Fontaine–Mazur modularity at the prime 2

We prove that every continuous, irreducible, odd two-dimensional 2-adic representation of $G_{\mathbb{Q}}$, unramified outside finitely many primes and de Rham at 2 with distinct Hodge–Tate weights, is modular up to Tate twist. This resolves the odd, regular two-dimensional Fontaine–Mazur conjecture over ℚ at 2, including all residual representations.