Finite-Closure Geometry (FCG) : Global Geometric Closure and the Cosmological Constant

Finite-Closure Geometry (FCG) V — Global Geometric Closure and the Cosmological Constant This paper explores the cosmological implications of the Finite-Closure Geometry (FCG) framework developed in Parts I–IV, proposing an interpretation of the cosmological constant as a global geometric invariant associated with the closure of large-scale spacetime. Finite-Closure Geometry models space as a system generated by finite rules whose large-scale structure emerges through asymptotic growth and renormalization. Earlier papers in the series established: • axiomatic foundations for discrete generative geometry• algebraic constraints imposed by polynomial growth• a renormalization interpretation for geometric constants such as π• the emergence of continuum volumetric invariants in arbitrary dimension Within this framework, classical geometric quantities appear not as primitive parameters but as large-scale invariants selected by structural constraints. The present work applies this perspective to cosmology. Instead of interpreting the cosmological constant as a local vacuum energy density, we analyze it as a parameter encoding the global closure properties of spacetime geometry when viewed as the large-scale limit of a finitary generative structure. In this interpretation: • local geometric dynamics remain governed by standard relativistic equations• the cosmological constant reflects global boundary or closure conditions• the observed small value of Λ may arise from large-scale geometric constraints rather than from microscopic quantum vacuum contributions This perspective reframes the well-known cosmological constant problem. Rather than requiring a delicate cancellation between quantum vacuum energy and gravitational dynamics, the small observed value of Λ can be interpreted as a manifestation of global geometric structure. The paper does not attempt a detailed dynamical derivation of Λ. Its purpose is conceptual: to situate the cosmological constant within the broader FCG program as a large-scale invariant emerging from the geometry of finite generative systems. Keywords finite-closure geometrycosmological constantemergent geometrydiscrete spacetimeglobal curvaturerenormalization