Positive Scalar Curvature Obstructions via Singular Dimension Descent

In light of recent advances in conformal blow-up methods for the positive mass theorem, including He–Shi–Yu, Bi–Hao–He–Shi–Zhu, and Brendle–Wang, we develop a Schoen–Yau type singular dimension descent method for positive scalar curvature obstructions in arbitrary dimensions. We prove obstructions to positive scalar curvature on enlargeable manifolds and establish the corresponding cubical width inequalities and two-systole estimates. The method also applies to enlargeable AM–PI spaces, giving a positive scalar curvature obstruction when the singular set has Assouad codimension greater than $3-2\mathrm{/}n$.