Packaging Systems: Intrinsic Structure, Recovery Ranks, and Globalization Theorems
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 ℋ. A geometric obstruction rank ν(ℋ) yields thresholds 2k+1<ν(ℋ) for empty-preserving closures of local Carathéodory number at most k≥1; d<ν(ℋ) for complete functional-dependency theories of local minimum Armstrong size at most d≥3; and 3<ν(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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BibTeX
@misc{ma2026packaging,
title = {{Packaging Systems: Intrinsic Structure, Recovery Ranks, and Globalization Theorems}},
author = {Rongchao Ma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33092},
eprint = {2610.33092},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33092},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Packaging Systems: Intrinsic Structure, Recovery Ranks, and Globalization Theorems
AU - Rongchao Ma
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.33092
UR - https://arxiv.org/abs/2610.33092
AB - 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 ℋ. A geometric obstruction rank ν(ℋ) yields thresholds 2k+1<ν(ℋ) for empty-preserving closures of local Carathéodory number at most k≥1; d<ν(ℋ) for complete functional-dependency theories of local minimum Armstrong size at most d≥3; and 3<ν(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.
ER -