosmological jerk field sourced by baryonic density and the Hubble constant.

Starting from a phenomenological rotation curve relation that allows mild logarithmic growth, we regularize divergences using the Hubble scale and introduce a velocity-cubed auxiliary field $\psi$. 

Differentiating the implied kinematics yields a jerk field with monopole-like $1/r^2$ radial dependence, leading to a Poisson-type equation $\nabla \cdot \mathbf{j} = -4\pi G H_0 \rho$.

The effective gravitational potential includes a correction $(\psi)^{2/3}$ (or equivalently $(j^2 r^4)^{1/3}$ up to constants), providing a first-principles interpretation of the MOND acceleration scale $a_0 \approx c H_0 / 2\pi$ without unknown dipoles or ad hoc interpolating functions.

Preliminary fits to galaxies in the SPARC database show high precision, capturing mild outer rises better than standard deep-MOND flatness.