Closure Mathematics Regime, Architecture, and Generative Unity A Structural Ontology of Mathematical Domains

This paper develops a unified framework in which closure is treated not as a property internal toparticular mathematical domains, but as a generative principle from which such domains arise.The central thesis is that number, algebra, topology, geometry, and analysis are most adequatelyunderstood as distinct regimes of closure, each characterized by a specific mode ofadmissibility: discrete, operational, relational, transport, and continuous completion. The paper proceeds in three stages. First, closure is established as primitive, functioning as thecondition of determinacy and lawful formation. Second, the principal mathematical domains arederived as closure regimes, each emerging through a principled extension of admissibility. Third,these regimes are shown to form a coherent architectural system governed by dependence,retention, and boundary transition, and unified by closure as a single generative source. The resulting framework avoids both reductionism and fragmentation. It does not reducemathematics to a single foundational domain, nor treat domains as independent. Instead, itshows how mathematical unity arises through structured differentiation within closure. Thepaper concludes by articulating closure as the unifying ontological principle of mathematics andindicating a broader program in which major mathematical frameworks are derived explicitly asclosure regimes.