Quadratic Scalar Curvature Decay and Uniform Positivity in Dimensions Four through Seven

Let $4\le n\le 7$ and let $(M^n,g)$ be a complete, connected, orientable, noncompact Riemannian manifold of positive scalar curvature. We prove that if the asymptotic quadratic scalar curvature coefficient of $g$ is greater than $(n-1)\mathrm{/}n$, then $M$ carries a complete smooth metric whose scalar curvature is at least one. The threshold $(n-1)\mathrm{/}n$ and the strict inequality are optimal. This confirms the second part of Gromov’s critical rate of decay conjecture in dimensions four through seven.