A DISCRETE NORMALIZATION APPROACH TO THE BIRCH–SWINNERTON–DYER CONJECTURE

We present a normalization-first approach to the Birch--Swinnerton--Dyer conjecture for elliptic curves over Q. By fixing from the outset a global minimal model, the Neron differential, the real Neron period, and the N\'eron--Tate canonical height pairing, all continuous and real-analytic contributions are isolated explicitly in the period and the regulator. With this canonical normalization in place, the conjectural formula separates naturally into continuous and purely discrete arithmetic components. We show that the resulting normalized formulation is rigid under Q-isogenies, forcing exact compensation among regulators, torsion, Tamagawa numbers, and the Tate--Shafarevich group. Quadratic twists are then used to vary arithmetic data in a controlled way, with all variation localized at finitely many places. Selmer groups provide the mechanism that translates these local conditions into global arithmetic information and allow precise control of the Tate-Shafarevich group. This reorganization introduces no new arithmetic invariants or conjectural inputs. Rather, it exposes the internal coherence of the Birch-Swinnerton-Dyer formula and reduces it to a finite, local arithmetic problem once canonical normalization is fixed. The prime-by-prime elimination of the remaining normalized defect, via S-admissible quadratic twists andone-place Selmer modifications, is proved in Appendix F. Appendix G provides an explicit audit protocollisting the normalization checks, dependency chain, and the standard external inputs used.