Hysteresis, Metastability, and Return Thresholds in Regime Transition

@misc{son2026hysteresis,
  title = {{Hysteresis, Metastability, and Return Thresholds in Regime Transition}},
  author = {Vien Nguyen Son},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23497195},
  url = {https://zenodo.org/records/23497195},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Hysteresis, Metastability, and Return Thresholds in Regime Transition
AU  - Vien Nguyen Son
PY  - 2026
DA  - 2026/10/20
PB  - Zenodo
DO  - 10.5281/zenodo.23497195
UR  - https://zenodo.org/records/23497195
AB  - Hysteresis, Metastability, and Return Thresholds in Regime Transition defines ReturnThresholdStructure(ThresholdClaim), a claim-relative architecture for relating an already valid forward threshold to a separately constructed return threshold. A return is evaluated by ReturnThresholdClaim(ThresholdClaim) under the foundational RegimeAdmissibleSet, ThresholdBoundary, crossing, and persistence semantics, with ReturnEvaluationFrame(ThresholdClaim), PriorRegimeReference(ThresholdClaim), PostTransitionRegimeReference(ThresholdClaim), ReturnReferenceCriterion(ThresholdClaim), ScaleSet, EvalWindow, Horizon, EvidenceRel, UncertaintyDeclaration, TransformClass, and RejectCond explicitly declared. The framework therefore preserves the distinction between a prior regime as a scientific reference and that regime as an event or endpoint achieved by a particular mathematical or physical process.  The paper constructs ForwardReturnComparisonMap(ThresholdClaim) and ForwardReturnAsymmetryRelation(ThresholdClaim) for type-valid comparison and assigns distinct HysteresisStatus(ThresholdClaim), MetastableResidenceStatus(ThresholdClaim), ReturnAccessibilityStatus(ThresholdClaim), ReturnCompletionStatus(ThresholdClaim), ReversibilityStatus(ThresholdClaim), ReturnRestrictionStatus(ThresholdClaim), and IrreversibilityStatus(ThresholdClaim). History-dependent hysteresis differs from asymmetry induced by changing a scale or a model; metastable residence differs from stable endpoint membership; an accessible return trajectory differs from a qualifying persistent return. Reversibility is relative to a declared regime criterion, while a positive irreversibility classification requires an adequate no-return search or a demonstrated necessary-condition obstruction in the stated model, horizon, and transformation class.  The formal analysis accommodates finite-state transition graphs, constrained continuous dynamics, stochastic transitions, and non-scalar regime classifications where the relevant predicates can be declared. Constructive countermodels distinguish parameter reversal from regime return, recovery of an observable from restoration of a regime predicate, accessibility from completion, and hysteresis from irreversibility. A particularly important countermodel preserves the identity and functional continuity of a system through a successor regime while its former ReturnReferenceCriterion(ThresholdClaim) remains false. Finite non-observation is also consistent with models that permit return and models that preclude it; absence of an observed return therefore leaves a strong non-return claim underdetermined unless independent model, search, or evidence conditions exclude all admissible return trajectories.  The work is intended for scientific classification in dynamical systems, viability and reachability theory, ecological regime shifts, alternative-stable-state models, dissipative synchronization, support-dependent systems, and mathematical model assessment. Its external literature includes foundational resilience and multiple-stability studies, bifurcation-, noise-, and rate-dependent tipping, constrained viability kernels, and experimental spatial recovery-length evidence. Each domain contributes its own state variables, dynamics, resolution, observation class, and evidence burden. Model-internal accessibility, model-event completion, and observed physical return carry different truth conditions, and their epistemic grades remain explicitly separated. The framework supports scalar and non-scalar thresholds, hysteretic and non-hysteretic transitions, and domain-specific restoration mechanisms under their respective declared conditions.  The constitutive Regime Threshold Series parent is Regime Thresholds Before Timelines: Admissibility Boundaries, Crossing Conditions, and Hitting Times, which defines the foundational threshold claim. Lawhood Before Laws: Invariance, Admissibility, and Regime Transformation supplies the public admissibility and rejection discipline; Time as Architecture: Temporal Points, Non-Linear Adjacency, and Readability Without Control preserves the chronology/readability distinction. Phase Coherence as an Order Parameter in Coupled Dissipative Systems, Ecological Phase Transitions in Coupled Planetary Networks, Support Mediation and Capacity Under Load, and Support Consumption and Renewal Failure contribute conditionally mapped scientific predicates for metastability, hysteresis, recovery, renewal, or continuation. Other threshold papers enter when their independent precedence, composite, or approach offices are activated. Philosophical reception and internal project governance remain outside the scientific proof structure of the paper.  The maximum public result is an exact, qualified classification of return structure: whether an independently declared return remains accessible, occurs as a qualifying completed event, exhibits hysteresis or metastable residence, is reversible, is restricted, is model-relatively excluded, or remains underdetermined. SAT, FAIL, UND, and FAIL_UND preserve positive, negative, unresolved, and mixed scientific outcomes. The analysis makes return intelligible as a structural relation while retaining the distinction among mathematical possibility, realized occurrence, empirical trace, and subsequent reconstruction. Operational composition requires separately validated actuator mappings, calibrated action-to-return relations, restoration procedures, intervention timing, target selection, and feedback architecture. The classification therefore remains an independent diagnostic result, distinct from prediction, intervention, and control.
ER  -