Finite-Horizon Structures XIII: Hybrid Jump Dynamics, Threshold Events, and Metastable Maintenance in Completed Propagation Domains

This article introduces a stochastic hybrid extension of the Finite-Horizon Structures framework. It does not modify the intrinsic projective core of the theory, nor the deterministic propagation, singular transition, stratified completion, or relational action layers developed in the preceding articles. Its purpose is more specific: to add an external Poisson-type jump mechanism on top of completed finite-horizon propagation domains. In this hybrid setting, states evolve continuously between jump events according to compatible deterministic transport. At marked jump times, they undergo finite discontinuous transitions determined by admissible jump maps. The resulting dynamics is piecewise deterministic, structurally compatible between jumps, and discontinuous only at externally specified stochastic events. The article develops the geometric and structural consequences of this jump layer. It introduces a finite logarithmic jump displacement as the projectively meaningful observable associated with jumps, studies jump-induced threshold crossing, scalar-level preservation, structural loss of tangency, regular-to-critical transitions, and the destruction or preservation of maintained domains. It then extends deterministic reachability to hybrid reachability by allowing admissible chains made of both continuous propagation segments and jump events. This leads to hybrid envelopes, stochastic fronts, exit times, threshold hitting times, survival laws, and a classification of maintained regimes into persistent, metastable, jump-fragile, flow-unstable, hybrid-fertile, and hybrid-sterile cases. The article also provides a measure-theoretic and generator-level formulation of the hybrid dynamics, clarifying which structures remain projectively meaningful, which require additional measure-admissibility assumptions, and which belong only to the external stochastic extension. Finally, local logarithmic normal forms are established near regular thresholds. These normal forms show that compatible deterministic transport acts continuously in the transverse threshold coordinate, while jumps act by finite displacement. This gives canonical local models for persistent, metastable, basin-escape, multi-basin, and scalar-tangential jump regimes. The resulting framework should be understood as a structural stochastic extension of compatible transport and maintained propagation. It is not presented as a physical theory of quantum collapse, measurement, or decoherence, and it does not derive a microscopic origin for Poissonian jump events. Rather, it identifies a new external layer of the Finite-Horizon Structures programme: a coherent theory of jump-extended structural propagation, threshold loss, metastable maintenance, and stochastic front geometry. This article forms part of the Ranesis framework developed by Alexandre Ramakers.