Mordell-Weil saturation ($H^{\mathrm{sat}}=\{P\in E(K):nP\in H\text{ for some }n\ge 1\}$)

The saturation of a subgroup $H$ of the Mordell-Weil group consists of all points with a positive multiple in $H$. Saturation includes the full torsion subgroup. For a full-rank $H$ containing that torsion, it equals $E(K)$, but the index of $H$ may exceed one. Checking that $\ell P\in H$ forces $P\in H$ for every prime divisor of a certified index bound proves that $H$ is already the whole group. A canonical height pairing proves independence of found points; saturation answers the separate question of whether a full-rank subgroup is the full group.