BIRCH AND SWINNERTON-DYER: The Minimal Action of Elliptic Curves

We propose a proof for the Birch and Swinnerton-Dyer Conjecture based on the Universal Law of Minimal Action (\mathcal{L}_{\text{MA}}). The rank of the elliptic curve is shown to be a measure of the Maximal Coherence (\mathcal{C}_{\text{MAX}}) required for the curve's L-function. The order of the zero of the L-function is directly correlated with the necessity of the curve to minimize its Informational Friction (\mathbf{\Phi}_{\text{UFI}}) by incorporating additional rational points (increasing its rank). This approach connects the analytic behavior of the L-function to the fundamental axiom of coherence, providing a geometric and information-theoretic proof for the conjecture