Strong BSD in Quadratic-Twist Families with Controlled Additive Reduction

Let E/Q be an optimal elliptic curve of analytic rank zero satisfying ex- plicit two-primary hypotheses, with multiplicative reduction away from a possible ad- ditive prime at 11. We prove, under stated residual, local, and period conditions, that infinitely many positive quadratic twists of E have analytic and algebraic rank zero, finite Tate–Shafarevich group, and satisfy the exact strong Birch–Swinnerton-Dyer for- mula at every prime. This extends recent semistable twist-family results to a controlled nonsemistable setting in which the reduction at 11 is potentially good supersingular. The proof constructs one common Chebotarev family, combines exact two-primary for- mulas with ordinary, supersingular, and reducible-residual results at primes away from the twist discriminant, and treats primes ramified in the twist through Kato’s zeta ele- ments. At those primes we verify the required ramified finite-character specialization in the Skinner–Urban Eisenstein datum, obtain primitivity of the finite Kato Kolyvagin system, and prove the finite descent independently once integral Neron normalization has been supplied. Primewise identities are finally assembled in Q>0.