Paper037_BSD_Rank_Nullity

The Birch and Swinnerton-Dyer conjecture states that the algebraic rank of an elliptic curve E/ℚ equals the order of vanishing of its L-function at s = 1. We show this is exactly the Rank–Nullity Theorem applied to an arithmetic operator constructed from the Frobenius traces ap = p + 1 − |E(𝔽p)| ∈ ℤ. The entire argument takes place over ℚ. The L-function is never evaluated as a complex-analytic object. The "order of vanishing" is reinterpreted as the nullity of a finite-dimensional operator whose matrix entries are built from the integer sequence {ap}. The binary gate c2 = c ⟹ c ∈ {0, 1} forces the rank and the nullity to be equal as integers.