TEBAC Birch--Swinnerton--Dyer Program: Expanded Annals/JAMS Master Architecture with Central-Germ Analytic Determinant and Selmer--Height--Sha Certificates

This preprint presents the expanded Annals/JAMS-facing master architecture of the TEBAC Birch--Swinnerton--Dyer program for elliptic curves over \( Q\). The target statements are\[ordₛ₌₁L(E,s)=rankE( Q)\]and, for\[r=rankE( Q),\]\[{L⁽ʳ⁾(E,1)}{r!}={Ω_E\,Reg(E)\,|(E)|∏_p c_p(E)}{|E( Q)ₜₒᵣₛ|^2}.\] The manuscript is written as a claim-safe master reduction paper rather than as a completed \(100%\) unconditional proof. It records the current state of the TEBAC determinant--Selmer--height--Sha architecture and isolates the remaining terminal certificates required for a full unconditional proof. The analytic side is organized around the central-germ determinant route near \(s=1\). The completed determinant is written as\[D_Eᶜᵒᵐᵖ(s)=D_Eʳᵉᶠ(s)\,D_Eᶜᵉⁿ((s-1)^2),\]and the central analytic target is the germ identity\[D_Eᶜᵒᵐᵖ(s)=Λ(E,s)\]near \(s=1\). This is reduced to central log-derivative ledger matching, no-spurious-center control, central unit normalization, and independent central spectral multiplicity. The arithmetic side develops a finite response-matrix bridge\[ρ_E(KE, Q)=κ_E(ME, Q)⊂ SʳᵉᵈE,cen,\]where\[ME, Q=E( Q)free⊗Z Q.\]The bridge is reduced to detector no-loss, primitive packet spanning, and the spectral full-rank certificate\[rank(T_EZ_E)=QΠᵖʳⁱᵐE,cen.\] The Selmer--Kummer comparison is organized place by place: good primes, bad primes, primes dividing the level, the Archimedean place, and the global Kummer image. The height/regulator block is reduced to local Néron-symbol calibration\[qʳᵃʷE,x,spec(u,v)= DC_Eu,DC_Ev_xNer\]for every place \(x\). The final BSD-V assembly records the remaining Sha, Tamagawa, torsion, and Cassels--Tate certificates needed to convert the formal coefficient ledger into the classical BSD leading-term formula. An earlier, incomplete architectural chain of this TEBAC BSD proof attempt received a desk rejection from the Journal of the American Mathematical Society because the proof architecture was not yet complete as a full unconditional proof. The present version is a substantially expanded and claim-safe master architecture. It is estimated by the author to contain approximately \(60%\) classical unconditional closure of the intended proof, with the remaining part reduced to explicitly stated terminal certificates. The remaining terminal certificates include:\[D_Eᶜᵒᵐᵖ(s)=Λ(E,s) central germs at s=1,\]\[rank(T_EZ_E)=QΠᵖʳⁱᵐE,cen,\]\[qʳᵃʷE,x,spec=\ ,\ _xNer all places x,\]and\[ O_Eˢᵖᵉᶜ(E),βˢᵖᵉᶜ=\ ,\ CT.\] Once these terminal blocks are completed to a fully unconditional proof, the manuscript is planned for submission to Annals of Mathematics around mid-October 2026, or earlier as a completed revised version for possible resubmission to JAMS. No claim in this upload should be interpreted as using the BSD rank formula, the BSD leading coefficient formula, or the equality\[|(E)|=Pf(T)^2\]as an input. These are intended terminal outputs, conditional on the explicitly listed certificates.