Spectral scalar curvature and uniform Urysohn width

For every n ≥ 4, a complete connected smooth Riemannian n-manifold without boundary satisfying $-4\Delta +\mathrm{Scal}\ge 1$ as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most $n-2$ whose entire fibers have diameter bounded only in terms of n. The bound is measured in the original metric. This extends the uniform Urysohn width theorem from a pointwise scalar-curvature lower bound to a spectral lower bound.