Proof of the satisfiability conjecture for large k

We establish the satisfiability threshold for random $k$-SAT for all $k\ge k_0$,with$k_0$anabsoluteconstant\mathrm{.}Thatis,thereexistsalimitingdensity$\alpha_*(k)$suchthatarandom$k$-SATformulaofclausedensity$\alpha$ is with high probability satisfiable for $\alpha <{\alpha }_{\ast }$, and unsatisfiable for $\alpha >{\alpha }_{\ast }$. We show that the threshold ${\alpha }_{\ast }(k)$ is given explicitly by the one-step replica symmetry breaking prediction from statistical physics. The proof develops a new analytic method for moment calculations on random graphs, mapping a high-dimensional optimization problem to a more tractable problem of analyzing tree recursions. We believe that our method may apply to a range of random CSPs in the 1-RSB universality class.