Closure Atlases and Local-to-Global Obstructions in Finite Closure Systems

2026-06-18

AI Summary

This study addresses the local-to-global consistency problem for finite closure systems over overlapping domains: given a collection of local closure operators, can they be coherently extended to a global conservative closure system? To this end, the paper introduces the “atlas-induced closure” construction and establishes a semantic correspondence between indexed truth spaces and closure systems, revealing an intrinsic connection between regional inclusion relations and closure derivations. The central contribution is a finite, directly computable criterion—termed “local visibility obstructions”—that precisely characterizes the necessary and sufficient conditions for the existence of a global conservative realization. In the absence of such obstructions, the atlas-induced closure yields the unique global conservative extension, and an efficient algorithm is provided for detecting these obstructions.

Abstract

This paper studies finite closure operators on overlapping finite universes and gives an exact local-to-global obstruction criterion for conservative globalization. Given a finite family of local closure systems, its atlas-generated closure is obtained by repeatedly applying the local closure operators to the parts visible in each chart. This closure is the least global closure operator extending all chart closures. A chart-visible obstruction is a consequence produced by this global propagation that lies inside a chart but is not validated by that chart's own closure operator. The main theorem proves that a finite closure atlas has a global conservative realization exactly when no such obstruction occurs; in that case the atlas-generated closure itself is the conservative realization. The obstruction condition is finite and directly computable. The paper also records the indexed representation layer motivating the terminology. For a finite closure system, an indexed truth space selects closed theories as contexts and represents each element by the region of selected closed theories containing it. Closure consequence is always sound for region inclusion, and the full indexed space of all closed theories recovers the original closure consequence exactly; reduced indexed spaces can therefore create spurious region consequences by deleting separating closed theories. A formal opposite gives a four-region membership decomposition - only one, only the other, both, and neither - unless additional separation assumptions are imposed. Finally, overlap-compatible local closed theories glue by canonical union under the atlas-generated closure. The framework is finite, structural, and closure-theoretic; the logical terminology is used only as an interpretation of the underlying closure data.

Problem

• closure systems
• local-to-global
• conservative realization
• obstruction
• finite universes

Innovation

• closure atlas
• local-to-global obstruction
• atlas-generated closure
• conservative globalization
• indexed truth space