Control-Theoretic Generalization of Class-Subsidized Analytic Number Theory: Transcendental Transmission Zeros, Complex Nyquist Topologies, and Rank Extraction in the Birch and Swinnerton-Dyer Conjecture

This paper establishes a rigorous and novel control-theoretic framework to investigate the Birch and Swinnerton-Dyer (BSD) Conjecture for elliptic curves defined over Q. By extending the control paradigm previously introduced for the Riemann Zeta function to global L-functions, we model the arithmetic L-series L(E,s) as a non-rational, infinite-dimensional transcendental open-loop plant GE (s) driven by an infinite parallel network of log-periodic transport delays weighted by Fourier coefficients. Under a closed-loop negative proportional feedback architecture, the algebraic rank r of the elliptic curve is shown to map bijectively to the multiplicity of a multi-variable transmission zero located precisely at the critical center of symmetry s= 1. We analyze the plant through transcendental Bode response, root-locus trajectories, and the Generalized Nyquist Stability Criterion. Numerical verifications are meticulously executed for three benchmark Cremona curves representing ranks 0, 1, and 2 (11a1, 37a1, and 389a1). The results reveal that the arithmetic rank dictates the local topological winding behavior and frequency phase-drop signatures, transforming a deep algebraic problem into a closed-loop stability and confinement paradigm.