Finite-Horizon Structures X: Singular Propagation and Transition Geometry of the Projective Y-Structure

This article develops the local singular propagation layer of the Finite-Horizon Structures programme. It studies what becomes of admissible local propagation near critical points, critical strata, and critical threshold levels, where the regular first-order propagation geometry is no longer sufficient. The paper introduces second-order singular classifiers, defines singular transition carriers, transition loci, and regular-singular interfaces, and classifies the resulting local mechanisms into crossing, blocking, trapping, bifurcation, neutral, and higher-order indeterminate types. The theory is developed for Morse, Morse–Bott, and degenerate singular regimes, and is related to the geometry of superlevel domains and to a local notion of singular reachability. The resulting framework remains local and intrinsic to the singular projective Y-structure, and prepares the next stage of the programme devoted to stratified gluing and singular completion. This article forms part of the Ranesis framework developed by Alexandre Ramakers.