Arithmetic K-Theory of Family Replication : No-Go Theorem on the Closed Cayley Plane

We present a rigorous, computation-ready spine for index-theoretic family generation on the closed Cayley plane X=\mathbb{O}P^{2}=F_{4}/Spin(9). Using the non-circular Borel-Hirzebruch normalization from the tangent bundle, we compute the A-genus and the Chern characters of the canonical homogeneous bundles induced by the fundamental Spin(9) representations. We demonstrate that the corresponding twisted Dirac indices identically vanish, establishing a strict "no-go" constraint for generating chiral families via these natural twists. To bypass this obstruction, we introduce a canonical generator x\in K^{0}(X) which allows for a rank-3 description of the arithmetic family index N_{fam}=q. Furthermore, we demonstrate that phenomenological completion necessitates a topological symmetry-breaking transition, explicitly detailing a minimal geometric extension X^{\prime}=\mathbb{O}P^{2}\times S^{2}, the derivation of a \mathbb{Z}_{4} discrete remnant, and the structural constraints imposed by FI-tail slope quantization and modulus stabilization.