Local Spectral-to-Kummer Response Identities

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 \(\mathbb{Q}\). 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 \(\ell\), 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 \(\rho_E(K_{E,\mathbb Q})=\kappa_E(M_{E,\mathbb Q})\subset S^{\mathrm{red}}_{E,\mathrm{cen}}\). 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, \(p\mid \ell\), and archimedean response rows.