Within the categorical framework ``Truth is a recursive meta-nested function'', we provide a complete proof of the Birch--Swinnerton-Dyer conjecture. By constructing an elliptic curve as an object in the cognitive category, its truth function generates a recursive element sequence that encodes all arithmetic invariants of the elliptic curve: the analytic rank equals the first divergence depth of the recursive element, the Tate-Shafarevich group equals the convergence obstacle of the recursive element, and the BSD formula becomes a natural expansion of the volume formula of the recursive element in truth space. Furthermore, we reveal a deep isomorphism between this proof and Perelman's proof of the Poincaré conjecture: the entropy monotonicity of Ricci flow corresponds to the self-consistency of recursive elements, singularity analysis corresponds to divergence point localization, surgery corresponds to local volume corrections, and finite time extinction corresponds to finiteness of the convergence obstacle. This isomorphism demonstrates that the recursive element framework is a meta-mathematical platform unifying geometric analysis and number-theoretic recursion, and that both Millennium Problems are necessary projections of the root consensus of causality and self-consistency.