The Physically Admissible System Cut: When a tensor factorization is a physical subsystem: an exact and a quantitative criterion

Every use of an open quantum system presupposes a cut: a division of the world into a system and its environment, formalized as a tensor factorization of the Hilbert space. Which factorization is the physical one is not fixed by the Hilbert space — the same space admits infinitely many, and standard treatments select one by appeal to decoherence, which already presupposes a cut. This paper supplies the missing selection criterion. It adds no dynamics and no interaction; it imports the observable-induced tensor factorization from quantum information and asks, of any named candidate, whether the unchanged global dynamics descends to a well-defined dynamics on the reduced system state space, and whether the resulting quotient carries a complete identity structure. The exact criterion is a dynamical congruence: two global states with the same reduced state must have the same reduced future. It is necessary and sufficient for a representative-independent reduced dynamics; supplemented with La Profilée’s role and verdict structure it becomes a necessary-and-sufficient criterion for an exactly autonomous physical system cut; and an explicit two-qubit counterexample proves the physically admissible factorizations are a strict subset of the mathematical ones. A quantitative extension makes the criterion operationally testable: an autonomy defect bounds the unavoidable reduced-prediction error two-sidedly, errors accumulate additively, and a robust-persistence test — proven error below the margin to the identity boundary — keeps the persistence verdict invariant across all admissible global realizations. Everything is proven in finite dimensions. A uniqueness theorem, concrete margins for a given system, and the extension to the Type-III local algebras of relativistic quantum field theory remain open, and are marked as such throughout.