Stability of Shifted Complexes via the Second-Moment Defect of the Up-Laplacian

Let $K$ be a finite pure $k$-dimensional simplicial complex, with $k\ge 1$, on the vertex set $[n]$ and with facet family $K_k$. Let ${\lambda }_1(K)\ge {\lambda }_2(K)\ge \cdots >0$ be the nonzero eigenvalues of its $(k-1)$-dimensional up-Laplacian, and, after ordering the vertices so that ${\mathrm{°}}_K(1)\ge \cdots \ge {\mathrm{°}}_K(n)$, let $\text{dvT}r(K)$ be the number of vertices contained in at least $r$ facets. A complex is \emph{shifted} if replacing a vertex of a face by a smaller vertex outside the face always yields another face. We prove that there is a shifted family $\text{HH}$ of $(k+1)$-element subsets of $[n]$, with the same number of members as $K_k$, such that $$\frac{1}{2}\mid K_k\mathrm{△}\text{HH}\mid \le \frac{1}{2}\left[\sum _{r\ge 1}\right(\text{dvT}r(K){)}^2-\sum _r{\lambda }_r(K{)}^2]\mathrm{.}$$ The left-hand side counts the facets that have to be exchanged to reach $\text{HH}$; thus one half of the gap between the second power sums of the two sequences bounds the distance of $K_k$ to a shifted family. The characterization $\lambda (K)=\text{dv}(K{)}^{\mathsf{T}}⟺K$ is isomorphic to a shifted complex was established in \cite{Gupta} from the identity that this gap equals twice the number of failed elementary shifts. The present paper converts that identity into a quantitative stability statement and recovers the equality characterization at zero defect. For $k=1$ this bounds the number of edge exchanges needed to reach a threshold graph with the same number of edges.