Effective Observer-Split Source Terms in Rotating Frames and Gravitomagnetic Backgrounds in Extended Aharonov-Bohm Electrodynamics

We examine whether rotating frames and stationary gravitomagnetic backgrounds can provide a meaningful link to extended Aharonov-Bohm electrodynamics without invoking microscopic charge non-conservation. For standard generally covariant, locally $U(1)$-invariant matter, the answer at the microscopic level is negative: the physical four-current remains covariantly conserved, so neither rotation nor stationary gravitomagnetism by themselves generate a genuine source for the scalar sector. A weaker but still useful connection nevertheless emerges after a $3+1$ decomposition with respect to a rotating observer congruence. In that description, the observer-measured transport current on the spatial slice obeys a projected continuity equation containing an exact split source term $I_{\mathrm{split}} \equiv \frac{1}{N} D_i(\rho\,\beta^i)$, which reduces in the weak-field regime to $I_G = D_i(\rho\,\beta^i)$. This term is not a frame-independent microscopic anomaly; it is the bookkeeping term that appears when covariant conservation is rewritten in transport variables adapted to a rotating slicing. We then propose a phenomenological AB-type closure in which this split source drives the scalar sector on finite-scale rotating systems. In the rigid-rotation weak-field limit, the source reduces to $I_G = (\boldsymbol{\Omega} \times \mathbf{r})\cdot \nabla \rho$, and for localized transients to $I_G = \Omega\partial_\phi (\delta\rho_s)$. The resulting framework is therefore effective rather than fundamental, observer-tied rather than local-inertial, and experimentally meaningful only at mesoscopic or macroscopic scales. It yields concrete operational signatures, including reversal under $\Omega \to -\Omega$, suppression for nearly axisymmetric charge distributions, and sensitivity to transient non-axisymmetric charge structure.