 Compact harmonic-spinor rigidity under nonnegative scalar curvature

On a compact boundaryless Riemannian spin manifold with $R\ge 0$, integrating the Lichnerowicz spinor-square formula against a harmonic spinor gives $\int (|\nabla \psi {|}^2+R|\psi {|}^2/4)=0$. Positivity forces the spinor to be parallel. Its norm is constant on each connected component, so a nontrivial solution is nowhere zero there. Strictly positive scalar curvature at any point on such a component excludes nonzero harmonic spinors.