Contact Rigidity and Comparison Kernels for Type A lci Schubert Varieties

Let $X_w$ be a Type A local complete intersection Schubert variety. We prove Contact Rigidity: the existence of two smooth singular components forces some pair of singular components to contain a common Schubert subvariety of codimension one in each. If the singular locus is a single smooth component $X_z$, the rational comparison kernel is $I{C}_z$, and $P_{u,w}(q)=1+q^{(\mathrm{\ell }(w)-\mathrm{\ell }(z)-1)\mathrm{/}2}$ for every $u\le z$. The first proof combines pattern avoidance with a computer-assisted finite overlap classification, rectangle inheritance, and extremal repairs. The second establishes the hypotheses of Woo’s theorem and uses Euler characteristics and Bruhat triangularity to identify the entire perverse kernel.