Residual Selmer, Cassels--Tate, Tamagawa, and Torsion Ledger

This preprint is Core VI of the TEBAC BSD program. It isolates the finite residual arithmetic ledger that remains after the Mordell--Weil rank bridge and the N\'eron height/regulator calibration have been separated in the determinant--Selmer approach to the Birch--Swinnerton--Dyer problem over \(\mathbb{Q}\). The module treats the finite part of the BSD leading coefficient ledger. Its main claim-safe purpose is to show that, once the residual Selmer quotient is identified with the Tate--Shafarevich/Cassels--Tate finite obstruction, the local component corrections are identified with the Tamagawa groups, and the torsion normalization is imposed, the finite residual contribution is exactly \(\frac{|\Sha(E)|\prod_p c_p(E)}{|E(\mathbb{Q})_{\mathrm{tors}}|^2}\). The paper keeps the finite Sha, Tamagawa, and torsion terms separate from the rank bridge and from the regulator comparison. In particular, these finite factors are not used to construct the Mordell--Weil bridge, the determinant central space, or the N\'eron--Tate regulator identity. They enter only at the finite leading-coefficient ledger stage. This is a claim-safe preprint module. It does not assert a standalone proof of the Birch--Swinnerton--Dyer conjecture. It does not prove the finiteness of \(\Sha(E)\) from first principles, nor does it prove the analytic determinant identity \(D_E^{\mathrm{comp}}(s)=\Lambda(E,s)\). Its role is to isolate the finite arithmetic routing required before the final BSD leading-coefficient assembly.