Category-Theoretic Architecture of Closure Mathematics Functors, Commutative Diagrams, and Coherence-Preserving Disclosure

Paper 3 in the Ontology of Mathematical Physics sequence Category theory provides the formal grammar of closure-disclosure. This paper is developed from an earlier prepaper on the ontology of mathematical physics. The source manuscript proposed that category theory may provide a formal pathway for representing the transition from invariant or absolute mathematical conditions into relational and derived mathematical structures. It identified morphisms, functors, commutative diagrams, natural transformations, and equivalences as possible tools for formalizing coherence-preserving emergence. The present paper narrows that material into a dedicated formal-architecture paper. It does not claim to complete a final proof of mathematical ontology. Instead, it defines the categorical scaffolding required for future proof-like development. The guiding claim is that closure mathematics can be represented through distinct but connected categorical domains: O -> H -> R -> D, where O represents invariant closure, H represents continuum coherence, R represents relational resonance, and D represents derived formal appearance. The aim is to show how mathematical structures may disclose across regimes without losing coherence and without collapsing all regimes into a single undifferentiated ontology.