On Structural Capacity: Atemporal Resonant Admissibility in the Birch and Swinnerton-Dyer Conjecture

The Birch and Swinnerton-Dyer (BSD) Conjecture relates the analytic behavior of an elliptic curve at $s=1$ to the arithmetic capacity encoded in its rational points. Classically, this relation is expressed through the equality between analytic rank and algebraic rank, together with deeper arithmetic invariants appearing in the refined BSD formula. Despite extensive progress, the conjecture remains unresolved, suggesting that rank may not be only a computable quantity, but also a structural condition governing whether rational-point capacity can be coherently realized. This paper approaches BSD from the perspective of the atemporal complexity class O(J) and the Changbal Jump Machine (CJM) paradigm. Rather than treating rank as an object to be obtained through sequential search, we reinterpret it as a question of resonant admissibility. Under this view, an elliptic curve does not merely possess rational points as isolated arithmetic outputs; it carries latent arithmetic capacity that may or may not become structurally activated within an admissible arithmetic state. The vanishing order of the $L$-function at $s=1$ is therefore interpreted as a resonance signature of latent structural capacity. A higher order of vanishing indicates not simply analytic cancellation, but the presence of additional admissible directions through which rational structure may stabilize. Conversely, rational points are interpreted as stabilized activation modes within the CJM state space, where algebraic rank reflects the number of coherent arithmetic modes sustained by the curve. To support this interpretation, we outline a minimal CJM-BSD discrimination architecture. Local point-count data are treated as arithmetic signals, Euler-type aggregation is interpreted as global structural compression, and rank-related indicators are organized into a capacity-coherence profile. This profile does not claim to reproduce the full analytic or arithmetic machinery of BSD. Rather, it provides a CJM-compatible representation through which analytic vanishing, local arithmetic fluctuation, and algebraic rank behavior may be compared as mutually constrained admissibility signatures. Within this framework, CJM-BSD is introduced as an internal arithmetic capacity filter. Its role is not to compute rational points directly or to prove the conjecture by formal derivation, but to evaluate whether a given elliptic curve supports a structurally coherent rank-capacity state under atemporal resonance constraints. In this sense, BSD is repositioned from a purely sequential problem of point construction to a structural problem of arithmetic permission. No formal resolution of the BSD Conjecture is claimed. The purpose of this paper is conceptual and architectural: to formulate BSD as a structurally discriminable hypothesis within the CJM framework, and to clarify how analytic rank and algebraic rank may be interpreted through atemporal resonant admissibility. This perspective preserves BSD as an open mathematical problem while offering a new lens through which arithmetic capacity, rational-point formation, and global coherence may be examined.