Threshold-Constrained Regime Trajectory Families Between Declared Endpoints

@misc{son2026thresholdconstrained,
  title = {{Threshold-Constrained Regime Trajectory Families Between Declared Endpoints}},
  author = {Vien Nguyen Son},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23570328},
  url = {https://zenodo.org/records/23570328},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Threshold-Constrained Regime Trajectory Families Between Declared Endpoints
AU  - Vien Nguyen Son
PY  - 2026
DA  - 2026/10/22
PB  - Zenodo
DO  - 10.5281/zenodo.23570328
UR  - https://zenodo.org/records/23570328
AB  - Threshold-Constrained Regime Trajectory Families Between Declared Endpoints defines RegimeTrajectoryFamily(RegimeEndpointPair, RegimeThresholdSequence), a domain-independent architecture for classifying families of state or regime trajectories between two independently declared regime descriptions. RegimeEndpointPair(InitialRegimeDescriptor, TerminalRegimeDescriptor) fixes the comparison endpoints before family adjudication, while TrajectoryFamilyEvaluationFrame(RegimeEndpointPair, RegimeThresholdSequence) fixes TrajectoryBasis, TrajectoryEquivalenceRule, RequiredThresholdSet, ExcludedThresholdSet, inherited sequence structure, support and regularity conditions, scale, evaluation window or horizon where required, evidence relations for empirical claims, transformation class, uncertainty declarations, and rejection conditions.  Candidate trajectories are first evaluated inside the declared TrajectoryBasis and then against endpoint membership, threshold-set conditions, sequence compatibility, support and regularity conditions, evidence, transformation, uncertainty, and rejection requirements. TrajectorySequenceCompatibilityStatus preserves the applicable RegimeThresholdSequence, including concurrency blocks, branch membership, exclusions, conditional relations, multiplicity structure, and inherited restrictions. TrajectoryAdmissibilityStatus classifies individual candidates. The admissible candidate set is then quotiented by the predeclared TrajectoryEquivalenceRule to construct RegimeTrajectoryFamily, allowing family existence, emptiness, singleton structure, multiplicity, restriction, and underdeclaration to receive separate scientific status.  The paper establishes two local structural results inside this trajectory-admissibility architecture. The Regime-Retention Obstruction classifies a candidate as inadmissible when it retains unchanged a constitutive initial-regime relation whose unchanged retention is independently incompatible with the declared terminal regime. Retention-Constraint Monotonicity establishes set inclusion when one declared retention constraint is removed while the remaining frame is held fixed. These results are membership and set-theoretic consequences inside the declared frame; they do not select a relation for intervention, supply a transition operator, or establish control.  Empirical reconstruction is treated as a downstream evidential layer. Observations can support reconstructed candidate trajectories, threshold events, threshold-order relations, and family-class assignment at a declared resolution. A completed surviving transition with the required identity-continuity witness additionally supports a retrospective successor-regime result: the post-transition constitutive structure defines a successor regime for the realized terminal state. The prospectively declared TerminalRegimeDescriptor and the a posteriori successor regime remain separately typed until an endpoint-equivalence relation is established.  The framework preserves separate scientific offices for threshold identity, threshold precedence, regime-sequence representation, complete state or regime trajectory classification, temporal-corridor analysis, empirical identification, external realization, forecasting, and directed intervention. CorridorInterfaceStatus supplies only the typed interface required to submit a trajectory representation to the independently defined temporal-corridor architecture. The paper is relevant to viability and reachability theory, dynamical systems, stochastic and hybrid systems, timed and hybrid automata, critical-transition and regime-transition analysis, formal classification, and machine-readable scientific routing.  The direct scientific parent is From Threshold Partial Orders to Regime Sequences. Regime Thresholds Before Timelines: Admissibility Boundaries, Crossing Conditions, and Hitting Times supplies foundational threshold identity and event semantics; Partial Ordering of Regime Thresholds: Necessary, Conditional, Concurrent, and Incomparable Transitions supplies the inherited relation field; Composite Admissibility and First-Boundary Exit in Coupled Systems and Threshold Identifiability: Observable Signatures of Approach and Crossing Without Prediction enter where composite or empirical claims are activated. External mathematical anchors include viability theory, viability-kernel approximation, reachable-set analysis, hybrid automata, and timed automata. The maximum public result is a structural classification of endpoint-constrained trajectory families satisfying the declared scientific frame and distinguishing whether those families are empty, singleton, multiple, restricted, or underdeclared. Operational composition requires additional independently validated relations connecting scientific structure to objectives, manipulable variables, actuators, calibrated causal transfer, intervention timing, and feedback.
ER  -