  In a broad class of sparse random constraint satisfaction problems(CSP), deep
heuristics from statistical physics predict that there is a condensation phase
transition before the satisfiability threshold, governed by one-step replica
symmetry breaking(1RSB). In fact, in random regular k-NAE-SAT, which is one of
such random CSPs, it was verified \cite{ssz22} that its free energy is
well-defined and the explicit value follows the 1RSB prediction. However, for
any model of sparse random CSP, it has been unknown whether the solution space
indeed condenses on O(1) clusters according to the 1RSB prediction. In this
paper, we give an affirmative answer to this question for the random regular
k-NAE-SAT model. Namely, we prove that with probability bounded away from zero,
most of the solutions lie inside a bounded number of solution clusters whose
sizes are comparable to the scale of the free energy. Furthermore, we establish
that the overlap between two independently drawn solutions concentrates
precisely at two values. Our proof is based on a detailed moment analysis of a
spin system, which has an infinite spin space that encodes the structure of
solution clusters. We believe that our method is applicable to a broad range of
random CSPs in the 1RSB universality class.
