Existence of a rank-one curve in quadratic twist families without BSD

Demonstrate, without assuming the Birch and Swinnerton–Dyer conjecture or parity, the existence of at least one parameter t in the quadratic twist family t y^2 = f(x) over a number field K such that the twist has Mordell–Weil rank 1, for base elliptic curves beyond the generic full rational 2-torsion case considered in this paper.

Background

The authors explain that, although 2^∞-Selmer rank results imply upper bounds and conditional rank-one statements given BSD, unconditional proofs of a single rank-one twist in quadratic families have been elusive. Their theorem addresses this for generic curves with full rational 2-torsion, showing infinitude of rank-one twists.

The broader unresolved question concerns proving such existence unconditionally for more general base curves and families, closing the gap between Selmer-based bounds and actual rank.