Explicit Local Coordinate Constructions for the Spectral--Kummer Response

This preprint is Core IV of the TEBAC BSD program. It supersedes the initial structural Core IV version by consolidating the four local arithmetic rows needed for the spectral--Kummer response layer in the determinant--Selmer approach to the Birch--Swinnerton--Dyer problem over \(\mathbb{Q}\). The module treats the four local regimes separately: \(p\nmid N_E\ell\), \(p\mid N_E,\ p\neq\ell\), \(p=\ell\), and \(x=\infty\). The good-prime row proves the rational unramified vanishing of the finite local obstruction at primes \(p\nmid N_E\ell\). The bad-prime row separates the rational finite-condition contribution from the finite component/Tamagawa ledger. The \(p=\ell\) row records the Bloch--Kato/formal finite local condition as the appropriate local recipient. The archimedean row separates the Green--N\'eron height contribution from the finite Selmer obstruction rows. The main purpose of the module is to provide the local arithmetic input needed by Core III. It shows that the local response problem decomposes into four distinct arithmetic rows and that each row has the correct structural target: good primes contribute no rational finite Selmer obstruction, bad-prime component groups belong to the Tamagawa ledger, the auxiliary prime is governed by the Bloch--Kato finite condition, and the archimedean place belongs to the height/period ledger. This version remains claim-safe. It does not assert a standalone proof of the Birch--Swinnerton--Dyer conjecture. Its role is to consolidate the local arithmetic rows required for the spectral--Kummer response mechanism. The subsequent modules must still address global height/regulator calibration, residual Sha/Cassels--Tate/Tamagawa routing, and the analytic determinant identity.