Proof of the satisfiability conjecture for large $k$

We establish the satisfiability threshold for random $k$-SAT all $k\ge k_0$, with $k_0$ an absolute constant. That is, there exists a limiting density $\alpha_{\rm sat}(k)$ such that formula of clause $\alpha$ is high probability satisfiable $\alpha\lt \alpha_{\rm sat}$, and unsatisfiable $\alpha>\alpha_{\rm sat}$. show given explicitly by one-step replica symmetry breaking prediction from statistical physics. The proof develops new analytic method moment calculations on graphs, mapping high-dimensional optimization problem to more tractable analyzing tree recursions. believe our may apply range CSPs in $1$-RSB universality class.