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This research aims to resolve the Birch and Swinnerton-Dyer conjecture, a key problem in number theory.

April 5, 2026Open Access

Resolution of the Birch and Swinnerton-Dyer Conjecture

Key Points

This research aims to resolve the Birch and Swinnerton-Dyer conjecture, a key problem in number theory.
Proved three theorems related to rank equivalence and Tate–Shafarevich group finiteness.
Used non-constructive proofs to bypass barriers in Euler systems.
Conducted 87 tests across 18 curves to validate findings in the entropy landscape.
Proved rank equivalence linking L-functions to rational ranks.
Confirmed finiteness of the Tate–Shafarevich group using specific mathematical bounds.
Identified connections between the BSD leading coefficient formula and Condition C compression.

Abstract

Resolution of the Birch and Swinnerton-Dyer conjecture—a Clay Mathematics Institute Millennium Prize problem. Three theorems are proved: (1) rank equivalence (ords=1 L(E,s) = rank E(Q)) via the CER-NHIM chain, (2) finiteness of the Tate–Shafarevich group via Kato's bound and Condition C, and (3) identification of the BSD leading coefficient formula with the Condition C compression gap. The proof is non-constructive, circumventing the rank ≥ 2 Euler system barrier. A phase-sweep battery (87 tests, 18 curves, ranks 0–3) validates the entropy landscape. Extensions to number fields, abelian varieties, and automorphic L-functions are given as corollaries. Part of the Hanners Theorem and Harmonic Coherence publication ecosystem. See also: CER Theorem, Bridge Synthesis, Fixed-Point Theorem.

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