Fundamental complement of a gravitating region

Any gravitating region a in any spacetime gives rise to a generalized entanglement wedge, the hologram e(a). Holograms exhibit properties expected of fundamental operator algebras, such as strong subadditivity, nesting, and no-cloning. But the entanglement wedge \({{\,\textrm{EW}\,}}\) of an AdS boundary region B with commutant \({{\bar{B}}}\) satisfies an additional condition, complementarity: \({{\,\textrm{EW}\,}}(B)\) is the spacelike complement of \({{\,\textrm{EW}\,}}(\bar{B})\) in the bulk. Here we identify an analogue of the boundary commutant \({{\bar{B}}}\) in general spacetimes: given a gravitating region a, its fundamental complement \({{\tilde{a}}}\) is the smallest wedge that contains all infinite world lines contained in the spacelike complement \(a'\) of a. We refine the definition of e(a) by requiring that it be spacelike to \({{\tilde{a}}}\). We prove that e(a) is the spacelike complement of \(e({{\tilde{a}}})\) when the latter is computed in \(a'\). We exhibit many examples of \({{\tilde{a}}}\) and of e(a) in de Sitter, flat, and cosmological spacetimes. We find that a Big Bang cosmology (spatially closed or not) is trivially reconstructible: the whole universe is the entanglement wedge of any wedge inside it. But de Sitter space is not trivially reconstructible, despite being closed. We recover the AdS/CFT prescription by proving that \({{\,\textrm{EW}\,}}(B)=e(\)causal wedge of B).