This preprint is Core III of the TEBAC BSD program. It isolates the local response-matching layer needed in the determinant--Selmer approach to the Birch--Swinnerton--Dyer problem over \ (Q\). The module formulates the place-by-place spectral-to-Kummer response identities at all relevant local regimes: good primes, bad primes, primes above the auxiliary prime \ (\), and the archimedean place. Its main structural result proves that if the local identities hold for every admissible local dual test, then their sum gives the required global detector-response matching. This supplies the local certificate input needed by Core II, where unreduced Selmer inclusion certificates imply the reduced central image equality \ (E (K₄, ₐ) =E (M₄, ₐ) S^red₄, ₂₄₍\). The preprint is claim-safe: it does not assert a standalone proof of the Birch--Swinnerton--Dyer conjecture. Its role is to reduce the next arithmetic burden to explicit local coordinate constructions. The subsequent module is expected to begin with the good-prime Frobenius-coordinate identity and then proceed to the bad-prime, \ (p \), and archimedean response rows.
Tosho Lazarov Karadzhov (Sat,) studied this question.