Consistency conditions for O-plane unsmearing from second-order perturbation theory

Scale-separated AdS compactifications of string theory can be constructed at the two-derivative supergravity level in the presence of smeared orientifold planes. The unsmearing corrections are known to leading order in the large volume, weak coupling limit. However, first-order perturbative approximations of non-linear problems can often produce spurious solutions, which are only weeded out by additional consistency conditions imposed at higher orders. In this work, we revisit the unsmearing procedure and present consistency conditions obtained from the second order warp factor and dilaton equations. This requires proper treatment of the near-source singularities. The resulting conditions appear as integral constraints on various non-linear combinations of the first order corrections, which we argue can generally be satisfied by appropriate choice of integration constants of the leading-order solutions. This provides a non-trivial consistency check for the perturbative unsmearing procedure and supports the existence of scale-separated AdS vacua in string theory.