Generative Geometry of Closure: Recursive Generation of Dimension from Operational Primitives

Abstract

This paper proposes a programme of generative geometry of closure, in which space, dimension, line, and surface are not assumed as a priori entities but reconstructed as stable configurations of distinction, neighborhood, traversal, and closure operations. Its point of departure is the distinction between formal and genetic formulations of axiomatics: formal theory begins from fixed sorts of objects and relations, whereas a generative programme asks how these objects and relations arise. A methodological reading of Hilbert's axiomatics is developed. It is not a criticism of Hilbert's aims: the task of the Hilbertian system was the logical articulation, independence, and consistency of geometry, not the ontological minimization of its initial sorts. The generative problem appears only after points, lines, and planes are no longer accepted as independent primitives. In this setting, the structural schema of Pasch's axiom is used as a source for a new Pasch-inspired operator of order-closure, PaschClR. A surface is defined as a stable fixed configuration of this operator rather than as a pre-given container. The paper distinguishes metric from ontological scale. Ontological scale does not characterize the size of a system but its regime of determinacy: givenness is a stabilized domain of relations, whereas becoming is a regime in which prior stability is lost and a new closure is formed. A metric change becomes structurally significant when it fractures a prior domain and produces a domain of the next rank. Finally, a limited generative hypothesis concerning the quantum regime is formulated: contextuality and incompatibility may be represented by families of contextual closure operators that do not admit a single global classical completion. The paper does not derive quantum mechanics, the Born rule, phase structure, or unitary dynamics. Rather, it specifies a language and a sequence of mathematical tasks needed to assess such a programme.