Adaptive Lorentz-Weighted Persistence Surfaces: Injectivity, Wasserstein Stability, and Scale Equivariance

This preprint presents a mathematical investigation of adaptive Lorentz-weighted persistence surfaces for topological data analysis. The study introduces a diagram-dependent saturating weight whose scale is determined by the total persistence of a finite persistence diagram. The proposed framework combines Lorentz-inspired weighting with Gaussian persistence surfaces to investigate information preservation, mathematical stability, and scale equivariance. The main theoretical contributions include: Injectivity: A proof that the proposed continuous representation uniquely determines finite persistence diagrams with strictly positive, finite lifetimes, including repeated points. Wasserstein Stability: A cardinality-independent Lipschitz bound with respect to the 1-Wasserstein distance between persistence diagrams. Scale Equivariance: Mathematical conditions under which the representation is preserved when filtration coordinates and dimensional parameters are consistently rescaled. Mathematical Foundations: Supporting derivations, counterexamples, and analyses of the limitations of scalar persistence summaries, thresholded weights, and finite-resolution representations. The research distinguishes the proposed adaptive weighting mechanism from established Gaussian persistence representations and classical persistent homology methods. The results are theoretical and do not establish universal computational superiority over existing topological data analysis techniques. The originality of the specific adaptive construction remains subject to further literature review and independent mathematical verification. Publication status: Preprint — not peer-reviewed. Author: Mahir SogutAffiliation: Independent Researcher, Çankırı, Türkiye