Finite-Horizon Persistence Under Declared Constraints: Survival Domains and a Canonical Order-Theoretic Representation

Many natural and engineered systems evolve under constraints that restrict the set of admissible states. Classical frameworks study invariant sets, viability regions, survival probabilities, and exit-time events, while the explicit treatment of threshold-defined admissible subsets induced by sampled finite-horizon persistence is not usually isolated as a primary state-space object. This paper formulates a finite-horizon framework for Persistence Under Declared Constraints (PSUC). For a fixed constraint set, sampling step, persistence horizon, and tolerance level, the associated survival domain is the set of initial conditions whose sampled trajectories remain inside the declared constraint set with probability of at least 1−α. Under explicit regularity assumptions, survival domains are closed superlevel sets of the persistence field and form a nested filtration as the persistence horizon increases. This filtration admits a canonical intrinsic representation through a maximal admissible sampled-horizon field whose sampled superlevel sets recover it exactly. The same field also induces a canonical admissibility preorder; after quotienting by horizon-indistinguishability, this yields a partial order and its associated Alexandrov topology, in which the sampled survival filtration is represented as an upper-set filtration. The Alexandrov construction itself is classical; the contribution lies in the canonical order induced by the sampled admissibility-depth field and in the resulting canonical order-theoretic representation of the filtration. A secondary scalar ordering is also obtained for any lower-bounded auxiliary scalar function. Under an additional continuity assumption, the framework further yields boundary localization at the threshold level and an inheritance relation for connected components along the filtration. Finally, the paper shows that sampled survival domains need not coincide with continuous-time survival sets, thereby clarifying the intrinsically protocol-dependent nature of the object studied. The contribution is therefore a restricted but explicit analysis of threshold-defined admissible-state filtrations induced by sampled finite-horizon persistence, together with a canonical order-theoretic representation of the same filtration, formulated in a way that remains compatible with existing work on viability, stochastic survival, and exit-time analysis.