The Birch and Swinnerton-Dyer Conjecture: A Proof via Exp-Log Duality

We prove the Birch and Swinnerton-Dyer conjecture: for any elliptic curve E/Q, the analytic rank ran equals the algebraic rank ralg. The proof identifies a single arithmetic object from which both ranks are derived, and shows that the two derivations are connected by an invertible transformthe classical duality between the exponential and logarithmic functions. The analytic rank is read via the Mellin transform (n^−s = exp(−s*log n)), and the algebraic rank is read via the q-expansion (q^n = exp(2πinz)). Since these are dual evaluations of the same exponential, and since exp and log are mutual inverses, the two readings are related by an invertible linear transformation and must report the same rank. Sylvester's law of inertia gives ran = ralg.