A Local Phase–Variational Formula for the BSD Rank

This short technical note introduces a concise phase–variational expression for the analytic rank of an elliptic curve over the rationals.It presents a purely local, topologically stable formula linking the Birch–Swinnerton–Dyer (BSD) rank to the phase–winding index of the completed L–function around the central point. The work demonstrates that the analytic rank coincides with the quantized winding number of the local phase field, establishing the phase–energy functional as a robust diagnostic of the BSD phenomenon.Numerical evaluations for several elliptic curves confirm the stability and correctness of the phase–winding index under contour deformations, supporting its interpretation as a local topological invariant rather than a numerical artifact. This note serves as a computational and conceptual supplement to the paper “A Phase–Balanced Interpretation of the Birch–Swinnerton–Dyer Conjecture” (DOI: 10.5281/zenodo.17333682), where the full variational and operator–theoretic framework is developed.Together, the two works outline a unified geometric interpretation of the BSD conjecture in terms of local phase equilibrium and topological degeneracy.