Localization of Singularities and Universal Geometric Rank Bounds in the Satake Correspondence

This article introduces a framework for the localization and isolation of singularities in the affine Grassmannian. Our primary result is a structural factorization of the transition matrix $C$ between the Mirković–Vilonen (MV) basis and the convolution basis into $C=P\cdot M\cdot A\cdot Q^{-1}$, where the four factors represent: equivariant localization ($Q$), fusion via nearby cycles ($A$), local intersection cohomology stalks ($M$), and diagonal normalization ($P$). Utilizing this factorization, and by introducing the Geometric Efficiency metric ($\eta$) we establish a Universal Geometric Rank Bound, proving that the rank of the transition matrix $C$ is bounded by the dimension of the local Braden–MacPherson (BMP) stalks.