This preprint is Core II of the TEBAC BSD program. It proves a claim-safe central-image criterion: two unreduced Selmer inclusion certificates imply the reduced central image equality \ (E (K₄, ₐ) =E (M₄, ₐ) S^red₄, ₂₄₍\). The module does not claim a standalone proof of the Birch--Swinnerton--Dyer conjecture. It supplies the structural bridge needed after Core I and identifies the remaining arithmetic burden as the construction of local spectral-to-Kummer response identities.
Tosho Lazarov Karadzhov (Sat,) studied this question.