Positive energy-momentum theorems for asymptotically AdS spin initial data sets with charge

For complete spin initial data sets with an asymptotically anti–de Sitter end, we introduce a charged energy–momentum defined as a linear functional arising from the Einstein–Maxwell constraints. Under a dominant energy condition adapted to the presence of a negative cosmological constant, we establish positive energy–momentum theorems, showing in particular that this functional is non–negative on a natural real cone. We place particular emphasis on the case where the manifold carries a compact inner boundary. In the time–symmetric setting, this yields a mass–charge inequality for asymptotically hyperbolic manifolds with charge.