Finite-Nucleation Cosmology from a Higher-Dimensional Braneworld

This paper develops a theoretical framework for finite-nucleation cosmology in which the observable four-dimensional universe emerges from a higher-dimensional braneworld geometry. The framework combines five-dimensional gravity, a stabilized extra dimension, a bulk scalar field, two branes, and the radion as the effective four-dimensional dynamical degree of freedom. The radion potential is generated through bulk-scalar stabilization rather than introduced as an independent four-dimensional inflaton potential. A central hypothesis is that the conventional Big Bang singularity may be replaced by a finite-size nucleation event. At the effective four-dimensional level, the nucleation geometry is described through Euclidean continuation and subsequent Lorentzian evolution, allowing an initial hypersurface with nonzero scale factor. The radion can then provide an inflationary phase before settling into a stable late-time configuration. The paper develops the corresponding five-dimensional action, field equations, brane junction conditions, radion stabilization mechanism, effective potential, Euclidean nucleation framework, inflationary dynamics, reheating, recovery of standard four-dimensional cosmology, perturbative stability, Kaluza--Klein effects, dark-radiation constraints, and possible gravitational-wave signatures. The work is presented as a theoretical framework and consistency program rather than as a completed exact five-dimensional solution. In particular, the simultaneous construction of a fully backreacted Euclidean nucleation solution satisfying all phenomenological and stability constraints remains an open problem. Phenomenological benchmark potentials and parameter sets are therefore clearly distinguished from results derived directly from the fundamental five-dimensional theory. The purpose of the framework is to provide a mathematically testable route connecting higher-dimensional geometry, finite cosmological nucleation, radion-driven inflation, and the subsequent evolution toward standard four-dimensional cosmology.