On the Unified Geometric Origin of Gravity and Cosmological Expansion

AbstractWe develop a geometric framework in which gravity and cosmological evolution areinterpreted as consequences of global spacetime closure. Assuming a compact,boundaryless Lorentzian manifold (∂ = Ø), we argue that curvature cannot be freely ℳcreated or eliminated, but is constrained by global geometric consistency — a conditionfrom which both the existence of gravity and the nature of cosmological expansionfollow as necessary consequences once closure is adopted as a postulate.Motivated by topological results such as the Gauss–Bonnet–Chern theorem, we introducean effective global curvature functional ₀, interpreted as a finite curvature budget. ℬWhile ₀ is not a strict topological invariant, closure implies that curvature must be ℬredistributed rather than generated, leading to measurable local effects. We show thatcurvature does not appear as an accessible direction but as a scalar field κ(x), whosegradient produces effective gravitational acceleration. In this sense, gravity emerges asthe necessary local manifestation of global geometric structure, without modifying theEinstein field equations.We further reinterpret cosmological expansion as internal geometric reconfigurationrather than literal spatial growth, introducing an effective observational scaling functionχ(t). The framework preserves the Einstein field equations and the standard FLRWformalism without modification, while revising their ontological reading. This separationallows the proposal to remain consistent with established gravitational dynamics whileaddressing foundational questions regarding their origin.To connect the framework with observations, we introduce a minimal phenomenologicalextension parameterized by a dimensionless quantity β — the effective measure ofcurvature redistribution across cosmic history. This yields small but systematiclogarithmic corrections to the Hubble relation, reducing exactly to standard ΛCDM for β= 0, with deviations in H(z) and supernova distance moduli potentially detectable bynext-generation surveys such as Euclid and the Rubin Observatory.This work is structural and interpretive: it adds a global explanatory layer to generalrelativity, offering a unified perspective on gravity, cosmology, and the relation betweenlocal dynamics and global geometry. Appendix A provides a formal derivation — viaaxioms, lemmas, and theorems grounded in the Gauss–Bonnet–Chern theorem —establishing that, given the closure postulate and a conserved positive curvature budget,gravity follows as a structural necessity rather than an independent assumption: the totalcurvature of any topologically non-trivial compact, boundaryless manifold (such as onewith S³ spatial slices) is then constrained to be non-zero, and this curvature, having notraversable direction, manifests locally as gravitational acceleration. Appendix Bprovides the variational motivation for the logarithmic correction term. The initialsingularity is identified as a special limiting case in which the global and local levels ofthe framework coincide, the curvature budget ₀ is maximally concentrated, and ℬquantum-gravitational and classical-geometric descriptions become indistinguishable —the origin point of the relaxation process that constitutes cosmic evolution.