TEBAC Birch–Swinnerton–Dyer Program: Journal Master Proof Spine for Elliptic Curves over Q

This preprint presents a journal-facing master proof spine for the TEBAC Birch--Swinnerton--Dyer program for elliptic curves over \ (Q\). The manuscript compresses the longer modular proof dossier into a shorter theorem-oriented structure organized around the chain-I-II-III-IV-V. \ BSD-I fixes the centered operator and parity package at the arithmetic center \ (s=1\), with centered variables\=s-1, = (s-1) ². -II supplies the determinant, parity/reference-factor, and completed comparison package through the \ (OC1--OC4\) route: corrected centered heat trace, Mellin/zeta regularization, centered determinant, normalized reference factor, completed comparison quotient, and no-spurious-center control. BSD-III extracts the analytic-rank identityₒ=₁L (E, s) = (D₁ₒ₃-1). -IV is the arithmetic bridge layer, organized around the \ (H1--H5\) gates: restricted Poitou--Tate detector saturation, support-boundary reciprocity cancellation, terminal exactness, generator-complete Mordell--Weil realization, and raw local Néron-symbol height normalization. BSD-V performs the terminal completed-to-classical leading-term assembly. The manuscript is intended as a compact journal-style proof spine. The longer 1228-page master dossier is cited as a verification archive containing the expanded module-level derivations, proof ledgers, and non-circularity checks. This journal version should be read together with that dossier for full source-level verification. Status statement: the current proof spine records the TEBAC modular route toward the Birch--Swinnerton--Dyer rank statement and leading-term formula. The remaining critical verification boundary for a fully unconditional external proof is the theorem-level closure of the BSD-IV arithmetic bridge, especially the \ (H1--H5\) hard gates, primitive spectral-to-Kummer bridge, and raw Néron--Tate height comparison, without importing the final BSD formula, pre-assumed rank equality, \ ( (E) =0\), or downstream BSD-V assembly.