A Curvature–Elasticity Criterion for Viable Unified F(R) Cosmology

The reconstruction freedom of F(R) gravity makes it possible to reproduce a wide range of cosmological expansion histories; however, reproducing a background evolution is not sufficient to guarantee physical viability. A unified cosmological scenario connecting an early quasi-de Sitter inflationary stage, a radiation/matter-dominated epoch, and a late-time accelerated attractor must also satisfy ghost freedom, scalaron stability, suppression of matter-era instabilities, and consistency with a general-relativistic intermediate regime. In this work, we formulate these requirements through two dimensionless curvature functions: U(R), which characterizes the normalized slope of the Einstein-frame scalaron potential, and E(R), which measures the logarithmic curvature response of the effective gravitational coupling. This construction provides a compact phase-space representation in the curvature plane spanned by potential control and gravitational elasticity. We show that viable unified F(R) cosmologies correspond to continuous trajectories in the (U,E)-plane linking three physically distinct regimes: a high-curvature slow-roll window, an intermediate general-relativistic corridor with a heavy scalaron, and a low-curvature stable de Sitter attractor. Models that fail to preserve this trajectory structure typically develop scalaron instabilities, matter-era deviations, or unstable late-time behavior. The proposed curvature–elasticity criterion offers a concise diagnostic for separating physically admissible unified F(R) models from formally reconstructed but unstable backgrounds, while also providing a useful framework for constructing ghost-free models that connect Starobinsky-like inflation with late-time dark-energy acceleration.