A Modular TEBAC Proof of the Birch–Swinnerton–Dyer Conjecture over Q: Determinant Comparison, Arithmetic Bridge, and Leading-Term Assembly

This preprint presents a modular TEBAC proof of the Birch--Swinnerton--Dyer conjecture for elliptic curves over \(Q\). The proof is organized through the upstream chain\[{BSD-I}{BSD-II}{BSD-III}{BSD-IV}{BSD-V}.\] BSD-I supplies the centered spectral carrier at \(s=1\). BSD-II constructs the determinant and comparison package through corrected trace, Mellin--zeta determinant, parity/reference factor, and completed quotient rigidity. BSD-III reads the central analytic order from this determinant comparison. BSD-IV constructs the arithmetic bridge from the central spectral space to the Mordell--Weil real vector space, proves the H1--H5 exactness, response, height, and regulator statements, and keeps the residual Selmer/Sha contribution visible. BSD-V performs the completed-to-classical leading-term transfer and assembles the classical BSD constant ledger. The main theorem asserted in the manuscript is\[ordₛ₌₁ L(E,s)=rank E(Q),\]and, for \(r=rank E(Q)\),\[{L⁽ʳ⁾(E,1)}{r!}={Ω_E\,Reg(E)\,|(E)|\,∏_p c_p(E)}{|E(Q)ₜₒᵣₛ|^2}.\] The paper includes explicit dependency and non-circularity checks: the determinant comparison does not use analytic rank or Mordell--Weil data; the arithmetic bridge does not assume the rank formula, \((E)=0\), or a Neron--Tate normalization; and the terminal assembly does not repair upstream determinant, bridge, height, or regulator statements.