Analytic Determinant Gate and Completed (L)-Function Ledger

This preprint is Core VII of the TEBAC BSD program. Version 2.0 supersedes the initial Core VII version by consolidating the analytic determinant gate into a single H1--H8 module. The paper isolates the analytic determinant identity layer in the determinant--Selmer approach to the Birch--Swinnerton--Dyer problem over \(\mathbb{Q}\). The central target is the completed determinant comparison \(D_E^{\mathrm{comp}}(s)=\Lambda(E,s)\), where \(D_E^{\mathrm{comp}}(s)\) is the completed determinant candidate on the spectral side and \(\Lambda(E,s)\) is the completed Hasse--Weil \(L\)-function of an elliptic curve \(E/\mathbb{Q}\). The consolidated H1--H8 chain includes: Euler