We present a pair of open smooth \(4\)-manifolds that are mutually homeomorphic. One of them admits a Riemannian metric that possesses quasi-cylindricity, and positivity of scalar curvature, and of dimension of certain \(L^2\) harmonic forms. By contrast, for the other manifold, no Riemannian metric can simultaneously satisfy these properties. Our method uses Seiberg-Witten theory on compact \(4\)-manifolds and applies \(L^2\) harmonic theory on non-compact, complete Riemannian \(4\)-manifolds. We introduce a new argument to apply Gauge theory, which arises from a discovery of an asymptotic property of the range of the differential.