Gromov's codimension-two Urysohn width conjecture for positive scalar curvature, with the macroscopic dimension conjecture for universal covers

Principal manuscript, Theorem 1.1: for every $n\ge4$ there is $C_n$ such that every connected complete boundaryless Riemannian $n$-manifold with $\mathrm{Scal}\ge1$ admits a continuous map to a simplicial complex of dimension at most $n-2$ whose entire fibers have diameter at most $C_n$; no orientability, spin, compactness or bounded-geometry assumption. Corollaries: Gromov's filling-radius bound $\mathrm{FillRad}\le C_n/(2\sigma)$ for closed manifolds with $\mathrm{Scal}\ge\sigma^2$, $\dim_{mc}\le n-2$ for universal covers of closed positive-scalar-curvature manifolds in every $n\ge2$, and an independent proof that closed aspherical manifolds of dimension $\ge4$ carry no positive scalar curvature. The October 5 companions extend the width bound to the spectral condition $-4\Delta+\mathrm{Scal}\ge1$ ($n\ge4$, and $n=3$ with a graph target and bound $500/\sqrt\lambda$). Not shown: the Lipschitz form $\dim_{MC}$, or any explicit constant.

Principal manuscript, Theorem 1.1: for every $n\ge4$ there is $C_n$ such that every connected complete boundaryless Riemannian $n$-manifold with $\mathrm{Scal}\ge1$ admits a continuous map to a simplicial complex of dimension at most $n-2$ whose entire fibers have diameter at most $C_n$; no orientability, spin, compactness or bounded-geometry assumption. Corollaries: Gromov's filling-radius bound $\mathrm{FillRad}\le C_n/(2\sigma)$ for closed manifolds with $\mathrm{Scal}\ge\sigma^2$, $\dim_{mc}\le n-2$ for universal covers of closed positive-scalar-curvature manifolds in every $n\ge2$, and an independent proof that closed aspherical manifolds of dimension $\ge4$ carry no positive scalar curvature. The October 5 companions extend the width bound to the spectral condition $-4\Delta+\mathrm{Scal}\ge1$ ($n\ge4$, and $n=3$ with a graph target and bound $500/\sqrt\lambda$). Not shown: the Lipschitz form $\dim_{MC}$, or any explicit constant.