Identity Continuity: Behavioural Invariants, Supervisory Closure, and the Geometry of Admissible Execution

Modern computational systems routinely undergo transformations—migration, retraining, re‑profiling, distributional shift—that unsettle representational anchors while leaving behavioural invariants partially intact. This work develops a unified axiomatic theory in which identity is grounded not in representation but in perturbation‑stable behavioural invariants extracted through a structural mapping and constrained by an admissibility boundary that preserves continuity under drift measured by a metric. The algebraic kernel supplies the primitive operators and structural axioms governing supervisory closure; the behavioural primitives (, , , ) instantiate invariant geometry and drift‑bounded dynamics; and the **Supervisory Cycle Theorem** formalises supervision as the internal law governing admissible composition. Identity continuity is realised through the behavioural quotient S /, , whose equivalence classes represent identity‑conditioned behavioural states preserved under admissible transitions. The categorical layer defines the admissible execution category C, the drift functor D: C Met, and the provenance functor Prov: C Seq, yielding a mathematical architecture in which identity continuity is expressed as bounded movement within the invariant space and supervision emerges as the constraint on morphism composition. Appendix A. 1 provides the minimal signature substitute (s), demonstrating computability of the quotient and operational realisation of the categorical layer. Appendix A. 2 supplies a behavioural instantiation through a 20‑bus microgrid model, illustrating drift‑bounded admissibility, supervisory closure, and identity continuity under controlled perturbations. Appendix B presents the commuting diagrams that formalise the categorical structure, showing how invariants, drift, admissibility, provenance, and composition interact functorially. Together these appendices provide operational, behavioural, and categorical realisations of the unified theory. (© 2026 Thomas Filsecker. All rights reserved. This work is provided solely for academic reading, citation, and non‑commercial theoretical evaluation. No licence is granted for implementation, operational use, reproduction, distribution, or derivative works. All source code, pipelines, and operational details referenced herein remain proprietary and are not licensed for any use.