Toward a K-Theoretic Deformation of BSD: A Framework for Fractional Rank and Regulator Influence

This paper explores an exploratory deformation of the concept of rank in the context of the Birch and Swinnerton-Dyer (BSD) Conjecture. Rather than treating rank strictly as a discrete invariant, we examine whether effective rank can be modeled as a continuous function influenced by the logarithm of the regulator. Two simple deformation models—additive and multiplicative—are introduced and evaluated on both real elliptic curves and synthetic surfaces. The results support the hypothesis that “soft” rank behavior may offer analytic insights that complement classical structure, especially in low-rank regions. We interpret this not as a refutation of BSD, but as a conceptual broadening. The work is situated at the interface of algebraic K-theory, arithmetic geometry, and motivic exploration.