Persistent Laplacians are matrix operators that track how the shape and structure of data transform across scales and are popularly adopted in biology, physics, and machine learning. Their eigenvalues are concise descriptors of geometric and topological features in a filtration. Although earlier work established global algebraic stability for these operators, the precise change in a single eigenvalue when one simplex, such as a vertex, edge, or triangle, is added has remained unknown. This is important because downstream tools, including heat-kernel signatures and spectral neural networks, depend directly on these eigenvalues. We close this gap by proving a uniform Lipschitz bound: after inserting one simplex, every up-persistent Laplacian eigenvalue can vary by at most twice the Euclidean norm of that simplex's boundary, independent of filtration scale and complex size. This result delivers the first eigenvalue-level robustness guarantee for spectral topological data analysis. It guarantees that spectral features remain stable under local updates and enables reliable error control in dynamic data settings.

We are particularly interested in the setting where neither ex plicit knowledge about f,g are available nor their unbiased stochastic derivatives. In this zeroth-order setting, we assume that only noisy evaluations of f and g are available upon query to an oracle. The BLP problem was first introduced by Bracken and McGill in t he 1970s [7] followed by a more general form of problem involving joint constraints of outer and inner variables. This is a fundamental problem in engineering and economics with dire ct applications in problems such as decision making [48], supply chain [61, 59], network design [51, 43], transportation and planning [16, 83], and optimal design [4, 32]. More recently, BLP has f ound applications in many areas of machine learning and artificial intelligence. Zeroth-order methods apply to many optimization problems ( including the BLP) where for various reasons such as complexity, lack of access to an accurat e model, or computational limitations, there is no or limited access to the objective gradient.