Finite-Closure Geometry (FCG) — Algebraic Structure of Geometric Closure: FCG Systems, Nilpotent Groups, and the Emergence of Dimension Two

This paper constitutes Paper II of the Finite-Closure Geometry (FCG) research program. This paper develops the algebraic foundations underlying homogeneous Finite Closure Generative (FCG) systems. We show that the large-scale geometry of these systems is controlled by their symmetry groups, with polynomial growth acting as a fundamental structural constraint. 1. Geometry of Symmetry A functorial correspondence is established between FCG systems and finitely generated groups. For systems with growth degree d: The symmetry group Γₐ is finitely generated and quasi-isometric to the system itself. By Gromov’s theorem, polynomial growth implies that Γₐ is virtually nilpotent. 2. Classification in Dimension Two Under the assumptions of a proper cocompact action with finite stabilizers, we define the asymptotic geometric constant: Πₐ = limᵣ→∞ βₐ(r) / r² This constant characterizes the asymptotic growth of metric balls and can be interpreted through the covolume structure of lattices in two-dimensional nilpotent Lie groups. 3. The Distinguished Role of π Within this classification, the Euclidean plane appears as the abelian limit of the algebraic family. In this case, the growth constant coincides with the classical Euclidean value π, corresponding to the area growth of metric balls in ℝ². Dimension two is structurally special: nilpotent geometries allow continuous deformation toward the abelian Euclidean structure, selecting the Euclidean growth constant as a distinguished large-scale limit. Key insight In the FCG framework, classical geometric constants do not appear as primitive inputs but arise as asymptotic invariants of finite generative systems with maximal symmetry.