Geometrical Continuity in Cosmologies with Dynamic Curvature

The standard Friedmann–Lemaître–Robertson–Walker (FLRW) metric describes homogeneous and isotropic cosmologies with constant spatial curvature k∈-1, 0, +1. In the classical formulation, the curvature sectors are treated separately and the spatial curvature is assumed to remain constant throughout cosmic evolution. Many fundamental cosmological relations, including null geodesics, cosmological distances, wave propagation, and optical equations, are derived under this assumption. In this work, we generalize the FLRW framework by introducing a scale dependent curvature function X (a). The key concept is the geometrical continuity of the curvature representation. Instead of describing positive, flat, and negative curvature as disconnected geometric cases, the continuous curvature formalism unifies all curvature sectors within a single analytic structure. This allows smooth and finite transitions through k=0 while preserving differentiability of the geometric relations. Geometrical continuity replaces the discrete separation of curvature sectors by a unified analytic structure. The geometrical continuity is achieved by expressing the curvature through the continuous function X, including its imaginary continuation for negative curvature. In this representation, trigonometric and hyperbolic geometries become analytically connected. The resulting formalism removes the need for separate treatments of the curvature sectors and provides a unified geometric language for dynamic curvature. The framework is independent of the physical mechanism responsible for the curvature evolution and therefore applies generally to any cosmological model with effective scale dependent curvature. The purpose of this work is not to introduce a specific cosmological model, but to derive the geometric consequences that follow directly from dynamic curvature itself. Allowing X to depend on the scaling factor modifies several fundamental cosmological relations. The null geodesic equation acquires additional terms generated directly by curvature evolution. These corrections propagate into wave phases, the cosmological redshift relation, and the Sachs optical equation. In particular, the enhanced Sachs equation contains new source terms proportional to curvature gradients and curvature acceleration. These contributions vanish identically in the classical FLRW limit of constant curvature. A central result of this work is that all modified relations remain finite and analytically continuous for transitions through k=0. Dynamic curvature can therefore be incorporated consistently into FLRW cosmology without introducing singular geometric behaviour at curvature transitions. Geometrical continuity thus provides a natural extension of the standard FLRW geometry beyond the assumption of static curvature.