The sphere theorem and its refinements show how curvature pinching forces strong topological rigidity on Riemannian manifolds. Building on this tradition, I will present a new curvature condition that is strictly weaker than the classical quarter-pinched sectional or flag curvature but still ensures spherical space form topology. I will also discuss applications to Yau’s 1990 pinching problem, where our results sharpen previous bounds of Gu–Xu and provide new partial results.