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 \ (Q\). The central target is the completed determinant comparison \ (DE^comp (s) = (E, s) \), where \ (DE^comp (s) \) is the completed determinant candidate on the spectral side and \ ( (E, s) \) is the completed Hasse--Weil \ (L\) -function of an elliptic curve \ (E/Q\). The consolidated H1--H8 chain includes: Euler