Admissibility Under Constraint: Dynamical Criterion Beyond Existence and Consistency

Abstract

Paper IV: Classical mathematical foundations evaluate structures primarily through existence and consistency. While sufficient for formal development, these criteria do not deter mine whether a mathematical object remains structurally meaningful under interaction, evolution, or transport. In this work we make explicit a principle of admissibility un der constraint, which characterizes whether a structure exhibits bounded, stable, and resolvable behavior when subjected to soft constraint dynamics. We formalize admissibility using continuous penalty functionals defined relative to constraint manifolds and show that this criterion naturally explains the emergence, persistence, and breakdown of structures across constrained Hamiltonian systems, spectral statistics, and transport processes. The framework clarifies longstanding foundational tensions without invoking prohibitions on infinity or constructability, and reframes them instead as questions of dynamical stability under constraint.

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.

Keywords: Admissibility, Constraint Dynamics, Structural Invariants, Continuation Criteria, Observability Limits, Finite Resolution, Non-existence / Refusal Logic, System Theory