Finite-Closure Geometry (FGC): Finite-Closure Geometry in General Dimension: Universal Fixed Points and the Emergence of Vᵈ

Finite-Closure Geometry (FCG) IV — General Dimension and Universal Fixed Points This paper extends the Finite-Closure Geometry (FCG) framework developed in Parts I–III to arbitrary integer dimension d ≥ 1, providing a general dimensional theory for the emergence of continuum geometric invariants from discrete generative systems. Building on the axiomatic structure, algebraic classification, and renormalization-group interpretation established in the previous parts, this work analyzes the asymptotic geometry of homogeneous FCG systems with polynomial growth. 1. Asymptotic Growth Constants in General Dimension For a homogeneous FCG system with polynomial growth of degree d, we define the asymptotic geometric constant: Πₐ = limᵣ→∞ βₐ(r) / rᵈ Under suitable regularity assumptions, this constant characterizes the large-scale growth of metric balls and can be interpreted in terms of the volume of unit balls associated with norms on ℝᵈ arising as asymptotic limits of the discrete geometry. 2. Universal Fixed Points and the Euclidean Limit For isotropic refining families of FCG systems, the renormalization flow selects a distinguished infrared fixed structure corresponding to Euclidean geometry. In this regime, the asymptotic growth constant converges to the classical Euclidean volume of the unit ball: Vᵈ = πᵈ/² / Γ(d/2 + 1) This provides an interpretation of the familiar volumetric constants of continuum geometry as universal large-scale invariants emerging from discrete closure rules. 3. Toward a Discrete Scalar Curvature The paper further proposes a higher-dimensional notion of discrete scalar curvature within the FCG framework, intended to distinguish asymptotically spherical, flat, or hyperbolic large-scale behavior in generative systems. Conclusion Together with the previous papers, this work situates Finite-Closure Geometry within the broader landscape of discrete geometry, geometric group theory, and renormalization-based approaches to the emergence of continuum structures.