Finite-Closure Geometry (FCG) - Renormalization of Geometric Constants: From Discrete Closure to the Continuous Constant π

This paper constitutes Paper III of the Finite-Closure Geometry (FCG) research program and studies the emergence of classical geometric constants through a renormalization framework applied to finite generative systems. Building on the discrete foundations introduced in Paper I and the algebraic classification developed in Paper II, we formulate a geometric renormalization group acting on families of homogeneous Finite Closure Generative (FCG) systems. Refinement and coarse-graining procedures induce a scale-dependent geometric constant that can be interpreted as an effective coupling associated with area growth. The central result identifies π\piπ as an infrared fixed point of this geometric renormalization flow. For refining families that are asymptotically homogeneous and isotropic in two dimensions, the effective geometric constant converges universally to π\piπ, independently of the microscopic discrete structure. The paper develops: • a formal definition of geometric renormalization maps for FCG systems• a flow relation connecting finite-scale growth constants across renormalization steps• the identification of the Euclidean plane as a scale-invariant fixed structure• a stability analysis of the infrared fixed point associated with π\piπ Connections are discussed with renormalization methods in statistical physics and with discrete approaches to continuum geometry. This work provides the renormalization-theoretic component of the FCG program, complementing the axiomatic and algebraic results developed in the previous papers. Keywords finite closure geometryrenormalization groupdiscrete geometrygeometric constantsinfrared fixed pointasymptotic geometry