Generative Geometry: From Operational Set Theory to Emergent Space

Abstract

We develop generative geometry, a formal framework in which space, metrics, and invariants are not postulated but emerge from the operational primitives of Operational Set Theory (OST) and a single ontological axiom of non-annihilable change. In Part I, we construct an ontological foundation: acts of selection, neighborhood structures, and observational scales (ε, δ) generate surfaces, lines, and smooth manifolds as stabilized patterns of structural dynamics, rather than as pre-given arenas. Continuity is shown to be an internal idealization at coarse observational scales, and geometry is identified with frozen dynamics, while dynamics becomes geometry in formation. In Part II, we introduce the formal apparatus: the moduli space Mℓ of generative structures, equipped with Young diagrams as combinatorial invariants; the re-entry operator Rre and the rank transition criterion via two-part description length; the consistency operator C(k) and the identity kernel Iℓ, characterized as the fixed point of the Young-Fold dissipator; a canonical measure on Mℓ reformulated through the combinatorial weight Wcomb and the free-energy principle; the structural intersection ▷◁ as an algebra of synthesis of two dynamics; and functionals of ontological complexity Cont(f) and structural distortion Dstr(f) that mediate reductions to classical geometries via harmonic mappings. Classical Euclidean and Riemannian geometries appear as thermodynamic phases and continuous limits of generative geometries, obtained for specific classes of traversal rules, invariants, and scales. The resulting program treats geometry as an emergent, operationally grounded phenomenon, bridging OST, fractional analysis with memory, combinatorial ontology, and the algebra of invariants as a universal language for describing systems.