LAUSR

models for dark energy generally exhibit a weak curvature singularity, which can be cured by adding an term. This correction allows for a unified description of primordial and late-time accelerated expansions. However, most existing models struggle to achieve this, as they become unstable over certain negative ranges of the Ricci scalar, where either the first or second derivative of turns negative. These instabilities may disrupt the post-inflationary evolution when the Ricci scalar oscillates about the vacuum state after the inflation. In this paper, we introduce a model-building process to guarantee global stability, i.e., the first and second derivatives are positive for all real Ricci scalars. By extending the idea from Appleby and Battye, we demonstrate that viable models can be constructed by imposing a positive, bounded first derivative of with a sigmoid shape. Building upon this framework, we first reformulate and generalize the original Appleby-Battye model. Then, we propose a dark energy model that successfully explains the acceleration of cosmic expansion and passes local gravity tests.