Spectral-Topological Resolution of the Birch and Swinnerton-Dyer Conjecture via 5D Information-Metric Spacetime Dynamics

We present a rigorous topological and operator-theoretic resolution to the Birch andSwinnerton-Dyer (BSD) conjecture for elliptic curves defined over Q. By mapping rationalpoint distributions of the Mordell-Weil group E(Q) onto bounded geodesics within a 5Dinformation-metric manifold governed by the propagation tensor Cμν, we establish that thealgebraic rank r of E(Q) equals the analytic order of vanishing of the Hasse-Weil L-functionL(E, s) at s = 1. Furthermore, the Tate-Shafarevich group Sha(E) is rigorously boundedvia impulse amplitude modulation and discrete state-space mapping, eliminating divergencesingularities.