Birch & Swinnerton-Dyer Resolution_PFF_Hill_June2025.

This manuscript presents a complete, constructive resolution of the Birch and Swinnerton–Dyer conjecture for all elliptic curves defined over the rational numbers . Utilizing a novel deterministic approach called the Prime-Function Framework (PFF), the proof integrates two core components: the Prime Discovery System (PDS), which systematically generates primes without probabilistic methods, and Structured Entropy Layering (SEL), a prime-indexed system of arithmetic transformations. These mechanisms deterministically construct rational points, regulators, and other key arithmetic invariants that align precisely with the analytic behavior of the associated -functions. The manuscript rigorously establishes the equality of algebraic and analytic ranks, verifies the leading coefficient formula predicted by the conjecture, and proves the finiteness of the Tate–Shafarevich group. Supported by explicit numerical examples and canonical height computations, this work offers a unified arithmetic-analytic proof that satisfies the full criteria set forth by the Clay Mathematics Institute.