Admissibility Under Constraint: A Dynamical Criterion Beyond Existence and Consistency

Paper IV: Formalizes admissibility enforcement as an operator-level principle — not control, not feedback, not optimization. Constraint Geometry Series track. Classical mathematical foundations evaluate structures primarily through existence and consistency. While sufficient for formal development, these criteria do not determine whether a structure remains 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 embedded in explicit constraint dynamics. Admissibility is treated as graded and contextual, expressed through continuous penalty models rather than categorical prohibition. A general framework is developed using soft constraint manifolds, and the paper describes the emergence of regime transitions separating stable transport from structural breakdown. The analysis clarifies why regularization, confinement, and stabilization recur across mathematics and physics without reducing them to ad hoc techniques or domain-specific exceptions. Admissibility is presented as orthogonal to both existence and consistency, functioning instead as a structural viability criterion. This document is observer-only and non-procedural: it provides no measurement procedure, estimator, calibration method, thresholds for deployment, or implementation guidance. Structural claims are stated at the level of formal consequence and interpretive unification within the series. CRL-0 · Stage-0 · observer-only · non-authoritative · no methods, thresholds, procedures, or operational guidance. Revisionv1.2: This revision aligns the document to CRL-0 licensing posture (observer-only, non-authoritative, non-procedural)and strengthens non-operational boundaries. Structural claims and results unchanged. Published: Jan 2, 2026 Revised: Jan 25, 2026 (v1.2) Revision v1.3 (June 2026): Title-page affiliation and series-position language updated for consistency with the Constraint Geometry Series. Structural claims, abstract, and body content unchanged.