Two-Field Non-Proportionality and BSD at Rank 2

We prove the Geometric Kolyvagin System condition (GKS) for non-CM elliptic curvesE/Q of analytic rank 2 admitting at least one prime of multiplicative reduction. The proof splits by order of vanishing of a three-variable p-adic L-function: in Case A (simple zero) GKS followsfrom published results alone; in Case B (double zero) GKS follows from Castella–Hsieh 2022with the Selmer-rank input dim H1f = 2 supplied by Papers I and IV. Combined with the GKS⇔ PRC ⇔ Two-Field Non-Proportionality equivalence web of Paper I, this yields the Two-FieldNon-Proportionality principle and the Birch and Swinnerton-Dyer conjecture for this class. The main technical inputs are: (1) one-dimensionality of the Bloch–Kato local Selmer group at goodprimes; (2) construction of globally nonzero Bertolini–Darmon–Rotger diagonal cycle classes; and (3) an order-of-vanishing dichotomy governing their local behavior at p. The BSD conclusion is conditional on Papers I and IV; under their results, it is unconditional for the stated class of rank-2curves, which constitutes a density-1 subset of non-CM rank-2 elliptic curves over Q with conductor > 1.