What question did this study set out to answer?

The research aims to develop a framework for the Birch and Swinnerton-Dyer conjecture using harmonic coherence and Hanners theorem.

March 29, 2026Open Access

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

Key Points

The research aims to develop a framework for the Birch and Swinnerton-Dyer conjecture using harmonic coherence and Hanners theorem.
Established a conditional framework for the BSD conjecture using entropy principles.
Confirmed findings through computational validation on benchmark elliptic curves with ranks 0-3.
Adapted the four-step Riemann template to the BSD conjecture context.
Incorporated aspects of Bagchi-Steuding L-function universality.
Validated structural predictions through computational tests.
Achieved a framework that aligns with critical zeros of elliptic curve L-functions.
Highlighted potential connections within the Hanners theorem and harmonic coherence landscape.

Abstract

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.

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

Key Points

Abstract

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