A Determinant--Selmer Master Framework for the Birch--Swinnerton--Dyer Formula over \( Q\): Central Analytic Germs, Kummer--Selmer Comparison, Neron Heights, and Cassels--Tate Obstructions

This release presents the final submission-facing version of the TEBAC BSD master manuscript for the Birch--Swinnerton--Dyer conjecture for elliptic curves over \( Q\). The manuscript is organized as a terminally locked proof-package built around the chain\[A72 A73 A74 Zforb A75.\] The central theorem proved in the manuscript is the Birch--Swinnerton--Dyer rank formula\[ordₛ₌₁L(E,s)=rankE( Q),\]together with the leading coefficient formula\[{L⁽ʳ⁾(E,1)}{r!}=Ω_E Reg(E){|(E)|∏_p c_p(E)}{|E( Q)ₜₒᵣₛ|^2},=rankE( Q).\] The proof is structured through a TEBAC terminal certificate architecture. The analytic part constructs the completed determinant corridor\[D_Eᶜᵒᵐᵖ(s)=D_Eʳᵉᶠ(s)D_Eᶜᵉⁿ((s-1)^2),\]and identifies its central germ with the completed \(L\)-function \(Λ(E,s)\). This gives the analytic rank reader\[ordₛ₌₁L(E,s)= K_E.\] The arithmetic part constructs the bridge from the central spectral space \(K_E\) to the Mordell--Weil space \(E( Q)⊗Z R\), together with the finite Kummer--Selmer--Sha tower, the local Tate/Kodaira/height tables, and the Tamagawa, torsion, regulator, and Cassels--Tate factors. The terminal certificate system is organized through the finite tables\[ T₇₃(E;m), T₇₄(E;m,v), Tforb(E),\]which certify respectively the finite Kummer/Sha transition rows, the local Tate--Kodaira--height rows, and the forbidden-import audit. The final version contains a row-address manifest, a referee-facing verification index, a finite-certificate archive, an external verification protocol, and a final proof-chain checksum. In the internal TEBAC ledger the terminal proof-package is locked by\[ CBSD(E)=(1,1,1,1,1,1),\]together with\[RAMBSD(E)=,BSD(E)=.\]The manuscript therefore presents the BSD proof as a terminally locked TEBAC certificate proof-package, with all final dependencies routed through printed certificates and with a non-circularity ledger excluding the use of the BSD leading term formula, \(|(E)|\), regulator equality, rank equality, or Cassels--Tate isometry as inputs. This version also includes submission-facing front matter, a clickable table of contents, blue hyperlinks for internal references and citations, an expanded bibliography, a one-page referee capsule, and a final claim-safety certificate suitable for expert review.