A Minimal and Stable Vacuum Bounce in Exponential $f(R)$ Gravity

We investigate the realization of a nonsingular cosmological bounce in metric $f(R)$ gravity using a controlled exponential deformation of the Starobinsky $R^{2}$ model. Adopting a smooth Gaussian-type bouncing scale factor, we first demonstrate a no-go result showing that a positive-curvature vacuum bounce cannot be supported by the model $f(R)=R+\alpha R^{2}(1-e^{-R/R_b})$ alone. We then show that a minimal extension obtained by introducing a constant term restores the bounce exactly, with the constant fixed algebraically by the bounce condition. A systematic parameter-space scan is performed to identify regions free of ghost and tachyonic instabilities. Working in the Einstein frame, we study the evolution of scalar and tensor perturbations across the bounce and show that both remain finite and well behaved. Our results establish a minimal, perturbatively stable realization of a vacuum bounce in $f(R)$ gravity that goes beyond background-level constructions.