TEBAC BSD-V: Terminal Assembly of the Birch--Swinnerton--Dyer Formula

This preprint presents Module BSD-V of the TEBAC Birch--Swinnerton--Dyer program. BSD-V is the terminal assembly module: it imports the upstream BSD-II determinant/comparison export, the BSD-III analytic-rank export, and the BSD-IV arithmetic-bridge export, and then performs the completed-to-classical leading-term assembly. The manuscript records the terminal dependency package\[ DBSDV(E)=( EII(E), EIII(E), EIV(E), Cₜₑᵣₘ(E)),\]where the BSD-IV export includes the arithmetic bridge\[B_E^{}:(-1){~}E( Q)⊗Z R,\]the rank bridge\[(-1)=rankE( Q),\]the raw height comparison\[q^{}E,=B_E,*\, ,\,_{},\]and the regulator identity\[_{}(E)=(E).\] The terminal BSD-V assembly derives the classical BSD rank identity\[ordₛ₌₁L(E,s)=rankE( Q)\]and the classical leading-term formula\[{L⁽ʳ⁾(E,1)}{r!}=Ω_E\,(E)\,{|(E)|∏_p c_p(E)}{|E( Q)ₜₒᵣₛ|^2}.\] The manuscript is status-explicit. BSD-V does not prove the upstream BSD-II determinant package, BSD-III analytic-rank package, or BSD-IV arithmetic-bridge hard inputs; it imports them as already established in the preceding modules. Its role is terminal assembly, completed-to-classical transfer, constant routing, and non-circularity verification.