Analytic Determinant Identity and Completed L-Function Ledger

This preprint is Core VII of the TEBAC BSD program. It isolates the analytic determinant identity layer in the determinant--Selmer approach to the Birch--Swinnerton--Dyer problem over \(\mathbb Q\). The main analytic target is the logarithmic-derivative identity\[\frac{d}{ds}\log D_E^{\mathrm{comp}}(s)=\frac{d}{ds}\log \Lambda(E,s),\]together with the required normalization at a base point. Once this identity and normalization are supplied, the quotient \(D_E^{\mathrm{comp}}(s)/\Lambda(E,s)\) is forced to be constant and equal to \(1\). The preprint also records the rank-transfer mechanism:\[D_E^{\mathrm{comp}}(s)=\Lambda(E,s),\qquad\operatorname{ord}_{s=1}D_E^{\mathrm{comp}}(s)=\dim_{\mathbb R}K_E\]imply\[\operatorname{ord}_{s=1}L(E,s)=\dim_{\mathbb R}K_E.\] Together with the previous core modules, this analytic layer is the bridge from the determinant central space to the classical BSD rank formula and leading-coefficient ledger. This is a claim-safe preprint module. It does not assert a standalone proof of the Birch--Swinnerton--Dyer conjecture and does not claim that the analytic determinant identity has already been constructed from first principles. Its role is to isolate the exact analytic determinant gate that must be closed before a full unconditional BSD assembly can be claimed.