Finite-Horizon Structures XIV: Symplectic Realizations and Poisson-Compatible Dynamics

This article develops the fourteenth layer of the Finite-Horizon Structures programme by introducing a symplectic and Poisson-geometric realization of the projective finite-horizon framework. The article treats symplectic forms, Poisson brackets, and Hamiltonian vector fields as external structures added on top of the inherited projective threshold geometry, not as objects derived from the intrinsic core of the theory. The central aim is to classify how Hamiltonian dynamics interact with the threshold structure determined by the finite-horizon representative and its coherence one-form. Hamiltonian transport is classified according to whether it preserves persistence leaves, moves toward higher maintained levels, moves toward threshold loss, or exhibits mixed behaviour. This leads to preservation and exit criteria for maintained superlevel domains, Hamiltonian reachability relations, propagation envelopes, threshold fronts, and local normal forms in threshold-adapted Darboux coordinates. The article also clarifies the relation between this continuous Hamiltonian layer and the stochastic hybrid jump layer introduced in FHS XIII. While FHS XIII studies discontinuous threshold transitions governed by finite jump displacements, the present article studies continuous Hamiltonian threshold transport governed by Poisson-geometric dynamics. The two layers are therefore complementary external realizations of the same inherited finite-horizon threshold geometry. The work does not claim to derive a canonical symplectic structure, a Lorentzian metric, a physical Hamiltonian, or a quantum measurement law. Its contribution is more precise: it identifies the structural interface through which an externally supplied symplectic or Poisson system may realize, preserve, cross, or obstruct the threshold geometry of the projective finite-horizon framework. This article forms part of the Ranesis framework developed by Alexandre Ramakers.