  We consider the random regular $k$-NAE-SAT problem with $n$ variables each
appearing in exactly $d$ clauses. For all $k$ exceeding an absolute constant
$k_0$, we establish explicitly the satisfiability threshold $d_*=d_*(k)$. We
prove that for $d<d_*$ the problem is satisfiable with high probability while
for $d>d_*$ the problem is unsatisfiable with high probability. If the
threshold $d_*$ lands exactly on an integer, we show that the problem is
satisfiable with probability bounded away from both zero and one. This is the
first result to locate the exact satisfiability threshold in a random
constraint satisfaction problem exhibiting the condensation phenomenon
identified by Krzakala et al. (2007). Our proof verifies the one-step replica
symmetry breaking formalism for this model. We expect our methods to be
applicable to a broad range of random constraint satisfaction problems and
combinatorial problems on random graphs.
