Explicit Local Coordinate Constructions for the Spectral--Kummer Response

This preprint is Core IV of the TEBAC BSD program. It gives a structural local-coordinate reduction for the spectral--Kummer response layer in the determinant--Selmer approach to the Birch--Swinnerton--Dyer problem over \(\mathbb{Q}\). The paper organizes the local coordinate problem into the four relevant local regimes: \(p\nmid N_E\ell\), \(p\mid N_E\), \(p=\ell\), and \(x=\infty\). These correspond respectively to good-prime Frobenius coordinates, bad-prime component/conductor coordinates, formal or Bloch--Kato coordinates at primes above the auxiliary prime, and the archimedean Green-kernel coordinate. The main result proves that if these four local coordinate certificates are supplied and satisfy the stated compatibility conditions, then the local spectral-to-Kummer response identities required in Core III follow. In particular, the module supplies the structural implication \(\text{local coordinate certificates} \Longrightarrow \text{local spectral-to-Kummer response identities}\). This preprint is claim-safe. It does not assert a standalone proof of the Birch--Swinnerton--Dyer conjecture and does not yet claim a full unconditional construction of all local coordinates from first principles. Its role is to isolate the exact local-coordinate data that must be hardened in later versions or follow-up modules, beginning with the good-prime Frobenius-coordinate identity.