Finite-Closure Geometry (FCG) — Mathematical Companions: Technical Notes Supporting the FCG Research Program

This record collects a set of technical companion notes supporting the Finite-Closure Geometry (FCG) research program. The main FCG results are developed in the core papers of the series, which introduce the axiomatic framework, the algebraic structure of generative systems, the renormalization interpretation of geometric constants, and the general-dimensional theory of asymptotic geometric invariants. The present companion documents provide additional mathematical context, clarifications, and structural connections that complement the main papers without duplicating their central results. In particular, these notes explore: • background results from geometric group theory relevant to polynomial growth and nilpotent structures• technical aspects of asymptotic geometry and scaling limits• auxiliary constructions related to renormalization and geometric invariants• conceptual bridges between discrete generative systems and continuum geometric descriptions These materials are intended to help situate Finite-Closure Geometry within the broader mathematical landscape and to provide readers with additional technical perspectives related to the program. The companions should be read in conjunction with the core FCG papers, which contain the formal definitions, theorems, and main results of the framework. Core Papers of the Finite-Closure Geometry Program • FCG I — Geometric Closure in Finite Generative Systems: Axioms, Growth, and Gromov–Hausdorff Convergence • FCG II — Algebraic Structure of Geometric Closure: FCG Systems, Nilpotent Groups, and the Emergence of Dimension Two • FCG III — Renormalization of Geometric Constants: From Discrete Closure to the Continuous Constant π • FCG IV — Finite-Closure Geometry in General Dimension: Universal Fixed Points and the Emergence of VdV_dVd • FCG V — Global Geometric Closure and the Interpretation of the Cosmological Constant Contained Companion Notes This record includes the following technical notes: • Companion A• Companion B• Companion C• Companion D• Companion E Each document addresses a specific mathematical aspect related to the Finite-Closure Geometry framework. Keywords finite-closure geometrydiscrete geometrygeometric group theoryasymptotic geometryemergent geometryrenormalization