Unified Generative Operating System (UGOS)

This paper presents an axiomatic formulation of the Unified Generative Operating System (UGOS), a 13‑layer hierarchical framework encompassing directionality, branching, selection, charts, spectra, curvature, stability landscapes, flows, generative units, finite subdivision trees, and partition‑based space. UGOS places the fourfold directionality F={D,E,W,H} at the highest level, introducing multivaluedness through branching A and selection fi, and defining directional structure via local charts S and hierarchical spectra S(L). Geometric and stability properties emerge from curvature K(L) and the stability landscape Φ, while the generative flow g provides a unified dynamical mechanism. Furthermore, generative units (GU), finite subdivision trees (SubTree), and partition‑glued spaces (PG‑Space) offer a framework in which space itself arises as the finite output of generative processes. As a finite, hierarchical generative model that does not rely on continuity or point‑set assumptions, UGOS integrates generative theory, subdivision geometry, and cell/partition geometry, providing a unified structural foundation for describing the entire process from primordial directionality to the emergence of space.