A pointwise 2-converse for elliptic curves with rational two-torsion

We prove a pointwise 2-converse for elliptic curves over $\mathbf{Q}$ with nonzero rational two-torsion: if the $2^{\infty }$-Selmer corank is zero or one, then the analytic rank and Mordell–Weil rank equal that corank, and the Shafarevich–Tate group is finite. The result allows arbitrary reduction at 2.