Finite Closure Geometry (FGC):} A Structural Program for Global Non-Closure

This document serves as Paper 0 of the Finite Closure Geometry (FCG) research program. It provides a global conceptual map, epistemological positioning, and guided reading of the foundational FCG papers already published.

Rather than introducing new technical results, this work clarifies what Finite Closure Geometry is, what it is not, why it is mathematically inevitable, and what it changes at the epistemological level. Its role is explicitly structural and interpretive: to organize the program as a coherent ecosystem rather than a collection of isolated papers.

The core thesis of the FCG program is that certain classical geometric and physical constants—most notably π—should not be treated as ontological primitives, but as effective descriptors emerging from finite, operationally closed generative systems in appropriate asymptotic limits. This paper explains how this idea is realized across the program, without duplicating proofs or technical constructions.

Specifically, this document:

• Provides a defensive clarification of what FCG does not claim (it is not a discretization of the continuum, nor a physical model per se).

• Explains why finite closure is mathematically unavoidable when working with generative systems and growth.

• Summarizes what is genuinely new in the program, particularly the treatment of geometric constants as scale-dependent invariants.

• Articulates the epistemological shift implied by finite closure: the reinterpretation of infinity, limits, and transcendental constants as properties of idealized descriptive limits rather than of underlying structures.

This paper is intended to be read before the technical works and acts as a stable entry point for mathematicians, physicists, and philosophers interested in discrete geometry, asymptotic limits, renormalization-like structures, and the foundations of geometry.

Related works in this program (Zenodo DOIs)

Geometric Closure in Finite Generative Systems: Axioms, Growth, and Gromov–Hausdorff Convergence
https://doi.org/10.5281/zenodo.17989932

Algebraic Structure of Geometric Closure: FCG Systems, Nilpotent Groups, and the Emergence of Dimension Two
https://doi.org/10.5281/zenodo.17990060

Finite-Closure Geometry (FCG): Renormalization of Geometric Constants
https://doi.org/10.5281/zenodo.17990196

Finite-Closure Geometry in General Dimension: Universal Fixed Points and the Emergence of Vᵈ
https://doi.org/10.5281/zenodo.17990513