Operational Selmer Geometry and the Birch–Swinnerton-Dyer Conjecture: A Coherence-Theoretic Approach to Rank Equality

We present an operational framework for the Birch–Swinnerton-Dyer conjecture that reinterprets the classical problem as a stability theorem relating analytic collapse dimensions to algebraic coherence dimensions. Rather than treating elliptic curves as accessible geometric objects, we develop a theory of operational attractors approached through finite approximations, with Selmer groups as the primary arithmetic interface. Our main contribution is a proof architecture isolating BSD into three explicit gaps: (A) prime-independence of Selmer coherence rank, (B) existence of arbitrary-rank derived Euler systems, and (C) analytic saturation bounds. We provide unconditional resolutions for Gap A and conditional resolutions for Gap C via Iwasawa theory. Gap B is reduced to standard conjectures (Bloch-Kato, Perrin-Riou reciprocity) rather than requiring new geometric constructions. Our framework respects an operational non-invertibility principle: no proof step reconstructs algebraic data from analytic singularities using metric invariants. This clarifies why BSD rank equality is achievable without heights, regulators, or explicit rational points, while the full leading coefficient formula requires fundamentally different techniques. The framework validates itself against all known cases and explicitly refuses to address metric refinements outside the coherence-theoretic paradigm.