Admissibility Under Constraint: A Dynamical Criterion Beyond Existence and Consistency

Boundary and Scope Notice (CRL-0): This document is observer-only and non-authoritative. Any method-like descriptions, contrasts, figures, or illustrations are interpretive and non-normative; they are not offered as deployment guidance, certification criteria, or operational decision rules. No recommended calibration, thresholds, protocols, or implementation guidance are provided. Any evaluable instantiation belongs to downstream evaluation artifacts. Structural claims and results are unchanged. Classical mathematical foundations evaluate structures primarily through existence and consistency. While sufficient for formal development, these criteria do not determine whether a structure remains operationally meaningful under interaction, evolution, or transport. This paper formalizes admissibility under constraint as a third evaluative criterion, characterizing whether a structure exhibits bounded, stable, and resolvable behavior when subjected to explicit constraint dynamics. Admissibility is treated as a graded, contextual, and dynamical property, evaluated through continuous penalty mechanisms rather than axiomatic prohibition. We introduce a general framework for admissibility using soft constraint manifolds and demonstrate the emergence of threshold behavior separating stable transport from structural breakdown. The analysis clarifies why many techniques in mathematics and physics—regularization, confinement, and stabilization—are necessary without restricting formal existence. Admissibility is shown to be orthogonal to both existence and consistency, functioning instead as an operational viability criterion. By making this principle explicit, the paper unifies phenomena previously treated as domain-specific methods and reframes foundational tensions as questions of dynamical stability under constraint.