Bottom spectrum, vertical $\widehat{A}$-cowaist and scalar curvature rigidity

We introduce the vertical $\widehat{A}$-cowaist, a codimension-one invariant for partitioned manifolds. It extends the concept of infinite vertical $\widehat{A}$-cowaist for bands to arbitrary partitioned manifolds, which may be noncompact and have compact boundary. We establish a sharp inequality relating the scalar curvature, the bottom spectrum of the Laplacian, and this invariant. As an application, we obtain a high-dimensional analogue of Munteanu-Wang's bottom spectrum estimate. We also prove a quantitative strengthening of Anghel's theorem together with a boundary version, as well as a Calabi-Yau type theorem that goes beyond the dimensional restrictions of the earlier $\mu$-bubble method. Our approach is based on deformed Dirac operators.