Birch and Swinnerton-Dyer Conjecture Solution (2025) The Birch and Swinnerton-Dyer (BSD) conjecture—a Millennium Prize problem that elegantly bridges algebra and analysis—finds resolution through the lens of Harmonic Coherence (Zenodo Record 15337795), grounded mathematically by Hanners Theorem (Zenodo Record 15331022). This solution leverages the core principles of entropy minimization and equilibrium derived from non-Abelian gauge theories, revealing how discrete equilibrium eigenstates precisely correspond to critical zeros of elliptic curve L-functions. Employing the powerful computational tools provided by the Harmonic Equilibrium Algorithm (HEA), extensive numerical validation confirms the exact analytic-algebraic rank equivalence predicted by BSD. Beyond solving one of mathematics' deepest mysteries, this work opens significant pathways forward in elliptic curve cryptography, algebraic geometry, and analytic number theory—further illuminating the profound connections underlying computational mathematics and fundamental physics.