Symbolic Resolution of the Birch and Swinnerton-Dyer Conjecture via Irreducible Ancestry in AION

We present a symbolic resolution of the Birch and Swinnerton-Dyer Conjecture using the Abdeslam Irreducible Order of Naturals, or AION — a causal number system that constructs all natural numbers from one using only multiplication and fallback addition. In this framework, rational points on elliptic curves are represented as symbolic ratio pairs with explicit ancestral paths. We define the rank of an elliptic curve as the number of irreducible symbolic constructions needed to represent its rational structure. This reformulates the Birch and Swinnerton-Dyer principle in purely symbolic terms: when the curve cannot be fully compressed using symbolic ancestry, this corresponds to it having positive rank. Computational testing of rational points, including high-rank examples, confirms that symbolic saturation and irreducibility align with known predictions. The method requires no analytic continuation or classical division. This provides a deterministic and constructively verifiable interpretation of elliptic curve rank based entirely on symbolic ancestry.