Finite-Closure Geometry (FCG) : Geometric Closure in Finite Generative Systems: Axioms, Growth, and Gromov-Hausdorff Convergence

This upload constitutes Paper I (Axioms) of the Finite-Closure Geometry (FCG) research program. Finite-Closure Geometry is an axiomatic framework in which geometric structure is derived from finite generative processes rather than from completed infinite constructions. In this setting, space is modeled as a system of discrete states acted upon by a finite set of generators; metric notions arise from combinatorial accessibility, and geometric measure is defined through structured counting procedures. This first paper introduces and formalizes the foundational axioms of FCG. The framework replaces infinite completion and limit-based constructions with the principle of finite closure: geometric entities are defined through generative procedures that terminate after finitely many operations. Within this setting, classical geometric quantities—such as circles, distances, and area ratios—are reconstructed as invariants associated with closed generative structures. At finite scale, geometric quantities defined within FCG appear as rational invariants determined by the combinatorial structure of the generative system. Classical constants such as π\piπ emerge only in appropriate asymptotic regimes when these finite constructions are interpreted through continuum limits. The axiomatic framework developed in this paper provides the basis for the subsequent works of the series: • Paper II studies the algebraic and group-theoretic structure of homogeneous FCG systems and relates growth properties to virtually nilpotent groups. • Paper III analyzes the emergence of Euclidean behavior through a geometric renormalization-type flow, interpreting classical constants as scale-dependent invariants. • Paper IV extends the framework to arbitrary dimension and studies universal fixed structures associated with isotropic growth. Together, these works develop Finite-Closure Geometry as a structural approach to geometric generation and asymptotic limits, situated at the intersection of discrete geometry, geometric group theory, and the study of continuum emergence from finite structures.