The Birch and Swinnerton-Dyer Conjecture: Toward an Entropy-Minimization Resolution via Harmonic Coherence and Hanners Theorem

A conditional framework for the Birch and Swinnerton-Dyer (BSD) conjecture—a Clay Mathematics Institute Millennium Prize problem—using Harmonic Coherence (HC) and Hanners Theorem. The framework establishes discrete equilibrium eigenstates corresponding to critical zeros of elliptic curve L-functions via entropy-minimization principles, conditional on a spectral gap axiom (A3). Computational validation across benchmark elliptic curves (ranks 0–3) confirms the structural predictions. The analytical extension strategy adapts the four-step Riemann template (domain symmetry, value distribution, persistence, defect instability) to the BSD setting using Bagchi–Steuding L-function universality. Part of the Hanners Theorem and Harmonic Coherence publication ecosystem. See also: CER Theorem, Bridge Note, Fixed-Point Theorem.