Relative Topology, Principal Bundle Reduction, and Index Theory on Perforated Information Manifolds: Unified Framework Toward S(U(3)\times U(2)), Three-Generation Index, and Yukawa--Winding

Full-rank density matrix manifold D^{\rm full}_N=\{\rho>0,\ tr\rho=1\} is open convex contractible, Uhlmann principal bundle admits global square-root section w=\rho on full domain, thus absolute integer-valued topological invariants are absent on full domain. This paper turns to perforated relative topology: In N=5 case, removing tubular neighborhood of three--two level gap closing set \Sigma_{3|2}=\{\lambda_3=\lambda_4\} from D^{\rm full}_5 yields perforated domain D^{\rm exc}. On D^{\rm exc}