A Structural Tension Within Operational Geometry: Two Conditional Approaches to BSD

Two conditional approaches to the Birch and Swinnerton-Dyer conjecture have been developed within the Operational Geometry framework: one via Arithmetic Shock Dynamics, the other via Operational Selmer Geometry. Both reject the classical framing of BSD as a counting problem. Both claim Operational Geometry as their parent framework. Yet they arrive at directly opposing conclusions about what constitutes an admissible proof step for rank equality. This note does not resolve the tension. It identifies it precisely, argues that it is productive rather than pathological, and records why collapsing it prematurely in either direction would be a mistake.