Homology constraint for replica fixed sets in general RT derivations

Prove that, in general bulk geometries beyond the near‑thermal regime, the codimension‑two fixed point set of the replica symmetry used in the gravitational replica construction satisfies the homology constraint required by the Ryu–Takayanagi formula.

Background

The RT prescription requires the extremal surface to be homologous to the boundary region whose entropy is being computed. In the bulk replica derivation, the codimension‑two fixed point (cosmic brane) is assumed to satisfy this homology condition.

While the condition holds in near‑thermal configurations where topology is controlled, its general validity for arbitrary states and geometries is not established, leaving a gap between the formal replica construction and the RT homology requirement.