Gravitational Charge from the Weinberg Cosmological Coincidence: A Phenomenological Coulomb-Like Formulation of Newtonian Gravity

We present a novel phenomenological framework that reinterprets the Newtonian gravitational field as arising from a signed ''gravitational charge'' $q_g \equiv dE/dt$, where the effective gravitational permittivity of spacetime $\mu_h$ is fixed by the Weinberg cosmological coincidence relating the proton mass $m_p$, reduced Planck constant $\hbar$, Newton's constant $G$, speed of light $c$, and Hubble constant $H_0$. Starting from $g=GM/R^2$ and the cosmological scaling $dM/dt = H_0 M$, we derive the inverse-square law $$g = \frac{1}{4\pi \mu_h} \frac{q_g}{R^2},$$ with $q_g$ positive for mass/energy growth (standard attractive gravity) and negative for mass/energy loss (effective repulsion). The parameter $\mu_h = \hbar/V_n$ is identified as the dynamical friction density (or angular-momentum modulus density) of spacetime evaluated at nuclear scales. Closing the loop with the Weinberg relation $m_p^3 \approx \hbar^2 H_0/(G c)$ yields the unique effective volume $V_n = 4\pi (\hbar/(m_p c))^3$, corresponding to the proton reduced-Compton-wavelength sphere---in excellent order-of-magnitude agreement with a neutron/proton-scale volume. Numerical consistency holds to within the typical factor $\sim$ few--10 of the underlying coincidence. We discuss physical interpretation, Gauss-law formulation, and possible implications for cosmology, black-hole evaporation, and quantum-gravity phenomenology. This effective theory unifies microphysical, gravitational, and cosmological scales in a remarkably simple manner while remaining fully compatible with general relativity on local scales.