Causal Number Fields: A Recursive Interpretation of the Birch and Swinnerton-Dyer Conjecture

This paper reframes the Birch and Swinnerton-Dyer conjecture (BSD) within the framework of causal mathematics. It proposes that BSD is not merely an analytic identity, but a manifestation of a recursive stability principle across entropic, semantic, and topological domains. The L-function is interpreted as a semantic compression filter, and its zeros as entropy singularities requiring repair by modular membranes—recursive stabilizers that span higher causal dimensions. The BSD relation emerges naturally as a feedback-invariant, revealing not a singular problem but a universal law of information preservation and collapse in recursive systems. The result is a shift from solving a conjecture to embedding it within the ontological architecture of causal mathematics.