arXiv:2610.33092v1 Announce Type: new Abstract: We define packaged structures through carriers, internal units, incidence, candidate spaces, and selected families. A realized packaging system additionally retains a context and formation datum related to its output by a possibly partial, multivalued realization relation. We establish comparison-relative recoverability and non-collapse criteria and an intrinsic refinement-completion theory. For atlas specializations, exact extension becomes a specialization-relative packaging completion problem. Under additional recovery and elimination axioms, we establish general globalization sufficiency criteria and sharp classifications for three classes of compatible finite atlases on a fixed cover $\mathcal H$. A geometric obstruction rank $\nu(\mathcal H)$ yields thresholds $2k+1<\nu(\mathcal H)$ for empty-preserving closures of local Carath\'eodory number at most $k\ge1$; $d<\nu(\mathcal H)$ for complete functional-dependency theories of local minimum Armstrong size at most $d\ge3$; and $3<\nu(\mathcal H)$ for transitive relations, including equivalence relations. For nonempty finite label sets and empty-preserving closures, minimum commutant realization dimension equals minimum Armstrong size. We also determine intrinsic recovery ranks and construct canonical clique completions under bounded conformality without chordality. Explicit obstructions establish sharpness; a transport hypothesis is essential for the general recovery--obstruction converse. Complexity bounds apply locally, not necessarily to global extensions.
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