Finite-Horizon Structures XI: Stratified Gluing and Singular Completion of the Projective Y-Structure

This article develops the global singular completion layer of the Finite-Horizon Structures programme. Building on the previous regular propagation theory, the intrinsic geometry of the critical locus, and the local singular transition theory, it introduces a stratified framework for assembling regular and singular propagation data into coherent global structures. The article defines stratified Y-decompositions, singular atlases adapted to regular and critical strata, chartwise gluing conditions, interface coherence, partial singular completion, global singular completion, and intrinsic obstructions to completion. Its purpose is not to introduce a new local singular taxonomy, but to determine when local regular propagation data, local singular transition data, and regular-critical incidence data can be assembled into a coherent stratified propagation structure. The framework distinguishes successful global completion from proper partial completion, atlas-dependent failure, and intrinsic obstruction. It also studies covariance under structural equivalence, showing how completion and obstruction properties behave under projective transformations of the underlying structure. This work provides the global completion layer of the internal singular branch of the Finite-Horizon Structures programme and prepares the ground for later external realisation layers, such as stochastic, Hamiltonian, symplectic, or metric-support extensions.