Closure Trivialization in Typed Separation Systems

Closure Trivialization in Typed Separation Systems: An Order-Theoretic Characterization of Degeneration under Jurisdictional Expansion. A broad class of systems — layered software architectures, protocols with consensus and execution planes, access control frameworks, institutional designs with separation of powers — defines its structural correctness through a separation criterion: differentiated domains, a declared relation stating which interactions between domains are legitimate, and purity when no observed interaction violates that declaration. If the jurisdiction relation can be enlarged, every violation can be eliminated by redefining what is permitted, without modifying the structure of the system at all. This work formalizes that mechanism and determines exactly under which condition a separation-based theory retains content. Four results are established. The absence of violations is exactly equivalent to the declared jurisdiction containing the observed support, so that the set of jurisdictions trivializing purity is the principal filter generated by that support, and coincides with the fixed points of the induced absorption operator; every jurisdiction admits a unique minimal enlargement taking it there. Non-expansion on the observed support is sufficient to preserve non-triviality — a local condition, verifiable transformation by transformation — while global non-expansion is not necessary: enlarging the jurisdiction outside the support is innocuous. The structure of violations depends only on the cut of the jurisdiction over the observed support, not on the full ontology. And the trivialization cone is determined by the isomorphism class of the observational core. As a corollary, every graded coherence measure satisfying the natural axioms inherits the degeneration of the binary criterion: it assigns the maximal value to every trivialized jurisdiction and cannot signal deterioration along the expansive path. The first three results are independent of finiteness and well-foundedness hypotheses; only the termination of restriction sequences requires an additional hypothesis, declared as such, with a dependency table stating which result uses which. The work introduces no new algebraic objects and claims no mathematical depth: the operator of union with a fixed set is a textbook example of a closure operator and its fixed-point system is order-theoretic folklore. What is claimed is the jurisdictional reading of that operator, the exact localization of the preservation discipline on the cut with the observed support, and the connection with a mechanism that exists in engineering practice without theory — the freeze mechanism of architectural conformance tools, which implements a finer-grained variant of the same absorption. Two areas are flagged as pending bibliographic verification: functional dependencies and formal concept analysis.