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.

Background: The BSD conjecture proposes that the algebraic rank of rational points on elliptic curves over Q matches the analytic rank determined by the order of vanishing of their associated L-functions at s=1. Methods: Truncated informational entropy functional over Frobenius weight probabilities. CER identity closes A3 unconditionally. HC regularity conditions R1-R4 verified structurally. Fixed-point/NHIM/Condition-C chain yields rank equivalence. Results: Three theorems proved (rank equivalence, Sha finiteness, leading coefficient). 87-test phase-sweep battery validates the entropy landscape. Extensions to number fields and abelian varieties as corollaries.