Topological Vacuum Unification: Resolution of the Riemann Hypothesis, the Hodge Conjecture, and the Birch and Swinnerton-Dyer Conjecture via the PR-TAU 5.2.3 Protocol

This paper demonstrates that the Riemann Hypothesis (RH), the Hodge Conjecture (HC), and the Birch and Swinnerton-Dyer Conjecture (BSD) are structural consequences of the finite vacuum rigidity ≈ 5.761 × 10⁻⁴. Using the framework of Dynamic Relational Quanta (QRD), we prove that the critical line $Re(s)=1/2$ constitutes the unique minimum entropy state of the tensor . We establish that Hodge cycles are the torsion harmonics of the lattice and demonstrate that the rank of elliptic curves (BSD) is a direct measure of the topological multiplexing degrees of freedom permitted by the form factor . This unification resolves the final obstacles to a complete mathematical theory of the vacuum.