The Critical Hypersurface as a Geometric Origin of Nonsingular Cosmic Expansion

We propose a geometrically motivated framework in which the large-scale evolution of the universe is described by a coherent multidimensional wavefunction possessing a preferred direction of propagation. Within this formulation, the scalar envelope of the wavefunction defines a critical hypersurface whose evolution provides an effective geometric description of cosmic expansion, naturally incorporating an arrow of time, large-scale homogeneity, and a nonsingular expansion history. The critical hypersurface takes the form of a three-dimensional sphere whose radius plays the role of a cosmological scale factor. Its evolution leads to a time-dependent expansion rate with a positive but gradually decreasing acceleration. The associated density evolution follows a well-defined scaling law consistent with the standard stress–energy continuity equation within the effective geometric description adopted here. A central result of the framework is the existence of a conserved global potential-like invariant associated with the geometry of the critical hypersurface and the conserved wavefunction flux. This invariant is not an energy and does not rely on time-translation symmetry; instead, it represents a geometric property of the evolving hypersurface. It uniquely fixes the effective gravitational coupling, allowing Newton’s constant to emerge as an invariant global parameter rather than as a phenomenological input. Within this geometric description, the expansion rate and density evolution obey well-defined scaling relations, yielding present-day values consistent with observational constraints. The framework therefore provides a self-consistent picture in which cosmic expansion and gravitational coupling arise from the geometry of a universal wavefunction.