Scalar curvature rigidity for products of spheres and tori

We prove Llarull-type rigidity for $S^{n-m}\times {\mathbb{T}}^m$ ($3\le n\le 7$, $1\le m\le n-2$). If a closed spin $(M^n,g)$ admits a degree-nonzero map to $S^{n-m}\times {\mathbb{T}}^m$ whose spherical projection is area non-increasing, and there exists $\psi \in C^{\mathrm{\infty }}(M)$ with $-{\mathrm{\Delta }}_M\psi -\frac{1}{2}\mathrm{\mid }{D}_M\psi {\mathrm{\mid }}^2+\frac{1}{2}(R_M-(n-m)(n-m-1))\ge 0$, then $(M,g)$ is isometrically covered by $S^{n-m}\times {\mathbb{R}}^m$. For bands, we extend Gromov’s torical inequality and obtain sharp width bounds: $\text{dist}({\mathrm{\partial }}_{-}M,{\mathrm{\partial }}_{+}M)\le 2\pi \sqrt{n\mathrm{/}((n+1)\sigma )}$ when $R_M\ge (n-m)(n-m-1)+\sigma$. The method combines stable weighted slicing with a spectral Dirac operator argument.